how to improve problem solving speed

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How can I get faster at doing math? [closed]

In general, my level of mathematics is good with respect to the competitions that I need to appear in.

However, no matter what the competition is, whether it is an easier level or a more difficult level of Olympiad, or JEE, or just some internal math test, I am always facing a time issue. I get almost all my answers correct, but I’m simply not able to get score beyond a threshold because of the time crunch. Even simple advices like how to get faster at calculations help.

What is it that you advice I should do to get faster at solving problems? Even simple suggestions on how to get faster at calculations help.

• contest-math

• $\begingroup$ It's possible that you have a form of dyscalculia (that I believe I suffer from) which is poorly understood. That is, you have high conceptual mathematical abilities but low calculation abilities. Consider computer science as a field. Computers are really good at calculations. $\endgroup$ –  JimmyJames Commented Jul 19, 2023 at 21:57
• $\begingroup$ The what-you-want question seems important here. I mean, if you truly just want to win competitions, then that'd seem to call for an aggressively simplistic mindset -- though that might be at-odds with other goals you might have. $\endgroup$ –  Nat Commented Jul 20, 2023 at 13:03
• $\begingroup$ @PM2Ring, OP mentions JEE. It is, by definition, a competition, but it is also the entrance exam for the top engineering universities in the country. Many students feel like they need to write the JEE for this reason. $\endgroup$ –  aqualubix Commented Jul 23, 2023 at 4:39

2 Answers 2

In my view, paradoxically you have to be slower in order to be faster. To be fast at maths you have to be less focused on getting the job in front of you right now done, and more focused during the learning stage on learning deeper insights into what you're doing. When you attain a high level of MASTERY , problems are faster to solve.

I'm slow at learning maths because I need deep insights in order to remember things, and I'm never satisfied until I know something inside out.

Taking a really simple example, to calculate $5 \times (10+4)$ you might say that's $5\times14=70$ and be done with it and move on.

But if you're focused on learning deeper insights you might think about it in multiple different ways:

Multiplication distributes over addition so I can multiply before I add, or after:

$(5\times10) + (5\times 4)=5\times(10+4)=70$

You might think about how multiplying is equivalent to adding the exponents of the prime factors:

$10+4$ has the prime factors $\{2^1,7^1\}$ and $5$ has the prime factor $\{5^1\}$ so the product is $2^{1+0}\times5^{1+0}\times7^{0+1}=70$

Since we write numbers in base $10$ you might think about multiplying by five as multiplying by $\frac{10}2$ so $5 \times (10+4) = \frac{10}2\times14$ then cancel the twos to get $10\times7$

You might think of $14$ as $20-6$ giving you $100-30=70$

Then you might think about the fact that this last example is

$5\times (20-6)=(5\times20-5\times6)$ and ask yourself whether this means that multiplication distributes over subtraction.

Then you might deduce that this is true because subtraction is simply the addition of a negative number.

Then you might think about how addition is an operation on the monoid of non-negative integers, and how subtraction is the extension of rightward transformations on the real line to leftward transformations on the line which are their inverse transformations. And by introducing subtraction you extend the closure of your algebraic operations to include the negatives, and this makes the integers a group according to the group axioms. Multiplying a negative number scales to the left rather than scaling to the right.

A good exercise, is to pay attention to how you solved a problem . Sometimes you do it automatically, and the process is subconscious. Once you become conscious of your method, ask yourself what other methods you might have used, and solve the same problem by those methods. You will learn shortcuts this way and multiple ways of understanding the same thing.

The more insight you build, the more visualisation tools you have at hand on which to pin memories. Then when you have to apply quickly e.g. in an exam, it's not a case of finding your one way to solve a problem. You can see multiple ways and one may instantly jump out as a quick solution. And you can move on with confidence, knowing you've arrived at your answer by multiple different methods.

This of course, requires an investment of more time in the learning stage. But by building deeper insights and a higher level of mastery, you can progress faster when it comes to applying what you learnt.

Improving your speed in solving mathematical problems requires a combination of practice, strategy, and efficiency. Here are some suggestions that may help you get faster at solving problems:

Practice regularly: Consistent practice is crucial for developing speed. Solve a variety of problems regularly, including both easier and more challenging ones. This will help you build familiarity with different problem types and increase your overall speed.

Develop mental math skills: Strengthen your mental math abilities by practicing mental calculations, such as addition, subtraction, multiplication, and division. Learn techniques like estimation, rounding, and simplification to quickly approximate and simplify calculations.

Focus on key concepts: Identify the key concepts and strategies relevant to the types of problems you frequently encounter. Mastering these core concepts will enable you to solve problems more efficiently and quickly.

Improve problem-solving techniques: Learn and practice various problem-solving techniques, such as visualization, pattern recognition, and logical reasoning. These techniques can help you approach problems in a more organized and efficient manner, saving you time in the process.

Time yourself: Set time limits for solving practice problems and exams to simulate the pressure of real test conditions. This will help you develop a sense of time management and train yourself to work efficiently within specific time constraints.

Utilize shortcuts and tricks: Learn and utilize shortcuts, tricks, and formulas specific to the types of problems you are solving. Familiarize yourself with commonly used formulas, properties, and identities in areas such as algebra, geometry, and trigonometry. This will help you save time by avoiding unnecessary calculations.

Analyze your mistakes: Review and analyze your mistakes and the solutions to the problems you have solved. Identify the areas where you are consistently making errors or spending too much time. Understanding your weaknesses will allow you to focus on those areas during your practice and improve your efficiency.

Build problem-solving intuition: Work on developing an intuition for problem-solving by solving a wide range of problems. With practice, you will start recognizing common problem-solving patterns and become more adept at choosing the most efficient approach for each problem.

Remember, improving speed takes time and effort. Be patient and persistent in your practice, and gradually you will see improvement.

• $\begingroup$ Upvoted for mentioning pattern matching . Generally speaking, humans are primarily pattern-matching machines, even to the point where we think we recognize a pattern where none exists. $\endgroup$ –  njuffa Commented Jul 19, 2023 at 22:13

Not the answer you're looking for? Browse other questions tagged contest-math advice .

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Research shows the best ways to learn math.

Students learn math best when they approach the subject as something they enjoy. Speed pressure, timed testing and blind memorization pose high hurdles in the pursuit of math, according to Jo Boaler, professor of mathematics education  at Stanford Graduate School of Education and lead author on a new working paper called "Fluency Without Fear."

"There is a common and damaging misconception in mathematics – the idea that strong math students are fast math students," said Boaler, also cofounder of YouCubed at Stanford, which aims to inspire and empower math educators by making accessible in the most practical way the latest research on math learning.

Fortunately, said Boaler , the new national curriculum standards known as the Common Core Standards for K-12 schools de-emphasize the rote memorization of math facts. Maths facts are fundamental assumptions about math, such as the times tables (2 x 2 = 4), for example. Still, the expectation of rote memorization continues in classrooms and households across the United States.

While research shows that knowledge of math facts is important, Boaler said the best way for students to know math facts is by using them regularly and developing understanding of numerical relations. Memorization, speed and test pressure can be damaging, she added.

Number sense is critical

On the other hand, people with "number sense" are those who can use numbers flexibly, she said. For example, when asked to solve the problem of 7 x 8, someone with number sense may have memorized 56, but they would also be able to use a strategy such as working out 10 x 7 and subtracting two 7s (70-14).

"They would not have to rely on a distant memory," Boaler wrote in the paper.

In fact, in one research project the investigators found that the high-achieving students actually used number sense, rather than rote memory, and the low-achieving students did not.

The conclusion was that the low achievers are often low achievers not because they know less but because they don't use numbers flexibly.

"They have been set on the wrong path, often from an early age, of trying to memorize methods instead of interacting with numbers flexibly," she wrote. Number sense is the foundation for all higher-level mathematics, she noted.

Role of the brain

Boaler said that some students will be slower when memorizing, but still possess exceptional mathematics potential.

"Math facts are a very small part of mathematics, but unfortunately students who don't memorize math facts well often come to believe that they can never be successful with math and turn away from the subject," she said.

Prior research found that students who memorized more easily were not higher achieving – in fact, they did not have what the researchers described as more "math ability" or higher IQ scores. Using an MRI scanner, the only brain differences the researchers found were in a brain region called the hippocampus, which is the area in the brain responsible for memorizing facts – the working memory section.

But according to Boaler, when students are stressed – such as when they are solving math questions under time pressure – the working memory becomes blocked and the students cannot as easily recall the math facts they had previously studied. This particularly occurs among higher achieving students and female students, she said.

Some estimates suggest that at least a third of students experience extreme stress or "math anxiety" when they take a timed test, no matter their level of achievement. "When we put students through this anxiety-provoking experience, we lose students from mathematics," she said.

Math treated differently

Boaler contrasts the common approach to teaching math with that of teaching English. In English, a student reads and understands novels or poetry, without needing to memorize the meanings of words through testing. They learn words by using them in many different situations – talking, reading and writing.

"No English student would say or think that learning about English is about the fast memorization and fast recall of words," she added.

Strategies, activities

In the paper, coauthored by Cathy Williams, cofounder of YouCubed, and Amanda Confer, a Stanford graduate student in education, the scholars provide activities for teachers and parents that help students learn math facts at the same time as developing number sense. These include number talks, addition and multiplication activities, and math cards.

Importantly, Boaler said, these activities include a focus on the visual representation of number facts. When students connect visual and symbolic representations of numbers, they are using different pathways in the brain, which deepens their learning, as shown by recent brain research.

"Math fluency" is often misinterpreted, with an over-emphasis on speed and memorization, she said. "I work with a lot of mathematicians, and one thing I notice about them is that they are not particularly fast with numbers; in fact some of them are rather slow. This is not a bad thing; they are slow because they think deeply and carefully about mathematics."

She quotes the famous French mathematician, Laurent Schwartz. He wrote in his autobiography that he often felt stupid in school, as he was one of the slowest math thinkers in class.

Math anxiety and fear play a big role in students dropping out of mathematics, said Boaler.

"When we emphasize memorization and testing in the name of fluency we are harming children, we are risking the future of our ever-quantitative society and we are threatening the discipline of mathematics," she said. "We have the research knowledge we need to change this and to enable all children to be powerful mathematics learners. Now is the time to use it."

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You don’t have to be a math teacher to know that a lot of students—and likely a lot of parents (it’s been awhile!)—are intimidated by math problems, especially if they involve large numbers. Learning techniques on how to do math quickly can help students develop greater confidence in math , improve math skills and understanding, and excel in advanced courses.

If it’s your job to teach those, here’s a great refresher.

Fast math tricks infographic

10 tricks for doing fast math

Here are 10 fast math strategies students (and adults!) can use to do math in their heads. Once these strategies are mastered, students should be able to accurately and confidently solve math problems that they once feared solving.

1. Adding large numbers

Adding large numbers just in your head can be difficult. This method shows how to simplify this process by making all the numbers a multiple of 10. Here is an example:

While these numbers are hard to contend with, rounding them up will make them more manageable. So, 644 becomes 650 and 238 becomes 240.

Now, add 650 and 240 together. The total is 890. To find the answer to the original equation, it must be determined how much we added to the numbers to round them up.

650 – 644 = 6 and 240 – 238 = 2

Now, add 6 and 2 together for a total of 8

To find the answer to the original equation, 8 must be subtracted from the 890.

890 – 8 = 882

So the answer to 644 +238 is 882.

2. Subtracting from 1,000

Here’s a basic rule to subtract a large number from 1,000: Subtract every number except the last from 9 and subtract the final number from 10

For example:

1,000 – 556

Step 1: Subtract 5 from 9 = 4

Step 2: Subtract 5 from 9 = 4

Step 3: Subtract 6 from 10 = 4

The answer is 444.

3. Multiplying 5 times any number

When multiplying the number 5 by an even number, there is a quick way to find the answer.

For example, 5 x 4 =

• Step 1: Take the number being multiplied by 5 and cut it in half, this makes the number 4 become the number 2.
• Step 2: Add a zero to the number to find the answer. In this case, the answer is 20.

When multiplying an odd number times 5, the formula is a bit different.

For instance, consider 5 x 3.

• Step 1: Subtract one from the number being multiplied by 5, in this instance the number 3 becomes the number 2.
• Step 2: Now halve the number 2, which makes it the number 1. Make 5 the last digit. The number produced is 15, which is the answer.

4. Division tricks

Here’s a quick way to know when a number can be evenly divided by these certain numbers:

• 10 if the number ends in 0
• 9 when the digits are added together and the total is evenly divisible by 9
• 8 if the last three digits are evenly divisible by 8 or are 000
• 6 if it is an even number and when the digits are added together the answer is evenly divisible by 3
• 5 if it ends in a 0 or 5
• 4 if it ends in 00 or a two digit number that is evenly divisible by 4
• 3 when the digits are added together and the result is evenly divisible by the number 3
• 2 if it ends in 0, 2, 4, 6, or 8

5. Multiplying by 9

This is an easy method that is helpful for multiplying any number by 9. Here is how it works:

Let’s use the example of 9 x 3.

Step 1 : Subtract 1 from the number that is being multiplied by 9.

3 – 1 = 2

The number 2 is the first number in the answer to the equation.

Step 2 : Subtract that number from the number 9.

9 – 2 = 7

The number 7 is the second number in the answer to the equation.

So, 9 x 3 = 27

6. 10 and 11 times tricks

The trick to multiplying any number by 10 is to add a zero to the end of the number. For example, 62 x 10 = 620.

There is also an easy trick for multiplying any two-digit number by 11. Here it is:

Take the original two-digit number and put a space between the digits. In this example, that number is 25.

Now add those two numbers together and put the result in the center:

2_(2 + 5)_5

The answer to 11 x 25 is 275.

If the numbers in the center add up to a number with two digits, insert the second number and add 1 to the first one. Here is an example for the equation 11 x 88

(8 + 1)_6_8

There is the answer to 11 x 88: 968

7. Percentage

Finding a percentage of a number can be somewhat tricky, but thinking about it in the right terms makes it much easier to understand. For instance, to find out what 5% of 235 is, follow this method:

• Step 1: Move the decimal point over by one place, 235 becomes 23.5.
• Step 2: Divide 23.5 by the number 2, the answer is 11.75. That is also the answer to the original equation.

8. Quickly square a two-digit number that ends in 5

Let’s use the number 35 as an example.

• Step 1: Multiply the first digit by itself plus 1.
• Step 2: Put a 25 at the end.

35 squared = [3 x (3 + 1)] & 25

[3 x (3 + 1)] = 12

12 & 25 = 1225

35 squared = 1225

9. Tough multiplication

When multiplying large numbers, if one of the numbers is even, divide the first number in half, and then double the second number. This method will solve the problem quickly. For instance, consider

Step 1: Divide the 20 by 2, which equals 10. Double 120, which equals 240.

Then multiply your two answers together.

10 x 240 = 2400

The answer to 20 x 120 is 2,400.

10. Multiplying numbers that end in zero

Multiplying numbers that end in zero is actually quite simple. It involves multiplying the other numbers together and then adding the zeros at the end. For instance, consider:

Step 1: Multiply the 2 times the 4

Step 2: Put all four of the zeros after the 8

200 x 400= 80,000

Practicing these fast math tricks can help both students and teachers improve their math skills and become secure in their knowledge of mathematics—and unafraid to work with numbers in the future.

You may also like to read

• Research-Based Math Teaching Strategies
• Tips in Teaching a Hands-On Math Curriculum
• 5 Tips to Help Get Students Engaged in High School Math
• 3 Tips for Running an Elementary School Math Workshop
• Seven Everyday Online Math Resources for Teachers
• Three Tips for Developing Elementary Math Tests

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How to Train Your Brain to Think Faster: 15 Simple Tips

It is clear that fast thinking is a desirable skill for both adults and kids. You can improve your thinking speed with brain exercises and small changes in your lifestyle. Let’s learn how to think faster together.

15 Simple Tips for Learning How to Think Faster

If you wonder how to think better and faster, the answer is balancing a healthy lifestyle and training the brain with the right exercises. Discover how to think faster and get smarter!

1. Practice with MentalUP

Mental and physical exercises are essential to improving fast thinking skills. As an effective way, the MentalUP app benefits overall mental performance. If you think now is the right time to support both your kids’ and your development, you can check out these boosting games.

If you want to improve your processing speed skills and start to think faster, this game is for you.

Memory Cards

Memory skills are important for training our brains to think faster. This game will help you in this.

If you need a fun game to improve both your and your kids’ cognitive skills, you can play this game.

When you develop your decision-making ability by playing this game, you can start to think faster.

You Are Different!

This game will improve your visual attention and comparison skills at the same time.

Colorful Blocks

You can develop your reasoning skills and have fun by playing this entertaining game.

Let's Find It

If you need to boost your conceptualization ability, this game is tailored for you.

Face Recognition

This challenging game will empower both your visual and verbal memory skills.

MentalUP’s memory, concentration, and logic games improve the overall learning ability of individuals and support essential skills, like attention, problem-solving, critical thinking, creativity, and concentration, that improve the brain’s processing speed and fast thinking skills.

Also, MentalUP fitness exercises and activities enhance cognitive function and support the physical development of individuals, which is essential for improved fast thinking skills.

As one of the best problem-solving apps , by including 150+ brain games for cognitive development and 240+ fitness exercises , MentalUP offers a great source for both kids and adults to spend quality time while improving their fast thinking skills . 🚀

In addition to all these benefits, MentalUP offers a safe learning environment by excluding pop-ups, ads, or any inappropriate content, and it can be used on any laptop, smartphone, or PC. It is time for you to learn how to think faster by having fun. 😎

2. Exercise Regularly

Physical exercise increases the blood flow to the brain and supports the release of essential hormones, like endorphins and serotonin, that enhance brain processing speed.

Thus, regular exercise can help you think faster, stay focused and improve your long and short-term memory. All types of exercise are essential for supporting fast thinking skills. Still, a daily aerobic exercise routine is one of the best ways to improve brain health and processing speed.

3. Eat Healthy

A balanced diet is essential for both mental and physical health, and consuming nutritional-rich brain boosting foods is especially important for improving mental performance.

Thus, to support your brain development and cognitive functioning, you should consume foods that include vitamins, healthy fats, and antioxidants.

Having a healthy diet might not teach you how to think faster, but it surely contributes to your fast thinking skills.

4. Get Enough Sleep

Sleep affects every single system of your body and mind. Thus, if you want to learn how to think faster and make a decision easily, you should ensure that you get enough sleep before training your brain.

For people over the age of 18, seven to nine hours of night sleep is optimal. For kids, this time can reach up to 11 hours of sleep. To ensure that you and your kids get enough sleep, you can create a nightly sleep routine and support it with calming activities.

5. Solve Puzzles

Puzzles are a great workout for your brain, and they are suitable for people of all ages. Plus, it is a great activity to spend quality time with as a family.

Puzzles can exercise both sides of your brain, improve your long and short-term memory, and enhance critical skills, like problem-solving, critical thinking, and visual and spatial thinking.

If you are looking for fun and beneficial puzzles to test your cognitive skills and maximize them to learn how to think faster, you can check out MentalUP’s brain puzzles . 🚀

The MentalUP app includes a variety of puzzles to support your and your kid’s mental development and help acquire better fast thinking skills. 🎉

It is time to play fun games right now to enjoy the benefits of solving puzzles !

GET STARTED

6. Meditate

Meditation enhances communication between brain cells and speeds mental processing in turn. Also, meditation improves the ability to learn and concentrate.

Regular meditation practice with your family members can be a great idea to keep you mentally sharp and build a more efficient brain by stimulating the formation of new brain cells and neural connections. Plus, wellness activities like meditation also contribute to stress management skills.

7. Play Word Games

Word games are great for sharpening the executive functioning of your brain and support essential skills like critical thinking and problem-solving.

Word games strengthen your executive functioning and control by supporting cognitive flexibility, concentration, and verbal fluency. So practicing your cognitive skills with word games, such as Scrabble, Sudoku, or Wordle, is a great idea to learn how to think and talk faster.

Reading stimulates every part of your brain as well as the neutral connections between them. As a result, reading boosts your intelligence and improves your dexterity.

As reading requires multiple cognitive functions like attention, predicting, memory, reasoning, and comprehension, it is essential for both kids and adults to improve processing speed and fast thinking skills.

9. Learn a New Skill

Pushing yourself in fresh directions is one of the most contributional things you can do to build better fast thinking skills.

You can learn a new language, start playing an instrument or find a different skill to hone. Having new experiences will help your brain grow new pathways and enable you to perform better on new tasks by improving your fast thinking skills.

10. Rest Your Mind

The human brain requires a great amount of energy to function properly. Thus, any lack of rest, relaxation, or overthinking can reduce the efficiency of your brain and processing speed.

Thus, resting your mind is as significant as resting your body to improve your fast thinking skills. To rest your mind and senses, you can arrange daily times in which you can sit and relax without any interruptions. Also, mindfulness can be very helpful for this purpose.

11. Get More Social

Interacting with others can actually train our brains by improving social motivation, memory formation, and recall processes. It is also good for protecting the brain from neurological diseases.

Getting social is not all about going out with friends. You can also participate in volunteering activities, invite your friends over more often, or join sports clubs or speaking groups.

Getting involved in different activities and environments and meeting new people will help your brain produce new learning pathways and improve your vision as well as your fast thinking skills.

12. Exercise Timed Writing

Timed exercises are great for learning how to think fast under pressure. Thus, this timed writing exercise is both good for your fast thinking and literacy skills.

Start by writing down a variety of writing prompts and tossing them into a jar. Then, you can set one out and start your timer for 3-5 minutes to write on that topic until the timer goes off. When you are finished, count how many words you used and how much your writing makes sense.

If you do this activity regularly, you will be surprised when you see how much it contributes to your fast thinking skills.

13. Practice Speed

We all have some things we are good at. It can be cooking, playing an instrument, or drawing. Whatever it is, adding another layer of challenge by speeding them up is beneficial to our fast thinking skills and overall mental performance.

You can start by timing yourself on completing your daily tasks or demonstrating your skills. All you need to do is set a timer and test how much time you spend. Then, you can gradually shorten it by improving your brain’s reaction times.

14. Play Card Games

Playing card games is a mentally stimulating activity for both adults and kids. Playing card games can lead to greater volume in several regions of the brain and improve memory and overall thinking skills.

You can go with Uno, Crazy Eights, Blitz, or Monopoly and play these card games with your friends and family to both improve your fast thinking skills and have a good time.

If traditional card games are boring for you or your kids, you can always benefit from online sources. But it is important to find a suitable alternative that is safe. 🎈

MentalUP is here to help you with its fun card games you can use to develop your cognitive skills instantly . People of all ages love these entertaining games. 🎮

15. Vary Your Routines

Practicing intentional variety is essential to supporting our brains to obtain new learning paths and improve mental performance, as completing the same routines every day can reduce the processing speed of our brains.

Intentional variety is not as hard as it sounds. You can simply take a new route when you are on the way back home, meet some new friends, change your grocery store, taste entirely different foods, or listen to a music genre that you have never heard before.

4 Factors That Cause Slow Thinking

Until now, we have learned useful tips to learn how to think better and faster. Now it is time to realize the factors that can reduce your overall mental performance and result in longer reaction times and inadequate fast thinking skills.

Let’s discover the 4 factors that slow your thinking!

1. A Busy Schedule

If you have an overflowing calendar, you probably don’t have enough time to just think. This is one of the essential factors that reduce mental performance and result in people having slower thinking skills.

So if you want to accelerate your thinking skills, you should create a daily thinking time for yourself. You don’t need hours to practice thinking. Including 30 minutes of daily thinking time in your schedule might be enough.

You can also provide this thinking time for your kids to help them stay mentally sharp when dealing with school tasks.

2. Poor Nutrition

The brain’s most important job is to communicate with other cells in the brain and the rest of the body. This communication allows us to perform both physical and cognitive tasks. To complete this process successfully, the brain needs to be supported with proper nutrition and exercise routines.

Thus, consuming nutritionally poor foods is a reducing factor for cognitive tasks like thinking, learning, memorizing, and planning. You can have some fast food and unhealthy snacks from time to time, but still, it is important to keep things balanced by having a healthy diet.

3. Lack of Lexicon

Without having a large vocabulary, it is not possible to build better fast thinking skills, as literacy is an important factor for cognitive development.

So if you feel you don’t think fast enough, the reason might be your lack of lexical knowledge. Still, you don’t need to worry too much. There are plenty of activities to practice your lexical knowledge, such as speed reading, speed writing, memory games, and using a dictionary.

Having high-stress levels can increase blood pressure, weaken the immune system, and cause brain fog, which is a condition that affects one’s cognitive function.

If you are experiencing chronic stress, you might be suffering from memory problems, lack of mental clarity, poor concentration, and inability to focus.

All these symptoms can be factors that slow your thinking. Thus, it is essential to learn how to manage stress to improve your mental performance.

Fast Thinking vs. Efficient, Accurate Thinking

Now, you might be thinking that fast thinking is a must-have skill, but actually, efficient and accurate thinking is as important as thinking quickly.

Therefore, thinking efficiently is not about panicking and coming up with quick decisions. It is more about going into action mode and making accurate decisions as fast as you can.

You already know how to think better and faster. Still, if you need some additional resources to improve your and your kid’s mental performance to think faster and more efficiently, MentalUP is here with its fun and educational games, activities, and exercises to support both you and your kids in this journey. 🏆

MENTALUP BRAIN GAMES

Benefits of Fast Thinking

Fast thinking enables people to solve problems more efficiently and easily, improves their academic performance, and creates a great advantage for having a successful career.

In addition, this essential skill benefits aspects of life in many ways:

• According to the Association for Psychological Science , when people are asked to think faster, they are more likely to feel happier. Also, this research shows that when people are made to think quickly, they feel more energetic, creative, powerful, and self-assured.
• Fast thinking enhances goal setting, planning, and problem-solving skills. Also, it enables people to stay focused and supports their ability to learn new subjects.
• Building faster-thinking skills during their youth can help people stay mentally sharp in their older years.
• It benefits one’s social life. When humans recognize that they have better fast-thinking skills than others, they feel more confident speaking in front of others and participating in group activities.
• Students with fast thinking skills spend their time more efficiently when completing homework and exams.
• For athletes, improving fast learning skills leads to better athletic performance by quickening their reaction times.

Now you might be thinking about how to think faster on your feet and train your brain to improve your mental performance. You are in luck because we are just about to discover 15 simple tips for learning how to train your brain to think faster.

Frequently Asked Questions

When it comes to thinking faster and improving mental performance, there are a lot of common questions that arise. To help you understand the concept better and answer all your questions about this topic, we have compiled all the frequently asked questions.

Get ready for all your questions to be answered!

How to think faster and clearer?

If you think about how to think faster and clearer, the answer is simple–by adopting a healthy lifestyle and training your brain in balance with right brain vs left brain exercises, games, activities, and daily routines.

How are some people able to critically think better or faster than others?

This is because they perform better mentally than others. Still, this doesn’t mean that these people are natural talents because mental performance and fast thinking skills are related to how you train your mind, eat, sleep, and exercise.

How to help your child think faster?

If you want to help your kid think faster and perform better mentally, you should ensure that they have a healthy lifestyle. In addition to that, you can utilize the tips we’ve shared with you through this blog and check out MentalUP games and activities.

How to train yourself to work and think faster?

Simply by testing your processing speed and practicing mental exercises such as riddles for adults to think more efficiently and faster. As thinking and working are highly related to each other, by improving your fast thinking skills with healthy habits and brain exercises, you can also complete your daily tasks faster and easier.

How to think faster in math?

If you want to improve when doing math especially, you should practice some math techniques as well as brain exercises. You can practice these techniques by playing math games, learning some short-cut tactics, and practicing specific math subjects.

If you are looking for some fun and educational games to satisfy your kid’s academic and cognitive needs , you can get the MentalUP app . Plus, you will see games and activities to improve math , language, visualization, memory skills, and much more!

Explore MentalUP’s age-appropriate games, activities, and fitness exercises now to develop both your kid’s and your mental performance and have fun as a family!

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Strategies to Improve Your Problem-Solving Skills

Got problems? We all do.

They’re something we encounter daily, both at work and at home. Tackling problems and finding solutions are useful skills that are in high demand.

At a basic level, there are three steps to solving any problem:

• Define the problem
• Generate ideas for solutions
• Implement solutions

You might be tempted to think that the first step is unnecessary. After all, that’s why you’re here in the first place, to solve a problem. However, defining the problem is arguably the most important step in problem-solving.

Albert Einstein is famously quoted as saying, “If I had an hour to solve a problem I’d spend 55 minutes thinking about the problem and 5 minutes thinking about solutions.”

When you can spend more time defining the real problem, and not just a symptom, it will be easier to find a lasting solution.

How to better define the problem:

• Ask “why” questions
• Talk it through with others
• Write down the problem in words
• Use graphs or flow charts

Now that we’ve talked about the steps of solving a problem and how to better define it, let’s dig into some strategies to help your brain perform at its best for solving problems.

“Thanks to a process called neuroplasticity, your brain is continually reorganizing itself by forming new neural connections throughout your life, which gives you the power to make your brain better.” – Daniel G. Amen, MD

Neuroplasticity enables your brain to continue to learn and grow throughout your life. Like your muscles, your brain needs exercise to become stronger. Becoming a life-long learner will not only strengthen your brain, but also sharpen your memory, boost confidence, and bring new knowledge and skills into your life.

3 Ways to Improve Your Problem-Solving Skills

1.    Regularly Engage in Brain Boosting Activities

There are a number of easy and fun ways to strengthen your brain. Adding one or more of these activities into your daily routines can help boost your brain and result in better problem-solving abilities.

• Work on a jigsaw puzzle – Puzzles can be done on your own, or as a social activity. Putting together a puzzle requires concentration and spatial awareness, activating multiple parts of the brain and improving short-term memory.
• Play a musical instrument – Research has shown that learning to play an instrument can improve neuroplasticity and help improve your memory. Playing music engages multiple regions of the brain, providing numerous benefits. Maybe it’s been a while since you last played, or maybe you’ve never learned an instrument. Either way, it’s never too late to tap into your musical side and begin making music.
• Try a new hobby – Remember the “use it or lose it” concept when it comes to the brain. It’s recommended to never stop learning new things. Challenge yourself, no matter your age! Trying new hobbies is a great mental exercise to sharpen your brain. You also may find a new activity that brings more joy to your life.
• Meditate – The practice of meditation has been around for thousands of years as a tool for reducing stress, clearing your mind, promoting relaxation, and improving focus. Meditation is a powerful tool that can boost your brain anytime, anywhere.
• Play brain games – Chess, crossword puzzles, and sudoku all fall under this category. Brain games are an easy and fun way to improve concentration and strengthen memory. The best part is that they only take a few minutes to play and offer a nice break during the day.
• Read a book with a book club – Reading a book offers many benefits, including stimulating different areas of your brain to process and analyze information. When you participate in a reading group , your brain will need to remember information for later recall. This information recall is highly beneficial to protecting short-term memory. Book clubs also can provide a fun and supportive social network.

2.    Spend time NOT looking for the solution

This is counterintuitive, but it’s an important strategy to use when working on a problem. Allow yourself some downtime after defining the problem.

Let your subconscious do some work. Setting a task aside for a time can actually improve your efforts later. When you return to the problem at hand, you’ll likely have a fresh perspective.

What should you do while giving your brain a break from active problem-solving? Enjoy a hobby, get some rest, or move your body with a walk or other form of exercise.

3.    Practice healthy habits

You guessed it, those healthy habits that affect so many areas of your life are also tied to a healthy brain. Exercise, a healthy diet, and quality sleep can all help your brain function better and improve your problem-solving skills overall.

• Exercise – Moving your body increases blood flow to the brain, which can improve your ability to think critically, clearly, and creatively. Additionally, physical activity is a known way to reduce stress and anxiousness. Research has shown creativity and problem-solving to be negatively affected by stress. Using exercise to combat stress can improve your ability to find solutions with a clear mind. By exercising regularly, your overall physical, emotional, and brain health may be positively impacted.
• Healthy Diet – Dr. Daniel Amen teaches that one of the secrets to a healthy brain is to focus on detoxification in your diet. This includes avoiding alcohol, drinking plenty of water, and consuming detoxifying vegetables . Some good vegetables to incorporate into your diet would be lettuce, spinach, kale, broccoli, and asparagus. You also may try increasing your protein intake for a healthy brain, or try adding in turmeric , which can increase neuroplasticity.
• Quality Sleep – Finally, don’t forget about the impact quality sleep, or the lack of it, can have on your brain function and problem-solving abilities. Getting a good night’s rest gives your brain time to recharge and that necessary downtime of not actively thinking about the problems needing solving. While you sleep, your subconscious has a chance to do some work for you!

When you engage in brain-boosting activities, take some downtime, and practice healthy habits you’ll be better prepared for the problems in your days. And, next time you’re faced with the inevitable problems that come with life and work, you can address them with more clarity and confidence.

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How to Improve Mental Math Skills

Last Updated: August 2, 2024 Approved

This article was co-authored by Daron Cam . Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary's College. There are 8 references cited in this article, which can be found at the bottom of the page. wikiHow marks an article as reader-approved once it receives enough positive feedback. In this case, 84% of readers who voted found the article helpful, earning it our reader-approved status. This article has been viewed 334,843 times.

Eventually, you'll find yourself in a situation where you'll have to solve a math problem without a calculator. Trying to imagine a pen and paper in your head often doesn't help much. Fortunately there are faster and easier ways to do calculations in your head—and they often break down a problem in a way that makes more sense than what you learned in school. Whether you're a stressed-out student or a math wizard looking for even faster tricks, there's something for everyone to learn.

Break addition and subtraction problems into parts.

• 712 + 281 → "700 + 200," "10 + 80," and "2 + 1"
• 700 + 200 = 9 00, then 10 + 80 = 9 0, then 2 + 1 = 3
• 900 + 90 + 3 = 993 .
• Thinking in "hundreds" or "tens" instead of single digits will make it easier to keep track when digits sum to more than ten. For example, for 37 + 45, think "30 + 40 = 70" and "7 + 5 = 12". Then add 70 + 12 to get 82.

Change the problem to make round numbers.

• Addition : For 596 + 380 , realize that you can add 4 to 596 to round it to 600, then add 600 + 380 to get 980. Undo the rounding by subtracting 4 from 980 to get 976 .
• Subtraction : For 815 - 521 , break it up into 800 - 500, 10 - 20, and 5 - 1. To turn the awkward "10 - 20" into "20 - 20", add 10 to 815 to get 825. Now solve to get 304, then undo the rounding by subtracting 10 to get 294 .
• Multiplication : For 38 x 3 , you can add 2 to 38 to make the problem 40 x 3, which is 120. Since the 2 you added got multiplied by three, you need to undo the rounding by subtracting 2 x 3 = 6 at the end to get 120 - 6 = 114 .

Learn to add many numbers at once.

• For example, 7 + 4 + 9 + 13 + 6 + 51 can be reorganized to (7 + 13) + (9 + 51) + (6 + 4) = 20 + 60 + 10 = 90.

Multiply from left to right.

• For 453 x 4 , start with 400 x 4 = 1600, then 50 x 4 = 200, then 3 x 4 = 12. Add them all together to get 1812 .
• If both numbers have more than one digit, you can break it into parts. Each digit has to multiply with each other digit, so it can be tough to keep track of it all. 34 x 12 = (34 x 10) + (34 x 2) , which you can break down further into (30 x 10) + (4 x 10) + (30 x 2) + (4 x 2) = 300 + 40 + 60 + 8 = 408 .

Try a fast multiplication trick best for numbers 11 through 19.

• Let's look at numbers close to 10, like 13 x 15 . Subtract 10 from the second number, then add your answer to the first: 15 - 10 = 5, and 13 + 5 = 18.
• Multiply your answer by ten: 18 x 10 = 180.
• Next, subtract ten from both sides and multiply the results: 3 x 5 = 15.
• Add your two answers together to get the final answer: 180 + 15 = 195 .
• Careful with smaller numbers! For 13 x 8, you start with "8 - 10 = -2", then "13 + -2 = 11". If it's hard to work with negative numbers in your head, try a different method for problems like this.
• For larger numbers, it will be easier to use a "base number" like 20 or 30 instead of 10. If you try this, make sure you use that number everywhere that 10 is used above. [3] X Research source For example, for 21 x 24, you start by adding 21 + 4 to get 25. Now multiply 25 by 20 (instead of ten) to get 500, and add 1 x 4 = 4 to get 504.

Simplify problems with numbers ending in zero.

• Addition : If all numbers have zeroes at the end, you can ignore the zeroes they have in common and restore them at the end. 85 0 + 12 0 → 85 + 12 = 97, then restore the shared zero: 97 0 .
• Subtraction works the same way: 10 00 - 7 00 → 10 - 7 = 3, then restore the two shared zeroes to get 3 00 . Notice that you can only remove the two zeroes the numbers have in common, and must keep the third zero in 1000.
• Multiplication : ignore all the zeroes, then restore each one individually. 3 000 x 5 0 → 3 x 5 = 15, then restore all four zeroes to get 15 0 , 00 0 .
• Division : you can remove all shared zeroes and the answer will be the same. 60, 000 ÷ 12, 000 = 60 ÷ 12 = 5 . Don't add any zeroes back on.

Easily multiply by 4, 5, 8, or 16.

• To multiply by 5, instead multiply by 10, then divide by 2.
• To multiply by 4, instead double the number, then double it again.
• For 8, 16, 32, or even higher powers of two, just keep doubling. For example, 13 x 8 = 13 x 2 x 2 x 2, so double 13 three times: 13 → 26 → 52 → 104 .

Memorize the 11s trick.

• What is 7 2 x 11?
• Add the two digits together: 7 + 2 = 9.
• Put the answer in between the original digits: 7 2 x 11 = 7 9 2 .
• If the sum is more than 10, place only the final digit and carry the one: 5 7 x 11 = 6 2 7 , because 5 + 7 = 12. The 2 goes in the middle and the 1 gets added to the 5 to make 6.

Turn percentages into easier problems.

• 79% of 10 is the same as 10% of 79. This is true of any two numbers. If you can't find the answer to a percentage problem, try switching it around.
• To find 10% of a number, move the decimal one place to the left (10% of 65 is 6.5). To find 1% of a number, move the decimal two places to the left (1% of 65 is 0.65).
• Use these rules for 10% and 1% to help you with more difficult percentages. For example, 5% is ½ of 10%, so 5% of 80 = (10% of 80) x ½ = 8 x ½ = 4 .
• Break percentages into easier parts: 30% of 900 = (10% of 900) x 3 = 90 x 3 = 270 .

Memorize advanced multiplication shortcuts for specific problems.

• For problems like 84 x 86 , where the tens place is the same and the ones place digits sum to exactly 10, the first digits of the answer are (8 + 1) x 8 = 72 and the last digits are 4 x 6 = 24, for an answer of 7224 . That is, for a problem AB x AC, if B + C = 10, the answer starts with A(A+1) and ends with BC. This also works for larger numbers if all digits besides the ones place are identical. [6] X Research source
• You can rewrite the powers of five (5, 25, 125, 625, ...) as powers of 10 divided by an integer (10 / 2, 100 / 4, 1000 / 8, 10000 / 16, ...). [7] X Research source So 88 x 125 becomes 88 x 1000 ÷ 8 = 88000 ÷ 8 = 11000 .

Memorize squares charts.

• Memorize the squares from 1 to 20 (or higher, if you're ambitious). (That is, 1 x 1 = 1; 2 x 2 = 4; 3 x 3 = 9, and so on.)
• To multiply two numbers, first find their average (the number exactly between them). For example, the average of 18 and 14 is 16.
• Square this answer. Once you've memorized the squares chart, you'll know that 16 x 16 is 256.
• Next, look at the difference between the original numbers and their average: 18 - 16 = 2. (Always use a positive number here.)
• Square this number as well: 2 x 2 = 4.
• To get your final answer, take the first square and subtract the second: 256 - 4 = 252 .

Find useful ways to practice your mental math.

• Flashcards are great for memorizing multiplication and division tables, or for getting used to tricks for specific kinds of problems. Write the problem on one side and the answer on the other, and quiz yourself daily until you get them all right.
• Online math quizzes are another way to test your ability. Look for a well-reviewed app or website made by an educational program.
• Practice in everyday situations. You could add together the total of items you buy as you shop, or multiply the gas cost per volume by your car's tank size to find the total cost. The more of a habit this becomes, the easier it will be.

Joseph Meyer

Exercise your mental math muscles. Improve your math skills by solving daily math problems without using calculators, paper, or counting aids. By solely using your mind and getting into math discussions with your classmates, you will refine your skills and discover new approaches to problem-solving.

Practice Problems and Answers

Community Q&A

• In the real world, you don't always need to know the exact answer. If you're at the grocery store and trying to add 7.07 + 8.95 + 10.09, you could round to the closest whole numbers and estimate that the total is roughly 7 + 9 + 10 = 26. Thanks Helpful 12 Not Helpful 3
• Some people find it easier to think in money than abstract numbers. Instead of 100 - 55, try thinking of a dollar minus a 50¢ coin and a 5¢ coin. Thanks Helpful 6 Not Helpful 9

You Might Also Like

• ↑ http://gizmodo.com/10-tips-to-improve-your-mental-math-ability-1792597814
• ↑ https://www.youtube.com/watch?v=Rgw9Ik5ZGaY
• ↑ https://www.youtube.com/watch?v=SV1dC1KAl_U
• ↑ https://www.youtube.com/watch?v=1JW9BA57aR8
• ↑ http://www.wired.co.uk/article/master-mental-maths
• ↑ https://www.youtube.com/watch?v=YCBTw8KAqkw
• ↑ https://www.scientificamerican.com/article/5-tips-faster-mental-multiplication/
• ↑ Daron Cam. Academic Tutor. Expert Interview. 29 May 2020.

About This Article

One way to improve your mental math skills is to memorize your multiplication and division tables, so you always have the answer to those problems instantly. If you have trouble memorizing the numbers, try creating your own flash cards with blank notecards and asking a friend to help you practice. Another good way to practice your mental math skills is to add up the prices of your items when you’re at the store, and check to make sure you added correctly once the cashier rings you up. You can also try downloading a mental math app like Luminosity to keep your math skills sharp. To learn how to visualize an equation in your head, read on! Did this summary help you? Yes No

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Struggling to overcome challenges in your life? We all face problems, big and small, on a regular basis.

So how do you tackle them effectively? What are some key problem-solving strategies and skills that can guide you?

Effective problem-solving requires breaking issues down logically, generating solutions creatively, weighing choices critically, and adapting plans flexibly based on outcomes. Useful strategies range from leveraging past solutions that have worked to visualizing problems through diagrams. Core skills include analytical abilities, innovative thinking, and collaboration.

Want to improve your problem-solving skills? Keep reading to find out 17 effective problem-solving strategies, key skills, common obstacles to watch for, and tips on improving your overall problem-solving skills.

Key Takeaways:

• Effective problem-solving requires breaking down issues logically, generating multiple solutions creatively, weighing choices critically, and adapting plans based on outcomes.
• Useful problem-solving strategies range from leveraging past solutions to brainstorming with groups to visualizing problems through diagrams and models.
• Core skills include analytical abilities, innovative thinking, decision-making, and team collaboration to solve problems.
• Common obstacles include fear of failure, information gaps, fixed mindsets, confirmation bias, and groupthink.
• Boosting problem-solving skills involves learning from experts, actively practicing, soliciting feedback, and analyzing others’ success.
• Onethread’s project management capabilities align with effective problem-solving tenets – facilitating structured solutions, tracking progress, and capturing lessons learned.

What Is Problem-Solving?

Problem-solving is the process of understanding an issue, situation, or challenge that needs to be addressed and then systematically working through possible solutions to arrive at the best outcome.

It involves critical thinking, analysis, logic, creativity, research, planning, reflection, and patience in order to overcome obstacles and find effective answers to complex questions or problems.

The ultimate goal is to implement the chosen solution successfully.

What Are Problem-Solving Strategies?

Problem-solving strategies are like frameworks or methodologies that help us solve tricky puzzles or problems we face in the workplace, at home, or with friends.

Imagine you have a big jigsaw puzzle. One strategy might be to start with the corner pieces. Another could be looking for pieces with the same colors. 

Just like in puzzles, in real life, we use different plans or steps to find solutions to problems. These strategies help us think clearly, make good choices, and find the best answers without getting too stressed or giving up.

Why Is It Important To Know Different Problem-Solving Strategies?

Knowing different problem-solving strategies is important because different types of problems often require different approaches to solve them effectively. Having a variety of strategies to choose from allows you to select the best method for the specific problem you are trying to solve.

This improves your ability to analyze issues thoroughly, develop solutions creatively, and tackle problems from multiple angles. Knowing multiple strategies also aids in overcoming roadblocks if your initial approach is not working.

Here are some reasons why you need to know different problem-solving strategies:

• Different Problems Require Different Tools: Just like you can’t use a hammer to fix everything, some problems need specific strategies to solve them.
• Improves Creativity: Knowing various strategies helps you think outside the box and come up with creative solutions.
• Saves Time: With the right strategy, you can solve problems faster instead of trying things that don’t work.
• Reduces Stress: When you know how to tackle a problem, it feels less scary and you feel more confident.
• Better Outcomes: Using the right strategy can lead to better solutions, making things work out better in the end.
• Learning and Growth: Each time you solve a problem, you learn something new, which makes you smarter and better at solving future problems.

Knowing different ways to solve problems helps you tackle anything that comes your way, making life a bit easier and more fun!

17 Effective Problem-Solving Strategies

Effective problem-solving strategies include breaking the problem into smaller parts, brainstorming multiple solutions, evaluating the pros and cons of each, and choosing the most viable option. 

Critical thinking and creativity are essential in developing innovative solutions. Collaboration with others can also provide diverse perspectives and ideas. 

By applying these strategies, you can tackle complex issues more effectively.

Now, consider a challenge you’re dealing with. Which strategy could help you find a solution? Here we will discuss key problem strategies in detail.

1. Use a Past Solution That Worked

This strategy involves looking back at previous similar problems you have faced and the solutions that were effective in solving them.

It is useful when you are facing a problem that is very similar to something you have already solved. The main benefit is that you don’t have to come up with a brand new solution – you already know the method that worked before will likely work again.

However, the limitation is that the current problem may have some unique aspects or differences that mean your old solution is not fully applicable.

The ideal process is to thoroughly analyze the new challenge, identify the key similarities and differences versus the past case, adapt the old solution as needed to align with the current context, and then pilot it carefully before full implementation.

An example is using the same negotiation tactics from purchasing your previous home when putting in an offer on a new house. Key terms would be adjusted but overall it can save significant time versus developing a brand new strategy.

2. Brainstorm Solutions

This involves gathering a group of people together to generate as many potential solutions to a problem as possible.

It is effective when you need creative ideas to solve a complex or challenging issue. By getting input from multiple people with diverse perspectives, you increase the likelihood of finding an innovative solution.

The main limitation is that brainstorming sessions can sometimes turn into unproductive gripe sessions or discussions rather than focusing on productive ideation —so they need to be properly facilitated.

The key to an effective brainstorming session is setting some basic ground rules upfront and having an experienced facilitator guide the discussion. Rules often include encouraging wild ideas, avoiding criticism of ideas during the ideation phase, and building on others’ ideas.

For instance, a struggling startup might hold a session where ideas for turnaround plans are generated and then formalized with financials and metrics.

3. Work Backward from the Solution

This technique involves envisioning that the problem has already been solved and then working step-by-step backward toward the current state.

This strategy is particularly helpful for long-term, multi-step problems. By starting from the imagined solution and identifying all the steps required to reach it, you can systematically determine the actions needed. It lets you tackle a big hairy problem through smaller, reversible steps.

A limitation is that this approach may not be possible if you cannot accurately envision the solution state to start with.

The approach helps drive logical systematic thinking for complex problem-solving, but should still be combined with creative brainstorming of alternative scenarios and solutions.

An example is planning for an event – you would imagine the successful event occurring, then determine the tasks needed the week before, two weeks before, etc. all the way back to the present.

4. Use the Kipling Method

This method, named after author Rudyard Kipling, provides a framework for thoroughly analyzing a problem before jumping into solutions.

It consists of answering six fundamental questions: What, Where, When, How, Who, and Why about the challenge. Clearly defining these core elements of the problem sets the stage for generating targeted solutions.

The Kipling method enables a deep understanding of problem parameters and root causes before solution identification. By jumping to brainstorm solutions too early, critical information can be missed or the problem is loosely defined, reducing solution quality.

Answering the six fundamental questions illuminates all angles of the issue. This takes time but pays dividends in generating optimal solutions later tuned precisely to the true underlying problem.

The limitation is that meticulously working through numerous questions before addressing solutions can slow progress.

The best approach blends structured problem decomposition techniques like the Kipling method with spurring innovative solution ideation from a diverse team. 

An example is using this technique after a technical process failure – the team would systematically detail What failed, Where/When did it fail, How it failed (sequence of events), Who was involved, and Why it likely failed before exploring preventative solutions.

5. Try Different Solutions Until One Works (Trial and Error)

This technique involves attempting various potential solutions sequentially until finding one that successfully solves the problem.

Trial and error works best when facing a concrete, bounded challenge with clear solution criteria and a small number of discrete options to try. By methodically testing solutions, you can determine the faulty component.

A limitation is that it can be time-intensive if the working solution set is large.

The key is limiting the variable set first. For technical problems, this boundary is inherent and each element can be iteratively tested. But for business issues, artificial constraints may be required – setting decision rules upfront to reduce options before testing.

Furthermore, hypothesis-driven experimentation is far superior to blind trial and error – have logic for why Option A may outperform Option B.

Examples include fixing printer jams by testing different paper tray and cable configurations or resolving website errors by tweaking CSS/HTML line-by-line until the code functions properly.

6. Use Proven Formulas or Frameworks (Heuristics)

Heuristics refers to applying existing problem-solving formulas or frameworks rather than addressing issues completely from scratch.

This allows leveraging established best practices rather than reinventing the wheel each time.

It is effective when facing recurrent, common challenges where proven structured approaches exist.

However, heuristics may force-fit solutions to non-standard problems.

For example, a cost-benefit analysis can be used instead of custom weighting schemes to analyze potential process improvements.

Onethread allows teams to define, save, and replicate configurable project templates so proven workflows can be reliably applied across problems with some consistency rather than fully custom one-off approaches each time.

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7. Trust Your Instincts (Insight Problem-Solving)

Insight is a problem-solving technique that involves waiting patiently for an unexpected “aha moment” when the solution pops into your mind.

It works well for personal challenges that require intuitive realizations over calculated logic. The unconscious mind makes connections leading to flashes of insight when relaxing or doing mundane tasks unrelated to the actual problem.

Benefits include out-of-the-box creative solutions. However, the limitations are that insights can’t be forced and may never come at all if too complex. Critical analysis is still required after initial insights.

A real-life example would be a writer struggling with how to end a novel. Despite extensive brainstorming, they feel stuck. Eventually while gardening one day, a perfect unexpected plot twist sparks an ideal conclusion. However, once written they still carefully review if the ending flows logically from the rest of the story.

8. Reverse Engineer the Problem

This approach involves deconstructing a problem in reverse sequential order from the current undesirable outcome back to the initial root causes.

By mapping the chain of events backward, you can identify the origin of where things went wrong and establish the critical junctures for solving it moving ahead. Reverse engineering provides diagnostic clarity on multi-step problems.

However, the limitation is that it focuses heavily on autopsying the past versus innovating improved future solutions.

An example is tracing back from a server outage, through the cascade of infrastructure failures that led to it finally terminating at the initial script error that triggered the crisis. This root cause would then inform the preventative measure.

9. Break Down Obstacles Between Current and Goal State (Means-End Analysis)

This technique defines the current problem state and the desired end goal state, then systematically identifies obstacles in the way of getting from one to the other.

By mapping the barriers or gaps, you can then develop solutions to address each one. This methodically connects the problem to solutions.

A limitation is that some obstacles may be unknown upfront and only emerge later.

For example, you can list down all the steps required for a new product launch – current state through production, marketing, sales, distribution, etc. to full launch (goal state) – to highlight where resource constraints or other blocks exist so they can be addressed.

Onethread allows dividing big-picture projects into discrete, manageable phases, milestones, and tasks to simplify execution just as problems can be decomposed into more achievable components. Features like dependency mapping further reinforce interconnections.

Using Onethread’s issues and subtasks feature, messy problems can be decomposed into manageable chunks.

10. Ask “Why” Five Times to Identify the Root Cause (The 5 Whys)

This technique involves asking “Why did this problem occur?” and then responding with an answer that is again met with asking “Why?” This process repeats five times until the root cause is revealed.

Continually asking why digs deeper from surface symptoms to underlying systemic issues.

It is effective for getting to the source of problems originating from human error or process breakdowns.

However, some complex issues may have multiple tangled root causes not solvable through this approach alone.

An example is a retail store experiencing a sudden decline in customers. Successively asking why five times may trace an initial drop to parking challenges, stemming from a city construction project – the true starting point to address.

11. Evaluate Strengths, Weaknesses, Opportunities, and Threats (SWOT Analysis)

This involves analyzing a problem or proposed solution by categorizing internal and external factors into a 2×2 matrix: Strengths, Weaknesses as the internal rows; Opportunities and Threats as the external columns.

Systematically identifying these elements provides balanced insight to evaluate options and risks. It is impactful when evaluating alternative solutions or developing strategy amid complexity or uncertainty.

The key benefit of SWOT analysis is enabling multi-dimensional thinking when rationally evaluating options. Rather than getting anchored on just the upsides or the existing way of operating, it urges a systematic assessment through four different lenses:

• Internal Strengths: Our core competencies/advantages able to deliver success
• Internal Weaknesses: Gaps/vulnerabilities we need to manage
• External Opportunities: Ways we can differentiate/drive additional value
• External Threats: Risks we must navigate or mitigate

Multiperspective analysis provides the needed holistic view of the balanced risk vs. reward equation for strategic decision making amid uncertainty.

However, SWOT can feel restrictive if not tailored and evolved for different issue types.

Teams should view SWOT analysis as a starting point, augmenting it further for distinct scenarios.

An example is performing a SWOT analysis on whether a small business should expand into a new market – evaluating internal capabilities to execute vs. risks in the external competitive and demand environment to inform the growth decision with eyes wide open.

12. Compare Current vs Expected Performance (Gap Analysis)

This technique involves comparing the current state of performance, output, or results to the desired or expected levels to highlight shortfalls.

By quantifying the gaps, you can identify problem areas and prioritize address solutions.

Gap analysis is based on the simple principle – “you can’t improve what you don’t measure.” It enables facts-driven problem diagnosis by highlighting delta to goals, not just vague dissatisfaction that something seems wrong. And measurement immediately suggests improvement opportunities – address the biggest gaps first.

This data orientation also supports ROI analysis on fixing issues – the return from closing larger gaps outweighs narrowly targeting smaller performance deficiencies.

However, the approach is only effective if robust standards and metrics exist as the benchmark to evaluate against. Organizations should invest upfront in establishing performance frameworks.

Furthermore, while numbers are invaluable, the human context behind problems should not be ignored – quantitative versus qualitative gap assessment is optimally blended.

For example, if usage declines are noted during software gap analysis, this could be used as a signal to improve user experience through design.

13. Observe Processes from the Frontline (Gemba Walk)

A Gemba walk involves going to the actual place where work is done, directly observing the process, engaging with employees, and finding areas for improvement.

By experiencing firsthand rather than solely reviewing abstract reports, practical problems and ideas emerge.

The limitation is Gemba walks provide anecdotes not statistically significant data. It complements but does not replace comprehensive performance measurement.

An example is a factory manager inspecting the production line to spot jam areas based on direct reality rather than relying on throughput dashboards alone back in her office. Frontline insights prove invaluable.

14. Analyze Competitive Forces (Porter’s Five Forces)

This involves assessing the marketplace around a problem or business situation via five key factors: competitors, new entrants, substitute offerings, suppliers, and customer power.

Evaluating these forces illuminates risks and opportunities for strategy development and issue resolution. It is effective for understanding dynamic external threats and opportunities when operating in a contested space.

However, over-indexing on only external factors can overlook the internal capabilities needed to execute solutions.

A startup CEO, for example, may analyze market entry barriers, whitespace opportunities, and disruption risks across these five forces to shape new product rollout strategies and marketing approaches.

15. Think from Different Perspectives (Six Thinking Hats)

The Six Thinking Hats is a technique developed by Edward de Bono that encourages people to think about a problem from six different perspectives, each represented by a colored “thinking hat.”

The key benefit of this strategy is that it pushes team members to move outside their usual thinking style and consider new angles. This brings more diverse ideas and solutions to the table.

It works best for complex problems that require innovative solutions and when a team is stuck in an unproductive debate. The structured framework keeps the conversation flowing in a positive direction.

Limitations are that it requires training on the method itself and may feel unnatural at first. Team dynamics can also influence success – some members may dominate certain “hats” while others remain quiet.

A real-life example is a software company debating whether to build a new feature. The white hat focuses on facts, red on gut feelings, black on potential risks, yellow on benefits, green on new ideas, and blue on process. This exposes more balanced perspectives before deciding.

Onethread centralizes diverse stakeholder communication onto one platform, ensuring all voices are incorporated when evaluating project tradeoffs, just as problem-solving should consider multifaceted solutions.

16. Visualize the Problem (Draw it Out)

Drawing out a problem involves creating visual representations like diagrams, flowcharts, and maps to work through challenging issues.

This strategy is helpful when dealing with complex situations with lots of interconnected components. The visuals simplify the complexity so you can thoroughly understand the problem and all its nuances.

Key benefits are that it allows more stakeholders to get on the same page regarding root causes and it sparks new creative solutions as connections are made visually.

However, simple problems with few variables don’t require extensive diagrams. Additionally, some challenges are so multidimensional that fully capturing every aspect is difficult.

A real-life example would be mapping out all the possible causes leading to decreased client satisfaction at a law firm. An intricate fishbone diagram with branches for issues like service delivery, technology, facilities, culture, and vendor partnerships allows the team to trace problems back to their origins and brainstorm targeted fixes.

17. Follow a Step-by-Step Procedure (Algorithms)

An algorithm is a predefined step-by-step process that is guaranteed to produce the correct solution if implemented properly.

Using algorithms is effective when facing problems that have clear, binary right and wrong answers. Algorithms work for mathematical calculations, computer code, manufacturing assembly lines, and scientific experiments.

Key benefits are consistency, accuracy, and efficiency. However, they require extensive upfront development and only apply to scenarios with strict parameters. Additionally, human error can lead to mistakes.

For example, crew members of fast food chains like McDonald’s follow specific algorithms for food prep – from grill times to ingredient amounts in sandwiches, to order fulfillment procedures. This ensures uniform quality and service across all locations. However, if a step is missed, errors occur.

The Problem-Solving Process

The problem-solving process typically includes defining the issue, analyzing details, creating solutions, weighing choices, acting, and reviewing results.

In the above, we have discussed several problem-solving strategies. For every problem-solving strategy, you have to follow these processes. Here’s a detailed step-by-step process of effective problem-solving:

Step 1: Identify the Problem

The problem-solving process starts with identifying the problem. This step involves understanding the issue’s nature, its scope, and its impact. Once the problem is clearly defined, it sets the foundation for finding effective solutions.

Identifying the problem is crucial. It means figuring out exactly what needs fixing. This involves looking at the situation closely, understanding what’s wrong, and knowing how it affects things. It’s about asking the right questions to get a clear picture of the issue. 

This step is important because it guides the rest of the problem-solving process. Without a clear understanding of the problem, finding a solution is much harder. It’s like diagnosing an illness before treating it. Once the problem is identified accurately, you can move on to exploring possible solutions and deciding on the best course of action.

Step 2: Break Down the Problem

Breaking down the problem is a key step in the problem-solving process. It involves dividing the main issue into smaller, more manageable parts. This makes it easier to understand and tackle each component one by one.

After identifying the problem, the next step is to break it down. This means splitting the big issue into smaller pieces. It’s like solving a puzzle by handling one piece at a time. 

By doing this, you can focus on each part without feeling overwhelmed. It also helps in identifying the root causes of the problem. Breaking down the problem allows for a clearer analysis and makes finding solutions more straightforward. 

Each smaller problem can be addressed individually, leading to an effective resolution of the overall issue. This approach not only simplifies complex problems but also aids in developing a systematic plan to solve them.

Step 3: Come up with potential solutions

Coming up with potential solutions is the third step in the problem-solving process. It involves brainstorming various options to address the problem, considering creativity and feasibility to find the best approach.

After breaking down the problem, it’s time to think of ways to solve it. This stage is about brainstorming different solutions. You look at the smaller issues you’ve identified and start thinking of ways to fix them. This is where creativity comes in. 

You want to come up with as many ideas as possible, no matter how out-of-the-box they seem. It’s important to consider all options and evaluate their pros and cons. This process allows you to gather a range of possible solutions. 

Later, you can narrow these down to the most practical and effective ones. This step is crucial because it sets the stage for deciding on the best solution to implement. It’s about being open-minded and innovative to tackle the problem effectively.

Step 4: Analyze the possible solutions

Analyzing the possible solutions is the fourth step in the problem-solving process. It involves evaluating each proposed solution’s advantages and disadvantages to determine the most effective and feasible option.

After coming up with potential solutions, the next step is to analyze them. This means looking closely at each idea to see how well it solves the problem. You weigh the pros and cons of every solution.

Consider factors like cost, time, resources, and potential outcomes. This analysis helps in understanding the implications of each option. It’s about being critical and objective, ensuring that the chosen solution is not only effective but also practical.

This step is vital because it guides you towards making an informed decision. It involves comparing the solutions against each other and selecting the one that best addresses the problem.

By thoroughly analyzing the options, you can move forward with confidence, knowing you’ve chosen the best path to solve the issue.

Step 5: Implement and Monitor the Solutions

Implementing and monitoring the solutions is the final step in the problem-solving process. It involves putting the chosen solution into action and observing its effectiveness, making adjustments as necessary.

Once you’ve selected the best solution, it’s time to put it into practice. This step is about action. You implement the chosen solution and then keep an eye on how it works. Monitoring is crucial because it tells you if the solution is solving the problem as expected. 

If things don’t go as planned, you may need to make some changes. This could mean tweaking the current solution or trying a different one. The goal is to ensure the problem is fully resolved. 

This step is critical because it involves real-world application. It’s not just about planning; it’s about doing and adjusting based on results. By effectively implementing and monitoring the solutions, you can achieve the desired outcome and solve the problem successfully.

Why This Process is Important

Following a defined process to solve problems is important because it provides a systematic, structured approach instead of a haphazard one. Having clear steps guides logical thinking, analysis, and decision-making to increase effectiveness. Key reasons it helps are:

• Clear Direction: This process gives you a clear path to follow, which can make solving problems less overwhelming.
• Better Solutions: Thoughtful analysis of root causes, iterative testing of solutions, and learning orientation lead to addressing the heart of issues rather than just symptoms.
• Saves Time and Energy: Instead of guessing or trying random things, this process helps you find a solution more efficiently.
• Improves Skills: The more you use this process, the better you get at solving problems. It’s like practicing a sport. The more you practice, the better you play.
• Maximizes collaboration: Involving various stakeholders in the process enables broader inputs. Their communication and coordination are streamlined through organized brainstorming and evaluation.
• Provides consistency: Standard methodology across problems enables building institutional problem-solving capabilities over time. Patterns emerge on effective techniques to apply to different situations.

The problem-solving process is a powerful tool that can help us tackle any challenge we face. By following these steps, we can find solutions that work and learn important skills along the way.

Key Skills for Efficient Problem Solving

Efficient problem-solving requires breaking down issues logically, evaluating options, and implementing practical solutions.

Key skills include critical thinking to understand root causes, creativity to brainstorm innovative ideas, communication abilities to collaborate with others, and decision-making to select the best way forward. Staying adaptable, reflecting on outcomes, and applying lessons learned are also essential.

With practice, these capacities will lead to increased personal and team effectiveness in systematically addressing any problem.

 Let’s explore the powers you need to become a problem-solving hero!

Critical Thinking and Analytical Skills

Critical thinking and analytical skills are vital for efficient problem-solving as they enable individuals to objectively evaluate information, identify key issues, and generate effective solutions. 

These skills facilitate a deeper understanding of problems, leading to logical, well-reasoned decisions. By systematically breaking down complex issues and considering various perspectives, individuals can develop more innovative and practical solutions, enhancing their problem-solving effectiveness.

Communication Skills

Effective communication skills are essential for efficient problem-solving as they facilitate clear sharing of information, ensuring all team members understand the problem and proposed solutions. 

These skills enable individuals to articulate issues, listen actively, and collaborate effectively, fostering a productive environment where diverse ideas can be exchanged and refined. By enhancing mutual understanding, communication skills contribute significantly to identifying and implementing the most viable solutions.

Decision-Making

Strong decision-making skills are crucial for efficient problem-solving, as they enable individuals to choose the best course of action from multiple alternatives. 

These skills involve evaluating the potential outcomes of different solutions, considering the risks and benefits, and making informed choices. Effective decision-making leads to the implementation of solutions that are likely to resolve problems effectively, ensuring resources are used efficiently and goals are achieved.

Planning and Prioritization

Planning and prioritization are key for efficient problem-solving, ensuring resources are allocated effectively to address the most critical issues first. This approach helps in organizing tasks according to their urgency and impact, streamlining efforts towards achieving the desired outcome efficiently.

Emotional Intelligence

Emotional intelligence enhances problem-solving by allowing individuals to manage emotions, understand others, and navigate social complexities. It fosters a positive, collaborative environment, essential for generating creative solutions and making informed, empathetic decisions.

Leadership skills drive efficient problem-solving by inspiring and guiding teams toward common goals. Effective leaders motivate their teams, foster innovation, and navigate challenges, ensuring collective efforts are focused and productive in addressing problems.

Time Management

Time management is crucial in problem-solving, enabling individuals to allocate appropriate time to each task. By efficiently managing time, one can ensure that critical problems are addressed promptly without neglecting other responsibilities.

Data Analysis

Data analysis skills are essential for problem-solving, as they enable individuals to sift through data, identify trends, and extract actionable insights. This analytical approach supports evidence-based decision-making, leading to more accurate and effective solutions.

Research Skills

Research skills are vital for efficient problem-solving, allowing individuals to gather relevant information, explore various solutions, and understand the problem’s context. This thorough exploration aids in developing well-informed, innovative solutions.

Becoming a great problem solver takes practice, but with these skills, you’re on your way to becoming a problem-solving hero. 

How to Improve Your Problem-Solving Skills?

Improving your problem-solving skills can make you a master at overcoming challenges. Learn from experts, practice regularly, welcome feedback, try new methods, experiment, and study others’ success to become better.

Learning from Experts

Improving problem-solving skills by learning from experts involves seeking mentorship, attending workshops, and studying case studies. Experts provide insights and techniques that refine your approach, enhancing your ability to tackle complex problems effectively.

To enhance your problem-solving skills, learning from experts can be incredibly beneficial. Engaging with mentors, participating in specialized workshops, and analyzing case studies from seasoned professionals can offer valuable perspectives and strategies. 

Experts share their experiences, mistakes, and successes, providing practical knowledge that can be applied to your own problem-solving process. This exposure not only broadens your understanding but also introduces you to diverse methods and approaches, enabling you to tackle challenges more efficiently and creatively.

Improving problem-solving skills through practice involves tackling a variety of challenges regularly. This hands-on approach helps in refining techniques and strategies, making you more adept at identifying and solving problems efficiently.

One of the most effective ways to enhance your problem-solving skills is through consistent practice. By engaging with different types of problems on a regular basis, you develop a deeper understanding of various strategies and how they can be applied. 

This hands-on experience allows you to experiment with different approaches, learn from mistakes, and build confidence in your ability to tackle challenges.

Regular practice not only sharpens your analytical and critical thinking skills but also encourages adaptability and innovation, key components of effective problem-solving.

Openness to Feedback

Being open to feedback is like unlocking a secret level in a game. It helps you boost your problem-solving skills. Improving problem-solving skills through openness to feedback involves actively seeking and constructively responding to critiques. 

This receptivity enables you to refine your strategies and approaches based on insights from others, leading to more effective solutions. 

Learning New Approaches and Methodologies

Learning new approaches and methodologies is like adding new tools to your toolbox. It makes you a smarter problem-solver. Enhancing problem-solving skills by learning new approaches and methodologies involves staying updated with the latest trends and techniques in your field. 

This continuous learning expands your toolkit, enabling innovative solutions and a fresh perspective on challenges.

Experimentation

Experimentation is like being a scientist of your own problems. It’s a powerful way to improve your problem-solving skills. Boosting problem-solving skills through experimentation means trying out different solutions to see what works best. This trial-and-error approach fosters creativity and can lead to unique solutions that wouldn’t have been considered otherwise.

Analyzing Competitors’ Success

Analyzing competitors’ success is like being a detective. It’s a smart way to boost your problem-solving skills. Improving problem-solving skills by analyzing competitors’ success involves studying their strategies and outcomes. Understanding what worked for them can provide valuable insights and inspire effective solutions for your own challenges. 

Challenges in Problem-Solving

Facing obstacles when solving problems is common. Recognizing these barriers, like fear of failure or lack of information, helps us find ways around them for better solutions.

Fear of Failure

Fear of failure is like a big, scary monster that stops us from solving problems. It’s a challenge many face. Because being afraid of making mistakes can make us too scared to try new solutions. 

How can we overcome this? First, understand that it’s okay to fail. Failure is not the opposite of success; it’s part of learning. Every time we fail, we discover one more way not to solve a problem, getting us closer to the right solution. Treat each attempt like an experiment. It’s not about failing; it’s about testing and learning.

Lack of Information

Lack of information is like trying to solve a puzzle with missing pieces. It’s a big challenge in problem-solving. Because without all the necessary details, finding a solution is much harder. 

How can we fix this? Start by gathering as much information as you can. Ask questions, do research, or talk to experts. Think of yourself as a detective looking for clues. The more information you collect, the clearer the picture becomes. Then, use what you’ve learned to think of solutions. 

Fixed Mindset

A fixed mindset is like being stuck in quicksand; it makes solving problems harder. It means thinking you can’t improve or learn new ways to solve issues. 

How can we change this? First, believe that you can grow and learn from challenges. Think of your brain as a muscle that gets stronger every time you use it. When you face a problem, instead of saying “I can’t do this,” try thinking, “I can’t do this yet.” Look for lessons in every challenge and celebrate small wins. 

Everyone starts somewhere, and mistakes are just steps on the path to getting better. By shifting to a growth mindset, you’ll see problems as opportunities to grow. Keep trying, keep learning, and your problem-solving skills will soar!

Jumping to Conclusions

Jumping to conclusions is like trying to finish a race before it starts. It’s a challenge in problem-solving. That means making a decision too quickly without looking at all the facts. 

How can we avoid this? First, take a deep breath and slow down. Think about the problem like a puzzle. You need to see all the pieces before you know where they go. Ask questions, gather information, and consider different possibilities. Don’t choose the first solution that comes to mind. Instead, compare a few options. 

Feeling Overwhelmed

Feeling overwhelmed is like being buried under a mountain of puzzles. It’s a big challenge in problem-solving. When we’re overwhelmed, everything seems too hard to handle. 

How can we deal with this? Start by taking a step back. Breathe deeply and focus on one thing at a time. Break the big problem into smaller pieces, like sorting puzzle pieces by color. Tackle each small piece one by one. It’s also okay to ask for help. Sometimes, talking to someone else can give you a new perspective. 

Confirmation Bias

Confirmation bias is like wearing glasses that only let you see what you want to see. It’s a challenge in problem-solving. Because it makes us focus only on information that agrees with what we already believe, ignoring anything that doesn’t. 

How can we overcome this? First, be aware that you might be doing it. It’s like checking if your glasses are on right. Then, purposely look for information that challenges your views. It’s like trying on a different pair of glasses to see a new perspective. Ask questions and listen to answers, even if they don’t fit what you thought before.

Groupthink is like everyone in a group deciding to wear the same outfit without asking why. It’s a challenge in problem-solving. It means making decisions just because everyone else agrees, without really thinking it through. 

How can we avoid this? First, encourage everyone in the group to share their ideas, even if they’re different. It’s like inviting everyone to show their unique style of clothes. 

Listen to all opinions and discuss them. It’s okay to disagree; it helps us think of better solutions. Also, sometimes, ask someone outside the group for their thoughts. They might see something everyone in the group missed.

Overcoming obstacles in problem-solving requires patience, openness, and a willingness to learn from mistakes. By recognizing these barriers, we can develop strategies to navigate around them, leading to more effective and creative solutions.

What are the most common problem-solving techniques?

The most common techniques include brainstorming, the 5 Whys, mind mapping, SWOT analysis, and using algorithms or heuristics. Each approach has its strengths, suitable for different types of problems.

What’s the best problem-solving strategy for every situation?

There’s no one-size-fits-all strategy. The best approach depends on the problem’s complexity, available resources, and time constraints. Combining multiple techniques often yields the best results.

How can I improve my problem-solving skills?

Improve your problem-solving skills by practicing regularly, learning from experts, staying open to feedback, and continuously updating your knowledge on new approaches and methodologies.

Are there any tools or resources to help with problem-solving?

Yes, tools like mind mapping software, online courses on critical thinking, and books on problem-solving techniques can be very helpful. Joining forums or groups focused on problem-solving can also provide support and insights.

What are some common mistakes people make when solving problems?

Common mistakes include jumping to conclusions without fully understanding the problem, ignoring valuable feedback, sticking to familiar solutions without considering alternatives, and not breaking down complex problems into manageable parts.

Final Words

Mastering problem-solving strategies equips us with the tools to tackle challenges across all areas of life. By understanding and applying these techniques, embracing a growth mindset, and learning from both successes and obstacles, we can transform problems into opportunities for growth. Continuously improving these skills ensures we’re prepared to face and solve future challenges more effectively.

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9 Ways to Improve Math Skills Quickly & Effectively

Written by Ashley Crowe

• Parent Resources

• The importance of understanding basic math skills
• 9 Ways to improve math skills
• How to use technology to improve math skills

Math class can move pretty fast. There’s so much to cover in the course of a school year. And if your child doesn’t get a new math idea right away, they can quickly get left behind.

If your child is struggling with basic math problems every day, it doesn’t mean they’re destined to be bad at math. Some students need more time to develop the problem-solving skills that math requires. Others may need to revisit past concepts before moving on. Because of how math is structured, it’s best to take each year step-by-step, lesson by lesson.

This article has tips and tricks to improve your child’s math skills while minimizing frustrations and struggles. If your child is growing to hate math, read on for ways to improve their skills and confidence, and maybe even make math fun! 

But first, the basics.

Math is a subject that builds on itself. It takes a solid understanding of past concepts to prepare for the next lesson. 

That’s why math can become frustrating when you’re forced to move on before you’re ready. You’re either stuck trying to catch up or you end up falling further behind.

But with a strong understanding of basic math skills, your child can be set up for school success. If you’re unfamiliar with the idea of sets or whole numbers , this is a great place to start. 

What are considered basic math skills?

The basic math skills required to move on to higher levels of math learning are: 

• Addition — Adding to a set.
• Subtraction — Taking away from a set.
• Multiplication — Adding equal sets together in groups (2 sets of 3 is the same as 2x3, or 6).
• Division — How many equal sets can be found in a number (12 has how many sets of two in it? 6 sets of 2).
• Percentages — A specific amount in relation to 100.
• Fractions & Decimals — Fractions are equal parts of a whole set. Decimals represent a number of parts of a whole in relation to 10. These both contrast with whole numbers. 
• Spatial Reasoning — How numbers and shapes fit together.

How to improve math skills 

People aren’t bad at math — many just need more time and practice to gain a thorough understanding.

How can you help your child improve their math abilities? Use our top 9 tips for quickly and effectively improving math skills .

1. Wrap your head around the concepts

Repetition and practice are great, but if you don’t understand the concept , it will be difficult to move forward. 

Luckily, there are many great ways to break down math concepts . The trick is finding the one that works best for your child.

Math manipulatives can be a game-changer for children who are struggling with big math ideas. Taking math off the page and putting it into their hands can bring ideas to life. Numbers become less abstract and more concrete when you’re counting toy cars or playing with blocks. Creating these “sets” of objects can bring clarity to basic math learning.

2. Try game-based learning

During math practice, repetition is important — but it can get old in a hurry. No one enjoys copying their times tables over and over and over again. If learning math has become a chore, it’s time to bring back the fun! 

Game-based learning is a great way to practice new concepts and solidify past lessons. It can even make repetition fun and engaging.

Game-based learning can look like a family board game on Friday night or an educational app , like Prodigy Math .

Take math from frustrating to fun with the right game, then watch the learning happen easily!

3. Bring math into daily life

You use basic math every day. 

As you go about your day, help your child see the math that’s all around them:  

• Tell them how fast you’re driving on the way to school
• Calculate the discount you’ll receive on your next Target trip
• Count out the number of apples you need to buy at the grocery store
• While baking, explain how 6 quarter cups is the same amount of flour as a cup and a half — then enjoy some cookies!

Relate math back to what your child loves and show them how it’s used every day. Math doesn’t have to be mysterious or abstract. Instead, use math to race monster trucks or arrange tea parties. Break it down, take away the fear, and watch their interest in math grow.

4. Implement daily practice

Math practice is important. Once you understand the concept, you have to nail down the mechanics. And often, it’s the practice that finally helps the concept click. Either way, math requires more than just reading formulas on a page.

Daily practice can be tough to implement, especially with a math-averse child. This is a great time to bring out the game-based learning mentioned above. Or find an activity that lines up with their current lesson. Are they learning about squares? Break out the math link cubes and create them. Whenever possible, step away from the worksheets and flashcards and find practice elsewhere.

5. Sketch word problems

Nothing causes a panic quite like an unexpected word problem. Something about the combination of numbers and words can cause the brain of a struggling math learner to shut down. But it doesn’t have to be that way.

Many word problems just need to be broken down, step by step . One great way to do this is to sketch it out. If Doug has five apples and four oranges, then eats two of each, how many does he have left? Draw it, talk it out, cross them off, then count. 

If you’ve been talking your child through the various math challenges you encounter every day, many word problems will start to feel familiar. 

6. Set realistic goals

If your child has fallen behind in math, then more study time is the answer. But forcing them to cram an extra hour of math in their day is not likely to produce better results. To see a positive change, first identify their biggest struggles . Then set realistic goals addressing these issues . 

Two more hours of practicing a concept they don’t understand is only going to cause more frustration. Even if they can work through the mechanics of a problem, the next lesson will leave them feeling just as lost. 

Instead, try mini practice sessions and enlist some extra help. Approach the problem in a new way, reach out to their teacher or try an online math lesson . Make sure the extra time is troubleshooting the actual problem, not just reinforcing the idea that math is hard and no fun. 

Set Goals and Rewards in Prodigy Math

Did you know that parents can set learning goals for their child in Prodigy Math? And once they achieve them, they'll unlock in-game rewards of your choice!

7. Engage with a math tutor

If your child is struggling with big picture concepts, look into finding a math tutor . Everyone learns differently, and you and your child’s teacher may be missing that “aha” moment that a little extra time and the right tutor can provide.

It’s amazing when a piece of the math puzzle finally clicks for your child. If you’re ready to get that extra help, try a free 1:1 online session from Prodigy Math Tutoring. Prodigy’s tutors are real teachers who know how to connect kids to math. With the right approach, your child can become confident in math — and who knows, they may even begin to enjoy it. 

8. Focus on one concept at a time

Math builds on itself. If your child is struggling through their current lesson, they can’t skip it and come back to it later. This is the time to practice and repeat — re-examining and reinforcing the current concept until it makes sense.

Look for other ways to approach new math ideas. Use math manipulatives to bring numbers off the page. Or try a learning app with exciting rewards and positive reinforcement to encourage extra practice. 

Take a step back when frustrations get high — but resist the temptation to just let it go. Once the concept clicks, they’ll be excited to forge ahead.

9. Teach others math you already know

Even if your child is struggling in math, they’ve still learned so much since last year. Focus on the improvements they’ve made and let them showcase their knowledge. If they have younger siblings, your older child can demonstrate addition or show them how to use a number line. This is a great way to build their confidence and encourage them to keep going.

Or let them teach you how they solve new problems. Have your child talk you through the process while you solve a long division problem . You’re likely to find yourself a little rusty on the details. Play it up and get a little silly. They’ll love teaching you the ropes of this “new math.”

Embracing technology to improve math skills

Though much of your math learning was done with pencil to paper, there are many more ways to build number skills in today’s tech world. 

Your child can take live, online math courses to work through tough concepts. Or play a variety of online games, solving math puzzles and getting consistent practice while having fun.

These technical advances can help every child learn math, no matter their preferred learning or study style. If your child is a visual learner, there’s an app for that. Do they process best while working in groups? Jump online and find one. Don’t keep repeating the same lessons from their math class over and over. Branch out, try something new and watch the learning click. 

Look online for more math help

There are so many online resources, it can be hard to know where to start. 

At Prodigy, we’re happy to help you get the ball rolling on your child’s math learning, from kindergarten through 8th grade. It’s free to sign up, fun to play and exciting to watch as your child’s math understanding grows.

Sign up for a free parent account and get instant data on your child’s progress as they build more math skills with Prodigy Math Game . It’s time to take the math struggle out of your home and enjoy learning together!

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Improving your problem-solving skills with learning games for adults

We face problems every day . Whether it's a complex problem at work or a personal issue that needs solving, good problem-solving skills are essential for success in both your personal and professional life. 

If you’re feeling a little rusty in the problem-solving department, there are many ways to enhance your problem-solving abilities, like cognitive training techniques and brain games. That’s right: Games can help improve your cognitive abilities like processing speed, reasoning, and working memory , which are essential for effective problem-solving. 

So if you’re ready to learn how to improve your problem-solving skills with some of our recommended cognitive training techniques and tips, keep reading. And you’ll be making quicker, more confident decisions in no time. 

What is problem solving, and why is it important?

Problem-solving is, well, the process of identifying, defining, and finding a solution to challenges or difficulties. It involves several steps, including recognizing the existence of a problem, understanding its nature, generating potential solutions, evaluating those solutions, and then implementing the best one. 

Problem-solving is an essential skill that enables you to navigate various aspects of your personal and professional lives effectively. In your workplace, for example, you can quickly identify issues and implement appropriate solutions, contributing to increased productivity and efficiency. In your personal life, good problem-solving skills can help you navigate relationships, make informed decisions, and cope with unexpected situations.

Good problem-solving skills not only help you make better decisions but also improve your critical thinking abilities, allowing you to find effective solutions to complex problems. And by developing and honing your problem-solving skills through cognitive training, you can become more adaptable and resourceful, capable of tackling a wide range of challenges that life throws your way. 

The science behind cognitive training for improving problem solving

So, what is cognitive training? And what does science have to say about it? 

Cognitive training involves a range of activities and exercises that target different cognitive functions. These may include puzzles, memory exercises, or brain games that require strategic thinking. The goal is to stimulate your brain and enhance its ability to process information, reason effectively, and retain information. By engaging in cognitive training , you can boost your mental capabilities and improve your overall problem-solving skills.

As you engage in cognitive training exercises, you’ll experience improvements in processing speed (the ability to absorb and process information quickly), reasoning (logical thinking and decision-making), and working memory (the capacity to hold and manipulate information over short periods). These enhanced cognitive abilities directly contribute to more effective problem-solving skills.

By understanding the principles behind cognitive training and consistently practicing these types of exercises, you can enhance your problem-solving abilities and apply these skills in various aspects of your lives. But not before you learn how to identify problems, which is a key first step to finding effective solutions. 

The problem-solving process

Effective problem identification is a crucial first step in the problem-solving process. Here’s how to do it: 

• Define the Problem: Clearly articulating the issue at hand is essential for understanding its scope and complexity. So take time to describe the problem in detail, considering the context, constraints, and possible repercussions.
• Gather Information: Collect relevant data and information about the problem. This may involve research, consulting with experts, or seeking input from those affected by the issue. Having accurate and comprehensive information is critical for informed decision-making during the problem-solving process.
• Involve Others: Collaborate with your team or other people to ensure diverse perspectives and insights are considered. A good idea can come from everywhere, and a collective approach can lead to more innovative and effective solutions.
• Identify Root Causes: Once the problem is defined, delve deeper to identify its underlying causes. Use techniques such as the "5 Whys" method or cause-and-effect analysis to pinpoint the factors contributing to the issue. Addressing these root causes is crucial for developing long-term, sustainable solutions.
• Select a Problem-Solving Strategy: Employ various problem-solving methods to devise a solution that tackles the root causes effectively. These may include brainstorming, evaluating pros and cons, or implementing a trial-and-error approach. The strategy you ultimately choose should be adaptable and considerate of potential challenges or obstacles.

By following these tips for problem identification and employing problem-solving techniques, you can increase your chances of finding effective and lasting solutions to the issues you face.

6 ways to improve your critical thinking skills

Here’s the truth: You can’t effectively solve a problem without using your critical thinking skills. 

Critical thinking is the process of objectively analyzing information, evaluating the credibility of arguments, and making informed decisions based on logic and reasoning. It involves things like questioning assumptions, considering multiple perspectives, and weighing evidence before reaching a conclusion.

Think about it: Having the ability to analyze information, evaluate arguments, and make reasoned decisions allows you to approach problems logically —and we have a few tips to help you improve your ability to do just that: 

• Break Down Information: To sharpen your critical thinking abilities, practice breaking down complex information into smaller components. Identify patterns, relationships, and underlying principles that can help you better understand the situation.
• Evaluate Arguments: Develop the habit of assessing the credibility and relevance of arguments presented to you. Consider the source of the information, identify any potential biases, and scrutinize the validity of the evidence provided.
• Make Reasoned Decisions: When faced with a decision, take time to gather all relevant information and consider possible outcomes. Weigh the pros and cons before arriving at a well-reasoned conclusion that takes into account both short-term and long-term consequences. (We love a good pros and cons list.) 
• Play Brain Games: Regularly engaging in brain games such as Sudoku, crosswords, chess, or logic puzzles can be an effective way to enhance critical thinking skills. These games require you to analyze information, evaluate potential moves or solutions, and make strategic decisions based on reasoning. We’ll go into more detail about this later, so hang tight. 
• Try Mindfulness Meditation: Mindfulness meditation involves focusing on the present moment while calmly acknowledging and accepting your thoughts and feelings. Practicing mindfulness can enhance attention, concentration, and emotional regulation, all of which are critical for effective problem-solving. And if you're interested, you can try it for free for an entire year with the Balance app .
• Consider Cognitive-Behavioral Therapy (CBT): CBT is a psychotherapy technique that helps you identify and change negative thought patterns and behaviors. By learning to recognize unproductive thinking habits, you can develop more constructive approaches to problem-solving.

By exploring these various cognitive training techniques and consistently incorporating them into your daily life, you’ll be well on your way to enhancing your problem-solving skills and tackling life's challenges more logically and effectively.

How to approach problems with a critical mindset

Approaching problems with a critical mindset is a great way to turn critical thinking into a habit. But what does that mean, and how do you do it? Let’s break it down:  

• Embrace Critical Thinking: Develop the habit of questioning assumptions and challenging conventional wisdom when faced with a problem. This will help to uncover hidden biases or overlooked factors that may influence the issue at hand.
• Consider Multiple Perspectives: Explore different viewpoints and perspectives when assessing a problem. This allows for a more comprehensive understanding of the situation and can lead to innovative solutions that might not have been apparent from a single viewpoint.
• Evaluate Evidence: Gather relevant information and carefully evaluate its credibility and reliability. Assess the strength of the evidence supporting various arguments or positions before making a decision.

By following these tips, you can develop a critical mindset that habitually enables you to approach problems more effectively, leading to well-informed decisions and lasting solutions.

Problem solving methods and techniques

Now that you know a bit about how to approach a problem, here’s how you can implement these problem-solving techniques in your daily life:

• Understand the Context: When applying problem-solving techniques in different settings, it's essential to consider the unique context and constraints of each situation. The approach that works well in a professional environment may not be suitable for a personal issue, so tailor your strategies accordingly.
• Adapt and Be Flexible: Effective problem-solving requires adaptability and flexibility. Be open to changing your approach if circumstances shift or new information emerges. This willingness to adapt will help you find solutions that are relevant and sustainable in the long term.
• Communicate and Collaborate: In both workplace and personal settings, communication and collaboration are key to successful problem-solving. Share your thoughts, ideas, and concerns with team members or stakeholders, and actively seek their input. A diverse array of perspectives can lead to more innovative and effective solutions.
• Learn from Experience: Reflect on past problem-solving experiences and learn from both successes and failures. Apply these lessons to future situations to continuously improve your problem-solving skills.
• Practice Regularly: To develop strong problem-solving abilities, practice regularly by tackling problems in various aspects of your life. The more you practice, the more adept you'll become at identifying problems, generating solutions, and making well-informed decisions.

How to practice effective decision-making

By now, you know how to approach a problem. But how do you solve one? 

Effective decision-making skills are closely related to problem-solving skills, and the two can work together to help you achieve better results. So the next time you have to make a decision, give these steps a try: 

• Gather Information: Just as you need to gather information to understand a problem, you also need to gather information to make informed decisions. This may involve conducting research into various options, consulting with experts, or seeking input from those affected by the issue. Comprehensive and accurate information is crucial for evaluating potential solutions.
• Evaluate Options: Once you have gathered enough information, carefully assess the different options available to address the problem. Consider factors such as feasibility, impact, costs, and potential risks when weighing the pros and cons of each alternative.
• Make a Decision: After evaluating the options, select a solution based on the available information and your assessment of its effectiveness in addressing the root causes of the problem. Ensure that your chosen solution is sustainable in the long term and takes into account any potential challenges or obstacles that may arise.
• Monitor Outcomes: Track the outcomes of your decision to gauge its effectiveness and learn from the results. Be prepared to reassess and adjust your approach if necessary, based on feedback or changing circumstances.
• Refine Your Decision-Making Skills: Continuously work on improving your decision-making abilities by reflecting on past decisions, learning from both successes and failures, and seeking opportunities to practice these skills in various aspects of your life.

The result of putting this into action? Better outcomes and greater success. That’s a win-win if we ever saw one. 

Benefits of brain games for improving problem-solving skills

One fun way to improve all of these problem-solving and decision-making skills we’ve discussed is by playing brain games. 

Brain games stimulate your mind and foster the development of various cognitive abilities like processing speed, reasoning, and working memory, which are all essential for effective problem-solving. 

These games challenge you to think critically and make decisions based on logic and strategy. And as a result, they help cultivate a more agile and adaptable mindset that is valuable for tackling real-life problems. (Did we mention they’re also fun?) 

One popular brain training app that incorporates a wide variety of games is Elevate. 

With more than 40 games spread across math , reading , writing , speaking , and memory skills , the Elevate app offers personalized training programs based on your goals, and it adapts to your skill level and performance over time. 

By incorporating brain games into daily routines or cognitive training programs, you’ll be able to make big improvements in your critical thinking and problem-solving skills, making it easier to tackle challenges in both personal and professional aspects of your life. Oh, and did we mention they’re also fun to play?

Improve your problem-solving skills with learning games for adults

By knowing how to identify a problem, approach it with a critical mindset, and implement a few key problem-solving techniques, you’ll be able to tackle your next challenge with ease. 

And if you’re ready to up-level your overall problem-solving skills with the help of brain training games, download the Elevate app on iOS or Android today and discover 40+ brain training games, personalized training programs, and expert guidance to help you optimize your cognitive abilities and improve your overall performance in daily life. 

With the Elevate app, you can take control of your cognitive function and become a more effective problem solver. It’s what we like to call a no-brainer decision! 

Enhancing your cognitive abilities

• Learn what brain training is, its benefits, and how you can easily get started training your brain. 

The science of cognitive training

• Training cognitive skills can improve brain function. Think of it like a workout for your mind. Read on to learn how brain games can help. 

The importance of mental fitness

• Mental fitness refers to your ability to sustain your overall well-being. Learn tips to improve yours. 

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How to Improve Problem Solving Skills [10 Ways]

While it might seem like some people are just born with stronger problem-solving skills, there are strategies that anyone can use to improve them.

That’s right, it’s possible to significantly enhance your abilities in this area — and the best part is, most of these activities are also pretty fun!

What Are Problem Solving Skills?

Before we get to the engaging activities, let’s refine our understanding of problem-solving skills, which are any techniques that help you consistently:

• Understand the causes of problems
• Overcome short-term crises
• Create strategies to solve longer-term problems
• Turn problems into opportunities

What Problem Solving Skills Should I Have?

You’ll be able to solve problems in your role better as you grow in your industry-specific knowledge. But there are also a few universal problem solving skills we all need:

• Defining the Problem: Deeply understanding a problem through research , leading to better solutions. Research can include interviewing, reading books and emails, analyzing financial data, searching your organization’s intranet, and organizing your findings.
• Brainstorming: Creating a myriad of new solutions quickly. In group brainstorms, allow everyone to state ideas. Appreciate all input, and avoid criticism. Then, organize solutions into groups around common themes.
• Analyzing: Using disciplined thought processes to evaluate each possible solution. Besides listing their costs and benefits, you might apply deductive reasoning, game theory, and the rules of logic (including fallacies) to them.
• Managing Risk: Anticipating and trying to avoid the downsides of key solutions. Your team can list potential risks, rate how likely each is, predict a date by which each might either happen or no longer be an issue, and devise ways to reduce those risks.
• Deciding: The ability to decide on a solution and move forward with it. After an appropriate amount of time, an analysis of possible solutions, and feedback from team members, a designated decider must choose and implement a solution.
• Managing Emotions: Applying emotional intelligence in order to improve your and your team members’ ability to think clearly. This requires you to recognize emotions in yourself and others, manage feelings, and channel emotions into useful work.

10 Exciting Ways to Improve Problem Solving Skills

Use these ten creative ways to improve problem solving skills, develop more strategic ways of thinking , and train your brain to do more.

1. Dance Your Heart Out

Did you know that dancing has a positive impact on neural processing, possibly developing new neural pathways to go around dopamine-depleted blockages in the brain?

This means that if you engage in ballet or another form of structured dance, doing so may facilitate convergent thinking . In other words, it may help you find a single, appropriate answer to a problem. If you need help with divergent thinking (finding multiple answers to a problem), engaging in more improvised types of dance such as hip-hop or tap might just do the trick.

2. Work out Your Brain with Logic Puzzles or Games

The winning strategy when playing chess, Sudoku, a Rubik’s Cube, or other brain-boosting games is actually to work the problem backward, not forward. The same strategy can apply to realistic strategic-thinking situations.

To build up your brain muscle and develop new problem-solving techniques, practice some logic puzzles and other games .

3. Get a Good Night’s Sleep

More than any other sleeping or awake state, Rapid Eye Movement (REM) sleep directly enhances creative processing in the brain. REM sleep helps “stimulate associative networks, allowing the brain to make new and useful associations between unrelated ideas” and are “not due to selective memory enhancements” such as memory consolidation, which occurs when awake.

4. Work out to Some Tunes

A study of cardiac rehabilitation patients tested verbal fluency after exercising with and without music. Results showed that when they listened to music while working out, participants more than doubled their scores on verbal fluency tests in contrast to when they worked out in silence. According to the study’s lead author, “The combination of music and exercise may stimulate and increase cognitive arousal while helping to organize the cognitive output.”

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5. keep an “idea journal” with you, 6. participate in yoga.

The powerful combination of body awareness, breathing, and meditation that is required during yoga practice has been shown to significantly raise cognitive test scores. Other results from a University of Illinois study include shorter reaction times, more accuracy, and increased attention.

7. Eat Some Cheerios (And Then Think About It)

The Cheerios Effect is the name physicists have given to the event that happens when the last few cheerios in a bowl always cling to each other. The cause of this occurrence is surface tension.

The takeaway is that when it comes to experiencing tension while trying to solve a problem, cling to those around you. Rely on others’ experiences and ideas, even those from different career fields. Draw connections. Brainstorm. Work together to get the job done.

8. Use Mind Maps to Help Visualize the Problem

Mind Maps , a visual snapshot of a problem and its possible solutions, can help focus the mind, stimulate the brain, increase the capacity for creative thinking, and generate more ideas for solutions.

Make a Mind Map by drawing your problem as the central idea. Add “main branches” consisting of all the reasons for the problem. Use “sub-branches” to explore further details.

Next, make a separate Mind Map of all possible solutions to the central problem. Add “main branches” showing all the ways that your problem can be solved, such as colleagues that can help, techniques you can apply, and other resources you can use. Add “sub-branches” to further explore the details. Make a final branch with the most suitable solution for the main problem. Use “sub-branches” for details.

Through this exercise, you should be able to see which “branch” or option is the most practical, time-saving, and cost-effective problem solving method .

9. Create “Psychological Distance”

What is psychological distance? According to the construal level theory (CLT), it’s “anything that we do not experience as occurring now, here, and to ourselves.” Some examples include taking another person’s perspective or thinking of the problem as unlikely.

Scientists have shown that by increasing the mental distance between us and our problem, we’ll have an increase in creative solutions. This happens because thinking more abstractly helps us form unexpected connections between seemingly unrelated concepts, thus allowing our minds to increase its problem solving capacity.

10. Play Some Soccer

A link has been found between our brain’s “executive functions” and sports success . When in action, our brains are quickly multitasking between moving, anticipating, strategizing, reacting, and performing. Doing all these things at once requires an enormous amount of brain activity.

This can be related to our working world when we plan, reason, monitor our actions and problem solve all at once. Therefore, it may be concluded that when you play soccer or any other fast-moving sport, you’re rewiring your brain to be quicker at thinking, processing, and reacting to problems.

To learn more about how to develop your problem-solving and decision making capabilities or to receive training on applied strategic thinking skills , contact CMOE today!

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7 Ways To Increase Accuracy in Quantitative Aptitude

Whenever you attempt to give an examination, be it for placement or competitive exams. You must have seen that Quantitative Aptitude is included in the exam syllabus. Well, it’s one of the most crucial elements of the placement examinations like TCS, Wipro, Capgemini, Accenture, Infosys, IBM, Cognizant, Adobe, Cisco, Amazon, Samsung, Google, Yahoo, Facebook, Ola, Paypal, Microsoft, Oracle, etc and is required for many competitive exams too such as AMCAT, TCS CodeVita, NIMCET, CoCubes, Govt. Bank Exams like SSC , SBI PO , IBPS Clerk, PO Clerk, JEE, and  NEET. Knowing Quantitative Aptitude allows candidates like you to exhibit to employers their critical thinking, problem-solving, and decision-making abilities.

However, because of the length and intricacy of the calculations, many students fail to finish the Quantitative Aptitude portions on time. This is why candidates often look out for ways or methods they can increase their Quantitative Aptitude Accuracy.

In this article, we’ll tell you how to improve your quantitative aptitude by understanding and identifying the approaches and techniques.

If you want to excel in competitive exams, and interview rounds, You can explore our Quantitative Aptitude Online Course .

How to Increase Accuracy In Quantitative Aptitude

Here are 7 ways that will help you improve your accuracy rate while attempting Quantitative Aptitude – 

1. Have a Thorough Understanding of the Exam Syllabus 

The first phase of every major project has always been the most important. Because the first governs the general course of the journey. The Quants examination follows the same path. Without a doubt, it is a demanding endeavor with many ups and downs as well as substantial problems.

The  Quantitative Aptitude syllabus is extensive, diverse, and, in some cases, open-ended. It is also critical to recognize that the curriculum is neither restricted nor exhaustive. If you properly comprehend the syllabus, you may develop a fair schedule for your preparation that will need fewer hours, cover more topics, and enhance your overall accuracy rate by giving yourself more practice time.

2. Analyze Your Weak Areas & Improve Them Accordingly

It’s quite normal to have weaker areas when it comes to attempting Quantitative Aptitude. After all, different problem-solving methodologies or approaches might be necessary for varied areas. This is where you might get into the dilemma of whether to focus on refining Quant’s topics or on strengthening weaker concepts.

In this case, it’s better if you focus on your weaker areas. It’s known that focusing on topics that you’re not well versed in will cause you to stress during exam time which in turn makes you commit mistakes that will lower your total accuracy rate.

You should start with refining and focusing on your weakest areas. By doing so, you will gain confidence as their answers to subsequent questions improve. The goal is to ensure that candidates like you can grasp their fundamentals and gain confidence as they answer each following question with more accurate outcomes.

3. Understand the Question Properly Before Attempting It

You’ll also agree that Quants takes a lot of practice to master. The reason being so that students don’t really understand the question and answer it without understanding what it’s conveying. If you’re one of those people who do this, then you need to follow this method:

Pay special attention to the question’s specifics. Understand it and try to picture it in your head. After that, decide which concepts you can use. Is it required, for example, to utilize addition or the Remainder theorem? 

If you comprehend the problem and know which idea to apply, you may go to the next phase. Attempting to handle it in a single step may be difficult. Instead, look for a method that divides the problem into smaller bits and then solves it.

4. Use Tricks & Shortcuts Where it’s Needed

Remember when 90s kids used to use cheat codes while playing the game GTA so they can win the game easily? Yes, you also need shortcuts and cheat codes or tricks to solve the Quantitative Aptitude section fast.

These strategies offer students confidence that they are getting closer to a solution to an issue. Students today have a plethora of shortcuts and tactics at their disposal, and knowing which to use is half the battle.

Depending on the conditions, students may use a variety of quick approaches to solving the problem. When students learn how to attain what they want, they develop problem-solving skills and become more comfortable pursuing new solutions.

5. Time Management During Solving Solution Increase Solving Speed

Most students face the challenge of trying to solve problems within the time frame given during exams. They struggle to manage their time well during tests and make mistakes easily. 

The time-constrained setting is one of the aspects of Quantitative Aptitude exams, and students must finish all of the problems in that time; failure to do so may result in making incorrect decisions to save time. Keep track of how much time you spend answering a single question when preparing for or giving mock tests.

The Quantitative portion obviously takes longer to finish than the other sections. As a result, many candidates run out of time and are unable to achieve the sectional cutoff. In such cases, students can practice boosting their solving solution speed by learning Vedic and speed mathematics and memorizing tables, squares, and cubes.

Related Article – Must Do Questions for Companies like TCS, CTS, HCL, IBM …

6. Evaluate Yourself by Creating Exam Like Situation

Students can easily get afraid when they’re thrust into exam like environment suddenly which lowers their confidence. If you’re one of those students, you need to create an exam-like environment yourself beforehand to get a feel of the real situation.

All of your exam tactics will be futile unless you regularly test yourself under genuine exam settings. You must train your brain to adjust to time constraints and deal with the stress that comes with it.

Create test-like situations at home and try to solve as many problems as you can in the allotted time. After taking the test, critically examine the answers and calculate an accuracy percentage based on them (the number of correct answers). If the proportion is greater than 70-80% at any given time, you are on the correct track. If your percentage is lower than that, you need to practice more.

7. Practice the Basics

When you see or read something only once, you don’t learn it, at least not enough to remember it forever. It may hold your attention for a few more encounters, but you will quickly forget about it and go on to something else.

While age has an effect on our memory, there is still a lot we can do to assist us to recall more when we want to study. Rote learning has been used as a memorizing strategy for decades. The appropriate type of repetition can considerably help your memory. To prepare for Quantitative Aptitude placement examinations, you must put what you’ve studied into practice.

The first stage is to practice a large variety of basic mathematical questions. You may feel apprehensive when visiting and reviewing these topics, but if you have a firm grasp of them, you will easily pass the test.

Here are some typical subjects to get you started:

• Divisibility tests
• Profit and loss statements
• Compound Interest and Simple Interest
• Interpretation of Data
• Arithmetic progressions, geometric progressions, and the Number System Sum
• LCM, HCF (Highest Common factor and Least Common Multiple)
• Probability

This will assist you in learning basic formulae and remembering concepts so that you may work more efficiently on their applications.

In India, quantitative aptitude is an essential and important aspect of aptitude exams. It assesses numeric abilities as well as logical and analytical abilities. By answering these Quants questions, one may put their handling tactics and problem-solving skills to the test. A well-planned preparation for the Quants exam would benefit in saving more time during the examination, as the expression says “A stitch in time saves nine.” Nothing is difficult to handle if you prepare ahead of time and utilize the proper method at the right time.

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40 problem-solving techniques and processes

All teams and organizations encounter challenges. Approaching those challenges without a structured problem solving process can end up making things worse.

Proven problem solving techniques such as those outlined below can guide your group through a process of identifying problems and challenges , ideating on possible solutions , and then evaluating and implementing the most suitable .

In this post, you'll find problem-solving tools you can use to develop effective solutions. You'll also find some tips for facilitating the problem solving process and solving complex problems.

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What is problem solving?

Problem solving is a process of finding and implementing a solution to a challenge or obstacle. In most contexts, this means going through a problem solving process that begins with identifying the issue, exploring its root causes, ideating and refining possible solutions before implementing and measuring the impact of that solution.

For simple or small problems, it can be tempting to skip straight to implementing what you believe is the right solution. The danger with this approach is that without exploring the true causes of the issue, it might just occur again or your chosen solution may cause other issues.

Particularly in the world of work, good problem solving means using data to back up each step of the process, bringing in new perspectives and effectively measuring the impact of your solution.

Effective problem solving can help ensure that your team or organization is well positioned to overcome challenges, be resilient to change and create innovation. In my experience, problem solving is a combination of skillset, mindset and process, and it’s especially vital for leaders to cultivate this skill.

What is the seven step problem solving process?

A problem solving process is a step-by-step framework from going from discovering a problem all the way through to implementing a solution.

With practice, this framework can become intuitive, and innovative companies tend to have a consistent and ongoing ability to discover and tackle challenges when they come up.

You might see everything from a four step problem solving process through to seven steps. While all these processes cover roughly the same ground, I’ve found a seven step problem solving process is helpful for making all key steps legible.

We’ll outline that process here and then follow with techniques you can use to explore and work on that step of the problem solving process with a group.

The seven-step problem solving process is:

1. Problem identification 

The first stage of any problem solving process is to identify the problem(s) you need to solve. This often looks like using group discussions and activities to help a group surface and effectively articulate the challenges they’re facing and wish to resolve.

Be sure to align with your team on the exact definition and nature of the problem you’re solving. An effective process is one where everyone is pulling in the same direction – ensure clarity and alignment now to help avoid misunderstandings later.

2. Problem analysis and refinement

The process of problem analysis means ensuring that the problem you are seeking to solve is  the   right problem . Choosing the right problem to solve means you are on the right path to creating the right solution.

At this stage, you may look deeper at the problem you identified to try and discover the root cause at the level of people or process. You may also spend some time sourcing data, consulting relevant parties and creating and refining a problem statement.

Problem refinement means adjusting scope or focus of the problem you will be aiming to solve based on what comes up during your analysis. As you analyze data sources, you might discover that the root cause means you need to adjust your problem statement. Alternatively, you might find that your original problem statement is too big to be meaningful approached within your current project.

Remember that the goal of any problem refinement is to help set the stage for effective solution development and deployment. Set the right focus and get buy-in from your team here and you’ll be well positioned to move forward with confidence.

3. Solution generation

Once your group has nailed down the particulars of the problem you wish to solve, you want to encourage a free flow of ideas connecting to solving that problem. This can take the form of problem solving games that encourage creative thinking or techniquess designed to produce working prototypes of possible solutions. 

The key to ensuring the success of this stage of the problem solving process is to encourage quick, creative thinking and create an open space where all ideas are considered. The best solutions can often come from unlikely places and by using problem solving techniques that celebrate invention, you might come up with solution gold. 

4. Solution development

No solution is perfect right out of the gate. It’s important to discuss and develop the solutions your group has come up with over the course of following the previous problem solving steps in order to arrive at the best possible solution. Problem solving games used in this stage involve lots of critical thinking, measuring potential effort and impact, and looking at possible solutions analytically. 

During this stage, you will often ask your team to iterate and improve upon your front-running solutions and develop them further. Remember that problem solving strategies always benefit from a multitude of voices and opinions, and not to let ego get involved when it comes to choosing which solutions to develop and take further.

Finding the best solution is the goal of all problem solving workshops and here is the place to ensure that your solution is well thought out, sufficiently robust and fit for purpose. 

5. Decision making and planning

Nearly there! Once you’ve got a set of possible, you’ll need to make a decision on which to implement. This can be a consensus-based group decision or it might be for a leader or major stakeholder to decide. You’ll find a set of effective decision making methods below.

Once your group has reached consensus and selected a solution, there are some additional actions that also need to be decided upon. You’ll want to work on allocating ownership of the project, figure out who will do what, how the success of the solution will be measured and decide the next course of action.

Set clear accountabilities, actions, timeframes, and follow-ups for your chosen solution. Make these decisions and set clear next-steps in the problem solving workshop so that everyone is aligned and you can move forward effectively as a group. 

Ensuring that you plan for the roll-out of a solution is one of the most important problem solving steps. Without adequate planning or oversight, it can prove impossible to measure success or iterate further if the problem was not solved. 

6. Solution implementation 

This is what we were waiting for! All problem solving processes have the end goal of implementing an effective and impactful solution that your group has confidence in.

Project management and communication skills are key here – your solution may need to adjust when out in the wild or you might discover new challenges along the way. For some solutions, you might also implement a test with a small group and monitor results before rolling it out to an entire company.

You should have a clear owner for your solution who will oversee the plans you made together and help ensure they’re put into place. This person will often coordinate the implementation team and set-up processes to measure the efficacy of your solution too.

7. Solution evaluation 

So you and your team developed a great solution to a problem and have a gut feeling it’s been solved. Work done, right? Wrong. All problem solving strategies benefit from evaluation, consideration, and feedback.

You might find that the solution does not work for everyone, might create new problems, or is potentially so successful that you will want to roll it out to larger teams or as part of other initiatives. 

None of that is possible without taking the time to evaluate the success of the solution you developed in your problem solving model and adjust if necessary.

Remember that the problem solving process is often iterative and it can be common to not solve complex issues on the first try. Even when this is the case, you and your team will have generated learning that will be important for future problem solving workshops or in other parts of the organization. 

It’s also worth underlining how important record keeping is throughout the problem solving process. If a solution didn’t work, you need to have the data and records to see why that was the case. If you go back to the drawing board, notes from the previous workshop can help save time.

What does an effective problem solving process look like?

Every effective problem solving process begins with an agenda . In our experience, a well-structured problem solving workshop is one of the best methods for successfully guiding a group from exploring a problem to implementing a solution.

The format of a workshop ensures that you can get buy-in from your group, encourage free-thinking and solution exploration before making a decision on what to implement following the session.

This Design Sprint 2.0 template is an effective problem solving process from top agency AJ&Smart. It’s a great format for the entire problem solving process, with four-days of workshops designed to surface issues, explore solutions and even test a solution.

Check it for an example of how you might structure and run a problem solving process and feel free to copy and adjust it your needs!

For a shorter process you can run in a single afternoon, this remote problem solving agenda will guide you effectively in just a couple of hours.

Whatever the length of your workshop, by using SessionLab, it’s easy to go from an idea to a complete agenda . Start by dragging and dropping your core problem solving activities into place . Add timings, breaks and necessary materials before sharing your agenda with your colleagues.

The resulting agenda will be your guide to an effective and productive problem solving session that will also help you stay organized on the day!

Complete problem-solving methods

In this section, we’ll look at in-depth problem-solving methods that provide a complete end-to-end process for developing effective solutions. These will help guide your team from the discovery and definition of a problem through to delivering the right solution.

If you’re looking for an all-encompassing method or problem-solving model, these processes are a great place to start. They’ll ask your team to challenge preconceived ideas and adopt a mindset for solving problems more effectively.

Six Thinking Hats

Individual approaches to solving a problem can be very different based on what team or role an individual holds. It can be easy for existing biases or perspectives to find their way into the mix, or for internal politics to direct a conversation.

Six Thinking Hats is a classic method for identifying the problems that need to be solved and enables your team to consider them from different angles, whether that is by focusing on facts and data, creative solutions, or by considering why a particular solution might not work.

Like all problem-solving frameworks, Six Thinking Hats is effective at helping teams remove roadblocks from a conversation or discussion and come to terms with all the aspects necessary to solve complex problems.

The Six Thinking Hats   #creative thinking   #meeting facilitation   #problem solving   #issue resolution   #idea generation   #conflict resolution   The Six Thinking Hats are used by individuals and groups to separate out conflicting styles of thinking. They enable and encourage a group of people to think constructively together in exploring and implementing change, rather than using argument to fight over who is right and who is wrong.

Lightning Decision Jam

Featured courtesy of Jonathan Courtney of AJ&Smart Berlin, Lightning Decision Jam is one of those strategies that should be in every facilitation toolbox. Exploring problems and finding solutions is often creative in nature, though as with any creative process, there is the potential to lose focus and get lost.

Unstructured discussions might get you there in the end, but it’s much more effective to use a method that creates a clear process and team focus.

In Lightning Decision Jam, participants are invited to begin by writing challenges, concerns, or mistakes on post-its without discussing them before then being invited by the moderator to present them to the group.

From there, the team vote on which problems to solve and are guided through steps that will allow them to reframe those problems, create solutions and then decide what to execute on. 

By deciding the problems that need to be solved as a team before moving on, this group process is great for ensuring the whole team is aligned and can take ownership over the next stages. 

Lightning Decision Jam (LDJ)   #action   #decision making   #problem solving   #issue analysis   #innovation   #design   #remote-friendly   It doesn’t matter where you work and what your job role is, if you work with other people together as a team, you will always encounter the same challenges: Unclear goals and miscommunication that cause busy work and overtime Unstructured meetings that leave attendants tired, confused and without clear outcomes. Frustration builds up because internal challenges to productivity are not addressed Sudden changes in priorities lead to a loss of focus and momentum Muddled compromise takes the place of clear decision- making, leaving everybody to come up with their own interpretation. In short, a lack of structure leads to a waste of time and effort, projects that drag on for too long and frustrated, burnt out teams. AJ&Smart has worked with some of the most innovative, productive companies in the world. What sets their teams apart from others is not better tools, bigger talent or more beautiful offices. The secret sauce to becoming a more productive, more creative and happier team is simple: Replace all open discussion or brainstorming with a structured process that leads to more ideas, clearer decisions and better outcomes. When a good process provides guardrails and a clear path to follow, it becomes easier to come up with ideas, make decisions and solve problems. This is why AJ&Smart created Lightning Decision Jam (LDJ). It’s a simple and short, but powerful group exercise that can be run either in-person, in the same room, or remotely with distributed teams.

Problem Definition Process

While problems can be complex, the problem-solving methods you use to identify and solve those problems can often be simple in design. 

By taking the time to truly identify and define a problem before asking the group to reframe the challenge as an opportunity, this method is a great way to enable change.

Begin by identifying a focus question and exploring the ways in which it manifests before splitting into five teams who will each consider the problem using a different method: escape, reversal, exaggeration, distortion or wishful. Teams develop a problem objective and create ideas in line with their method before then feeding them back to the group.

This method is great for enabling in-depth discussions while also creating space for finding creative solutions too!

Problem Definition   #problem solving   #idea generation   #creativity   #online   #remote-friendly   A problem solving technique to define a problem, challenge or opportunity and to generate ideas.

The 5 Whys 

Sometimes, a group needs to go further with their strategies and analyze the root cause at the heart of organizational issues. An RCA or root cause analysis is the process of identifying what is at the heart of business problems or recurring challenges. 

The 5 Whys is a simple and effective method of helping a group go find the root cause of any problem or challenge and conduct analysis that will deliver results. 

By beginning with the creation of a problem statement and going through five stages to refine it, The 5 Whys provides everything you need to truly discover the cause of an issue.

The 5 Whys   #hyperisland   #innovation   This simple and powerful method is useful for getting to the core of a problem or challenge. As the title suggests, the group defines a problems, then asks the question “why” five times, often using the resulting explanation as a starting point for creative problem solving.

World Cafe is a simple but powerful facilitation technique to help bigger groups to focus their energy and attention on solving complex problems.

World Cafe enables this approach by creating a relaxed atmosphere where participants are able to self-organize and explore topics relevant and important to them which are themed around a central problem-solving purpose. Create the right atmosphere by modeling your space after a cafe and after guiding the group through the method, let them take the lead!

Making problem-solving a part of your organization’s culture in the long term can be a difficult undertaking. More approachable formats like World Cafe can be especially effective in bringing people unfamiliar with workshops into the fold. 

World Cafe   #hyperisland   #innovation   #issue analysis   World Café is a simple yet powerful method, originated by Juanita Brown, for enabling meaningful conversations driven completely by participants and the topics that are relevant and important to them. Facilitators create a cafe-style space and provide simple guidelines. Participants then self-organize and explore a set of relevant topics or questions for conversation.

Discovery & Action Dialogue (DAD)

One of the best approaches is to create a safe space for a group to share and discover practices and behaviors that can help them find their own solutions.

With DAD, you can help a group choose which problems they wish to solve and which approaches they will take to do so. It’s great at helping remove resistance to change and can help get buy-in at every level too!

This process of enabling frontline ownership is great in ensuring follow-through and is one of the methods you will want in your toolbox as a facilitator.

Discovery & Action Dialogue (DAD)   #idea generation   #liberating structures   #action   #issue analysis   #remote-friendly   DADs make it easy for a group or community to discover practices and behaviors that enable some individuals (without access to special resources and facing the same constraints) to find better solutions than their peers to common problems. These are called positive deviant (PD) behaviors and practices. DADs make it possible for people in the group, unit, or community to discover by themselves these PD practices. DADs also create favorable conditions for stimulating participants’ creativity in spaces where they can feel safe to invent new and more effective practices. Resistance to change evaporates as participants are unleashed to choose freely which practices they will adopt or try and which problems they will tackle. DADs make it possible to achieve frontline ownership of solutions.

Design Sprint 2.0

Want to see how a team can solve big problems and move forward with prototyping and testing solutions in a few days? The Design Sprint 2.0 template from Jake Knapp, author of Sprint, is a complete agenda for a with proven results.

Developing the right agenda can involve difficult but necessary planning. Ensuring all the correct steps are followed can also be stressful or time-consuming depending on your level of experience.

Use this complete 4-day workshop template if you are finding there is no obvious solution to your challenge and want to focus your team around a specific problem that might require a shortcut to launching a minimum viable product or waiting for the organization-wide implementation of a solution.

Open space technology

Open space technology- developed by Harrison Owen – creates a space where large groups are invited to take ownership of their problem solving and lead individual sessions. Open space technology is a great format when you have a great deal of expertise and insight in the room and want to allow for different takes and approaches on a particular theme or problem you need to be solved.

Start by bringing your participants together to align around a central theme and focus their efforts. Explain the ground rules to help guide the problem-solving process and then invite members to identify any issue connecting to the central theme that they are interested in and are prepared to take responsibility for.

Once participants have decided on their approach to the core theme, they write their issue on a piece of paper, announce it to the group, pick a session time and place, and post the paper on the wall. As the wall fills up with sessions, the group is then invited to join the sessions that interest them the most and which they can contribute to, then you’re ready to begin!

Everyone joins the problem-solving group they’ve signed up to, record the discussion and if appropriate, findings can then be shared with the rest of the group afterward.

Open Space Technology   #action plan   #idea generation   #problem solving   #issue analysis   #large group   #online   #remote-friendly   Open Space is a methodology for large groups to create their agenda discerning important topics for discussion, suitable for conferences, community gatherings and whole system facilitation

Techniques to identify and analyze problems

Using a problem-solving method to help a team identify and analyze a problem can be a quick and effective addition to any workshop or meeting.

While further actions are always necessary, you can generate momentum and alignment easily, and these activities are a great place to get started.

We’ve put together this list of techniques to help you and your team with problem identification, analysis, and discussion that sets the foundation for developing effective solutions.

Let’s take a look!

Fishbone Analysis

Organizational or team challenges are rarely simple, and it’s important to remember that one problem can be an indication of something that goes deeper and may require further consideration to be solved.

Fishbone Analysis helps groups to dig deeper and understand the origins of a problem. It’s a great example of a root cause analysis method that is simple for everyone on a team to get their head around. 

Participants in this activity are asked to annotate a diagram of a fish, first adding the problem or issue to be worked on at the head of a fish before then brainstorming the root causes of the problem and adding them as bones on the fish. 

Using abstractions such as a diagram of a fish can really help a team break out of their regular thinking and develop a creative approach.

Fishbone Analysis   #problem solving   ##root cause analysis   #decision making   #online facilitation   A process to help identify and understand the origins of problems, issues or observations.

Problem Tree 

Encouraging visual thinking can be an essential part of many strategies. By simply reframing and clarifying problems, a group can move towards developing a problem solving model that works for them. 

In Problem Tree, groups are asked to first brainstorm a list of problems – these can be design problems, team problems or larger business problems – and then organize them into a hierarchy. The hierarchy could be from most important to least important or abstract to practical, though the key thing with problem solving games that involve this aspect is that your group has some way of managing and sorting all the issues that are raised.

Once you have a list of problems that need to be solved and have organized them accordingly, you’re then well-positioned for the next problem solving steps.

Problem tree   #define intentions   #create   #design   #issue analysis   A problem tree is a tool to clarify the hierarchy of problems addressed by the team within a design project; it represents high level problems or related sublevel problems.

SWOT Analysis

Chances are you’ve heard of the SWOT Analysis before. This problem-solving method focuses on identifying strengths, weaknesses, opportunities, and threats is a tried and tested method for both individuals and teams.

Start by creating a desired end state or outcome and bare this in mind – any process solving model is made more effective by knowing what you are moving towards. Create a quadrant made up of the four categories of a SWOT analysis and ask participants to generate ideas based on each of those quadrants.

Once you have those ideas assembled in their quadrants, cluster them together based on their affinity with other ideas. These clusters are then used to facilitate group conversations and move things forward. 

SWOT analysis   #gamestorming   #problem solving   #action   #meeting facilitation   The SWOT Analysis is a long-standing technique of looking at what we have, with respect to the desired end state, as well as what we could improve on. It gives us an opportunity to gauge approaching opportunities and dangers, and assess the seriousness of the conditions that affect our future. When we understand those conditions, we can influence what comes next.

Agreement-Certainty Matrix

Not every problem-solving approach is right for every challenge, and deciding on the right method for the challenge at hand is a key part of being an effective team.

The Agreement Certainty matrix helps teams align on the nature of the challenges facing them. By sorting problems from simple to chaotic, your team can understand what methods are suitable for each problem and what they can do to ensure effective results. 

If you are already using Liberating Structures techniques as part of your problem-solving strategy, the Agreement-Certainty Matrix can be an invaluable addition to your process. We’ve found it particularly if you are having issues with recurring problems in your organization and want to go deeper in understanding the root cause. 

Agreement-Certainty Matrix   #issue analysis   #liberating structures   #problem solving   You can help individuals or groups avoid the frequent mistake of trying to solve a problem with methods that are not adapted to the nature of their challenge. The combination of two questions makes it possible to easily sort challenges into four categories: simple, complicated, complex , and chaotic .  A problem is simple when it can be solved reliably with practices that are easy to duplicate.  It is complicated when experts are required to devise a sophisticated solution that will yield the desired results predictably.  A problem is complex when there are several valid ways to proceed but outcomes are not predictable in detail.  Chaotic is when the context is too turbulent to identify a path forward.  A loose analogy may be used to describe these differences: simple is like following a recipe, complicated like sending a rocket to the moon, complex like raising a child, and chaotic is like the game “Pin the Tail on the Donkey.”  The Liberating Structures Matching Matrix in Chapter 5 can be used as the first step to clarify the nature of a challenge and avoid the mismatches between problems and solutions that are frequently at the root of chronic, recurring problems.

Organizing and charting a team’s progress can be important in ensuring its success. SQUID (Sequential Question and Insight Diagram) is a great model that allows a team to effectively switch between giving questions and answers and develop the skills they need to stay on track throughout the process. 

Begin with two different colored sticky notes – one for questions and one for answers – and with your central topic (the head of the squid) on the board. Ask the group to first come up with a series of questions connected to their best guess of how to approach the topic. Ask the group to come up with answers to those questions, fix them to the board and connect them with a line. After some discussion, go back to question mode by responding to the generated answers or other points on the board.

It’s rewarding to see a diagram grow throughout the exercise, and a completed SQUID can provide a visual resource for future effort and as an example for other teams.

SQUID   #gamestorming   #project planning   #issue analysis   #problem solving   When exploring an information space, it’s important for a group to know where they are at any given time. By using SQUID, a group charts out the territory as they go and can navigate accordingly. SQUID stands for Sequential Question and Insight Diagram.

To continue with our nautical theme, Speed Boat is a short and sweet activity that can help a team quickly identify what employees, clients or service users might have a problem with and analyze what might be standing in the way of achieving a solution.

Methods that allow for a group to make observations, have insights and obtain those eureka moments quickly are invaluable when trying to solve complex problems.

In Speed Boat, the approach is to first consider what anchors and challenges might be holding an organization (or boat) back. Bonus points if you are able to identify any sharks in the water and develop ideas that can also deal with competitors!   

Speed Boat   #gamestorming   #problem solving   #action   Speedboat is a short and sweet way to identify what your employees or clients don’t like about your product/service or what’s standing in the way of a desired goal.

The Journalistic Six

Some of the most effective ways of solving problems is by encouraging teams to be more inclusive and diverse in their thinking.

Based on the six key questions journalism students are taught to answer in articles and news stories, The Journalistic Six helps create teams to see the whole picture. By using who, what, when, where, why, and how to facilitate the conversation and encourage creative thinking, your team can make sure that the problem identification and problem analysis stages of the are covered exhaustively and thoughtfully. Reporter’s notebook and dictaphone optional.

The Journalistic Six – Who What When Where Why How   #idea generation   #issue analysis   #problem solving   #online   #creative thinking   #remote-friendly   A questioning method for generating, explaining, investigating ideas.

Individual and group perspectives are incredibly important, but what happens if people are set in their minds and need a change of perspective in order to approach a problem more effectively?

Flip It is a method we love because it is both simple to understand and run, and allows groups to understand how their perspectives and biases are formed. 

Participants in Flip It are first invited to consider concerns, issues, or problems from a perspective of fear and write them on a flip chart. Then, the group is asked to consider those same issues from a perspective of hope and flip their understanding.  

No problem and solution is free from existing bias and by changing perspectives with Flip It, you can then develop a problem solving model quickly and effectively.

Flip It!   #gamestorming   #problem solving   #action   Often, a change in a problem or situation comes simply from a change in our perspectives. Flip It! is a quick game designed to show players that perspectives are made, not born.

LEGO Challenge

Now for an activity that is a little out of the (toy) box. LEGO Serious Play is a facilitation methodology that can be used to improve creative thinking and problem-solving skills. 

The LEGO Challenge includes giving each member of the team an assignment that is hidden from the rest of the group while they create a structure without speaking.

What the LEGO challenge brings to the table is a fun working example of working with stakeholders who might not be on the same page to solve problems. Also, it’s LEGO! Who doesn’t love LEGO! 

LEGO Challenge   #hyperisland   #team   A team-building activity in which groups must work together to build a structure out of LEGO, but each individual has a secret “assignment” which makes the collaborative process more challenging. It emphasizes group communication, leadership dynamics, conflict, cooperation, patience and problem solving strategy.

What, So What, Now What?

If not carefully managed, the problem identification and problem analysis stages of the problem-solving process can actually create more problems and misunderstandings.

The What, So What, Now What? problem-solving activity is designed to help collect insights and move forward while also eliminating the possibility of disagreement when it comes to identifying, clarifying, and analyzing organizational or work problems. 

Facilitation is all about bringing groups together so that might work on a shared goal and the best problem-solving strategies ensure that teams are aligned in purpose, if not initially in opinion or insight.

Throughout the three steps of this game, you give everyone on a team to reflect on a problem by asking what happened, why it is important, and what actions should then be taken. 

This can be a great activity for bringing our individual perceptions about a problem or challenge and contextualizing it in a larger group setting. This is one of the most important problem-solving skills you can bring to your organization.

W³ – What, So What, Now What?   #issue analysis   #innovation   #liberating structures   You can help groups reflect on a shared experience in a way that builds understanding and spurs coordinated action while avoiding unproductive conflict. It is possible for every voice to be heard while simultaneously sifting for insights and shaping new direction. Progressing in stages makes this practical—from collecting facts about What Happened to making sense of these facts with So What and finally to what actions logically follow with Now What . The shared progression eliminates most of the misunderstandings that otherwise fuel disagreements about what to do. Voila!

Journalists  

Problem analysis can be one of the most important and decisive stages of all problem-solving tools. Sometimes, a team can become bogged down in the details and are unable to move forward.

Journalists is an activity that can avoid a group from getting stuck in the problem identification or problem analysis stages of the process.

In Journalists, the group is invited to draft the front page of a fictional newspaper and figure out what stories deserve to be on the cover and what headlines those stories will have. By reframing how your problems and challenges are approached, you can help a team move productively through the process and be better prepared for the steps to follow.

Journalists   #vision   #big picture   #issue analysis   #remote-friendly   This is an exercise to use when the group gets stuck in details and struggles to see the big picture. Also good for defining a vision.

Problem-solving techniques for brainstorming solutions

Now you have the context and background of the problem you are trying to solving, now comes the time to start ideating and thinking about how you’ll solve the issue.

Here, you’ll want to encourage creative, free thinking and speed. Get as many ideas out as possible and explore different perspectives so you have the raw material for the next step.

Looking at a problem from a new angle can be one of the most effective ways of creating an effective solution. TRIZ is a problem-solving tool that asks the group to consider what they must not do in order to solve a challenge.

By reversing the discussion, new topics and taboo subjects often emerge, allowing the group to think more deeply and create ideas that confront the status quo in a safe and meaningful way. If you’re working on a problem that you’ve tried to solve before, TRIZ is a great problem-solving method to help your team get unblocked.

Making Space with TRIZ   #issue analysis   #liberating structures   #issue resolution   You can clear space for innovation by helping a group let go of what it knows (but rarely admits) limits its success and by inviting creative destruction. TRIZ makes it possible to challenge sacred cows safely and encourages heretical thinking. The question “What must we stop doing to make progress on our deepest purpose?” induces seriously fun yet very courageous conversations. Since laughter often erupts, issues that are otherwise taboo get a chance to be aired and confronted. With creative destruction come opportunities for renewal as local action and innovation rush in to fill the vacuum. Whoosh!

Mindspin  

Brainstorming is part of the bread and butter of the problem-solving process and all problem-solving strategies benefit from getting ideas out and challenging a team to generate solutions quickly. 

With Mindspin, participants are encouraged not only to generate ideas but to do so under time constraints and by slamming down cards and passing them on. By doing multiple rounds, your team can begin with a free generation of possible solutions before moving on to developing those solutions and encouraging further ideation. 

This is one of our favorite problem-solving activities and can be great for keeping the energy up throughout the workshop. Remember the importance of helping people become engaged in the process – energizing problem-solving techniques like Mindspin can help ensure your team stays engaged and happy, even when the problems they’re coming together to solve are complex. 

MindSpin   #teampedia   #idea generation   #problem solving   #action   A fast and loud method to enhance brainstorming within a team. Since this activity has more than round ideas that are repetitive can be ruled out leaving more creative and innovative answers to the challenge.

The Creativity Dice

One of the most useful problem solving skills you can teach your team is of approaching challenges with creativity, flexibility, and openness. Games like The Creativity Dice allow teams to overcome the potential hurdle of too much linear thinking and approach the process with a sense of fun and speed. 

In The Creativity Dice, participants are organized around a topic and roll a dice to determine what they will work on for a period of 3 minutes at a time. They might roll a 3 and work on investigating factual information on the chosen topic. They might roll a 1 and work on identifying the specific goals, standards, or criteria for the session.

Encouraging rapid work and iteration while asking participants to be flexible are great skills to cultivate. Having a stage for idea incubation in this game is also important. Moments of pause can help ensure the ideas that are put forward are the most suitable. 

The Creativity Dice   #creativity   #problem solving   #thiagi   #issue analysis   Too much linear thinking is hazardous to creative problem solving. To be creative, you should approach the problem (or the opportunity) from different points of view. You should leave a thought hanging in mid-air and move to another. This skipping around prevents premature closure and lets your brain incubate one line of thought while you consciously pursue another.

Idea and Concept Development

Brainstorming without structure can quickly become chaotic or frustrating. In a problem-solving context, having an ideation framework to follow can help ensure your team is both creative and disciplined.

In this method, you’ll find an idea generation process that encourages your group to brainstorm effectively before developing their ideas and begin clustering them together. By using concepts such as Yes and…, more is more and postponing judgement, you can create the ideal conditions for brainstorming with ease.

Idea & Concept Development   #hyperisland   #innovation   #idea generation   Ideation and Concept Development is a process for groups to work creatively and collaboratively to generate creative ideas. It’s a general approach that can be adapted and customized to suit many different scenarios. It includes basic principles for idea generation and several steps for groups to work with. It also includes steps for idea selection and development.

Problem-solving techniques for developing and refining solutions 

The success of any problem-solving process can be measured by the solutions it produces. After you’ve defined the issue, explored existing ideas, and ideated, it’s time to develop and refine your ideas in order to bring them closer to a solution that actually solves the problem.

Use these problem-solving techniques when you want to help your team think through their ideas and refine them as part of your problem solving process.

Improved Solutions

After a team has successfully identified a problem and come up with a few solutions, it can be tempting to call the work of the problem-solving process complete. That said, the first solution is not necessarily the best, and by including a further review and reflection activity into your problem-solving model, you can ensure your group reaches the best possible result. 

One of a number of problem-solving games from Thiagi Group, Improved Solutions helps you go the extra mile and develop suggested solutions with close consideration and peer review. By supporting the discussion of several problems at once and by shifting team roles throughout, this problem-solving technique is a dynamic way of finding the best solution. 

Improved Solutions   #creativity   #thiagi   #problem solving   #action   #team   You can improve any solution by objectively reviewing its strengths and weaknesses and making suitable adjustments. In this creativity framegame, you improve the solutions to several problems. To maintain objective detachment, you deal with a different problem during each of six rounds and assume different roles (problem owner, consultant, basher, booster, enhancer, and evaluator) during each round. At the conclusion of the activity, each player ends up with two solutions to her problem.

Four Step Sketch

Creative thinking and visual ideation does not need to be confined to the opening stages of your problem-solving strategies. Exercises that include sketching and prototyping on paper can be effective at the solution finding and development stage of the process, and can be great for keeping a team engaged. 

By going from simple notes to a crazy 8s round that involves rapidly sketching 8 variations on their ideas before then producing a final solution sketch, the group is able to iterate quickly and visually. Problem-solving techniques like Four-Step Sketch are great if you have a group of different thinkers and want to change things up from a more textual or discussion-based approach.

Four-Step Sketch   #design sprint   #innovation   #idea generation   #remote-friendly   The four-step sketch is an exercise that helps people to create well-formed concepts through a structured process that includes: Review key information Start design work on paper,  Consider multiple variations , Create a detailed solution . This exercise is preceded by a set of other activities allowing the group to clarify the challenge they want to solve. See how the Four Step Sketch exercise fits into a Design Sprint

Ensuring that everyone in a group is able to contribute to a discussion is vital during any problem solving process. Not only does this ensure all bases are covered, but its then easier to get buy-in and accountability when people have been able to contribute to the process.

1-2-4-All is a tried and tested facilitation technique where participants are asked to first brainstorm on a topic on their own. Next, they discuss and share ideas in a pair before moving into a small group. Those groups are then asked to present the best idea from their discussion to the rest of the team.

This method can be used in many different contexts effectively, though I find it particularly shines in the idea development stage of the process. Giving each participant time to concretize their ideas and develop them in progressively larger groups can create a great space for both innovation and psychological safety.

1-2-4-All   #idea generation   #liberating structures   #issue analysis   With this facilitation technique you can immediately include everyone regardless of how large the group is. You can generate better ideas and more of them faster than ever before. You can tap the know-how and imagination that is distributed widely in places not known in advance. Open, generative conversation unfolds. Ideas and solutions are sifted in rapid fashion. Most importantly, participants own the ideas, so follow-up and implementation is simplified. No buy-in strategies needed! Simple and elegant!

15% Solutions

Some problems are simpler than others and with the right problem-solving activities, you can empower people to take immediate actions that can help create organizational change. 

Part of the liberating structures toolkit, 15% solutions is a problem-solving technique that focuses on finding and implementing solutions quickly. A process of iterating and making small changes quickly can help generate momentum and an appetite for solving complex problems.

Problem-solving strategies can live and die on whether people are onboard. Getting some quick wins is a great way of getting people behind the process.   

It can be extremely empowering for a team to realize that problem-solving techniques can be deployed quickly and easily and delineate between things they can positively impact and those things they cannot change. 

15% Solutions   #action   #liberating structures   #remote-friendly   You can reveal the actions, however small, that everyone can do immediately. At a minimum, these will create momentum, and that may make a BIG difference.  15% Solutions show that there is no reason to wait around, feel powerless, or fearful. They help people pick it up a level. They get individuals and the group to focus on what is within their discretion instead of what they cannot change.  With a very simple question, you can flip the conversation to what can be done and find solutions to big problems that are often distributed widely in places not known in advance. Shifting a few grains of sand may trigger a landslide and change the whole landscape.

Problem-solving techniques for making decisions and planning

After your group is happy with the possible solutions you’ve developed, now comes the time to choose which to implement. There’s more than one way to make a decision and the best option is often dependant on the needs and set-up of your group.

Sometimes, it’s the case that you’ll want to vote as a group on what is likely to be the most impactful solution. Other times, it might be down to a decision maker or major stakeholder to make the final decision. Whatever your process, here’s some techniques you can use to help you make a decision during your problem solving process.

How-Now-Wow Matrix

The problem-solving process is often creative, as complex problems usually require a change of thinking and creative response in order to find the best solutions. While it’s common for the first stages to encourage creative thinking, groups can often gravitate to familiar solutions when it comes to the end of the process. 

When selecting solutions, you don’t want to lose your creative energy! The How-Now-Wow Matrix from Gamestorming is a great problem-solving activity that enables a group to stay creative and think out of the box when it comes to selecting the right solution for a given problem.

Problem-solving techniques that encourage creative thinking and the ideation and selection of new solutions can be the most effective in organisational change. Give the How-Now-Wow Matrix a go, and not just for how pleasant it is to say out loud. 

How-Now-Wow Matrix   #gamestorming   #idea generation   #remote-friendly   When people want to develop new ideas, they most often think out of the box in the brainstorming or divergent phase. However, when it comes to convergence, people often end up picking ideas that are most familiar to them. This is called a ‘creative paradox’ or a ‘creadox’. The How-Now-Wow matrix is an idea selection tool that breaks the creadox by forcing people to weigh each idea on 2 parameters.

Impact and Effort Matrix

All problem-solving techniques hope to not only find solutions to a given problem or challenge but to find the best solution. When it comes to finding a solution, groups are invited to put on their decision-making hats and really think about how a proposed idea would work in practice. 

The Impact and Effort Matrix is one of the problem-solving techniques that fall into this camp, empowering participants to first generate ideas and then categorize them into a 2×2 matrix based on impact and effort.

Activities that invite critical thinking while remaining simple are invaluable. Use the Impact and Effort Matrix to move from ideation and towards evaluating potential solutions before then committing to them. 

Impact and Effort Matrix   #gamestorming   #decision making   #action   #remote-friendly   In this decision-making exercise, possible actions are mapped based on two factors: effort required to implement and potential impact. Categorizing ideas along these lines is a useful technique in decision making, as it obliges contributors to balance and evaluate suggested actions before committing to them.

If you’ve followed each of the problem-solving steps with your group successfully, you should move towards the end of your process with heaps of possible solutions developed with a specific problem in mind. But how do you help a group go from ideation to putting a solution into action? 

Dotmocracy – or Dot Voting -is a tried and tested method of helping a team in the problem-solving process make decisions and put actions in place with a degree of oversight and consensus. 

One of the problem-solving techniques that should be in every facilitator’s toolbox, Dot Voting is fast and effective and can help identify the most popular and best solutions and help bring a group to a decision effectively. 

Dotmocracy   #action   #decision making   #group prioritization   #hyperisland   #remote-friendly   Dotmocracy is a simple method for group prioritization or decision-making. It is not an activity on its own, but a method to use in processes where prioritization or decision-making is the aim. The method supports a group to quickly see which options are most popular or relevant. The options or ideas are written on post-its and stuck up on a wall for the whole group to see. Each person votes for the options they think are the strongest, and that information is used to inform a decision.

Straddling the gap between decision making and planning, MoSCoW is a simple and effective method that allows a group team to easily prioritize a set of possible options.

Use this method in a problem solving process by collecting and summarizing all your possible solutions and then categorize them into 4 sections: “Must have”, “Should have”, “Could have”, or “Would like but won‘t get”.

This method is particularly useful when its less about choosing one possible solution and more about prioritorizing which to do first and which may not fit in the scope of your project. In my experience, complex challenges often require multiple small fixes, and this method can be a great way to move from a pile of things you’d all like to do to a structured plan.

MoSCoW   #define intentions   #create   #design   #action   #remote-friendly   MoSCoW is a method that allows the team to prioritize the different features that they will work on. Features are then categorized into “Must have”, “Should have”, “Could have”, or “Would like but won‘t get”. To be used at the beginning of a timeslot (for example during Sprint planning) and when planning is needed.

When it comes to managing the rollout of a solution, clarity and accountability are key factors in ensuring the success of the project. The RAACI chart is a simple but effective model for setting roles and responsibilities as part of a planning session.

Start by listing each person involved in the project and put them into the following groups in order to make it clear who is responsible for what during the rollout of your solution.

• Responsibility  (Which person and/or team will be taking action?)
• Authority  (At what “point” must the responsible person check in before going further?)
• Accountability  (Who must the responsible person check in with?)
• Consultation  (Who must be consulted by the responsible person before decisions are made?)
• Information  (Who must be informed of decisions, once made?)

Ensure this information is easily accessible and use it to inform who does what and who is looped into discussions and kept up to date.

RAACI   #roles and responsibility   #teamwork   #project management   Clarifying roles and responsibilities, levels of autonomy/latitude in decision making, and levels of engagement among diverse stakeholders.

Problem-solving warm-up activities

All facilitators know that warm-ups and icebreakers are useful for any workshop or group process. Problem-solving workshops are no different.

Use these problem-solving techniques to warm up a group and prepare them for the rest of the process. Activating your group by tapping into some of the top problem-solving skills can be one of the best ways to see great outcomes from your session.

Check-in / Check-out

Solid processes are planned from beginning to end, and the best facilitators know that setting the tone and establishing a safe, open environment can be integral to a successful problem-solving process. Check-in / Check-out is a great way to begin and/or bookend a problem-solving workshop. Checking in to a session emphasizes that everyone will be seen, heard, and expected to contribute. 

If you are running a series of meetings, setting a consistent pattern of checking in and checking out can really help your team get into a groove. We recommend this opening-closing activity for small to medium-sized groups though it can work with large groups if they’re disciplined!

Check-in / Check-out   #team   #opening   #closing   #hyperisland   #remote-friendly   Either checking-in or checking-out is a simple way for a team to open or close a process, symbolically and in a collaborative way. Checking-in/out invites each member in a group to be present, seen and heard, and to express a reflection or a feeling. Checking-in emphasizes presence, focus and group commitment; checking-out emphasizes reflection and symbolic closure.

Doodling Together  

Thinking creatively and not being afraid to make suggestions are important problem-solving skills for any group or team, and warming up by encouraging these behaviors is a great way to start. 

Doodling Together is one of our favorite creative ice breaker games – it’s quick, effective, and fun and can make all following problem-solving steps easier by encouraging a group to collaborate visually. By passing cards and adding additional items as they go, the workshop group gets into a groove of co-creation and idea development that is crucial to finding solutions to problems. 

Doodling Together   #collaboration   #creativity   #teamwork   #fun   #team   #visual methods   #energiser   #icebreaker   #remote-friendly   Create wild, weird and often funny postcards together & establish a group’s creative confidence.

Show and Tell

You might remember some version of Show and Tell from being a kid in school and it’s a great problem-solving activity to kick off a session.

Asking participants to prepare a little something before a workshop by bringing an object for show and tell can help them warm up before the session has even begun! Games that include a physical object can also help encourage early engagement before moving onto more big-picture thinking.

By asking your participants to tell stories about why they chose to bring a particular item to the group, you can help teams see things from new perspectives and see both differences and similarities in the way they approach a topic. Great groundwork for approaching a problem-solving process as a team! 

Show and Tell   #gamestorming   #action   #opening   #meeting facilitation   Show and Tell taps into the power of metaphors to reveal players’ underlying assumptions and associations around a topic The aim of the game is to get a deeper understanding of stakeholders’ perspectives on anything—a new project, an organizational restructuring, a shift in the company’s vision or team dynamic.

Constellations

Who doesn’t love stars? Constellations is a great warm-up activity for any workshop as it gets people up off their feet, energized, and ready to engage in new ways with established topics. It’s also great for showing existing beliefs, biases, and patterns that can come into play as part of your session.

Using warm-up games that help build trust and connection while also allowing for non-verbal responses can be great for easing people into the problem-solving process and encouraging engagement from everyone in the group. Constellations is great in large spaces that allow for movement and is definitely a practical exercise to allow the group to see patterns that are otherwise invisible. 

Constellations   #trust   #connection   #opening   #coaching   #patterns   #system   Individuals express their response to a statement or idea by standing closer or further from a central object. Used with teams to reveal system, hidden patterns, perspectives.

Draw a Tree

Problem-solving games that help raise group awareness through a central, unifying metaphor can be effective ways to warm-up a group in any problem-solving model.

Draw a Tree is a simple warm-up activity you can use in any group and which can provide a quick jolt of energy. Start by asking your participants to draw a tree in just 45 seconds – they can choose whether it will be abstract or realistic. 

Once the timer is up, ask the group how many people included the roots of the tree and use this as a means to discuss how we can ignore important parts of any system simply because they are not visible.

All problem-solving strategies are made more effective by thinking of problems critically and by exposing things that may not normally come to light. Warm-up games like Draw a Tree are great in that they quickly demonstrate some key problem-solving skills in an accessible and effective way.

Draw a Tree   #thiagi   #opening   #perspectives   #remote-friendly   With this game you can raise awarness about being more mindful, and aware of the environment we live in.

Closing activities for a problem-solving process

Each step of the problem-solving workshop benefits from an intelligent deployment of activities, games, and techniques. Bringing your session to an effective close helps ensure that solutions are followed through on and that you also celebrate what has been achieved.

Here are some problem-solving activities you can use to effectively close a workshop or meeting and ensure the great work you’ve done can continue afterward.

One Breath Feedback

Maintaining attention and focus during the closing stages of a problem-solving workshop can be tricky and so being concise when giving feedback can be important. It’s easy to incur “death by feedback” should some team members go on for too long sharing their perspectives in a quick feedback round. 

One Breath Feedback is a great closing activity for workshops. You give everyone an opportunity to provide feedback on what they’ve done but only in the space of a single breath. This keeps feedback short and to the point and means that everyone is encouraged to provide the most important piece of feedback to them. 

One breath feedback   #closing   #feedback   #action   This is a feedback round in just one breath that excels in maintaining attention: each participants is able to speak during just one breath … for most people that’s around 20 to 25 seconds … unless of course you’ve been a deep sea diver in which case you’ll be able to do it for longer.

Who What When Matrix 

Matrices feature as part of many effective problem-solving strategies and with good reason. They are easily recognizable, simple to use, and generate results.

The Who What When Matrix is a great tool to use when closing your problem-solving session by attributing a who, what and when to the actions and solutions you have decided upon. The resulting matrix is a simple, easy-to-follow way of ensuring your team can move forward. 

Great solutions can’t be enacted without action and ownership. Your problem-solving process should include a stage for allocating tasks to individuals or teams and creating a realistic timeframe for those solutions to be implemented or checked out. Use this method to keep the solution implementation process clear and simple for all involved. 

Who/What/When Matrix   #gamestorming   #action   #project planning   With Who/What/When matrix, you can connect people with clear actions they have defined and have committed to.

Response cards

Group discussion can comprise the bulk of most problem-solving activities and by the end of the process, you might find that your team is talked out! 

Providing a means for your team to give feedback with short written notes can ensure everyone is head and can contribute without the need to stand up and talk. Depending on the needs of the group, giving an alternative can help ensure everyone can contribute to your problem-solving model in the way that makes the most sense for them.

Response Cards is a great way to close a workshop if you are looking for a gentle warm-down and want to get some swift discussion around some of the feedback that is raised. 

Response Cards   #debriefing   #closing   #structured sharing   #questions and answers   #thiagi   #action   It can be hard to involve everyone during a closing of a session. Some might stay in the background or get unheard because of louder participants. However, with the use of Response Cards, everyone will be involved in providing feedback or clarify questions at the end of a session.

Tips for effective problem solving

Problem-solving activities are only one part of the puzzle. While a great method can help unlock your team’s ability to solve problems, without a thoughtful approach and strong facilitation the solutions may not be fit for purpose.

Let’s take a look at some problem-solving tips you can apply to any process to help it be a success!

Clearly define the problem

Jumping straight to solutions can be tempting, though without first clearly articulating a problem, the solution might not be the right one. Many of the problem-solving activities below include sections where the problem is explored and clearly defined before moving on.

This is a vital part of the problem-solving process and taking the time to fully define an issue can save time and effort later. A clear definition helps identify irrelevant information and it also ensures that your team sets off on the right track.

Don’t jump to conclusions

It’s easy for groups to exhibit cognitive bias or have preconceived ideas about both problems and potential solutions. Be sure to back up any problem statements or potential solutions with facts, research, and adequate forethought.

The best techniques ask participants to be methodical and challenge preconceived notions. Make sure you give the group enough time and space to collect relevant information and consider the problem in a new way. By approaching the process with a clear, rational mindset, you’ll often find that better solutions are more forthcoming.  

Try different approaches  

Problems come in all shapes and sizes and so too should the methods you use to solve them. If you find that one approach isn’t yielding results and your team isn’t finding different solutions, try mixing it up. You’ll be surprised at how using a new creative activity can unblock your team and generate great solutions.

Don’t take it personally 

Depending on the nature of your team or organizational problems, it’s easy for conversations to get heated. While it’s good for participants to be engaged in the discussions, ensure that emotions don’t run too high and that blame isn’t thrown around while finding solutions.

You’re all in it together, and even if your team or area is seeing problems, that isn’t necessarily a disparagement of you personally. Using facilitation skills to manage group dynamics is one effective method of helping conversations be more constructive.

Get the right people in the room

Your problem-solving method is often only as effective as the group using it. Getting the right people on the job and managing the number of people present is important too!

If the group is too small, you may not get enough different perspectives to effectively solve a problem. If the group is too large, you can go round and round during the ideation stages.

Creating the right group makeup is also important in ensuring you have the necessary expertise and skillset to both identify and follow up on potential solutions. Carefully consider who to include at each stage to help ensure your problem-solving method is followed and positioned for success.

Create psychologically safe spaces for discussion

Identifying a problem accurately also requires that all members of a group are able to contribute their views in an open and safe manner.

It can be tough for people to stand up and contribute if the problems or challenges are emotive or personal in nature. Try and create a psychologically safe space for these kinds of discussions and where possible, create regular opportunities for challenges to be brought up organically.

Document everything

The best solutions can take refinement, iteration, and reflection to come out. Get into a habit of documenting your process in order to keep all the learnings from the session and to allow ideas to mature and develop. Many of the methods below involve the creation of documents or shared resources. Be sure to keep and share these so everyone can benefit from the work done!

Bring a facilitator 

Facilitation is all about making group processes easier. With a subject as potentially emotive and important as problem-solving, having an impartial third party in the form of a facilitator can make all the difference in finding great solutions and keeping the process moving. Consider bringing a facilitator to your problem-solving session to get better results and generate meaningful solutions!

Develop your problem-solving skills

It takes time and practice to be an effective problem solver. While some roles or participants might more naturally gravitate towards problem-solving, it can take development and planning to help everyone create better solutions.

You might develop a training program, run a problem-solving workshop or simply ask your team to practice using the techniques below. Check out our post on problem-solving skills to see how you and your group can develop the right mental process and be more resilient to issues too!

Design a great agenda

Workshops are a great format for solving problems. With the right approach, you can focus a group and help them find the solutions to their own problems. But designing a process can be time-consuming and finding the right activities can be difficult.

Check out our workshop planning guide to level-up your agenda design and start running more effective workshops. Need inspiration? Check out templates designed by expert facilitators to help you kickstart your process!

Save time and effort creating an effective problem solving process

A structured problem solving process is a surefire way of solving tough problems, discovering creative solutions and driving organizational change. But how can you design for successful outcomes?

With SessionLab, it’s easy to design engaging workshops that deliver results. Drag, drop and reorder blocks  to build your agenda. When you make changes or update your agenda, your session  timing   adjusts automatically , saving you time on manual adjustments.

Collaborating with stakeholders or clients? Share your agenda with a single click and collaborate in real-time. No more sending documents back and forth over email.

Explore  how to use SessionLab  to design effective problem solving workshops or  watch this five minute video  to see the planner in action!

Over to you

The problem-solving process can often be as complicated and multifaceted as the problems they are set-up to solve. With the right problem-solving techniques and a mix of exercises designed to guide discussion and generate purposeful ideas, we hope we’ve given you the tools to find the best solutions as simply and easily as possible.

Is there a problem-solving technique that you are missing here? Do you have a favorite activity or method you use when facilitating? Let us know in the comments below, we’d love to hear from you! 

James Smart is Head of Content at SessionLab. He’s also a creative facilitator who has run workshops and designed courses for establishments like the National Centre for Writing, UK. He especially enjoys working with young people and empowering others in their creative practice.

thank you very much for these excellent techniques

Certainly wonderful article, very detailed. Shared!

Your list of techniques for problem solving can be helpfully extended by adding TRIZ to the list of techniques. TRIZ has 40 problem solving techniques derived from methods inventros and patent holders used to get new patents. About 10-12 are general approaches. many organization sponsor classes in TRIZ that are used to solve business problems or general organiztational problems. You can take a look at TRIZ and dwonload a free internet booklet to see if you feel it shound be included per your selection process.

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How to Effectively Practice CP + Problem Solving Guide

This is a slight tweak of a practice guide I wrote a while ago on USACO reddit since I thought it could be helpful to people here. Some USACO specific sections or extra clutter I left out here that aren't needed for a general audience. This should cover all general cp advice I have so I never have to retype.

Introduction

This is a post on how I believe is the best method to practice modern day competitive programming based on my experiences. I assume you already have some knowledge and know simple things like binary search and dfs/bfs, but read the footnote if you are complete beginner (never code, solved <50 problems, div2 A/B too difficult, grey or stuck low pupil).

First, a quick tl;dr of the practice strategy before a bunch of specifics and explanation:

In short, mostly you only need to use codeforces (no matter what contest you're training for), find a rating range where you can solve around ~30-40% of the time on your own, and just grind down the problem set tab in reverse order of id (the default sorting). Also take part in every live contest you can, and virtual any live contests you miss. Also, if your primary goal is some goal outside of codeforces (let's say USACO, but could replace with any OI or if ICPC replace instance of OI with ICPC) Approximately once per week (probably on each weekend), I recommend you virtual an OI contest then upsolve the ones you understand the editorial for after. This should be old USACO contests until you finish all in the past 5 or so years, then use OI checklist to find new contests. Make sure you go for subtasks just as you would in real contests when doing so.

Some parts of this method may seem strange to you, so I'll explain in more detail and comment on why I believe it is the best method, and give some proof. If you're too lazy to read all of it, the most important parts of this article are bolded . Also, I am assuming you are able to practice somewhat regularly (at least a few days of practice done each week for multiple months), and this practice is unlikely to work if you don't. However if you really want to improve fast, ideally practice should be daily, no breaks. .

Goal of Practice

First off, what is the main goal in practicing efficiently? I would argue you want to come across as many subtle ideas and concepts as quickly as possible and learn to intuitively realize when to apply them and what they have in common. This is what my practice method is centered around.

Another important goal is you should also feel discomfort in effort of trying to think new ideas as much as possible, but don't mistake this as time being confused with discomfort having no idea what to do. Actively making new insights as fast as possible is the state you should be in a lot during live contests and need to endure actively thinking new ideas while trying to not repeat same ideas in your mind. But when you have no clue how to approach/understand a solution to a problem, you are more likely to lose focus and are not helping yourself, so you want to minimize this.

Why Codeforces?

So, why only codeforces? Well, recent codeforces problems do a decently good job of introducing a large variety of concepts, particularly in the 2000+ rating range. Thanks to the large standards of wanting non-standard problems each contests, many small math tricks and greedy techniques are introduced, along with standard algorithms and data structure appearing decently enough. This is why I think they are the best collection of problems, as opposed to many other judges that are more standard and less diverse or innovative. Recent codeforces contests are by far better than old contests however, so that is why you should grind down the problems from most to least recent in the problem set tab. If you have done all contests later than contest 450, you should probably start using another judge and start doing more virtual contests, but at that point you probably don't need this guide.

How to Approach Problems in Practice

Alright, so codeforces seems good. Why only a rating range where you can solve ~30-40% of the time? Shouldn't you be practicing coming up with solutions on your own? Well, like I said earlier, you want to come across as many concepts as quickly as possible. If you're able to solve ~80%+ of the problems you're doing on your own, even if it takes a while, or in fact especially if it takes a while, you are not using your time most effectively, as you were already able to come up with the concept on your own. It is OK to read editorials often , that is where you actually learn new things. Binary search on the problem set tab to find a rating range of problems that fits the ~30-40% specification, and I recommend the rating range to a few hundred points wide. You can just shift range upward whenever lower end feels easier and you're solving more.

Well, the next natural question is how long should you take before reading editorials? I will argue only spend 15m thinking, after that if you're still having ideas keep thinking, but if you're just stuck read the editorial. However, if reading the editorial gives you new ideas continue thinking again. Sure, you may discover a trick you came up with yourself you can use later after a long time thinking, but was it worth spending 3h coming up with the solution on your own when you could've gone through 2 or 3 more problems if you read the editorial instead. However, going through too hard problems is just as bad is going through too easy problems. It is not worth spending 4h understanding a 3000 rated problem when you could learn much more concepts from 4 2300 rated problems in the same amount of time (if that's good for your skill level). That's why I say ~30-40%, this is usually the point where you can understand the editorial relatively quickly but aren't able to see the concepts on your own. Also, this is another reason to use codeforces instead of other sources, the problems are shorter so you can get through more faster and it is easier to find many problems of similar skill level .

Some important notes, however, are to take the 15m of thinking very seriously and implement every problem . This is extremely important!!! you should only be looking at editorial when you are really out of ideas and trying to think longer will just make you unfocused or reiterate old ideas. In other words you should feel mentally exhausted!! (or you're not working thinking hard enough). Don't be lazier than you would be in a contest, don't give up because you don't want to think harder on details, don't think/implement leisurely. It is important to practice making observations on your own, and you should be solving problems in the range more and more often as you go down the problem list, that's how you know you're improving. If you're not improving, you are likely not exhausting yourself thoroughly. You may think you can get through more concepts earlier without implement too, and this would fit the main goal of practice better, however, it's important to always implement every problem that isn't completely trivial, even if you mind solve it on your own, as you will remember it better and often you will realize you didn't understand the details as well as you thought before implementing. Always implement before reading editorial if you think you have idea, even when not sure, and don't look at others implementation before you solve even if you read editorial except for last resort.

I also recommend timing yourself when doing problems, at least while implementing. This will help you stay focused and improve your implement speed (which is important so you don't waste time implementing in contest). If you record your times you should hopefully see yourself getting faster for a fixed problem difficulty :).

When you finish a problem, make sure you reflect on techniques and mindset used and how you could generalize thought process to solve other problems more efficiently (imagine you were teaching someone else best way to approach similar problems). Similarly do this when you learn new algorithms or tricks and imagine how you would come up with on your own (most are old and not hard to use same process as contest problems to derive from scratch). Try to come up with your own list similar to one I have in "extra advice how to think" section, keep track of it and update it as needed after each problem you solve. The goal is to find short fundamental list of questions to ask yourself that will always lead to the solution, not just categorizing by algorithms or techniques used. Ask heavily how one mindset applies to many contexts. You are trying to prepare a system to solve new problems, not just memorizing ones you've seen. Similarly reflect on what can go wrong and how to consciously avoid mental traps. Also, it can be good to look at others solutions after you finish a problem quickly to see if there are any implementation tricks you don't know, and similarly reflect how you could make your code more concise.

I think your future self will appreciate it if you put all your code solutions with a commented one sentence summary of the problem takeaways on github, but it is not too important.

When to Learn Algorithms/Data structures

Next thing to come up is when in this am I supposed to learn new standard algorithms and data structures? I advise when you come across an algorithm or any other concept (maybe math idea) in an editorial you don't know about to immediately find and read an article about it, implement in the context of this problem, and then continue just moving down the problem set tab. You can usually find an article on USACO guide, cp-algorithms, or a codeforces blog. The idea behind this is that algorithms should come up at a rate according to their relevance, so if the algorithm really is important you should see it in more problems soon, and you don't need to go looking for more problems with the topic. Similarly, it is important to see algorithms in context, which is why you should not practice by topic , as you will likely miss out on many more subtle techniques and tricks not in a topics list and get too used to knowing the algorithm used ahead time when you should be trying to figure that out in the 15m thinking time.

However, if you want a break or have other extra time when you can't do problems, reading through random algorithm articles in the locations listed above is a good way to expose you to some new ideas. But it is still more important to be actively solving problems when you can.

Live Contests

The number one thing that probably looks wrong with this practice method, despite the reasonings I gave earlier, is that you seem like you are not practicing solving problems on your own often enough. This is where live contests come in. It is important to take part in as many live contests as possible from every judge you can (except ones where every problem feels too easy) . This is where you practice thinking on your own, and if you look enough there are tons of contests all the time, particularly high quality ones from atcoder and codeforces. You should also upsolve the easiest problem you didn't solve during the contest, however, after that you should just go back to the codeforces problem set grind unless there are more problems from the contest within your practice rating range on codeforces. Lastly, to make sure you're taking enough contest, take every codeforces contest you miss that would be rated for you as a virtual contest.

Also, if your primary objective is some other contest (say USACO/OI but can replace with ICPC), you should do OI virtual about once per week as subtasks are becoming more important in USACO plus probably good to have more extended focus practice anyway. You also want to shift practice to doing mostly OI virtuals the week before a USACO contest begins. Make sure for these virtuals you are going for maximum points like in a real contest which may mean implementing subtasks, not just implementing full solves (or whatever other contest specific traits that differ from codeforces). If you aren't practicing a ton or you feel virtuals are taking too much time away from doing codeforces practice maybe do every other week instead of every week.

Scheduling Practice

This is less important but more just some pointers on scheduling time to practice consistently. I think it is obviously best to practice daily, and it isn't as hard as you may think it is if you build up good habits. I think it is good to have a regularly scheduled time where you can practice each day , as this makes it more of a consistent habit. Similarly, if you can set aside a specific location to practice as well that would be good , as this can give your mindset the habit that a specific time and place is for practicing only, and you build focus**.** Try to practice at least 90m for your scheduled time , but preferably longer. And get off discord!!! when you're practicing in the designated time :clown:.

Besides scheduled practice time, you can probably fit in more practice time in some or many days in different ways as well if you are serious. For example, I think it is good to memorize some problems at the beginning of each day, maybe a bit harder than you'd normally practice, and think about them all day during school, shower, eating, etc., or maybe the same problems for a few days. This helps you practice thinking more on your own. Also, when you have free time in class or while in car and someone else is driving or something, this is a good time to read algorithm articles. When I went to public school I also bought a portable keyboard to practice in class and spent most school lunch days in the library doing problems, but this might be overkill. Point is find all times of day to practice any way possible when you can, but most import is the scheduled practice time.

Adjustments Closer to Big Contest

If you are training for some main goal (hopefully for the past several months at least, following above methods), when you are within a few weeks away of big contest, start spending more practice time on vc's for that contest, and look over the syllabus/relevant ideas for that contest if list exists . Also consider if you are in these pitfalls:

You are too slow at working out ideas or implementation => do more fast-paced contest vc's, time yourself in other practice.

You are bad at allocating time in OI/ICPC style => focus on more relevant vc's and practice subtask allocation, figuring out which problems to work on, and when to move on like in real contest.

Still not able to make big insights that seem to come out of nowhere => try more guessing and some atcoder lol.

Hopefully this was somewhat useful to some of you, and gives you a comprehensive guide on how to practice for USACO and competitive programming in general. Please share this with others if you think it is useful.

For any more experienced people, let me know if there is anything you strongly disagree with what I said, I'd be interested to hear your viewpoint, though you're unlikely to change my mind :).

**I recommend the beginning of the usaco training page to complete beginners . I think it is a good way to start out as it guides you on the basics, and you should be able to start as soon as you know the very basics to a programming language, preferably c++ (you can use codeacademy to learn basics, it should take only a couple days max. you learn other parts about the language as you solve more problems and googling as needed). However, as soon as you finish chapter 1 or the problems feel easy (or if codeforces is still too intimidating maybe hard max finish chapter 2), that is when I recommend you start using this practice method, and perhaps also try some problems from the cses sorting and searching section. However most people reading this should already have some experience.

Sources mentioned: USACO — http://www.usaco.org Codeforces — https://codeforces.com Atcoder — https://atcoder.jp CSES — https://cses.fi/problemset/ Training gate — https://usaco.training OI Checklist — https://oichecklist.pythonanywhere.com Cp-Algorithms — https://cp-algorithms.com USACO Guide — https://usaco.guide Codeacademy — https://www.codecademy.com/catalog/language/c-plus-plus

Extra Advice How to Think to Solve Problems

Overall, just make sure you are always thinking new ideas and repeatedly combining old observations to make new ones. Don't worry about solving all at once, just think one small step at a time! Usually this means think what do you know for sure, then use to guess ideas on properties and direction and check if you can prove, combine your previous observations, then repeat. When really stuck, guess more extreme (it is another thing people who aren't improving don't do enough). Actually write down you're observations and make sure you're writing new things as fast as possible, even when seems small or irrelevant. But for some more direct tips, try going through the following checklist when approaching a problem:

• look at everything from perspective of binary (both bit representation and splitting things in halves) and graph (pairs in input), or sometimes as geometry coordinates
• think how information you have can be reused (like dp but more general, eg 2ptr or extending construction, sweepline, split query into reusable known parts), ask what is dependent and how, order by dependency. also try making one choice then and get same problem then induct (eg greedy, mst, dp, decision tree like trie, ask "what do i know for sure"), or combine smaller problems to get answer (eg range dp, d&q, mitm), so can reuse info of smaller problem.
• reduce things to as simple as possible, compact representation of info (eg bitmask), get rid of redundant transitions/states/etc. what is minimum needed for condition to be true? when something changes or decision made what is minimum that actually matters? sometimes combine operations into simpler one (eg try turning operation into something can binary exponentiate). bound everything as tight as possible and use to reduce states to consider. is answer/construction equivalent to bounds/minimum conditions (guess when stuck)?
• make formulas out of everything, expand/rewrite as many ways possible (even simple like |x| -> +-x). think about related formulas to transform (eg combo) and other representations (don't forget matrix/polynomial).
• visualize everything, draw things out
• look for structures like montonicity, concavity, etc. (eg bsearch/dp opt) along with new conditions/constraints implied (eg sqrtn distinct of n total), and do this for every part of problem, whether specific part or entire structure of solution
• go through testcases by hand (both initially with brute force and with your current best ideas), maybe also make generator/brute force checker if stuck to further look for patterns.
• don't think same things over and over, write down everything you think and try to always write down new ideas, every small new observation is progress and may be able to be combined with other ideas eventually, but rethinking same things will not help
• think of simplified cases then extend/reduce to them (reduce a[i] = 0/1, array->tree->graph, 2^x->k, lower constraint) or imagine assuming something you wish exists already exists (like data structure often range query, constraint eg for bsearch, previous knowledge, etc.) and solving from there, chances are thing then does exist if helps
• reverse/change ordering of process (eg change order to simpler like change general add/delete to add/[delete most recent op] offline, swap dependency order) or look at inverse (especially for counting) or just view problem in different way in general, restate problem/conditions in as many different ways as you can to get new perspectives. nice transformation usually means right direction (eg difference array).
• if something reminds you of standard algorithm, or you find too slow solution for some part, think of every way you know how to do that standard thing and see if any modification relates to what you are doing, and think deeply what parts can be changed for specific problem
• if something seems random in statement like any abnormal constraint or is similar to known problem but different in some way, is probably key to solving so consider why it is put in statement
• don't forget sometimes can brute force small choices or if too many choices can pick random one or something that stands out (like max/min, only closest on left/right, etc.), extremals is often key. think carefully and guess what not matter if problem seems too hard initially. in constructive/interactive with many options can likely solve with only small subset of options.
• don't overcomplicate. try multiple directions, if too many steps or edge cases probably not right direction. almost always a nice easily provable solution. guess nice things (eg simplest greedy/construction), hope they work, then check but don't get stuck forcing path. take step back when clowning on small details even if you know it is right general direction
• try focusing on answer for one element at time instead of entire process (like in counting or creating bsearch condition, local easier to update for queries), or sometimes opposite (eg graph out all solutions, know ahead of time offline). in general change scope of thinking
• believe you can solve every problem, but also treat every problem as a challenge that you take one step at a time. even most standard ideas you can learn on your own if you treat same way as any other problem
• if something you remember very vaguely seems similar but you don't remember source and barely remember details, don't waste time trying to remember old thing, just start resolving from scratch
• as stated by Perpetually_Purple in comment below, sometimes can try to cheese with random/heuristic if running out of time. especially true for OI contests with subtasks
• sometimes can split problem into parts which can be solved differently based on constraints (eg sqrt decomp, small to large, upd and qry compute different parts, even/odd).
• also break into independent problems (eg intervals that don't affect each other, solve x and y coordinate separately). when dependent on multiple things, process in order that gets rid of thinking about one and only worry about others (eg sweep one dimension, query other).
• map things to a canonical form (eg lexographically minimal) or map representations that are equivalent to help with counting or alternate way to view solution. (eg think of greedy idea to get specific configuration then have counting dp mimic the greedy method to not overcount, find simple idea for single query then speed up multiple queries by precomputing conditions when add during greedy to speed up).
• imagine assuming you know solution ahead of time and analyze or fix choices ahead of time and solve rest, can use this to prove things equivalent, choices not to consider, or properties of optimal configuration.
• try only computing minimum necessary at each point of time, especially for update/query. can sometimes use amortized/lazy arguments (eg keep track of covered intervals in set, lazy prop on segtree).
• ask what stays the same and what changes. how does a single operation affect properties of a problem (sum/difference of elements, always progress towards goal, reversable, etc.)? when doing testcases by hand guess these types of things then prove/disprove. use these properties to prove things like which choices are optimal or what is bottleneck to bound on answer.
• Similar to 3 and 10, try compressing groups of things and solve over those group when relations within them are irrelevant, and keep updating when you can simplify further throughout process (eg compress cycles, scc, biconnected components, directed mst).'
• When guessing idea, make sure you are listing through all assumptions being made and that those assumptions you know for sure hold true and completely encompass the problem. Also make samples around idea of what you think could go wrong, and use that to help you prove or disprove idea. If you're taking too long disproving wrong ideas, you likely need to go more one step at a time, don't guess extreme until more stuck.
• If stuck working out details when have main idea, work out more testcases by hand and/or write detailed pseudocode and find what steps you are not entirely sure what they work and think harder. Don't be lazy about writing details!
• Can you prove something is true most of the time? Try probabilistic techniques (eg hashing, testing set of samples, etc.) or also try 19.

Also it is good to use problem constraints to guide your initial direction of thinking, but don't let it constrain you to specific ideas. And whatever you do don't misread the problem , better to spend slightly longer reading and understanding correctly than solve wrong thing.

Implementation Tips

First check briefly that you are not missing easier idea/method to implement. That will save most time.

Try to have clear idea of each segment of code you will write, then write as fast as possible. Sometimes you don't have clear idea of entire code you write and only general outline, and that's ok, but in your mind have different parts of code in small chunks and have each small chunk planned out clearly before you write then think if needed before writing next part. Try to plan your code to be as concise as possible while still easily readable and make it where you are not rewriting same thing multiple times. If you keep rewriting, you need to step back and plan out better, check your ideas.

Also for debugging, just make a bunch of print statements in code and look for problems. Try to binary search and figure out where in the code the outputs are first not what you'd expect. If you realize some part is not right, don't get stuck making small edits trying to fix, go back to planning and rewrite when clear. Also try working through some examples by hand following steps of code, and read through every single line of code. It is likely the mistake is somewhere where you were sure you couldn't mess up lol.

Also adding one sentence comment to code on main idea of every problem might be nice if you ever need to refer back.

Allocating Time in OI Contest

I'm assuming 3 problems in 4 hours (adjust scale as needed). I usually read all 3 problems in first 15 minutes, then spend about 15 minutes each to think about each problem and decide order of difficulty I find easiest. If I fully mind solve one in that time I immediately implement, otherwise I do as follows. I then try to divide the next three hours to be roughly even among the three problems, and try the problems in order from easiest to hardest.

While focusing on a problem, it is very important to stay focused on only that problem. For most of hour on problem should implement as soon as you full solve but only implement subtasks to test ideas, if you think it help you towards full solution, or you are completely out of new ideas (in which case move on after implementing subtask if u still don't have new ideas). However, if you already use up ~50min for that problem and still don't know full solution and won't reach in next 5min, even if you think you could make more progress, just implement what subtasks you know and move on. It is important to actually move on as you may have wrongly assessed which problem was easiest so you want to have time to try all the problems (this has been my downfall multiple times in past). This means once you move on don't have more lingering thoughts usually and fully focus on next problem.

Math + CS Practice

If you are practicing math olympiad and cs olympiad, or just want some reading material that might help you, try reading some of and doing some problems from this combo book . Overall it will be better for you to just do be actively solving more problems for cp practice, but if you have some other free time it is a pretty good read and cp is basically olympiad combo + data structures + implementation anyway.

Practicing for math olympiad in general will also help you with competitive programming, but if you are only focusing on cp it is better to just work on cp problems.

Extra Motivation

In everything in life, the key to success is learning to find fulfillment in every small step you make towards progress. Related to cp, every problem solved and every day of practice is one step closer to your competitive programming goals. When solving a problem every new observation is one step closer to finding the solution.

Also, make sure you know your priorities and what you really want out of life, don't have regrets. If you really want to be good in cp, stop wasting time, stop taking days off, start solving problems as much as you can and you will find success. Obsess over what you want most until you achieve it.

Cognitive Remediation Therapy: 13 Exercises & Worksheets

This can result in concentration, organizational, and planning difficulties that impact their quality of life and independent living.

Cognitive Remediation Therapy (CRT) helps by increasing awareness of intellectual difficulties and improving thinking skills. While originally designed for people with thinking problems associated with schizophrenia, it has also proven successful for those with other diagnoses (Bristol Mental Health, n.d.).

CRT works by encouraging a range of exercises and activities that challenge memory, flexible thinking, planning, and concentration problems.

This article explores CRT and its potential to help clients and includes techniques, activities, and worksheets to build effective therapy sessions.

Before you continue, we thought you might like to download our three Positive CBT Exercises for free . These science-based exercises will provide you with detailed insight into Positive CBT and give you the tools to apply it in your therapy or coaching.

This Article Contains:

What is cognitive remediation therapy (crt), how does cognitive remediation work, 8 techniques for your sessions, 7 exercises, activities, & games, 6 helpful worksheets and manuals, implementing online crt programs, 3 best software programs for helping your clients, a take-home message.

“Cognitive remediation is a behavioral treatment for people who are experiencing cognitive impairments that interfere with daily functioning” (Medalia, Revheim, & Herlands, 2009, p. 1).

Successful cognitive functions, including memory, attention, visual-spatial analysis, and abstract reasoning, are vital for engaging with tasks, the environment, and healthy relationships.

CRT improves cognitive processing and psychosocial functioning through behavioral training and increasing individual confidence in people with mental health disorders (Corbo & Abreu, 2018). Training interventions focus on the skills and supports required to “improve the success and satisfaction people experience in their chosen living, learning, working, and social environments” (Medalia et al., 2009, p. 2).

Exercises typically focus on specific cognitive functions, where tasks are repeated (often on a computer) at increasing degrees of difficulty. For example:

• Paying attention
• Remembering
• Being organized
• Planning skills
• Problem-solving
• Processing information

Based on the principles of errorless learning and targeted reinforcement exercises , interventions involve memory, motor dexterity, and visual reading tasks. Along with improving confidence in personal abilities, repetition encourages thinking about solving tasks in multiple ways (Corbo & Abreu, 2018).

While initially targeted for patients with schizophrenia, CRT is an effective treatment for other mental health conditions , including mood and eating disorders (Corbo & Abreu, 2018).

CRT is particularly effective when the cognitive skills and support interventions reflect the individual’s self-selected rehabilitation goals. As a result, cognitive remediation relies on collaboration, assessing client needs, and identifying appropriate opportunities for intervention (Medalia et al., 2009).

Cognitive remediation vs cognitive rehabilitation

CRT is one of several skill-training psychiatric rehabilitation interventions. And yet, cognitive remediation is not the same as cognitive rehabilitation (Tchanturia, 2015).

Cognitive rehabilitation typically targets neurocognitive processes damaged because of injury or illness and involves a series of interventions designed to retrain previously learned cognitive skills along with compensatory strategies (Tsaousides & Gordon, 2009).

While initially done in person, they can subsequently be performed remotely as required (Corbo & Abreu, 2018; Bristol Mental Health, n.d.).

Well-thought-out educational software provides multisensory feedback and positive reinforcement while supporting success, choice, and control of the learning process. Its design can target either specific cognitive functions or non-specific learning skills and mechanisms (Medalia et al., 2009).

CRT successfully uses the brain’s neuroplasticity and is often more effective in younger age groups who haven’t experienced the effects of long-term psychosis. It works by increasing activation and connectivity patterns within and across several brain regions involved in working memory and high-order executive functioning (Corbo & Abreu, 2018).

The Neuropsychological Educational Approach to Cognitive Remediation (NEAR) is one of several approaches that provide highly individualized learning opportunities. It allows each client to proceed at their own pace on tasks selected and designed to engage them and address their cognitive needs (Medalia et al., 2009).

NEAR and other CRT techniques are influenced by learning theory and make use of the following (Medalia et al., 2009):

• Errorless learning Encouraging the client to learn progressively, creating a positive experience without relying on trial and error.
• Shaping and positive feedback Reinforcing behaviors that approximate target behaviors (such as good timekeeping) and offering rewards (for example, monthly certificates for attendance).
• Prompting Using open-ended questions that guide the client toward the correct response.
• Modeling Demonstrating how to solve a problem.
• Generalizing Learning how to generalize learned skills to other situations.
• Bridging Understanding how to apply skills learned inside a session outside  in everyday life.

Encouraging intrinsic motivation (doing the tasks for the satisfaction of doing them rather than for external rewards) and task engagement are also essential aspects of successful CRT programs (Medalia et al., 2009).

Therapy is most effective when it successfully supports clients as they transfer learning skills into the real world.

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Cognitive remediation techniques must be selected according to the skills and needs of the client and typically fall into one of three major intervention categories (Medalia et al., 2009):

• Planning exercises, such as planning a trip to the beach to practice cognitive strategies
• Cueing and sequencing , such as adding signs or placing reminder notes at home to encourage completing everyday tasks (for example, brushing teeth)

Such techniques rely on several key principles, including “(1) teaching new, efficient, information processing strategies; (2) aiding the transfer of cognitive gains to the real world; and (3) modifying the local environment” (Medalia et al., 2009, p. 5).

• Restorative approaches Directly target cognitive deficits by repeating task practices and gradually increasing difficulty and complexity; along with regular feedback, they encourage accurate and high levels of performance.

Practice is often organized hierarchically, as follows:

• Elementary aspects of sensory processing (for example, improving auditory processing speed and accuracy)
• High-order memory and problem-solving skills (including executive functioning and verbal skills)

This technique assumes a degree of neuroplasticity that, with training, results in a greater degree of accuracy in sensory representations, improved cognitive strategies for grouping stimuli into more meaningful groups, and better recall.

• Repetition and reaching for increasing levels of task difficulty
• Modeling other people’s positive behavior
• Role-play  to re-enact experienced or imagined behavior from different perspectives
• Corrective feedback to improve and correct unwanted or unhelpful behavior

Complex social cognitive processes are typically broken down into elemental skills for repetitive practice, role-play, and corrective feedback.

Professor Dame Til Wykes: cognitive remediation therapy

It is vital that activities within CRT are interesting and engaging for clients. They must foster the motivation required to persevere to the end of the task or game.

The following three games and puzzles are particularly valuable for children and adolescents (modified from Tchanturia, 2015):

SET is a widely available card game that practices matching based on color, shape, shading, etc.

Clients must shift their thinking to identify multiple ways of categorizing and grouping cards, then physically sort them based on their understanding.

It may be helpful to begin with a limited set of cards to reduce the likelihood of the clients becoming overwhelmed by the game or finding it less enjoyable.

2. Rush Hour

Rush Hour is another fun game that balances problem-solving skills with speed.

Puzzles start simple and increase in complexity, with additional elements involved. Skills developed include problem-solving and abstract thinking, and the game requires a degree of perseverance.

Other activities require no specialist equipment and yet can be highly engaging and support clients in learning transferable skills (modified from Tchanturia, 2015).

• Bigger picture thinking This involves the client picturing a shape in their minds or looking at one out of sight of the therapist. They then describe the shape (without naming it), while the therapist attempts to draw it according to the instructions. This practice is helpful with clients who get overwhelmed by detail and cannot see the bigger picture.
• Word searches Word searches encourage the client to focus on relevant information and ignore everything else – an essential factor in central coherence. Such puzzles also challenge memory, concentration, and attention.
• Last word response Last word response is a challenging verbal game promoting cognitive flexibility. The first player makes up and says a sentence out loud. Each subsequent player makes up a new sentence, starting with the last word of the previous player’s sentence. For example, ‘ I like cheese’ may be followed by the next player saying, ‘ Cheese is my favorite sandwich ingredient ,’ etc.
• Dexterity Using your non-dominant hand once a week (for example, combing your hair or brushing your teeth) stimulates different parts of your brain, creating alternative patterns of neuron firing and strengthening cognitive functions.

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The following therapy worksheets help structure Cognitive Remediation Therapy sessions and ensure that the needs of clients are met using appropriately targeted CRT interventions (modified from Medalia et al., 2009; Medalia & Bowie, 2016):

Client referral to CRT

The Cognitive Remediation Therapy Referral Form captures valuable information when a client is referred from another agency or therapist so that the new therapist can identify and introduce the most appropriate CRT interventions. The form includes information such as:

Primary reasons

Secondary reasons

• Self-confidence
• Working with others
• Time management
• Goal-directed activities

Cognitive Appraisal for CRT

The Cognitive Appraisal for CRT form is helpful for identifying and recording areas of cognitive processing that cause difficulty for the client and require focus during Cognitive Remediation Therapy sessions.

Clients are scored on their degree of difficulty with the following:

• Paying attention during conversation
• Maintaining concentration in meetings
• Completing tasks once started
• Starting tasks
• Planning and organizing tasks and projects
• Reasoning and solving problems

Software Appraisal for CRT

The Software Appraisal for CRT form helps assess which software would be most helpful in a specific Cognitive Remediation Therapy session. It provides valuable input for tailoring treatment to the needs of the client.

For example:

• Level of reading ability required
• Cognitive deficits addressed by the software
• What is the multimedia experience like?
• How much input is required by the therapist?

Appraisal records become increasingly important as more software is acquired for clients with various cognitive deficits from multiple backgrounds.

Software Usage for CRT

The Software Usage for CRT form helps keep track of the software clients have tried and how effectively it supports them as they learn, develop, and overcome cognitive deficits.

The client considers the software they use and whether they practiced the following areas of cognition:

• Concentration
• Processing speed
• Multitasking
• Logic and reasoning
• Organization
• Fast responses
• Working memory

Thought Tracking During Cognitive Remediation Therapy

Thought Tracking During Cognitive Remediation Therapy is valuable for identifying and recording the client’s goals for that day’s Cognitive Remediation Therapy session and understanding how it relates to their overall treatment goals.

Planning to Meet Goals in CRT

The Planning to Meet Goals in CRT worksheet is for clients requiring support and practice in planning, goal-setting, and goal achievement.

Working with the client, answer the following prompts:

• What goal or project are you working toward?
• What date should it be completed by?
• Are there any obstacles to overcome to complete the goal?
• Are there any additional resources required?
• Then consider the steps needed to achieve the goal.

Other free resources

Happy Neuron provides several other free resources that are available for download .

Consider the five Cs when selecting online CRT programs (modified from Medalia et al., 2009):

• Cognitive – What target deficits are being addressed?
• Client – What interests and level of functioning does the client have?
• Computer – What computing requirements and compatibility factors need to be considered?
• Context – Does the software use real-world or fantasy activities and environments? Are they age and cognitive ability appropriate?
• Choice – Is the learner given choice and options to adapt the activity to their preferences?

Once you’ve ordered the software, give it a thorough review to understand when it is most appropriate to use and with whom.

For online CRT programs to be effective as teaching tools and activities, they should include the following features (modified from Medalia et al., 2009, p. 53):

• Intrinsically motivating
• Active use of information
• Multisensory strategies
• Frequent feedback
• Control over the learning process
• Positive reinforcement
• Application of newly acquired skills in appropriate contexts
• Errorless learning – challenging yet not frustrating

Therapists must become familiar with each program’s content and processes so that targeted deficits are fully understood and clients are engaged without confusion or risk of failure.

17 Science-Based Ways To Apply Positive CBT

These 17 Positive CBT & Cognitive Therapy Exercises [PDF] include our top-rated, ready-made templates for helping others develop more helpful thoughts and behaviors in response to challenges, while broadening the scope of traditional CBT.

Created by Experts. 100% Science-based.

A great deal of software “targets different skills and offers a variety of opportunities for contextualization and personalization” (Medalia et al., 2009, p. 43).

We focus on three suppliers of extensive CRT software resources below (recommended by Medalia et al., 2009).

1. Happy Neuron

Happy Neuron provides a wide variety of online brain training exercises and activities to stimulate cognitive functioning in the following areas:

• Visual-spatial

When you’re performing well, the exercises become increasingly difficult.

The exercises are grouped into the following areas:

• Brain speed
• People skills
• Intelligence

3. Games for the Brain

Cognitive difficulties, such as challenges with paying attention, planning, remembering, and problem-solving, can further compound and exacerbate mental health issues

While initially created for schizophrenia, CRT is also valuable for other mental health problems, including eating and mood disorders. Treatments are effective in one-to-one and group sessions, and lessons can be transferred to the outside world, providing crucial gains for a client’s mental wellbeing and social interaction.

Through repeated and increasingly challenging skill-based interventions, CRT benefits cognitive functioning and provides confidence gains to its users. The treatment adheres to learning theory principles and targets specific brain processing areas such as motor dexterity, memory, and visual-spatial perception, along with higher-order functioning.

Involving clients in treatment choices increases the likelihood of ongoing perseverance, engagement, and motivation as activities repeat with increasing degrees of difficulty.

This article offers a valuable starting point for exploring CRT and its benefits, with several worksheets and forms to encourage effective treatment.

We hope you enjoyed reading this article. For more information, don’t forget to download our three Positive CBT Exercises for free .

• Bristol Mental Health. (n.d.). Cognitive remediation therapy: Improving thinking skills . Retrieved December 15, 2021, from http://www.awp.nhs.uk/media/424704/cognitive-remediation-therapy-022019.pdf
• Corbo, M., & Abreu, T. (2018). Cognitive remediation therapy: EFPT psychotherapy guidebook . Retrieved December 15, 2021, from https://epg.pubpub.org/pub/05-cognitive-remediation-therapy/release/3
• Medalia, A., & Bowie, C. R. (2016). Cognitive remediation to improve functional outcomes . Oxford University Press.
• Medalia, A., Revheim, N., & Herlands, T. (2009). Cognitive remediation for psychological disorders: Therapist guide . Oxford University Press.
• Tchanturia, K. (2015). Cognitive remediation therapy (CRT) for eating and weight disorders . Routledge.
• Tsaousides, T., & Gordon, W. A. (2009). Cognitive rehabilitation following traumatic brain injury: Assessment to treatment. Mount Sinai Journal of Medicine: A Journal of Translational and Personalized Medicine , 76 (2), 173-181.

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To my surprise this is a treatment that has not been discussed in the area I live and work. I just stumbled upon this when I was researching cognitive impairments with schizophrenia. I currently work on a team with multiple mental health professionals that go out into the community, to work with people diagnosed with Schizophrenia. It seems like most of what we do is manage and monitor symptoms. Are you aware of anyone or any agency in Buffalo, NY that uses this method of treatment? I am trying to figure out how to get trained and use it in practice, if that is possible. Any help will be greatly appreciated.

This looks like the treatment my daughter needs. She has struggled for years with the cognitive problems associated with depression. How do we find a therapist near us who can use these techniques?

I’m sorry to read that your daughter is struggling. You can find a directory of licensed therapists here (and note that you can change the country setting in the top-right corner). You’ll also find that there are a range of filters to help you drill down to the type of support you need: https://www.psychologytoday.com/us/therapists

I hope you find the help you need.

– Nicole | Community Manager

I clicked the link you provided. Unfortunately there is not a filter for Cognitive Remediation Therapy (CRT). Do you have a suggestion of how to find a provider who is trained in CRT?

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3 Positive CBT Exercises (PDF)

How To Improve Speed And Accuracy in JEE

Speed and accuracy have a really complex relationship. Perhaps they are the most vital things to take care of while attending a highly competitive exam like JEE Main or Advanced. If you don't have the required pace, you will be snatched off your opportunity and left with the regret of how you could not attempt an easy question. On the other hand, accuracy is quite subtle when it comes to making you regret. You are going to realize your mistake after leaving the hall, or sometimes after the result comes out. But speed and accuracy -- they just do not get along. Precision requires time, and speed is all about reducing the amount of time. So let me be honest here, it's not easy to manage both of them at once, in fact, it's one of the hardest jobs - but maybe not as hard as cracking the exam with an acceptance rate of 2% and 12 lakh candidates.

Did I Make You Nervous? Don't Be.

Nervousness and stress are the arch-enemies of speed and accuracy. When you are nervous, your brain cannot function it its expected ability. On the other hand, when you are confident and stress-free, your brain produces Dopamine, which improves your concentration and problem-solving skills, eventually improving your speed and accuracy.

The Art Of Time Management

Time management is necessary for learning or revising as well as for acing the exam. When done correctly, it is nothing short of an art. During revision, you need to make a compact plan for how you are going to handle each and every topic. You have to stick to your own routine. Procrastination is not an option. When you have a specific deadline for finishing a topic, your brain will start working faster. Your subconscious mind will be trained to respond to the urgency, and nothing will be able to distract you at that time.

During your exam, you have to take a similar approach, only less aggressive. This is not a training for your brain, so precision matters. Your strategy should be well-planned, but cautious. Start with the easier problems and make your way up to the harder ones. If you practiced time management during your mock tests and revisions, speed should not be an issue for you, and the extra cautiousness would take care of the accuracy.

The Cognitive Approach To Problem-solving

During your revision, gradually increase the level of problems' difficulty, and try to reduce the usage of pen and paper. Initially, you might face problems, the answers might not be accurate, but as you continue, the improvement in speed and accuracy will be noticeable.

Soon you will be confident enough to do short mind calculations and skip steps to solve the problems quickly. But don't be overconfident. Always be extra cautious during your exam.

Creative Thinking Makes A Difference

No one will check if you solved the problem in a traditional step by step method, so you have the chance to be creative while solving a problem. During revision and mock tests, always try to find a better and quicker way to solve a question. You will discover a lot of reliable shortcut methods that were never taught to you. Your problem-solving speed would increase as you keep thinking creatively.

Now, here is the important part, don't try this during the exam, unless really necessary. Stick to the methods that you are 100% sure about. Your brain has already been trained to function faster, don't be deliberate to prove that during your exam.

Smartness Is As Important As Knowledge

You have to be smart while solving the problems. Unconventional ways are not discouraged at all. Learn some calculations beforehand. Memorise some square roots, cube roots, derivatives, and integrations. Skim through the shortcut techniques, memorize multiplication table up to 20, apply elimination technique when necessary. These techniques might not be good for learning, but they will prove really helpful in JEE. So practice these during your revision and apply only when they are the most viable options.

Don't Get Stuck

Never, and I repeat, never get stuck in a single problem during the exam. Once you make a time management strategy at the beginning, you would have an idea how much time you can spend on a single problem. So stick to the plan and avoid being stuck at a particularly difficult problem. Move on, solve the other problems, and come back to it later if you are left with enough time for a revision.

Practice - The Ultimate Advice

Okay, don't skip the paragraph - I know you are tired of hearing people advising you to “practice, practice and practice”. But trust me, this reminder is necessary, as practicing problems on the same concepts for 2 years is a really frustrating job. Just think about it for a moment, how much have your problem-solving speed improved since the first day you started studying for JEE? No matter how much boring this advice may sound, it always proves to be the most important one.

Prepare through mock tests with proper timing, and every time try to finish quicker than the last time. If you are taking offline mock tests, and your plan is to appear for the online JEE exam, be careful with the time management. The shading process does eat up a lot of time. Be careful about silly mistakes. Try to identify if there is a pattern in what kinds of mistakes you are making, and rectify them.

Double-checking, Something We All Love To Hate

After answering the 90 questions, you will not feel too enthusiastic about the idea of going through all of those questions and rough works once again, searching for some silly mistakes which you are sure you didn't make. But double checking is the key to achieving accuracy. The options given for an MCQ can often be misleading and too close to reason out. Keep your rough work organized to make the process less frustrating. You will thank yourself later for saving at least 8 to 12 marks, which is enough to make a significant difference in your rank.

So, that was all for this article. Do your best to establish a friendship between speed and accuracy; the result will be worth the effort. Best of luck.

Harshita Srivastava IIT Kanpur 13 September 2019

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• Open access
• Published: 03 September 2024

An improved Coati Optimization Algorithm with multiple strategies for engineering design optimization problems

• Zhang Qi 1 , 3 ,
• Dong Yingjie 2 ,
• Ye Shan 3 ,
• He Dongcheng 3 &
• Xiang Guoqi 3  

Scientific Reports volume  14 , Article number:  20435 ( 2024 ) Cite this article

Metrics details

• Engineering
• Mathematics and computing

Aiming at the problems of insufficient ability of artificial COA in the late optimization search period, loss of population diversity, easy to fall into local extreme value, resulting in slow convergence and lack of exploration ability; In this paper, an improved COA algorithm based on chaotic sequence, nonlinear inertia weight, adaptive T-distribution variation strategy and alert updating strategy is proposed to enhance the performance of COA (shorted as TNTWCOA). The algorithm introduces chaotic sequence mechanism to initialize the position. The position distribution of the initial solution is more uniform, the high quality initial solution is generated, the population richness is increased, and the problem of poor quality and uneven initial solution of the Coati Optimization Algorithm is solved. In exploration phase, the nonlinear inertial weight factor is introduced to coordinate the local optimization ability and global search ability of the algorithm. In the exploitation phase, adaptive T-distribution variation is introduced to increase the diversity of individual population under low fitness value and improve the ability of the algorithm to jump out of the local optimal value. At the same time, the alert update mechanism is proposed to improve the alert ability of COA algorithm, so that it can search within the optional range. When Coati is aware of the danger, Coati on the edge of the population will quickly move to the safe area to obtain a better position, while Coati in the middle of the population will randomly move to get closer to other Coatis. IEEE CEC2017 with 29 classic test functions were used to evaluate the convergence speed, convergence accuracy and other indicators of TNTWCOA algorithm. Meanwhile, TNTWCOA was used to verify 4 engineering design optimization problems, such as pressure vessel optimization design and welding beam design. The results of IEEE CEC2017 and engineering design Optimization problems are compared with Improved Coati Optimization Algorithm (ICOA), Coati Optimization Algorithm (COA), Golden Jackal Optimization Algorithm (GJO), Osprey Optimization Algorithm (OOA), Sand Cat Swarm Optimization Algorithm (SCSO), Subtraction-Average-Based Optimizer (SABO). The experimental results show that the improved TNTWCOA algorithm significantly improves the convergence speed and optimization accuracy, and has good robustness. Three‑bar truss design problem, The Gear Train Design Problem, Speed reducer design problem shows a strong solution advantage. The superior optimization ability and engineering practicability of TNTWCOA algorithm are verified.

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Introduction.

Coati Optimization Algorithm (COA) is a novel metaheuristic algorithm created by Dehghani in 2023 1 , COA algorithm, which simulates Coati's hunting behavior, has the characteristics of strong searching ability and fast convergence speed. But, due to the random generation of the initial population, it lacks diversity, random strategy was adopted in the foraging stage of dung beetles, and the adaptive ability was lacking, location update of dung beetles' stealing behavior depended on the current optimal value, and the population diversity was lacking, those results in the unbalanced global exploration and local development ability of DBO algorithm, which is easy to fall into local optimal solution, weak global exploration ability, and low convergence accuracy efficiency. So, COA algorithm, as one of metaheuristic algorithms (MAs), has the commonality of all meta-heuristic algorithms. The search process can be divided into two stages: exploration and exploitation. The exploration stage refers to the algorithm's ability to search the global space, which determines whether the algorithm can get the optimal solution. Exploitation phase refers to the ability to search local space, which determines the speed of the algorithm to obtain the optimal solution. The better the balance between exploration and exploitation, the better the performance of the algorithm. But according to the “No Free Lunch” theory 2 , there is no metaheuristic algorithm can solve all optimization problems, the most important factor is that is metaheuristic algorithms existed the problem of falling into local optima, most MAs algorithms suffer from being trapped in local optima 3 .

In order to solve this problem, experts and scholars have begun to propose different types of strategies to solve this problem. Fan et al. proposed an improved African vultures optimization algorithm based on tent chaotic mapping and time-varying mechanism (TAVOA) to further improve the African vultures optimization algorithm (AVOA), and the tent chaos strategy to initialize the positions of 4 . Zhang et al. proposes a chaotic adaptive sailfish optimizer with genetic characteristics (CASFO). The CASFO algorithm first introduces the Tent chaos strategy to initialize the positions of sailfish and sardines to increase the diversity of the population 5 . Ding et al. proposed an improved WOA algorithm based on the concepts of chaos initialization, nonlinear convergence factor, and chaotic inertial weight to enhance its exploration abilities 6 . Li et al. proposed to integrate Tent chaotic mapping, nonlinear inertia weights and Gauss-Cauchy hybrid mutation strategy into the Gull algorithm to improve the computational accuracy and convergence speed of the algorithm 7 . Liu et al. proposed an improved butterfly optimization algorithm based on nonlinear inertial weight, bidirectional differential mutation strategy with decision coefficient and disturbance factor, bidirectional differential mutation strategy with decision coefficient and disturbance factor to improve the convergence speed and optimization accuracy of the butterfly optimization algorithm 8 . Cao et al. proposed an improved moth algorithm, which combines the adaptive crossover operator with the Lévy flight strategy, introduces an adaptive t-distribution variation in flight straight strategy, and uses the greedy strategy to enhance the global search capability and speed 9 . Zhu et al. proposed an improved SMA based on adaptive t-distributed variation strategy and chaotic opposition-based learning strategy to enhance the convergence speed, solution accuracy, and robustness of the SMA 10 , Liu proposed an improved Sparrow search algorithm (SSA) based on the Circle chaos mapping, T-distribution variation to enhanced the global optimization ability and convergence precision of the SSA 11 .

Aiming at the weakness of COA, Scholars have conducted in-depth analysis of this COA algorithm and put forward some improved methods. Yin et al. proposed an improved COA algorithm based on chaotic mapping, opposition learning mechanism and sine and cosine algorithm, which enhanced the flexibility and convergence speed of the algorithm 12 . Kaishi et al. involves a triple approach incorporating chaos mapping, Gaussian walk, and random walk to mitigate the randomness of the initial solution in the conventional Coati Optimization Algorithm (COA) and enhance the search capabilities through a dual population strategy, adaptive factors, and a stochastic differential variation strategy 13 . Fatma added some operators to improve the Coati Optimization Algorithm, such as adaptive s-best mutation operator to enhance the balance between exploration and exploitation, the directional mutation rule is used to open the way to discover the search space thoroughly, and the search direction is controlled toward the global best 14 .

Through the above analysis, we can find that in order to solve the problems of slow convergence speed and easy to fall into local optimization in metaheuristic algorithms, scholars mainly improve population initialization methods by introducing chaotic mapping and integrating Levy flight strategy. t-distribution variation, Gaussian walk, s-best mutation operator and other methods improve the exploration and exploitation ability of the algorithm, and obtain good results.

In order to improve the optimization speed and performance of Coati algorithm, a multi-strategy improved Coati algorithm is proposed, which combines chaotic sequence, nonlinear inertia weight, adaptive T-distribution variation strategy, alert update and other strategies to improve the optimization performance of the algorithm. The algorithm introduces chaotic sequence mechanism to initialize the position. The position distribution of the initial solution is more uniform, the high quality initial solution is generated, the population richness is increased, and the problem of poor quality and uneven initial solution of Coati Optimization Algorithm is solved. In the exploration phase, the nonlinear inertial weight factor is introduced to coordinate the local optimization ability and global search ability of Coati algorithm. In the exploitation phase, adaptive T-distribution variation is introduced to increase the diversity of individual population under low fitness value and improve the ability of the algorithm to jump out of the local optimal value. At the same time, the Coati alert update mechanism is proposed to improve the alert capability of the Coati algorithm, so that it can search within the optional range. When Coati is aware of the danger, the Coati on the edge of the population will quickly move to the safe area to obtain a better position, while the Coati in the middle of the population will move randomly to be close to others Coati. At the same time, we use IEEE CEC2017 benchmark test function to verify the optimization effect of TNTWCOA algorithm, and compare it with ICOA 3 , COA 1 , GJO 15 , OOA 16 , SCSO 17 and SABO 18 algorithms respectively in different dimensions (Dim = 30, Dim = 50, Dim = 100). The convergence curves, Friedman ordering test, boxplot and Wilcoxon rank sum test results of 6 algorithms in different dimensions are discussed. Finally, in order to verify the engineering practicability of TNTWCOA algorithm, this paper selects 4 engineering problems to test the optimization performance of TNTWCOA. Through the analysis of engineering problem data, it can be determined that the algorithm has application value in engineering optimization problems. In the experimental part, IEEE CEC2017 benchmark experiment is selected to test the optimization performance of TNTWCOA. In the experiment, TNTWCOA algorithm is compared with ICOA, COA, GJO, OOA, SCSO and SABO and other 7 algorithms, and the results show that TNTWCOA algorithm is better than other algorithms in terms of optimization performance. In addition, the TNTWCOA algorithm is applied to three bar truss design, The Gear Train Design, Speed reducer and other four practical constraint projects to verify the actual optimization effect of ICOA on engineering problems. In future work, we will continue to improve TNTWCOA's exploration capabilities and convergence rate.

The main contributions of this paper are as follows:

Chaotic sequence mechanism is introduced to initialize the position to solve the problem of poor quality and uneven initial solution of the Coati Optimization Algorithm

The nonlinear inertial weight factor is introduced to coordinate the local optimization ability and global search ability of the algorithm in exploration phase.

Adaptive T-distribution variation is introduced to increase the diversity of individual population under low fitness value and improve the ability of the algorithm to jump out of the local optimal value in the exploitation phase

A Coati alert update mechanism is proposed to improve the alerting ability of the Coati algorithm.

The rest of the paper is organized as follows. “ Coati Optimization Algorithm ” section briefly describes the theory and major steps of conventional COA algorithm. “ Improved Coati Optimization Algorithm ” section describes the proposed coati optimization algorithm in detail. “ Experimental studies and results ” section discusses the simulation results of the improved algorithm TNTWCOA and evaluates its performance. " TNTWCOA for engineering optimization problems " section discusses the results of the improved algorithm TNTWCOA, when used for solving 4 classical engineering design problems. Finally, “ Conclusion ” section summarizes the whole paper.

Coati Optimization Algorithm

Coati Optimization Algorithm (COA) is a novel metaheuristic algorithm created by Dehghani, M that mimicked coati behavior in nature in 2023 1 . The core of the COA is to mimic natural actions of two coatis, including exploration (hunting and attacking iguanas), exploration (fleeing from predators) two actions 19 . The key steps for COA are introduced in the following subsection.

Algorithm initialization process

In the initialization stage of the COA, the position of the coatis in search space is randomly generated for the COA by using the expression in Eq. ( 1 )

In Eq. ( 1 ): \(X_{i}\) stand for the position in the search space of the \(i_{th}\) coati, \(x_{i,j}\) represents the value of the \(j_{th}\) decision variable. \(b_{j}^{{\text{L}}}\) and \(b_{j}^{{\text{U}}}\) represent the upper and lower bound of the decision variables, respectively. \(N\) is the coatis’ number, \(m\) is the number of decision variables.

Mathematical model of COA

Phase 1: hunting and attacking strategy on iguana (exploration phase).

The first phase of updating the coatis’ population in the search space is modeled based on simulating their strategy when attacking iguanas. In this strategy, a group of coatis climbs the tree to reach an iguana and scare it. Several other coatis wait under a tree until the iguana falls to the ground. After the iguana falls to the ground, the coatis attack it and hunt it. This strategy leads coatis to move to different positions in the search space, which demonstrates the COA’s exploration ability in global search in the problem-solving space. In exploration phase, Coati's position update strategy mainly simulates Coati's hunting and attacking iguana behavior. The behavior of Coati is divided into two steps to complete the hunting and attacking of the iguana. (1) Fright. A group of Coatis climb a tree to approach an iguana and scare it. (2) Several other Coatis wait under the tree, waiting for the frightened iguana to fall to the ground, and after the iguana lands, complete the attack and hunt it. As shown in Fig.  1 .

Pattern diagram of the first phase of COA.

This strategy causes Coati to move to different locations in the search space, which in turn shows that the COA optimization algorithm has exploration capability in solving the global search of the problem space. In the COA algorithm, it is assumed that the location of the best member of the population is that of the iguana. It is also assumed that the number of Coatis completing steps (1) and (2) is each half of the total number of Coatis. Thus, the mathematical expression of position is:

where, \(X_{i,j}^{{{\text{P}}1}}\) is the new position of the ith Coati in the jth dimension; r is the random number between [0,1]; \(G_{j}\) is the iguana's position in the jth dimension, which actually refers to the position of the best member; I is a number randomly selected from the set {1,2}; N is the number of Coati; [N/2] is the largest integer not exceeding [N/2]; m is the number of decision variables.

After the iguana falls to the ground, it is placed in a random position in the search space. Based on this random location, the Coati on the ground moves through the search space. This step is simulated by two formulas.

where: \(G_{j}^{{\text{g}}}\) is the position of the iguana on the ground in the j dimension.

where: \(F_{{{\text{G}},j}}^{{\text{g}}}\) is the objective function value of the jth dimension iguana after it falls to the ground; \(F_{i,j}\) is the objective function value of the ith Coati in jth dimension. If the updated individual is better, the current individual is updated. Otherwise, leave it as it is.

where: \(F_{i}^{{{\text{P}}1}}\) is the objective function value of the ith Coati at the new position; \(F_{i}\) is the objective function value of the ith Coati at the previous position.

Phase 2: The process of escaping from predators (exploitation phase)

In exploitation phase, Coati's location-updating strategy mainly mimics Coati's natural behavior when encountering predators and when fleeing from predators. As shown in Fig.  2 .

Pattern diagram of coati escaping from a predator in the second phase of COA.

In exploitation phase, When a predator attacks Coati, Coati flees its position. Coati's move on this strategy resulted in it being in a safe position close to its current position. This demonstrates the exploitation of COA algorithms in local search. To simulate this behavior, a random location is generated near the location of each Coati based on the following equation.

where: \(b_{{j, {\text{L}}}}^{{{\text{loc}}}}\) is the local lower bound of the jth decision variable, \(b_{{j,{\text{U}}}}^{{{\text{loc}}}}\) is the local upper bound of the jth decision variable, and t is the number of iterations; T is the maximum number of iterations.

where: \(X_{i,j}^{{{\text{P}}2}}\) is the new position of the ith Coati in the jth dimension. If the updated individual is better, update the current individual, otherwise leave it as it is.

Improved Coati Optimization Algorithm

Chaotic mapping strategy for algorithm initialization process.

Because the individual positions of the original Coati Optimization Algorithm are generated randomly, the diversity of the population is likely to be lost, and its uniform distribution in the solution space cannot be guaranteed, this makes the algorithm easily fall into local optimization. Uniform population can speed up convergence 20 , 21 . Therefore, it is necessary to improve the population initialization method of the algorithm. Chaotic mapping is ergodic and stochastic. If chaotic mapping function is used to generate chaotic sequence as the initial position of population individuals to make the population distribution more uniform and avoid population uniformity, thus improving the search efficiency. Commonly used chaotic mappings are as follows: Chebyshev map, Circle map, Gauss map, Iterative map, Logistic map, Sine map, Singer map, Tent map, Cubic map. The population distribution results generated by the commonly used chaotic mappings methods is shown in Fig.  3 a, and the histogram is shown in Fig.  3 b.

Results of population samples under different chaotic maps.

As shown in Fig.  3 , the population distribution generated by tent chaos has the best uniformity among the above major chaotic maps. Therefore, this paper chooses TENT Chaos to improve the distribution quality of the initial population in search space, and the global search ability can be strengthened, so as to improve the solution efficiency of the algorithm, and Eq. ( 1 ) can be rewritten as:

The expression of the tent mapping was shown in Eq. ( 9 ), \(\alpha = 0.5\) .

Nonlinear inertia weight factor for hunting and attacking strategy on iguana

The local optimization ability and global search ability of the coordinated meta-heuristic algorithm are the key factors that affect the optimization accuracy and optimization speed of the algorithm. Since the update of Coati individual position is closely related to the current Coati position, the nonlinear inertia weight factor is used to adjust the correlation between the update of Coati position and the current Coati position information. The calculation method of the nonlinear inertia weight factor is as follows.

where, \(t\) is the current iteration number and \(T\) is the maximum iteration number. The maximum inertia weight \(\omega_{\max } = 1\) , with the progress of iteration, the inertia weight factor will increase nonlinear and eventually reach a large value.

The improved Coati calculation formula is as follows 22 :

During iteration, the variation of Nonlinear Inertia Weight Factor is shown in Fig.  4 . Nonlinear Inertia Weight Factor starts from 0 and increases to 1. According to Fig.  4 and Eq. ( 3 ). At the beginning of the iterative solution, the Nonlinear Inertia Weight Factor is small, and the update of the position of the seeking individual is less affected by the current Coati position, which is conducive to the search of the algorithm in a larger scope and improve the global development ability of the algorithm. With the advancement of the optimization process, the value of \(\omega\) gradually increases to 1, and the update of the position of the optimization individual becomes more influenced by the current Coati position. Narrowing the optimization range of the algorithm helps the algorithm search for the optimal solution, which not only improves the local exploration ability of the algorithm, but also improves the convergence speed of the algorithm.

Change of nonlinear inertia weight factor.

Adaptive T-distribution variation strategy for process of escaping from predators

In the exploitation phase of Coati algorithm, an adaptive T-distribution variation strategy is introduced. In each iteration, the relationship between the current Coati fitness value and the average fitness value of the population was compared. When the Coati fitness value is higher than the average fitness value of the population, it indicates that the current Coati is in an aggregation state. In this case, adaptive T-distribution variation strategy is adopted to increase Coati diversity. When the Coati fitness value is lower than the average fitness value of the population, the original Coati location updating method is used. The improved formula is as follows:

The T-distribution contains a degree of freedom parameter, and its probability density is shown as follows 23 :

When t(n → 1), t-distribution is Cauchy(0,1):

When t(n → ∞), t-distribution is Norm(0,1):

where, t is the degree of freedom parameter, n is the degree of freedom, Γ() is the gamma function. When t(n → ∞) → N(0,1), t(n → 1) = C(0,1), N(0,1)as the Gaussian distribution, C(0,1) for the Cauchy distribution.

In TNTWCOA, the position of each COATI is perturbed using t-distribution mutation with adaptive parameters. t-distribution mutation operator is mathematically formulated as:

At the beginning of iteration, the T-distribution mutation is similar to Cauchy mutation, and the algorithm has a good global exploration ability, which increases the diversity of the population, and the ability to jump out of the local optimal is also enhanced. With the increase of the number of iterations, the T-distribution mutation is similar to the Gaussian mutation, which improves the local development ability of the algorithm, and the disturbance strength of the whole population changes from strong to weak. By introducing adaptive T-distribution mutation as an improved search strategy, the optimization performance of the algorithm can be effectively enhanced, and the ability of the algorithm to escape local optimal can be improved.

Alert mechanism for process of escaping from predators

The first half of the Coati algorithm is updated using the formula of improvement point 3, and the second half is updated by introducing sparrow alert mechanism. Introducing the Coati alert update mechanism in the second stage of Coati can improve the alert capability of Coati algorithm and enable it to search within the optional range. When Coati is aware of danger, Coati on the edge of the group will quickly move to the safe area to obtain a better position. Coati in the middle of the group will randomly move around to get closer to other Coatis. The mathematical expression is as follows:

where: G is the current global optimal location. β, as a step control parameter, is a random number with a normal distribution of mean 0 and variance 1. K  ∈  [− 1,1] indicates the direction of Coati's movement and the step size. The control parameter is a random number, and F i is the fitness value of the current Coati individual. F g and F w are the best and worst global fitness values respectively. ε is a constant to avoid zeros in the denominator.

In short, F i  > F g indicates that the Coati is at the edge of the group and is vulnerable to predators, and F i  = F g indicates that the Coati in the middle of the group is aware of the danger and needs to stay close to other Coatis to avoid predation.

Pseudocode and flowchart

The flowchart of the proposed TNTWCOA technique is shown in Fig.  5 . Different improvement strategies are proposed in the initialization process, exploration phase and exploitation phase. In initialization process. The algorithm introduces chaotic sequence mechanism to initialize the position. The position distribution of the initial solution is more uniform, the high quality initial solution is generated, the population richness is increased, and the problem of poor quality and uneven initial solution of the Coati Optimization Algorithm is solved. In exploration phase, the nonlinear inertial weight factor is introduced to coordinate the local optimization ability and global search ability of the algorithm. In the exploitation phase, adaptive T-distribution variation is introduced to increase the diversity of individual population under low fitness value and improve the ability of the algorithm to jump out of the local optimal value. At the same time, the alert update mechanism is proposed to improve the alert ability of COA algorithm, so that it can search within the optional range. When Coati is aware of the danger, Coati on the edge of the population will quickly move to the safe area to obtain a better position, while Coati in the middle of the population will randomly move to get closer to other Coatis. Furthermore, Algorithm 1 defines the TNTWCOA technique’s pseudocode.

Optimization flowchart of the TNTWCOA algorithm.

Pseudocode of the TNTWCOA algorithm.

Computational complexity

Assume that the number of iterations is T, the number of populations is N, and the dimension of the problem is M. According to the algorithm flow, the computational complexity of the ICOA algorithm to initialize the population and compute the initial value of the objective function is \(O(N \times M)\) ; In the Exploration phase, the computational complexity of updating the population is \(O(N/2 \times M \times T)\) , respectively. \(O(N/2 \times M \times T)\) , the computational complexity of Calculate random position for the G is \(O(N/2 \times M \times T)\) . In the Exploitation phase, the computational complexity required to update the raccoon's position is \(O(N/2 \times M \times T)\) and \(O(N/2 \times M \times T)\) , respectively. Therefore, the comprehensive complexity of the ICOA algorithm proposed in this paper is \(O(N \times M) + 5 \times O(N/2 \times M \times T) = O(N \times M(1 + 5 \times N/2))\) . According to the references, the complexity of the original COA algorithm is \(O(N \times M(1 + 5 \times N/2))\) , so the improved algorithm does not increase the computational complexity of the algorithm.

Experimental studies and results

To verify the effectiveness of the Improved Coati Optimization Algorithm with Multiple strategies proposed in this paper. In this section, the well-known IEEE CEC2017 benchmark functions are used to validity of the Improved Coati Optimization Algorithm with Multiple strategies proposed in this paper in 30, 50 and 100 dimensions and compared with the Improved Coati Optimization Algorithm (ICOA), Coati Optimization Algorithm (COA), Golden Jackal Optimization Algorithm (GJO), Osprey Optimization Algorithm (OOA), Sand Cat Swarm Optimization Algorithm (SCSO), Subtraction-Average-Based Optimizer (SABO). The evaluation involves using statistical measurements, such as best values, mean values, worst values, and standard deviation (STD). All statistical measurements were obtained after 30 runs of each algorithm. The number of iterations was 10,000 and the individual was 50. The analyzes and discusses the optimization of IEEE CEC2017 benchmark functions between the algorithm proposed in this paper and other algorithms in the case of 30, 50 and 100 dimensions respectively as followings. The details of CEC2017 was shown in Table 1 .

Experiments on IEEE CEC2017 benchmark functions (Dim = 30)

Statistics analysis.

Table 2 shows the statistical results of IEEE CEC2017 benchmark functions in 30 dimensions optimized by TNTWCOA ICOA, COA, GJO, OOA, SCSO and SABO algorithms. Also the Friedman values based on the average value of the IEEE CEC2017 benchmark functions optimized by the TNTWCOA, ICOA, COA, GJO, OOA, SCSO and SABO algorithms were statistically analyzed. As can be seen from Table 2 , When optimizing the other 28 test functions in the 30-dimensional case, More or less evaluation index of the TNTWCOA proposed in this paper is superior to that of the ICOA, GJO, COA, SCSO, OOA, and SABO algorithms; Among them, when F1, F3, F4, F11, F12, F13, F14, F15, F18, F19, F22, F25, F28 and F30 functions are optimized, all the evaluation indexes of TNTWCOA algorithm are optimal, showing excellent performance; the performance of std value is inferior to that of SABO algorithm for F5, that of GJO algorithm for F6, that of ICOA algorithm for F9 and F20, the performance of std and worse value are inferior to that of SABO algorithm for F7, the performance of min value is inferior to that of SCSO algorithm and std value is inferior to COA algorithm for F8, the performance of min value is inferior to that of SBSO algorithm, avg, median and worse value are inferior to COA algorithm for F10, the performance of std and worse value are inferior to that of ICOA algorithm for F16, F17, the performance of std and worse value are inferior to that of COA algorithm for F21. All values are inferior to that of ICOA algorithm for F23, Std, avg, median and worse inferior to that of COA algorithm for F24, The std and worse values when optimizing F26 are not as good as GJO algorithm, The std and worse values when optimizing F27 and F29 are not as good as ICOA algorithm.

In general, the improved algorithm shows stronger optimization ability in the case of 30 dimensions, especially for F3, F4, F6, F11, F17 functions, whose optimization value is close to the theoretical value of the function. Friedman's overall order is TNTWCOA > ICOA > GJO > SABO > OOA > SCSO > COA. Therefore, from the statistical results of evaluation index, when Dim = 30, the algorithm proposed in this paper shows excellent performance compared with other six algorithms. Compared with the original COA algorithm, the statistical results of evaluation index of the improved algorithm have been significantly improved.

Convergence analysis

Figure  6 illustrates the convergence curves of GJO, SCSO, OOA, SABO, original COA, ICOA and TNTWCOA on 29 benchmark functions of the IEEE CEC2017 throughout the iterations with 10000times.

The convergence curves and search history of the proposed technique and other five algorithms for IEEE CEC2017 benchmark functions (Dim = 30).

As can be seen from Fig.  6 , except for functions F10 and F29, the convergence speed of the improved TNTWCOA algorithm is optimal. Among them, when the function F1, F3, F4, F6, F11, F12, F13, F14, F15, F16, F17, F18, F19, F20, F22, F25, F26, F27, F28 and F30 are optimized, it can quickly converge to the best value and be stable. When optimizing functions F5, F7, F8, F9, F21, they can rapidly converge to the best value and remain stable, but the best value will be surpassed by other algorithms in the later period, for example, when optimizing F5, F7, F8, F9, it will be surpassed by GJO algorithm in the later period, when optimizing F21, it will be exceeded by ICOA, GJO algorithms in the later period, when optimizing F23 and F24, it will be exceeded by ICOA algorithms in the later period.

Analysis of box plot results

A box chart is a statistical chart consisting of the smallest number (minimum value), the first quartile (25% locus value); The middle number (median value); The third quartile (75% locus value); The largest number (maximum value) constitutes. Figure  7 is the box graph obtained after 30 runs of the algorithm. The largest number (maximum value) and the smallest number (minimum value) in Fig.  5 constitute the variation range of the optimal values of 29 functions in CEC 2017 optimized by GJO, SCSO, OOA, SABO, COA, ICOA and TNTWCOA algorithms after running 30 times. That is, the narrower the box graph, the smaller the fluctuation range of the optimal value of the function running 30 times, and the more stable the optimization; The lower the position of the box diagram, the smaller the function optimization value and the closer it is to the theoretical value. The "o" in the diagram indicates the existence of singularity. As can be seen from Fig.  5 , Except for the optimized the F10, F23, F24, F26, functions, all of the other has the lower the position of the box diagram, when optimized F6, F7, F9, F16, F17, F20, F27, the box diagram which is not the most narrower, when optimized F1, F2, F4, F11, F12, F13, F14, F15, F18, F19, F21, F22, F25, F28, F30, they have the smallest number (minimum value), the first quartile (25% point value) of the function box diagram; Middle digit (median value); The third quartile (75% locus value); The largest numbers (maximum) almost overlap.

Boxplots for the proposed technique and other five algorithms for IEEE CEC2017 benchmark functions (Dim = 30).

Analysis of Wilcoxon rank sum test results

Wilcoxon rank sum test is a non-parametric statistical test. The difference between different algorithms is found by comparison. Table 3 shows the statistical results of TNTWCOA and five other algorithms over 30 runs. When the Wilcoxon comparison result is less than 0.05, it indicates that there is a significant deviation in the function optimization results between the two comparison algorithms; if the comparison result is greater than 0.05, it indicates that there is no significant deviation in the function optimization results between the two comparison algorithms. From the comparison results of TNTWCOA algorithm with ICOA, GJO, COA, SCSO, OOA and SABO algorithms, there is a significant deviation between TNTWCOA algorithm and most algorithms in function optimization results. However, when optimizing functions F6, F8, F9, F19, F16, F17, F23 and F26, the Wilcoxon rank sum test of TNTWCOA algorithm and GJO algorithm is greater than 0.05, indicating that there is no significant difference in the optimization values of the functions obtained. At the same time, when optimizing function F10, F16, F17, F21, F26 the Wilcoxon rank sum test of TNTWCOA algorithm and ICOA algorithm is greater than 0.05, indicating that there is no significant difference between the obtained function values, when optimizing function F10, F16, the Wilcoxon rank sum test of TNTWCOA algorithm and SCSO algorithm is greater than 0.05, indicating that there is no significant difference between the obtained function values, when optimizing function F9, the Wilcoxon rank sum test of TNTWCOA algorithm and SASBO algorithm is greater than 0.05, indicating that there is no significant difference between the obtained function values.

Experiments on CEC2017 benchmark functions (Dim = 500)

Table 4 shows the statistical results of IEEE CEC2017 benchmark functions in 50 dimensions optimized by TNTWCOA, ICOA, GJO, COA, SCSO, OOA and SABO algorithms. Also the Friedman values based on the average value of the IEEE CEC2017 benchmark functions optimized by the TNTWCOA, ICOA, GJO, COA, SCSO, OOA and SABO algorithms were statistically analyzed. As can be seen from Table 2 . More or less evaluation indexes of TNTWCOA proposed in this paper is superior to that of the ICOA, GJO, COA, SCSO, OOA, and SABO algorithms; Among them, when F1, F4, F11, F12, F13, F14, F15, F18, F19, F20, F25, F27, F28 and F30 functions are optimized, all the evaluation indexes of TNTWCOA algorithm are optimal, showing excellent performance. When optimizing F3, F9 the performance of std value is inferior to that of ICOA algorithm, for F5, which is inferior to that of COA algorithm, for F8, F22, F26, which is inferior to that of OOA algorithm, When optimizing F10 and F27, only worse values performed worse than ICOA algorithm; When optimizing F6, std value performance is inferior to COA algorithm, avg, median, worse value performance is inferior to GJO algorithm, for F7 std value performance is inferior to OOA algorithm, worse value performance is inferior to GJO algorithm, for F16 std, median, worse value performance is inferior to GJO algorithm, for F17 std value performance is inferior to SCSO algorithm, avg value performance is inferior to GJO algorithm, for F21 std and worse value performance are inferior to ICOA algorithm, for F23 std value performance is inferior to GJO algorithm, avg, median, worse value performance are inferior to ICOA algorithm, for F24 std, avg, median, worse value performance are inferior to ICOA algorithm, For f29 std, avg, median, value performance are inferior to ICOA algorithm. In general, the improved algorithm shows stronger optimization ability in the case of 50 dimensions. Friedman's overall order is ICOA > SCSO > ICOA > GJO > SABO > COA > OOA. Therefore, from the statistical results of evaluation index, when Dim = 50, the algorithm proposed in this paper shows excellent performance compared with other six algorithms. Compared with the original COA algorithm and ICOA, the statistical results of evaluation index of the improved algorithm have been significantly improved.

Figure  8 illustrates the convergence curves of GJO, SCSO, OOA, SABO, original COA, and ICOA on 29 benchmark functions of the IEEE CEC2017 throughout the iterations with 10000times.

The convergence curves of the proposed technique and other five algorithms for IEEE CEC2017 benchmark functions (Dim = 50).

As can be seen from Fig.  8 , the convergence speed of the improved TNTWCOA algorithm is optimal; Among them, when the optimization functionF1, F3, F4, F8, F10, F11, F13, F15, F18, F19, F20, F21, F22, F25, FF26, F27, F28, F29, F30, it can quickly converge to the best value and remain stable. When the optimization function F12, it can quickly converge to the best value and will continue to optimize to make the best better. When optimizing functions F6, F7, F16, F23, F24, it can quickly converge to the best value and remain stable, but the best value will be surpassed by other algorithms in the later period, such as optimizing F6, F7, F16, it will be surpassed by GJO algorithm in the later period, and for F23, F24 it will be surpassed by ICOA algorithm in the later period. When optimizing functions F3, F9, F14, it cannot converge to the best value quickly, but it will gradually converge to the best and better and surpass other algorithms.

The largest number (maximum value) and the smallest number (minimum value) in Fig.  7 constitute the variation range of the optimal values of 29 functions in CEC 2017 optimized by GJO, SCSO, OOA, SABO, COA, and ICOA algorithms after running 50 times. That is, the narrower the box graph, the smaller the fluctuation range of the optimal value of the function running 30 times, and the more stable the optimization; The lower the position of the box diagram, the smaller the function optimization value and the closer it is to the theoretical value. The "o" in the diagram indicates the existence of singularity. As can be seen from Fig.  9 , all of them have the lower the position of the box diagram, when optimized F22, F27, the narrow of the box diagram is worst, when optimized F1, F4, F11, F12, F13, F14, F15, F17, F18, F19, F20, F25, F28, F29, F30, they have the smallest number (minimum value), the first quartile (25% point value) of the function box diagram; Middle digit (median value); The third quartile (75% locus value); The largest numbers (maximum) almost overlap.

Boxplots for the proposed technique and other five algorithms for IEEE CEC2017 benchmark functions (Dim = 50).

Table 5 shows the Wilcoxon rank sum test results for ICOA and 5 different algorithms over 50 runs. From the comparison of TNTWCOA algorithm with ICOA, GJO, COA, SCSO, OOA, SABO algorithm, there is a significant deviation between TNTWCOA algorithm and most algorithms. However, when optimizing functions F7, F16, F17, F20 and F23, the Wilcoxon rank sum test of TNTWCOA algorithm and COA algorithm is greater than 0.05, indicating that there is no significant difference between the obtained function values.

Experiments on IEEE CEC2017 (Dim = 100)

Table 6 shows the statistical results of IEEE CEC2017 benchmark functions in 100 dimensions optimized by TNTWCOA, ICOA, GJO, COA, SCSO, OOA and SABO algorithms. Also the Friedman values based on the average value of the IEEE CEC2017 benchmark functions optimized by the TNTWCOA, ICOA, GJO, COA, SCSO, OOA and SABO algorithms were statistically analyzed. As can be seen from Table 2 . More or less evaluation indexes of TNTWCOA proposed in this paper is superior to that of the ICOA, GJO, COA, SCSO, OOA, and SABO algorithms; Among them, when F1, F4, F11, F12, F13, F14, F15, F16, F18, F19, F25, F28, F29 and F30 functions are optimized, all the evaluation indexes of TNTWCOA algorithm are optimal, showing excellent performance; When F3, F7, F8, F27 are optimized, std value only is inferior to ICOA algorithm,F5 are optimized, min and std value only is inferior to ICOA algorithm, and for F21,std and worse value only is inferior to ICOA algorithm; for 24, avg, median, worse value only is inferior to ICOA algorithm, F6, F22 are optimized, std value only is inferior to COA algorithm, min value only is inferior to GJO algorithm and std, worse value only is inferior to ICOA algorithm for F20, std value only is inferior to COA algorithm and worse value only is inferior to SCSO algorithm for F9, std value only is inferior to COA algorithm and worse value only is inferior to ICOA algorithm for F10, F26 MIN ICOA STD GJO, min value only is inferior to ICOA algorithm and std value only is inferior to GJO algorithm for F26. In general, the improved algorithm shows stronger optimization ability in the case of 50 dimensions.

Friedman's overall order is TNTWCOA > SCSO > ICOA > GJO > SABO > OOA > COA. Therefore, from the statistical results of evaluation index, when Dim = 50, the algorithm proposed in this paper shows excellent performance compared with other five algorithms. Compared with the original COA and ICOA algorithm, the statistical results of evaluation index of the improved algorithm have been significantly improved.

Figure  10 illustrates the convergence curves of GJO, SCSO, OOA, SABO, original COA, ICOA and TNTWCOA on 29 benchmark functions of the IEEE CEC2017 throughout the iterations with 10000times.

The convergence curves of the proposed technique and other five algorithms for IEEE CEC2017 benchmark functions (Dim = 100).

As can be seen from Fig.  10 , except F3, F17 and F24, the convergence speed of the improved TNTWCOA algorithm is optimal. Among them, when the optimization function F4, F5, F6, F7, F8, F9, F10, F11, F14, F15, F16, F18, F21, F22, F25, F26, F28, F29, F30, it can quickly converge to the best value and maintain the stability. When optimizing the functions F1, F12, F19, F20, it can quickly converge to the best value and will continue to optimize to make the best better. When optimizing functions F23, they can quickly converge to the best value and remain stable, but the best value will be surpassed by ICOA and GJO in the later period, and F27 will be surpassed by ICOA. When optimizing functions F13, it can not quickly converge to the best value, but it will gradually converge to the best and better.

The largest number (maximum value) and the smallest number (minimum value) in Fig.  9 constitute the variation range of the optimal values of 29 functions optimized by GJO, SCSO, OOA, SABO, COA, and ICOA algorithms in CEC 2017 after running 30 times. That is, the narrower the box graph, the smaller the fluctuation range of the optimal value of the function running 30 times, and the more stable the optimization; The lower the position of the box diagram, the smaller the function optimization value and the closer it is to the theoretical value. The "o" in the diagram indicates the existence of singularity. As can be seen from Fig.  11 , except for F23 and F 24, the other have the lower the position of the box diagram, when optimized F9, F20, the narrow of the box diagram is worst, when optimized F1, F3, F11, F12, F13, F14, F15, F16, F17, F19, F25, F27, F28, F29, F30, they have the smallest number (minimum value), the first quartile (25% point value) of the function box diagram; Middle digit (median value); The third quartile (75% locus value); The largest numbers (maximum) almost overlap.

Boxplots for the proposed technique and other five algorithms for IEEE CEC2017 benchmark functions (Dim = 100).

Table 7 shows the Wilcoxon rank sum test results (Dim = 100). As can be seen from Table 7 . There is a significant deviation between TNTWCOA and the function optimization results of most algorithms. However, when optimizing functions F20, the Wilcoxon rank sum test of TNTWCOA algorithm and GJO and ICOA algorithm is greater than 0.05, indicating that the obtained function optimization values are not significantly different. At the same time, when optimizing function F21, the Wilcoxon rank sum test of TNTWCOA algorithm and ICOA algorithm is greater than 0.05, indicating that there is no significant difference between the obtained function optimization values. F24, the Wilcoxon rank sum test of TNTWCOA algorithm and SCSO algorithm is greater than 0.05, indicating that there is no significant difference between the obtained function optimization values.

TNTWCOA for engineering optimization problems

To verify the actual optimization effect of TNTWCOA in solving engineering problems, the optimization performance of TNTWCOA is tested by using the selected four classic engineering problems, and the specific performance of the TNTWCOA on each engineering problem is as follows:

Three‑bar truss design problem6

The main purpose of studying the design of a three-bar truss is to reduce the structure’s weight under the action of the total sup- porting load p. The geometry of this problem is given in Fig.  12 . In the benchmark suite, the total number of decision variables \(D = 2\) , the number of inequality constraints \(g = 3\) , the number of equality constraints \(h = 0\) , and the best known feasible objective function value \(f(x*) = 2.6389584338E + 02\) 24 , 25 .

Schematic of three-bar truss problem.

The mathematical formula of the three-bar truss problem is as follows:

Consider: \(\vec{x} = \left[ {\begin{array}{*{20}l} {x_{1} } \hfill & {x_{2} } \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {A_{1} } \hfill & {A_{2} } \hfill \\ \end{array} } \right]\)

Object : \(Minimize\;f\left( {\vec{x}} \right) = \left( {2\sqrt 2 x_{1} + x_{2} } \right)*l\)

Subject to: \(g_{1} \left( {\vec{x}} \right) = \frac{{\sqrt 2 x_{1} + x_{2} }}{{\sqrt 2 x_{1}^{2} + 2x_{1} x_{2} }}P - \sigma \le 0\) , \(g_{2} \left( {\vec{x}} \right) = \frac{{x_{2} }}{{\sqrt 2 x_{1}^{2} + 2x_{1} x_{2} }}P - \sigma \le 0\) , \(g_{3} \left( {\vec{x}} \right) = \frac{{x_{1} }}{{\sqrt 2 x_{2} + x_{1} }}P - \sigma \le 0\)

Variables range: \(0 \le x_{1} ,x_{2} \le 1\)

Where: \(l = 100,\;P = 2\frac{{{\text{kN}}}}{{{\text{cm}}^{2} }},\;\sigma = 2\frac{{{\text{kN}}^{2} }}{{{\text{cm}}^{2} }}\)

The iterative process of finding the optimal solution of the six algorithms is shown in Fig.  13 a, and its box-plots is shown in Fig.  13 b. What is clear from the simulation results is that TNTWCOA has fast provided the optimal solution to the three-bar truss problem and the objective function value equal to 2.63896E+02 . The statistical results obtained from TNTWCOA and competitor algorithms implementation are released in Table 8 . These results show that TNTWCOA has superior performance over competitor algorithms due to better values of statistical indicators.

The convergence curves and boxplots of the proposed and others five algorithms for three-bar truss problem.

Welded beam design5

Welding beam design is a common and challenging problem in structural engineering. The goal is to achieve the best structural performance and minimize the weight of the beam by optimizing parameters such as the shape, size, and layout of the weld under given constraints. In the benchmark suite, the total number of decision variables \(D = 4\) , the number of inequality constraints \(g = 5\) , the number of equality constraints \(h = 0\) , and the best known feasible objective function value \(f(x*) = 1.6702177263\) 25 , 26 . The specific structure of the welded beam design is shown in Fig.  14 .

Schematic of welded beam design problem.

The mathematical formula of the Welded beam design is as follows:

Consider: \(\vec{x} = \left[ {\begin{array}{*{20}c} {x_{1} } & {x_{2} } & {x_{3} } & {x_{4} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} h & l & t & b \\ \end{array} } \right]\)

Object: \(Minimize\;f\left( {\vec{x}} \right) = 1.10471x_{1}^{2} x_{2} + 0.04811x_{3} x_{4} \left( {14.0 + x_{2} } \right)\)

Subject to: \(g_{1} \left( {\vec{x}} \right) = x_{1} - x_{4} \le 0\) , \(g_{2} \left( {\vec{x}} \right) = \sigma \left( {\vec{x}} \right) - \sigma_{{{\text{max}}}} \le 0\) , \(g_{3} \left( {\vec{x}} \right) = \delta \left( {\vec{x}} \right) - \delta_{{{\text{max}}}} \le 0\) , \(g_{4} \left( {\vec{x}} \right) = \tau \left( {\vec{x}} \right) - \tau_{{{\text{max}}}} \le 0\) , \(g_{7} \left( {\vec{x}} \right) = 1.10471x_{1}^{2} + 0.04811x_{3} x_{4} \left( {14.0 + x_{2} } \right) - 5.0 \le 0\)

Variables range: \(0.1 \le x_{1}\) , \(x_{4} \le 2\) , \(0.1 \le x_{2}\) , \(x_{3} \le 10\)

Where: \( \tau \left( {\vec{x}} \right) = \sqrt {\left( {r^{\prime}} \right)^{2} + 2\tau ^{\prime}r^{\prime\prime}\frac{{x_{2} }}{{2R}} + \left( {r^{\prime\prime}} \right)^{2} } \) , \(\tau^{\prime} = \frac{p}{{\sqrt 2 x_{1} x_{2} }}\) , \(\tau^{*} = \frac{MR}{J}\) , \(M = p\left( {L + \frac{{x_{2} }}{2}} \right)\) , \(R = \sqrt {\frac{{x_{2}^{2} }}{4} + \left( {\frac{{x_{1} + x_{3} }}{2}} \right)^{2} }\) , \(J = 2\left\{ {\sqrt 2 x_{1} x_{2} \left[ {\frac{{x_{2}^{2} }}{4} + \left( {\frac{{x_{1} + x_{3} }}{2}} \right)^{2} } \right]} \right\}\) , \(\sigma \left( {\vec{x}} \right) = \frac{6PL}{{x_{4} x_{3}^{2} }}\) , \(\delta \left( {\vec{x}} \right) = \frac{{6PL^{3} }}{{Ex_{3}^{2} x_{4} }}\) , \(P_{c} \left( {\vec{x}} \right) = \frac{{4.013E\sqrt {x_{3}^{2} x_{4}^{6} /36} }}{{L^{2} }}\left( {1 - \frac{{x_{3} }}{2L}\sqrt{\frac{E}{4G}} } \right)\) , \(P = 6000{\text{lb}}\) , \(L = 14{\text{in}}\) , \(\delta_{{{\text{max}}}} = 0.25{\text{in}}\) , \(\tau_{\max } = 13,600\;\;{\text{psi}}\) , \(\sigma_{\max } = 30,000\;\;{\text{psi}}\)

The iterative process of finding the optimal solution of the six algorithms is shown in Fig.  15 a, and its box-plots is shown in Fig.  15 b, What is clear from the simulation results is that TNTWCOA has fast provided the optimal solution to the welded beam design problem and the objective function value equal to 1.67022E+00 . The statistical results obtained from TNTWCOA and competitor algorithms implementation are released in Table 9 . These results show that TNTWCOA has superior performance over competitor algorithms due to better values of statistical indicators, except SCSO.

The convergence curves and boxplots of the proposed and others five algorithms for he Welded beam design problem.

The gear train design problem

The gear train design problem aims to minimize the transmission ratio. In the benchmark suite, the total number of decision variables \(D = 4\) , the number of inequality constraints \({\text{g}} = 1\) , the number of equality constraints h = 1, and the best known feasible objective function value \(f(x*) = 0\) 25 , 27 . The gear train design problem structural diagram is shown in Fig.  16 .

Schematic of gear train design problem.

The mathematical formula of the gear train design problem is as follows:

Consider: \(\vec{x} = \left[ {\begin{array}{*{20}c} {x_{1} } & {x_{2} } & {x_{3} } & {x_{4} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {n_{A} } & {n_{B} } & {n_{C} } & {n_{D} } \\ \end{array} } \right]\)

Object: \(Minimize\;f(\vec{x}) = \left( {\frac{1}{6.931} - \frac{{x_{1} x_{2} }}{{x_{3} x_{4} }}} \right)^{2} ,\)

Variables range: \(x_{1} ,x_{2} ,x_{3} ,x_{4} \in \{ 12,13,14, \ldots ,60\} .\)

The iterative process of finding the optimal solution of the six algorithms is shown in Fig.  17 a, and its box-plots is shown in Fig.  17 b, What is clear from the simulation results is that TNTWCOA has fast provided the optimal solution to the gear train design problem and the objective function value equal to 0. The statistical results obtained from TNTWCOA and competitor algorithms implementation are released in Table 10 . These results show that TNTWCOA has superior performance over ICOA, GJO, OOA, SCSO, SABO algorithms due to better values of statistical indicators.

The convergence curves and box-plots of the proposed and others five algorithms for gear train design problem.

Speed reducer design problem1

Speed reducer design is an optimization challenge in engineering sciences. The goal is to minimize the weight of the speed reducer. The total number of decision variables \(D = 7\) , the number of inequality constraints g = 11, the number of equality constraints \(h = 0\) , and the best known feasible objective function value \(f(x*) = 2.9944E + 03\) 25 , 28 . The specific structure of the speed reducer design is shown in Fig.  18 .

Schematic of speed reducer design problem.

The mathematical formulation of the speed reducer design optimization problem is as follows:

Consider: \(\vec{x} = \left[ {\begin{array}{*{20}c} {x_{1} } & {x_{2} } & {x_{3} } & {\begin{array}{*{20}c} {x_{4} } & {x_{5} } & {\begin{array}{*{20}c} {x_{6} } & {x_{7} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} b & m & {z_{2} } & {\begin{array}{*{20}c} {l_{1} } & {l_{2} } & {\begin{array}{*{20}c} {d_{1} } & {d_{2} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right]\)

Object: \(\begin{gathered} Minimize\;f\left( X \right) = 0.7854x_{1} x_{2}^{2} \left( {3.3333x_{3}^{2} + 14.9334x_{3} - 43.0934} \right) \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - 1.508x_{1} \left( {x_{6}^{2} + x_{7}^{2} } \right) + 7.4777\left( {x_{6}^{3} + x_{7}^{3} } \right) + 0.7854\left( {x_{4} x_{6}^{2} + x_{5} x_{7}^{2} } \right) \hfill \\ \end{gathered}\)

Subject to: \(g_{1} \left( {\vec{x}} \right) = 27 - x_{1} x_{2}^{2} x_{3} \le 0\) , \(g_{2} \left( {\vec{x}} \right) = 397.5 - x_{1} x_{2}^{2} x_{3}^{2} \le 0\) , \(g_{3} \left( {\vec{x}} \right) = 1.93x_{4}^{3} - x_{2} x_{6}^{4} x_{3} \le 0\) , \(g_{4} \left( {\vec{x}} \right) = 1.93x_{5}^{3} - x_{2} x_{7}^{4} x_{3} \le 0\) , \(g_{5} \left( {\vec{x}} \right) = 10x_{6}^{ - 3} \sqrt {\left( {745x_{4} /x_{2} x_{3} } \right)^{2} + 16.9 \times 10^{6} } - 1100 \le 0\) , \(g_{6} \left( {\vec{x}} \right) = 10x_{7}^{ - 3} \sqrt {\left( {745x_{5} /x_{2} x_{3} } \right)^{2} + 157.5 \times 10^{6} } - 850 \le 0\) , \(g_{7} \left( {\vec{x}} \right) = x_{2} x_{3} - 40 \le 0\) , \(g_{8} \left( {\vec{x}} \right) = 5x_{2} - x_{1} \le 0\) , \(g_{9} \left( {\vec{x}} \right) = x_{1} - 12x_{2} \le 0\) , \(g_{10} \left( {\vec{x}} \right) = 1.5x_{6} + 1.9 - x_{4} \le 0\) , \(g_{11} \left( {\vec{x}} \right) = 1.1x_{7} + 1.9 - x_{5} \le 0\)

Variables range: \(2.6 \le x_{1} \le 3.6\) , \(0.7 \le x_{2} \le 0.8\) , \(x_{3} \in \left\{ {17,18,19, \ldots ,28} \right\}\) , \(7.3 \le x_{4}\) , \(x_{5} \le 8.3\) , \(2.9 \le x_{6} \le 3.9\) , \(5 \le x_{7} \le 5.5\)

The iterative process of finding the optimal solution of the six algorithms is shown in Fig.  19 a, and its box-plots is shown in Fig.  19 b, what is clear from the simulation results is that TNTWCOA has fast provided the optimal solution to the speed reducer design problem and the objective function value equal to 2.99442E+03. The statistical results obtained from TNTWCOA and competitor algorithms implementation are released in Table 11 . These results show that TNTWCOA has superior performance over competitor algorithms due to better values of statistical indicators.

The convergence curves and boxplots of the proposed and others five algorithms for speed reducer design problem.

Wilcoxon rank sum test results on engineering optimization problems

Table 12 shows the Wilcoxon statistics for TNTWCOA and 6 different algorithms over 30 runs. From the comparison of TNTWCOA algorithm with ICOA, GJO, COA, SCSO, OOA, SABO algorithm, there is a significant deviation between TNTWCOA algorithm and most of the optimization results. However, when Welded beam design Problem are optimized, the Wilcoxon rank sum test of TNTWCOA algorithm and SCSO, GJO algorithm is greater than 0.05, indicating that there is no significant difference between the obtained results.

In order to improve the optimization speed and performance of Coati algorithm, a multi-strategy improved Coati algorithm is proposed, which combines chaotic sequence, nonlinear inertia weight, adaptive T-distribution variation strategy, alert update and other strategies to improve the optimization performance of the algorithm. The algorithm introduces chaotic sequence mechanism to initialize the position. The position distribution of the initial solution is more uniform, the high quality initial solution is generated, the population richness is increased, and the problem of poor quality and uneven initial solution of Coati Optimization Algorithm is solved. In the exploration phase, the nonlinear inertial weight factor is introduced to coordinate the local optimization ability and global search ability of Coati algorithm. In the exploitation phase, adaptive T-distribution variation is introduced to increase the diversity of individual population under low fitness value and improve the ability of the algorithm to jump out of the local optimal value. At the same time, the Coati alert update mechanism is proposed to improve the alert capability of the Coati algorithm, so that it can search within the optional range. When Coati is aware of the danger, the Coati on the edge of the population will quickly move to the safe area to obtain a better position, while the Coati in the middle of the population will move randomly. To be close to other Coati. In the experimental part, IEEE CEC2017 benchmark experiment is selected to test the optimization performance of TNTWCOA. In the experiment, TNTWCOA algorithm is compared with ICOA, GJO, COA, SCSO, OOA, SABO and other 6 algorithms, and the results show that TNTWCOA algorithm is better than other algorithms in terms of optimization performance. In addition, the TNTWCOA algorithm is applied to Three bar truss design, The Gear Train Design, Speed reducer and other four practical constraint projects to verify the actual optimization effect of TNTWCOA on engineering problems.

In future work, we will continue to improve TNTWCOA 's exploration capabilities and convergence rate, and apply it to the optimization design of mechanical structure components, and at the same time, optimize the accuracy of surrogate models such as KRIGING and SVR, and establish prediction models of performance and defects of mechanism structural components as well as reliability analysis models of mechanism structural components. On the basis of the prediction model or reliability analysis model, COA algorithm is used to optimize the parameters and reliability analysis of the structural components.

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Acknowledgements

This work was supported by the Sichuan Science and Technology Program (Project No. 2023ZYD01396), the Natural Science Foundation of Sichuan Province (Project Nos. 2022NSFSC0454, 2022NSFSC1975), Panzhihua City Science and Technology Program with Targeted financial transfer payment (No. 222Y2F-GG-04), Sichuan Technology & Engineering Research Center for Vanadium Titanium Materials (No. 2020FTGC-Z-02), the University Key Laboratory of Sichuan in Process Equipment and Control Engineering (No. GK201905), Key Laboratory of Fluid and Power Machinery, Ministry of Education (No. LTDL2020-006).

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Qi, Z., Yingjie, D., Shan, Y. et al. An improved Coati Optimization Algorithm with multiple strategies for engineering design optimization problems. Sci Rep 14 , 20435 (2024). https://doi.org/10.1038/s41598-024-70575-4

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Improved multi-strategy adaptive Grey Wolf Optimization for practical engineering applications and high-dimensional problem solving

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• Published: 05 September 2024
• Volume 57 , article number  277 , ( 2024 )

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• Mingyang Yu 1 ,
• Jing Xu 1 ,
• Weiyun Liang 1 ,
• Sixu Bao 1 &
• Lin Tang 2  

The Grey Wolf Optimization (GWO) is a highly effective meta-heuristic algorithm leveraging swarm intelligence to tackle real-world optimization problems. However, when confronted with large-scale problems, GWO encounters hurdles in convergence speed and problem-solving capabilities. To address this, we propose an Improved Adaptive Grey Wolf Optimization (IAGWO), which significantly enhances exploration of the search space through refined search mechanisms and adaptive strategy. Primarily, we introduce the incorporation of velocity and the Inverse Multiquadratic Function (IMF) into the search mechanism. This integration not only accelerates convergence speed but also maintains accuracy. Secondly, we implement an adaptive strategy for population updates, enhancing the algorithm's search and optimization capabilities dynamically. The efficacy of our proposed IAGWO is demonstrated through comparative experiments conducted on benchmark test sets, including CEC 2017, CEC 2020, CEC 2022, and CEC 2013 large-scale global optimization suites. At CEC2017, CEC 2020 (10/20 dimensions), CEC 2022 (10/20 dimensions), and CEC 2013, respectively, it outperformed other comparative algorithms by 88.2%, 91.5%, 85.4%, 96.2%, 97.4%, and 97.2%. Results affirm that our algorithm surpasses state-of-the-art approaches in addressing large-scale problems. Moreover, we showcase the broad application potential of the algorithm by successfully solving 19 real-world engineering challenges.

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1 Introduction

The rapid advancement of science, technology, and industry has given rise to a multitude of intricate optimization problems. These problems frequently entail numerous variables, constraints, and objectives. Their solution spaces are huge and complex, and it is difficult for traditional deterministic optimization methods to obtain satisfactory solutions in acceptable time (Deng et al. 2022 ; Guo et al. 2023 ; Zhou et al. 2022 ). To cope with these challenges, researchers in the field of computational intelligence have started to search for new approaches. Among them, metaheuristic algorithms have attracted much attention due to their high efficiency, universal applicability and powerful global search capability (Aldosari et al. 2022 ; Chauhan et al., 2024; Chen et al. 2023 ).

When dealing with engineering problems, constraints are a crucial consideration. Constraints may be physical limitations or requirements and restrictions of the project. In the field of engineering, there are various techniques available for handling these constraints to ensure that projects proceed as expected and achieve their intended goals. One common technique for constraint handling is optimization algorithms. Optimization algorithms assist engineers in finding the best solution given certain constraints. These algorithms can be mathematical optimization methods such as linear programming, integer programming, or nonlinear programming, or they can be heuristic algorithms such as genetic algorithms, simulated annealing, or particle swarm optimization. By leveraging these algorithms, engineers can find the optimal design or decision solution while taking into account various constraints (Fu et al. 2024b ; Li et al. 2023 ).

MH algorithms are inspired by certain phenomena in nature, such as PSO (Kennedy and Eberhart 1995b ), Firefly Algorithm (FA) (Yang 2009 ), Sine Cosine Algorithm (SCA) (Mirjalili 2016 ), Wind Driven Optimization (WDO) (Bayraktar et al. 2010), Fruit Fly Optimization Algorithm (FOA) (Pan 2012 ), Competitive Swarm Optimizer (Chauhan et al. 2024 ), Fox optimizer(FOX) (Mohammed and Rashid 2023 ), Fitness Dependent Optimizer(FDO) (Abdullah and Ahmed 2019 ) and so on. These algorithms often do not rely on the specific nature of the problem, but instead draw on nature's strategies for stochastic search, which can effectively avoid falling into local optima (Abdel-Basset et al. 2023 ). With the development of deep learning, neural networks and other machine learning techniques, researchers have begun to try to combine these techniques with metaheuristic algorithms to further improve the efficiency of solving complex optimization problems (Garg et al. 2023 ). In recent years, with the wide application of heuristic intelligent optimization algorithms in numerical optimization solving, various swarm intelligence algorithms have been proposed (Fu et al. 2022 ).

The popularity of MH algorithms has four distinct advantages: practicality, generalizability, non-leading properties and avoidance of local optima (Fu et al. 2023a ). First, based on their natural theoretical framework, these strategies are relatively intuitive to construct and deploy, thus allowing engineers and researchers to rapidly integrate them into concrete applications (Havaei & Sandidzadeh 2023 ). Next, since these algorithms treat the problem as an unknown mechanism, they can be applied to a wide range of different tasks such as selection (Said, Elarbi, Bechikh, Coello Coello, & Said, 2023), shop visit balancing (Xia et al. 2023 ) and engineering problems (Nadimi-Shahraki et al. 2022 ). Further, these methods do not rely on derivative information and are particularly good for dealing with nonlinear problems (Aldosari et al. 2022 ). Ultimately, with the help of a global search strategy and a stochastic strategy for updating the location, they can efficiently jump out of the local optimum, which is particularly effective for those scenarios where there are multiple locally optimal solutions.

The existing MH algorithm mainly include: Physics-based algorithms (PhA), Swarm Intelligence Algorithms (SI), Natural Evolutionary Algorithms (EA), and Human-based algorithms (Abualigah et al. 2021 ). In the course of evolution, cooperative behavior between individuals has been gradually formed through natural selection over a long period of time. For example, Trojovský and Dehghani, 2022 proposed a pelican optimization algorithm (POA) inspired by pelican predation. The Genetic Algorithm (GA) is a method of optimization based on the principles of natural evolution. It was proposed by John Holland and his colleagues. is a typical example inspired by Darwinian evolution (Bäck & Schwefel 1993 ). Differential Evolution (DE), based on the concepts of natural selection and reproduction in Darwinian evolution(Storn and Price 1997 ); Genetic Programming (GP), inspired by biological evolution processes; and Evolution Strategies (ES) (Wei 2012 ). Among these, Genetic Algorithms (GA) and Differential Evolution (DE) are widely considered to be the most popular evolutionary algorithms, having garnered significant attention and being applied in numerous applications. Physical method is the result of the interaction of physical law and chemical variation. For example, the Chernobyl Disaster Optimizer (CDO) is an optimization algorithm inspired by the core explosion at the Chernobyl nuclear power plant. (Shehadeh 2023 ). The Galaxy Swarm Optimization (GSO) (Muthiah-Nakarajan and Noel 2016 ) algorithm, inspired by the motion of galaxies; the Firefly Algorithm (FFA), drawing inspiration from soil fertility in agriculture (Shayanfar and Gharehchopogh 2018 ); the Firefly Algorithm (FFA), drawing inspiration from soil fertility in agriculture (Eskandar et al. 2012 ); and the Gravitational Search Algorithm (GSA), derived from Newton's law of universal gravitation and kinematic laws (Rashedi et al. 2009 ).In sharp contrast, a variety of human behaviors are simulated based on human behavior patterns, such as the "Alpine skiing Optimization (ASO)" proposed by Professor Yuan, a new idea influenced by the competitive behavior of athletes. Each of these different Metaheuristic algorithms has its own characteristics. According to different problems and requirements, appropriate algorithms can be selected to solve the optimal problems (Yuan et al. 2022 ). Particle Swarm Optimization (PSO) (Kennedy and Eberhart 1995a ) is inspired by the foraging behavior of bird flocks and fish schools. The Ant Colony Optimization (ACO) algorithm (Dorigo et al. 2006 ) is inspired by the social behavior of ant colonies during foraging. The Pathfinder Algorithm (PFA) (Yapici & Cetinkaya 2019 ) is inspired by the collective action of animal populations in finding optimal food areas or prey. The Harris Hawk Optimization algorithm (HHO)(Heidari et al. 2019 ) is based on the predatory process of Harris hawks hunting rabbits. The Sparrow Search Algorithm (SSA) (Xue and Shen 2020 ) is inspired by the foraging and anti-predatory behavior of sparrows. The Dung Beetle Optimization algorithm (DBO) (Xue and Shen 2022 ) is inspired by the rolling, dancing, foraging, stealing, and reproductive behaviors of dung beetles. The Remora Optimization Algorithm (ROA) (Jia et al. 2021 ) is inspired by the behavior of remoras adhering to different-sized hosts to facilitate foraging. The Black Widow Optimization algorithm (BWO) (Hayyolalam and Kazem 2020 ) is inspired by the unique reproductive behavior of black widow spiders. Dikshit Chauhan et al. proposed the Artificial Electric Field Algorithm (AFFEA) based on a series of learning strategies (Chauhan and Yadav 2024a , b ). Additionally, the Secretary Bird Optimization Algorithm (SBOA) was introduced based on the survival behavior of secretary birds in their natural environment (Fu et al. 2024b ), while the Red-Billed Blue Magpie Optimizer (RBMO) was proposed by simulating the search, chase, prey attack, and food storage behaviors of red-billed blue magpies (Fu et al. 2024a ).

Generally, the optimization process of the MH algorithm can be divided into two main steps (Saka et al. 2016 ): exploration and exploitation. In the exploration phase, the algorithm mainly focuses on searching all corners of the solution space to ensure that no possible optimal solution area is missed; while in the exploitation phase, the algorithm will focus on known high-quality solutions and further deepen the search in order to find the real the optimal solution. These two phases complement each other and ensure that the algorithm has both breadth and depth. GWO is inspired by the hunting behavior of grey wolves (Mirjalili et al. 2014 ). GWO effectively balances the two stages of exploration and exploitation by combining the social behavior of grey wolves with a dynamically adjusted location update strategy, thereby ensuring good global and local search capabilities.

Since its introduction in 2014, GWO has received widespread attention from scholars at home and abroad for its simplicity and efficiency, and has become an important tool for solving complex optimization problems (Fan and Yu 2022 ). However, similar to other optimization algorithms, the GWO algorithm does have some limitations although it has shown quite good performance in many optimization problems. In particularly, it is prone to suffer from prematurity and local optimality when dealing with multimodal function problems. As the iterative process of the GWO progresses, the inherent social hierarchy mechanism within the wolf population leads to a decrease in diversity. This mechanism prioritizes the positions and decisions of the leading wolves (Alpha, Beta, and Delta), influencing the entire pack's movement. As a result, the population tends to converge towards the leaders’ positions. However, this strong convergence driven by the social hierarchy can also lead to a drawback. The population may start to aggregate too closely or blindly around the leaders’ current positions. This phenomenon, often referred to as premature convergence, limits the algorithm’s ability to thoroughly explore the solution space. Consequently, the algorithm might struggle to escape local optima, as the current best solutions (guided by the leading wolves) might not always represent the global optimum. The pack, following the leaders too closely, can get trapped in these local optima, lacking the diversity or exploratory behavior needed to venture out and discover better solutions elsewhere in the search space. (Wang et al. 2018 ). In addition, when global exploration transitions to local mining, the algorithm may lose the ability to explore a wider solution space and overly concentrate on a specific region for detailed search. Such a centralized strategy, although helpful in finding the optimal solution in the local region accurately, may also lead the algorithm to ignore other promising regions (Wolpert and Macready 1997 ). Despite there has been various GWO variants, such as Advanced Grey Wolf Optimizer (AGWO) (Meng et al. 2021 ), Exponential Neighborhood Grey Wolf Optimization (EN-GWO) (Mohakud and Dash 2022 ), Hybrid Grey Wolf Optimizer with Mutation Operator (DE-GWO) (Gupta and Deep 2017 ), and others (Ambika et al. 2022; Biabani et al. 2022 ). These improved versions do not break through in solving the large-scale global optimization problems of CEC 2022 and CEC 2013. Moreover, their performance in dealing with complex problems remains unsatisfactory.

To improve the performance of the GWO, this study incorporates several key enhancements. Firstly, the search mechanism from PSO is employed to increase population diversity. This addition helps in broadening the search scope of the algorithm. Secondly, the IMF is used to adjust inertia weights, a strategy that aids in fine-tuning the balance between exploration and exploitation. Lastly, an adaptive mechanism based on the Sigmoid function is introduced for updating the positions of individuals within the population. This adaptive update strategy strengthens the group's ability to escape local optima, enhancing the overall effectiveness of the GWO algorithm in finding optimal solutions.

An improved adaptive grey wolf optimization (IAGWO) is proposed to address the shortcomings of the GWO algorithm. The main contributions are as follows.

The PSO search mechanism is introduced to enhances the algorithm's search efficiency and robustness by updating grey wolf positions early in each iteration. Additionally, the dynamic adjustment of inertia weights through the IMF boosts global search capability initially and local search effectiveness later.

Adaptive position updating strategy based on Sigmoid function to balance the exploration and exploitation of IAGWO.

To evaluate the exploration and exploitation capabilities of IAGWO, extensive experimentation is conducted using a suite of 67 test functions, which includes benchmarks from the CEC 2014, CEC 2017, CEC 2020, CEC 2022, and CEC 2013 for large-scale global optimization problems.

The effectiveness and accuracy of IAGWO in solving practical engineering design challenges are thoroughly assessed through its application to 19 diverse engineering design challenges.

The paper is organized as follows: Sect.  2 provides a brief review of the previous enhancements and potential application directions of the GWO. Section  3 details the original GWO algorithm and the proposed improvement strategy. Section  4 evaluates IAGWO performance through relevant experiments and in-depth analysis. Finally, Sect.  5 concludes this paper with a summary of the results and an outlook on future research directions.

2 Related work

In recent years, there has been a significant focus among researchers on enhancing the GWO. These improvements are aimed at boosting the algorithm's search performance and effectiveness. Scholars have explored various approaches to achieve this, including aspects such as adjusting the algorithm parameters, improving the speed and position equations, and combining it with other algorithms.

Yu et al. ( 2023 ) adopted a new update search mechanism, improved control parameters, mutation driven strategy and greedy selection strategy to improve GWO in the search process. (Singh and Bansal 2022a ) proposed a hybrid GWO and Differential Evolution (HGWODE) algorithm and applied it to UAV path planning. (Cuong-Le et al. 2022 ) introduced an equation to control the moving strategy of the algorithm in each iteration and proposed New Balance Grey Wolf Optimizer (NB-GWO), which was used to optimize the hyperparameters of the deep neural network for damage detection of two-dimensional concrete frames. Liu et al. ( 2023 ) proposed a hybrid differential evolution GWO (DE-GWO) algorithm and applied it to gas emission identification and localization. Luo et al. ( 2023 ) introduced butterfly optimization algorithm and opposition-based learning method based on elite strategy, adaptive nonlinear inertia weight strategy and random walk law to improve the shortcomings of slow convergence speed and low accuracy of GWO algorithm when dealing with high-dimensional complex problem. To address the issue of premature convergence encountered by the classic GWO in some situations due to the stagnation of sub-optimal solutions, Gupta et al. introduced an enhanced leadership-inspired grey wolf optimizer for global optimization problems (GLF-GWO)(Gupta and Deep 2020 ), Addressing the issues of slow convergence speed and insufficient global exploration in GWO, which can lead to settling in local optimal states and failure to achieve global optimal solutions, Singh et al. proposed a novel mutation-driven modified grey wolf optimizer (MDM-GWO) (Singh and Bansal 2022b ). MDM-GWO integrates new update search mechanisms, modified control parameters, mutation-driven schemes, and greedy selection methods into the search process of GWO. Addressing the issues of slow convergence speed and susceptibility to local optima in the Grey Wolf Optimizer (GWO) algorithm, Zhang et al. proposed a nonlinear control parameter strategy based on a sinusoidal function (GWO-SIN) and a nonlinear control parameter combination strategy (GWO-COM) (Zhang et al. 2019 ).

Soliman et al. ( 2022 ) proposed a novel hybrid African vultures–grey wolf optimizer (AV–GWO) approach to precisely estimate the electrical parameters of such TDM. Nadimi-Shahraki et al. ( 2021 ) introduced an enhanced variant of the Grey Wolf Optimization algorithm, termed I-GWO. The algorithm, based on a dimensionally learned hunting and searching (DLH) strategy, uniquely constructs hunting domains for each Wolf and enables them to share information about neighboring domains with each other. This enhances the algorithm's local and global search capabilities for more balanced performance, while also helping to maintain population diversity. A. Abushawish and A. Jarndal (Abushawish and Jarndal 2021 ) jointly proposed a new hybrid algorithm named GWO-CS that combines the advantages of Cuckoo Search (CS) algorithm and GWO algorithm. This algorithm primarily incorporates the position update equation from the CS to further refine the global search process of the GWO. Aimed at solving the problems of poor stability and easily falling into the local optimal solution in the grey wolf optimizer (GWO) algorithm, Liu et al. proposed an improved GWO algorithm based on the differential evolution (DE) algorithm and the OTSU algorithm is proposed (DE-OTSU-GWO) (Liu, Sun, Yu, Wang, & Zhou, 2020). The multithreshold OTSU, Tsallis entropy, and DE algorithm are combined with the GWO algorithm. The multithreshold OTSU algorithm is used to calculate the fitness of the initial population. The population is updated using the GWO algorithm and the DE algorithm through the Tsallis entropy algorithm for crossover steps. Multithreshold OTSU calculates the fitness in the initial population and makes the initial stage basically stable. Tsallis entropy calculates the fitness quickly. The DE algorithm can solve the local optimal solution of GWO. To address the issues of GWO's susceptibility to local optima and its low exploration capabilities, Hardi Mohammed et al. proposed the Enhanced GWO (EGWO) (Mohammed et al. 2024 ). EGWO employs diverse methods to improve the performance of GWO, utilizing gamma, z-position, and the golden ratio.

Liu et al. ( 2022 ) introduced a novel improvement strategy for the GWO algorithm, known as the exponential convergence factor improvement strategy. This strategy is designed to more accurately simulate the actual search process of grey wolves. It incorporates dynamic weighting factors and enhances control parameters to reduce the likelihood of the GWO algorithm getting stuck in local optima. However, despite these improvements, experimental findings indicate that GWO still faces challenges in accurately handling high-dimensional functions. Şenel et al. ( 2019 ) integrated a differential disturbance operator into the GWO algorithm. This addition brought an element of exploration into the exploitation phase, thereby enhancing the GWO algorithm's overall optimization capabilities. Jangir and Jangir ( 2018 ) proposed a multi-objective version of the GWO algorithm, named NSGWO. This algorithm utilizes a crowding distance mechanism to select the optimal solution from a set of Pareto optimal solutions. This approach helps guide the search towards the dominant region in multi-objective search spaces. NSGWO was tested on a variety of standard unconstrained, constrained, and engineering design challenges, demonstrating its efficiency and effectiveness in diverse optimization scenarios.

3 Methodology Overview: Standardized GWO and Proposed Enhancements

This section offers an overview of the hunting behavior and the mathematical model that forms the foundation of the original GWO. Additionally, we introduce the IAGWO, our proposed enhancement to GWO. IAGWO integrates the PSO search mechanism, the IMF strategy for inertia weighting, and an adaptive strategy for updating positions. These additions aim to refine and boost the efficiency of the original GWO algorithm.

3.1 The standardized GWO

3.1.1 inspiration of grey wolf packs’ hunting activity behavior.

The GWO algorithm draws inspiration from the hunting behavior of grey wolf packs. It mathematically simulates the way a group of grey wolves hunts, encircles, and targets their prey while adhering to a well-defined social hierarchy. In this hierarchy, the pack is led by three primary wolves: the Alphas ( α ), Betas ( β ), and Deltas ( δ ), each playing a crucial role in guiding the pack's movements and decisions. These wolves are considered the leaders, showcasing significant leadership abilities. Below them are the Omega ( w ) wolves, who occupy a subordinate role and follow the directives of the leading wolves. This hierarchical structure, integral to the functioning of the GWO algorithm, is depicted in Fig.  1 .

Hierarchy of the grey Wolf Pack

3.1.2 Mathematical model: GWO

GWO simulates grey wolf leadership and hunting mechanisms by dividing grey wolves based on their characteristics into a leader, α , who rules over the entire grey wolf; a facilitator, β , who helps α to make decisions and replaces α when α dies; and an enforcer, δ , who follows α 's and β 's orders (Fan and Yu 2022 ). GWO searches for excellence by modeling the wolf hunting process. In addition to the social hierarchy of wolves, group hunting is another interesting social behavior of grey wolves. The main phases of grey wolf hunting are as follows: The Grey Wolf Optimizer algorithm mimics the hunting behavior of grey wolf packs. Initially, in the "tracking, chasing, and approaching prey" phase, each wolf searches for potential solutions in the solution space and adjusts its position through certain search strategies to get closer to possible candidate solutions. Subsequently, in the "chasing, surrounding, and harassing prey until it stops moving" phase, the wolf pack collaborates to try to corner the prey into a smaller area and prevent its escape, involving behaviors such as encircling and harassing the prey to prevent its escape. Finally, in the "attacking prey" phase, once the prey is cornered and unable to escape, the wolves concentrate their attack on the prey, gradually optimizing the position of candidate solutions through strategies such as linear or leap searches until finding the optimal solution or meeting specific optimization criteria. These three phases represent the Grey Wolf Optimization algorithm's process of searching, chasing, and optimizing in the solution space, analogous to the behavior of a grey wolf pack during hunting, progressing from search to attack, gradually optimizing and approaching the optimal solution.

Now, this paper shows the calculation steps of the basic grey Wolf Optimization algorithm and the pseudo-code as follows (Algorithm 1). The GWO algorithm process is as follows:

1) Each member is initialized using Eq. ( 1 ), determine the population size N , the maximum number of iterations M , the single grey wolf dimension dim , and ɑ , A and C ;

where, LB and UB are the lower and upper boundaries of the solution space, respectively. X represent the positions of the current solution. phi is a random number between [0,1].

2) Calculate the fitness value of each individual using the test function. Then, based on the magnitude of the fitness values, select the best-fit individual as the α -wolf, the second-best individual as the β -wolf, and the third-best individual as the δ -wolf;

3) The mathematical model of Wolf pack leader tracking prey is shown in Eq. ( 2 ), which calculates the traction direction of the entire pack according to the distance difference between the Wolf leader and the pack, that is, the movement direction information of the pack, can be calculated as shown in Eqs. ( 3 ) and ( 4 ). Update the current grey wolf position according to Eqs. ( 2 )– ( 4 ).

where D α , D β , and D δ denote the distance difference between α -wolf, β -wolf, and δ -wolf and other individuals, respectively. X α , X β , and X δ indicate the current positions of α -wolf, β -wolf, and δ -wolf respectively, X indicate the current positions C 1 , C 2 , and C 3 satisfy the constraints of Eq. ( 6 ). A 1 , A 2 , and A 3 are random vectors satisfying the constrain of Eq. ( 5 ); X 1 , X 2 and X 3 are the traction directions of the three leading wolves; and X ( t  + 1) represents the next collective movement position of the wolf pack. As shown in Fig.  2 , the final orientation of the wolves in the search space will be randomly positioned within a circle defined by the locations of the α , β , and δ in the search space. This graphical representation illustrates how the wolves’ positions influence the movement and direction of the entire pack in the pursuit of their prey.

Position update of wolf groups in GWO algorithm

4) Update ɑ , A and C according to Eqs. ( 5 )–( 7 );

where, the parameter ɑ plays a crucial role in balancing global search and local exploration. Its value is set to decrease linearly from 2 to 0 over the course of the algorithm's iterations. Initially, a higher value of ɑ aids in the global convergence of the algorithm, guiding the wolf pack swiftly towards the region where the optimal solution might be found. As the algorithm progresses through its later iterations, the gradual decrease in the value of ɑ facilitates more refined exploration in the area of the optimal solution. This helps improve the convergence accuracy of the GWO algorithm, ensuring a more precise final result. r 1 and r 2 are random vectors and r 1 , r 2  ∈ [0, 1].

5) Update the positions of other individuals, calculate the updated fitness value based on the new position, and update the α -wolf, β -wolf, δ -wolf and global optimal solution, \(R\) represents the position vector of the optimization target;

6) Judge whether the specified stopping condition is reached (e.g., the maximum number of iterations is reached), if not, repeat steps 2 to 5. Otherwise, output the optimal result: the position of the α -wolf obtained at the end is the optimal solution, and the corresponding fitness value is the degree of superiority or inferiority of the optimal solution.

3.2 Improved grey wolf optimization algorithm

3.2.1 pso search mechanism.

The GWO (Grey Wolf Optimizer) exhibits a weak exploratory capability in its early stages and lacks diversity within the population, consequently resulting in suboptimal solution quality. In order to enhance exploration capabilities, improve population diversity (Hu et al. 2022 ), and increase the quality of solutions (Hakli and Kiran 2020 ), this study integrates the PSO, introducing a velocity concept to provide a new search mechanism for the GWO. The individual grey wolves are updated in terms of position during the early iterations, and the application in the velocity update introduces additional randomness. This prevents the algorithm from converging prematurely and encourages exploration of new areas, thereby increasing population diversity. By dynamically adjusting the velocity and position of each individual, this method may help in more effectively balancing global exploration and local exploitation. Leading to a wider search in the early stage of the iteration, this assists in identifying potential high-quality solutions. The computation Eq. ( 8 ) is as follows:

where t represents the current number of iterations, X and \({X}_{best}\) represent the positions of the current solution and the best-performing solution, respectively. \({v}_{rand}\left(t\right)\) is the velocity vector of the current solution at time t of iteration. phi is a random number between [0,1]. \({X}_{selfbest}\) is the best position vector in the history of the current solution.

In this study, at the start of each iteration, a PSO updating strategy is employed, along with the addition of extra randomness to stimulate a more extensive global search. This approach helps avoid local optimization and increases population diversity. This approach not only accumulates a more diverse and high-quality search experience for the GWO but also more effectively balances global exploration and local exploitation by dynamically adjusting the search behavior.

3.2.2 IMF inertia weighting strategy

Inverse Multiquadric Function is a decreasing function based on the principle of inverse multiple squares. It is often used as a regularization method in neural networks, such as a kernel function in support vector machines (Hu et al. 1998 ; Rathan et al. 2023). In accordance with the characteristics of the IMF, this paper incorporates it into the population position update mechanism within the framework of the GWO as delineated in Eq. ( 3 ). The IMF inertia weight ω , along with the revised formulae for the wolf pack updating process, are elucidated in Eqs. ( 9 )–( 10 ).

where, the parameter groups [a, b, c, d] are taken as [0.6, 0.02, 0.05, 0.3] and the graph of ω is shown in Fig.  3 . As indicated by Fig.  3 , during the early to mid-phases of the algorithm's iteration, the inertia weight ω is set to a higher value. This larger influence of the α -wolf, β -wolf, and δ -wolf on the updated positions is beneficial for the pack to quickly converge towards the optimal solution, effectively preventing the waste of search resources due to blind searching and thus enhancing the quality of the pack. As the development progresses to the mid and late stages and the pack becomes densely concentrated, if the higher-ranking wolves get trapped in a local optimum, the lower-ranking wolves led by them are also unable to escape this local optimum. At this juncture, the value of ω should be reduced to a lower level, thereby enlarging the pack's autonomous search capability and avoiding premature convergence.

IMF Inertia Weight Graph

3.2.3 Adaptive updating mechanism

The population updating mechanism based on IMF inertia weight effectively reduces the density of population clustering to a certain extent. However, due to the intrinsic dynamics of the GWO, the newly generated wolf packs are still inevitably concentrated and migrate towards the positions directed by the α -wolf, β -wolf, and δ -wolf during the iterative process. In response to this, the present study defines the aggregation coefficient as the ratio of an individual's fitness value to the average population fitness value, which serves to quantify the degree of divergence between the current solution and the optimal solution. In minimization problems, the smaller the fitness value, the better the solution. A smaller aggregation coefficient indicates a more favorable current solution, thus allowing for minor updates in the vicinity of the individual's current position. Conversely, a larger aggregation coefficient suggests a poor location of the individual, warranting a significant perturbation to facilitate a jump to other positions. Based on this analysis, this paper introduces a Sigmoid function to construct the adaptive updating amplitude of the population under different aggregation coefficients, as depicted in Eqs. ( 11 )-( 12 ).

where f i represents the fitness value of the i th individual, and f ave denotes the average fitness value of the population. θ is the exponential coefficient, which is taken as 0.5 in this paper.

In comparison to the standard GWO, the IAGWO brings several significant advancements. Firstly, it introduces a novel search mechanism by incorporating velocity concepts. This addition helps in preventing premature convergence and allows for a more thorough exploration of the search space. The integration of velocity updates also adds an element of randomness, which in turn increases the diversity within the population of solutions. Moreover, the implementation of the IMF inertia weight strategy in IAGWO improves the balance between exploring the global search space and exploiting local solutions. This strategic enhancement significantly boosts the convergence speed of the algorithm. Furthermore, IAGWO differentiates itself from the standard GWO through its adaptive updating mechanism. This mechanism combines the aggregation coefficient with the Sigmoid function, enhancing the algorithm's ability to switch between broad search patterns and detailed solution refinement. This results in improved performance in maintaining diversity and achieving faster convergence rates. This adaptive approach enables IAGWO to search and optimize more efficiently within the solution space of the problem. For an in-depth comprehension of the workings of IAGWO, the procedural flow is visually depicted in Fig.  4 , its pseudocode is meticulously detailed in Algorithm 2, and the proposed IAGWO workflow(Chauhan & Yadav 2023b ) is shown in Fig.  5 .

Implementation Process for IAGWO

Working procedure of the proposed IAGWO algorithm

3.3 Time complexity analysis

CEC17 (Competition on Evolutionary Computation) defines algorithm complexity as a measure of the computational resources required by an algorithm to solve a given problem instance. This section explains the computational complexity of IAGWO. The complexity of IAGWO is primarily influenced by two main factors: the initialization of solutions and the execution of the algorithm's core functions. These core functions involve calculating fitness functions and updating solutions. The computational complexity is determined by considering several variables: the count of solutions \(\left(N\right)\) , the upper limit of iterations \((T)\) , and the problem's dimension \((D)\) being tackled. Specifically, the complexity of initializing solutions in the IAGWO algorithm can be represented as \(O(N)\) , indicating the order of complexity in relation to the number of solutions. This gives an understanding of the computational resources required for the initial setup phase of the algorithm. This means that as the number of solutions \(N\) increases, the computational complexity of the initial solution of the algorithm will also increase accordingly. This represents the order of complexity for the initial setup phase of the algorithm. The time complexity of the original GWO algorithm is \(O(T\times N\times D)\) . IAGWO modifies this with Eqs. ( 8 ), ( 9 ), and ( 10 )–( 11 ), including enhancements to population diversity using the PSO position updating strategy, integration of IMF weights to reduce the excessive influence of higher-level wolves on lower-level ones, and the introduction of a population adaptive update based on the sigmoid function. The PSO position updating strategy requires calculations for each individual and each dimension, with a complexity of \(O(T\times N\times D)\) . The update from Eq. ( 10 ) is independent of population size and search dimensions, correlating only with the maximum number of iterations, resulting in a time complexity of \(O(T)\) . The time complexity for Eq. ( 11 ) is \(O(N\times D)\) . Consequently, the overall time complexity of IAGWO is \(O(\text{IAGWO})=O\left(T\times N\times D\right)+O\left(T\right)+O\left(N\times D\right)=O(T\times N\times D)\) , consistent with the original algorithm.

4 Results and comprehensive analysis

The simulation for this study was carried out on a Windows 11 platform, operating on a 64-bit system. The analysis was performed using MATLAB 2023b, running on a machine equipped with an AMD Ryzen 7 4800H CPU at 2.30 GHz and 16 GB of RAM.

4.1 Test functions and parameter settings

In this paper, the CEC 2017 (Dim = 30) (Mallipeddi and Suganthan 2010 ), CEC 2020 (Dim = 10 and 20) (Liang et al. 2019 ), and CEC 2022 (Dim = 10 and 20) (Ahrari et al. 2022) test suites were employed to evaluate the performance of the proposed IAGWO algorithm. The test suite for evaluating algorithms covers four different functional types: single-modal, multimodal, mixed, and combined. These different types of test suites are designed to comprehensively evaluate the performance and applicability of algorithms. Additionally, for assessing the scalability of the IAGWO algorithm, we employed the CEC 2013 Large-scale Global Optimization suite (800-dimensional) for simulation analysis (Li et al. 2013 ). The suite contains 15 highly complex reference functions that are grouped into four groups: fully separable, partially additively separable, overlapping, and completely indivisible. These different types of benchmark functions provide a comprehensive experimental framework for evaluating the scalability of optimization algorithms, so that we can more accurately evaluate the performance of IAGWO algorithm on different types of problems.

4.2 Comparison with other algorithms and parameter settings

The performance of the Improved Adaptive Grey Wolf Optimization (IAGWO) is benchmarked against 12 well-known algorithms, grouped into three categories for comparison:

High-citation algorithms: These include the Gravitational Search Algorithm (GSA) (Rashedi et al. 2009 ), Dolphin Echolocation Optimization (DMO) (Kaveh and Farhoudi 2013 ), Whale Optimization Algorithm (WOA) (Mirjalili and Lewis 2016 ), and Harris Hawks Optimization (HHO) (Tripathy et al. 2022 ).

Advanced algorithms: This category includes Combined Particle Swarm Optimization and Gravitational Search Algorithm (CPSOGSA), Crow Optimization Algorithm (COA) (Jia et al. 2023 ), African Vulture Optimization Algorithm (AVOA) (Abdollahzadeh et al. 2021 ), Optical Microscope Algorithm (OMA) (Cheng and Sholeh 2023 ), and Adaptive Artificial Electric Field Algorithm (iAEFA) (Chauhan and Yadav 2023a ).

GWO and its variants: This includes the original Grey Wolf Optimization (GWO), the Adaptive GWO (AGWO) (Meidani et al. 2022 ), the Enhanced GWO (ENGWO)(Mohammed et al. 2024 )and the Revised GWO (RGWO) (Banaie-Dezfouli et al. 2021 ).

Table 1 offers a comprehensive summary of the parameters for 14 different MH algorithms. For each of these algorithms, 30 independent runs were conducted, with each run limited to a maximum of 500 iterations and population size is set to 30 and a maximum of 30,000 evaluations. The outcomes of these runs were meticulously recorded, capturing the average values (denoted as Ave) and the standard deviations (Std) for each algorithm. To facilitate an easy comparison of their performance, the table highlights the best results among these 14 algorithms by formatting them in bold text. This highlighting method provides a clear visual indicator of which algorithms performed most effectively under the given testing conditions.

4.3 Qualitative assessment of IAGWO

4.3.1 exploring convergence patterns.

To verify the convergence performance of IAGWO, we plotted its convergence performance evaluation on the 30-dimensional CEC2017 test functions, as shown in Fig.  6 . It presents the corresponding results of different test functions in the form of nine images involving three instances selected from the same suite. In the presentation of images, the first column distinctly illustrates the two-dimensional profiles of the reference functions being analyzed. The visuals presented in the first column accurately depict the characteristics and contours of each function being optimized. These graphical representations offer a clear understanding of the challenges and intricacies of each function. In the second column, images depict the final positions of the search agents at the end of the optimization process. Within these visuals, the optimal solution's location is distinctly marked with a red dot. This not only illustrates the end point of the search agents' journey but also visually highlights the spot where they successfully identified the most favorable solution. This layout effectively communicates the results of the optimization process, making it easier to comprehend the behavior and efficacy of the search agents in navigating the solution space. This layout offers a clear and informative view of both the nature of the functions and the outcomes of the optimization process. By observing the second column of images, we can clearly find that the search agent is close to the optimal solution in most cases, which fully reflects the powerful ability of IAGWO algorithm in the process of exploration and development. In addition, the third column image accurately tracks the change of the average fitness value during the iteration. Initially high, these values decrease and stabilize after 100 iterations, albeit with minor fluctuations. These fluctuations are normal in complex optimization problems, indicate ongoing detailed searches for improvement and the maintenance of population diversity to prevent premature convergence to local optima. The fourth column reveals the search agents' trajectories in the first dimension, there were marked fluctuations in the early iterations, which then leveled off, and then fluctuations again at intervals, which leveled off again, signifying a balance between exploration and exploitation. Finally, the convergence curve, smooth for unimodal functions, suggests optimal values are achievable through iteration. For multimodal functions, however, the step-like curve reflects the need for continual avoidance of local optima to reach global optima. These four metrics collectively affirm IAGWO's robust convergence.

The convergence behavior of IAGWO

4.3.2 Analyzing the diversity of population

In optimization algorithms, the importance of population diversity is a matter of balance. Moderate population diversity can help the algorithm avoid falling into local optima, thereby increasing search space coverage and global search capability, improving convergence speed, and the quality of optimization results. However, excessive population diversity may lead to overly dispersed search, making it difficult for the algorithm to explore local regions deeply, thereby reducing convergence speed and the quality of final solutions. Therefore, when designing optimization algorithms, it is necessary to consider a balance between population diversity and search efficiency. This can be achieved through appropriate parameter settings or suitable strategies to maintain population diversity, thus effectively solving optimization problems. A population with high diversity indicates significant differences among individuals, allowing for broader exploration in the search space and avoiding premature convergence to local optima. Hence, maintaining good population diversity is a crucial objective in metaheuristic algorithms. Typically, we use Eq. ( 13 ) and Eq. ( 14 ) to measure the population diversity of the algorithm. This calculation method was proposed by Morrison in 2004. Where, \({I}_{C}\) represents the moment of inertia, \({x}_{id}\) denotes the ith search agent's value in the \({d}^{th}\) dimension at iteration t . Furthermore, \({c}_{d}\) represents the spread of the population from its center of mass, denoted by ' c ', in every iteration, as illustrated in Eq. ( 14 ) (Fu et al. 2023a , b).

Figure  7 displays the comparative experimental outcomes regarding population diversity for both IAGWO and GWO. The measurement of population diversity is conducted through \({I}_{C}\) . Observations from Fig.  7 reveal that IAGWO demonstrates an initial marked increase in diversity during the early phases of iteration, which then transitions to a state of relative stability at an elevated level. This indicates an increase in the variance among individuals within the IAGWO population during the early iterations, effectively exploring a vast search space. As iterations progress, the population diversity tends to stabilize, which aids in averting premature convergence to local optima. The minor fluctuation are normal and beneficial for the algorithm to adapt to dynamically changing search spaces and prevent premature convergence. In contrast, GWO shows insufficient population diversity, highlighting IAGWO’s effectiveness in maintaining diversity, crucial for exploring complex search spaces and avoiding local optima. These experimental outcomes demonstrate IAGWO's substantial potential in optimization.

The population diversity of IAGWO and GWO

4.3.3 Exploration and exploitation analysis

In optimization algorithms, managing the balance between exploration and exploitation is key for optimal performance (Saka et al. 2016 ). Exploration involves searching through the solution space, while exploitation focuses on refining known good solutions. This section deals with quantifying the extent of exploration and exploitation in the algorithm. To do this, we use Eq. ( 15 ) to calculate the percentage of exploration and Eq. ( 16 ) for the percentage of exploitation. Additionally, the parameter \(Div\left(t\right)\) used for measuring dimension diversity is calculated using Eq. ( 17 ). The parameter \({\rm Div}_{max}\) reflects the peak diversity noted throughout the entire course of iterations,, which is essential for understanding how broadly and effectively the algorithm explores the solution space(Li et al. 2023 ) (Nadimi-Shahraki et al. 2023 ).

Figure  8 depicts the results of the experiments conducted. It shows that for various function types, as the number of iterations progresses, GWO consistently demonstrates a higher rate of exploration and a comparatively lower rate of exploitation. In contrast, IAGWO shows a changing pattern, with exploration decreasing and exploitation increasing as iterations progress. This observation suggests that GWO tends towards a broad search across the entire space, with less focus on local search and weaker performance in thoroughly exploiting the optimal regions found. In comparison, IAGWO demonstrates the ability to dynamically adjust its search strategy. This implies that the algorithm initially identifies potential good solution areas through extensive exploration and then finely tunes these solutions in the later stages through focused exploitation, potentially enhancing both the efficiency of the algorithm and the quality of solutions. Overall, while GWO shows commendable exploration capabilities, it lacks effective exploitation. In contrast, IAGWO effectively strikes a balance between exploration and exploitation. This balance is well-maintained across a variety of benchmark functions, showcasing IAGWO's adaptability and efficiency in different optimization scenarios. This attribute is particularly important as it ensures the algorithm can thoroughly search the solution space while also honing in on the most promising solutions.

The exploration and exploitation of IAGWO and GWO

4.3.4 Ablation experiments

In this section, a detailed analysis is conducted on the impact of three proposed improvement strategies on the GWO. These strategies include the PSO position updating mechanism, the introduction of IMF inertia weight strategy, and the adoption of a Sigmoid adaptive updating strategy. Based on these improvements, three new algorithm variants are named: PGWO for the PSO search mechanism, IGWO for the IMF inertia weight, and SGWO for the Sigmoid adaptive updating strategy. According to the experimental results in Fig.  9 , all three strategies significantly enhance the convergence accuracy and speed of GWO, with IAGWO showing particularly notable performance.

Comparison of different improvement strategies

Specifically, when dealing with unimodal and multimodal functions, the results of PGWO and IAGWO are relatively consistent, showing a more significant improvement over GWO compared to SGWO and IGWO. However, when dealing with more complex hybrid modal functions, the enhancement of PGWO on GWO diminishes, while the IAGWO algorithm, integrating all three strategies, continues to exhibit exceptional optimization performance. Overall, the IAGWO algorithm successfully overcomes challenges of local optima and premature convergence, significantly boosting the algorithm's convergence speed and accuracy. These findings provide valuable insights for the further development and application of the GWO.

4.4 Quantitative evaluation

In this section, the efficacy of IAGWO is scrutinized using a series of test suites: CEC 2017, CEC 2020, and CEC 2022. Moreover, its proficiency in handling large-scale problems is assessed with the CEC 2013 suite. To clearly compare performance, the best results among the algorithms are highlighted in bold in the tables. The parameters are standardized with a population size of 100, a maximum iteration limit of 500, and a total of 30 independent runs. The performance outcomes are systematically presented in Tables  2 to 7 , which illustrate the average values (Ave) and the standard deviations (Std) for each competing algorithm. A thorough statistical analysis is conducted to highlight the superiority of IAGWO. This includes an initial evaluation represented by three indicators (W|T|L) in the first line of the results, denoting the algorithms' performance as best (win), comparable (tie), or least effective (loss) for specific functions. The second row compiles the mean performance of all algorithms, while the third row offers insights into the overall standings through the final Friedman ranking. The tables distinctly highlight the top results, emphasizing their significance. Furthermore, the comparative analysis of the convergence curves for each algorithm is depicted in Fig.  10 . This visual representation aids in understanding the progression and efficiency of each algorithm in finding optimal solutions over the course of iterations. This detailed evaluation underscores the robustness and adaptability of IAGWO in varied optimization contexts.

Convergence curves of different algorithms

4.4.1 Assessing performance with CEC 2017 test suite

This section examines the efficacy of IAGWO using the CEC 2017 test suite with a dimensionality of 30, as detailed in Table  2 . The results are quite telling: IAGWO recorded the highest number of best performances, leading in 16 out of the 30 functions tested. Notably, it did not register as the least effective in any of the functions. In terms of statistical standing, IAGWO's Friedman mean ranking is 3.00, earning it the top position. Further, a diverse range of functions from CEC 2017 (Dim = 30) were chosen for a more comprehensive evaluation. The comparative analysis of the convergence trends, depicted in Fig.  10 , reveals that IAGWO consistently achieved the quickest convergence rate and maintained the highest level of accuracy in convergence. These results underscore IAGWO's exceptional proficiency in both global exploration and local exploitation. Collectively, these findings solidify the effectiveness and superiority of IAGWO as an optimization tool.

4.4.2 Assessing performance with CEC 2020 test suite

This section is dedicated to evaluating 13 algorithms with the utilization of the CEC 2020 test suite, which includes tests with dimensions of 10 and 20. The outcomes of this evaluation are systematically presented in Table  3 and Table  4 . On the CEC 2020 tests, IAGWO mirrors the impressive results observed in the CEC 2017 suite, achieving the highest number of best performances while not being the least effective in any function. To provide a visual representation of these results, representative functions are chosen to illustrate the convergence curves, as depicted in Fig.  10 . IAGWO consistently shows the quickest convergence speed and the highest accuracy in convergence, reaffirming its efficiency. Additionally, it's important to note the contrasting performance of GWO on the CEC 2020 suite. Despite its lower ranking in the Friedman rankings, indicating a comparatively poor performance, its improved variants, namely AGWO, ENGWO, and RGWO, show marked improvements. Remarkably, RGWO secures the second-highest ranking, closely following IAGWO, underscoring the substantial research value in enhancing the GWO algorithm. A comprehensive statistical analysis among the 13 algorithms tested places IAGWO at the forefront in the Friedman rankings. This achievement highlights its superiority not only over the original GWO but also over other well-regarded algorithms. These results collectively demonstrate the robustness and effectiveness of IAGWO in a competitive algorithmic landscape.

4.4.3 Assessing performance with CEC 2022 test suite

This section is dedicated to a thorough evaluation of the proposed IAGWO and 12 other comparative algorithms, utilizing the CEC 2022 test suite. The primary objective of this evaluation is to gauge the exploration and exploitation capabilities of these algorithms and assess their proficiency in avoiding local optima traps. The experiments are conducted under 10-dimensional and 20-dimensional scenarios, with corresponding results displayed in Tables  5 and 6 , respectively. IAGWO ranks first in Friedman mean ranking in both dimensional settings, with ranking values of 1.75 and 2.25 respectively. Similarly, while GWO shows subpar performance, its variants enhance GWO's performance, emphasizing the research significance of GWO. The analysis of results depicted in Fig.  10 leads to a conclusive observation that IAGWO successfully evades getting stuck in local optima and avoids premature convergence. These findings serve not just as a testament to the excellence and robustness of IAGWO, but they also highlight its substantial performance benefits and the capability to yield enhanced solutions. This analysis underscores IAGWO's effectiveness in navigating complex optimization landscapes, further establishing its potential as a superior tool in optimization tasks.

4.4.4 Scalability evaluation using the CEC 2013 test suite

In real-world scenarios, solving optimization problems often requires adjusting multiple parameters at once. To test the scalability of the IAGWO for high-dimensional problems, we utilized the CEC 2013 suite for large-scale global optimization. The results of this testing are detailed in Table  7 . This suite includes 15 highly complex test functions, each with up to 1000 dimensions, providing a robust challenge for assessing algorithmic performance. In our experiments, IAGWO was compared with 12 other algorithms. The population size was fixed at 100, and we limited the maximum number of iterations to 10 for each run. After conducting 30 independent runs for each algorithm, IAGWO achieved a Friedman mean rank value of 2.63. This score signifies a higher level of performance relative to the other algorithms in the competition. The findings from these experiments demonstrate that the IAGWO algorithm has significant scalability, effectively handling complex, high-dimensional optimization challenges. This capability distinguishes IAGWO from other algorithms, highlighting its suitability for practical, large-scale optimization applications.

4.5 Wilcoxon rank sum test

This study utilizes the non-parametric Wilcoxon rank sum test (Wilcoxon 1945 ) to conduct comparative performance assessments of various algorithms, setting the significance level at 0.05. To succinctly represent the performance of IAGWO relative to its competitors, the symbols “ + / = /-” are used to denote whether IAGWO is superior to, equivalent to, or inferior to the competing algorithms. As shown in Table  8 , these statistical results clearly indicate significant performance differences between IAGWO and other competing algorithms in most cases. Specifically, the statistical data show the following comparative results: 344/0/46、119/0/11、111/13/6、150/0/6、152/0/4 and 175/0/5. The analysis presented above demonstrates that the IAGWO method, as introduced in this study, shows exceptional overall performance when compared to the traditional GWO and other rival algorithms, thereby underscoring its distinct advantages.

4.6 Time comparison analysis of IAGWO and GWO

Building on the findings from previous chapters, it's clear that IAGWO significantly surpasses the original GWO in terms of overall performance. In this section, we focus on a more detailed comparison, specifically looking at the computational costs of both algorithms, with a particular emphasis on the differences in computational time. To facilitate this comparison, we standardized the settings for both IAGWO and GWO. For this evaluation, the population size was configured to 50, the maximum iterations were limited to 1000, and each algorithm underwent 30 independent runs. Table 9 presents the total time (in seconds) each algorithm took to complete all 30 runs. This data provides a clear basis for comparing the efficiency of the two algorithms in terms of how long they take to execute, offering insights into their time-based performance efficiency.

Analysis of the experimental data on the CEC 2017 test suit (Dim = 30) indicates that under the same experimental parameters, IAGWO and GWO perform almost equally in terms of execution time on unimodal functions and some simpler multimodal functions, but when dealing with more complex multimodal and hybrid functions, IAGWO generally consumes significantly less computational time than GWO. This suggests that in handling highly complex problems, IAGWO demonstrates greater computational efficiency. Compared to the original GWO, IAGWO's improved search strategies are more efficient, possessing better global search capabilities or faster local convergence speeds. Overall, IAGWO not only excels in benchmark tests but also exhibits higher computational efficiency and better adaptability when addressing more complex optimization problems that may arise in practical applications.

However, on the CEC 2020 test suite (Dim = 10 and 20) and CEC 2022 test suite (Dim = 10 and 20), IAGWO generally exhibits higher computational times compared to GWO. This may indicate that the types of problems or characteristics included in CEC 2020 are not entirely compatible with the strategies of IAGWO, leading to a higher computational load.

4.7 Evaluating performance against CEC 2014 and CEC 2017 competition-winners

This section evaluates the performance of the proposed IAGWO using the CEC 2014 test suite with a dimensionality of 30 (Liang et al. 2013 ) and CEC 2017 (Dim = 30) test suites. Additionally, we compare the performance of IAGWO with the competition-winners of these two suites in previous CEC competitions, including L-SHADE (Tanabe & Fukunaga 2014 ) and AL-SHADE (Li et al. 2022 ) from CEC 2014, and LSHADE-SPACMA (Mohamed, Hadi, Fattouh, & Jambi, 2017) and LSHADE-cnEpSin (Awad, Ali, & Suganthan, 2017) from CEC 2017. In the experimental setup, the population size is fixed at 30, the maximum iterations are limited to 500, and a total of 30 independent runs are performed.

Table 10 presents the results from testing IAGWO using the CEC 2014 suite. In these tests, IAGWO surpassed other algorithms in six different scenarios, though it showed slightly weaker performance in one. Notably, IAGWO achieved a Friedman mean ranking value of 1.71, which places it second after L-SHADE but ahead of AL-SHADE. Table 11 focuses on the performance of IAGWO in the CEC 2017 suite. Here, IAGWO showed strong results in 8 of the test cases, but its performance was less impressive in 10 others. In terms of the Friedman mean ranking, IAGWO scored 1.99, which is slightly better than LSHADE-SPACMA, but not quite as good as LSHADE-cnEpSin. These results provide a detailed comparison of IAGWO's performance relative to other algorithms in these specific test environments.

Combining experimental outcomes, IAGWO can be positioned as a high-performing optimizer in test functions. These results not only demonstrate IAGWO's strong capability in handling different types of optimization problems but also indicate its competitive standing against existing top-tier algorithms. These findings emphasize the potential application value of IAGWO in the field of evolutionary computing and optimization. This simultaneously demonstrates the effectiveness of the three improvement strategies we introduced: the PSO Search Mechanism, the IMF Inertia Weighting Strategy, and the Adaptive Updating Mechanism, enhancing the optimization performance of the algorithm.

4.8 IAGWO for 19 engineering design challenges

The specific constrained handling technique used in engineering design challenges is called "constraint relaxation." Constraint relaxation involves temporarily easing or loosening certain constraints within the design problem to explore alternative solutions. This allows designers to generate a wider range of potential solutions without being overly restricted by strict constraints. Once various solutions have been identified, designers can then reintroduce and refine the constraints to ensure that the final design meets all necessary requirements. Intelligent optimization algorithms can efficiently explore the design space and uncover potential solutions. By integrating constraint relaxation techniques, these algorithms can dynamically handle constraints during the search process, allowing for a broader exploration of the design space and enhancing the efficiency of finding optimal solutions.

In this section, the proficiency of IAGWO is meticulously evaluated through a set of 19 engineering design challenges (EDC) sourced from the CEC 2020 real-world optimization benchmarks, as outlined by Kumar et al., 2020., 2020. A concise summary of these engineering challenges is presented in Table  12 , which includes key details such as their dimensions ( D ), the count of inequality constraints ( g ), equality constraints ( h ), and the known optimal cost ( f min ). The evaluation parameters are defined as follows: a population size of 50, a maximum of 1000 iterations, and 30 independent runs for each challenge.

Table 13 is dedicated to enumerating the performance metrics of IAGWO. This table encompasses various metrics including the best cost achieved (Best), the average cost (Ave), the cost's standard deviation (Std), and performance symbols (W|T|L), representing the number of wins, ties, and losses, respectively. Additionally, the evaluation includes a comprehensive analysis of the mean performance of all the algorithms involved in the testing. It also presents a ranking of these methods, providing a clear and structured comparison of their overall effectiveness and highlights instances where IAGWO achieves optimal results.

The statistical analysis drawn from these results clearly demonstrates IAGWO's superior ability in solving these real-world engineering design challenges, effectively outshining other methods. In terms of overall effectiveness, other algorithms like OMA, DMO, RGWO, and ENGWO trail behind IAGWO. This comprehensive analysis accentuates the robustness and efficacy of IAGWO.

5 Summary and future directions

In this study, we introduced an enhanced version of the GWO, aiming to tackle its inherent limitations and elevate its efficacy for addressing contemporary optimization challenges. The original GWO, while promising, exhibited deficiencies, notably in its convergence speed and its adaptability to intricate, high-dimensional problem landscapes. To fortify its capabilities, we embarked on an innovative path, culminating in the birth of an enhanced variant dubbed the Improved Adaptive Grey Wolf Optimizer (IAGWO). Central to our enhancement strategy was the infusion of concepts borrowed from Particle Swarm Optimization (PSO), introducing a velocity component to expedite convergence. This integration of velocity mechanics injected dynamism into the algorithm, enabling it to traverse solution spaces with greater agility. Moreover, a novel search mechanism was devised, augmenting the algorithm's exploration and exploitation capabilities to navigate complex problem domains more efficiently. In addition to these fundamental alterations, we devised novel strategies for Inertia Weighting and Position Updating, leveraging Nonlinear Inertia Weighting for Intermediary Fitness (IMF) and employing Sigmoid adaptive techniques. These refinements were meticulously crafted to synergize with the core algorithm, amplifying its prowess in navigating diverse optimization landscapes with finesse.

To validate the prowess of IAGWO, rigorous experimentation ensued, wherein 52 test functions sourced from prestigious benchmark suites were scrutinized. Comparative analysis against eight prominent Metaheuristic (MH) algorithms, including the original GWO and three of its variants, underscored IAGWO's supremacy in terms of convergence speed and solution precision. Furthermore, the algorithm's mettle was tested against 15 formidable large-scale global optimization challenges, affirming its adeptness in grappling with high-dimensional complexities. The litmus test for IAGWO's efficacy extended to competitive arenas, where it stood toe-to-toe against previous champions of the renowned CEC competitions across various iterations. Notably, IAGWO's performance surpassed expectations, firmly establishing its dominance and resilience in the face of formidable adversaries. Beyond the realm of academia, the real-world applicability of IAGWO was validated through its deployment in 19 diverse engineering design challenges. Here, its versatility and competitive edge shone brightly, outperforming established algorithms and offering tangible solutions to practical problems.

Despite its commendable achievements, the journey of IAGWO is far from over. While it has emerged as a potent force in the optimization landscape, ongoing efforts are directed towards fine-tuning its computational efficiency. Time comparison analyses have revealed areas for optimization, particularly concerning computational overhead in certain test suites. Future endeavors will thus focus on streamlining computational complexity without compromising search efficacy, ensuring that IAGWO remains at the forefront of optimization methodologies. Looking ahead, the horizon for IAGWO is brimming with promise. Beyond academic benchmarks, its utility extends to a myriad of real-world applications, ranging from feature extraction to operations research, classification, and logistical challenges. As we embark on this journey, our aim is clear: to harness the full potential of IAGWO in unraveling the complexities of the modern world, one optimization problem at a time.

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This work was supported by Natural Science Foundation of Tianjin Municipality (21JCYBJC00110) and China Postdoctoral Science Foundation (2023M731803).

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Yu, M., Xu, J., Liang, W. et al. Improved multi-strategy adaptive Grey Wolf Optimization for practical engineering applications and high-dimensional problem solving. Artif Intell Rev 57 , 277 (2024). https://doi.org/10.1007/s10462-024-10821-3

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