project on assignment problem

Assignment Problem: Meaning, Methods and Variations | Operations Research

After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.

Meaning of Assignment Problem:

An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.

The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.

Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.

Definition of Assignment Problem:

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Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.

The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table:

• Shahriar Tanvir Alam   ORCID: orcid.org/0000-0002-0567-3381 5 ,
• Eshfar Sagor 5 ,
• Tanjeel Ahmed 5 ,
• Tabassum Haque 5 ,
• Md Shoaib Mahmud 5 ,
• Salman Ibrahim 5 ,
• Ononya Shahjahan 5 &
• Mubtasim Rubaet 5  

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The aim of a supply chain team is to formulate a network layout that minimizes the total cost. In this research, the lowest production cost of the final product has been determined using a generalized plant location model. Furthermore, it is anticipated that units have been set up appropriately so that one unit of input from a source of supply results in one unit of output. The assignment problem is equivalent to distributing a job to the appropriate machine in order to meet customer demand. This study concentrates on reducing the cost of fulfilling the overall customer demand. Many studies have been conducted, and various algorithms have been proposed to achieve the best possible result. The purpose of this study is to present an appropriate model for exploring the solution to the assignment problem using the “Hungarian Method.” To find a feasible output of the assignment problem, this study conducted a detailed case study. The computational results indicate that the “Hungarian Method” provides an optimum solution for both balanced and unbalanced assignment problems. Moreover, decision-makers can use the study’s findings as a reference to mitigate production costs and adopt any sustainable market policy.

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Military Institute of Science and Technology, Department of Industrial and Production Engineering, Dhaka, Bangladesh

Shahriar Tanvir Alam, Eshfar Sagor, Tanjeel Ahmed, Tabassum Haque, Md Shoaib Mahmud, Salman Ibrahim, Ononya Shahjahan & Mubtasim Rubaet

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Department of Computer Science and Engineering, Sri Eshwar College of Engineering, Coimbatore, Tamil Nadu, India

Anandakumar Haldorai

Department of Computer Science and Engineering, CMR University, Bengaluru, Karnataka, India

Arulmurugan Ramu

Sri Eshwar College of Engineering, Coimbatore, Tamil Nadu, India

Sudha Mohanram

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Alam, S.T. et al. (2023). A Comparative Analysis of Assignment Problem. In: Haldorai, A., Ramu, A., Mohanram, S. (eds) 5th EAI International Conference on Big Data Innovation for Sustainable Cognitive Computing. BDCC 2022. EAI/Springer Innovations in Communication and Computing. Springer, Cham. https://doi.org/10.1007/978-3-031-28324-6_11

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Job Assignment Problem using Branch And Bound

Let there be N workers and N jobs. Any worker can be assigned to perform any job, incurring some cost that may vary depending on the work-job assignment. It is required to perform all jobs by assigning exactly one worker to each job and exactly one job to each agent in such a way that the total cost of the assignment is minimized.

Let us explore all approaches for this problem.

Solution 1: Brute Force  

We generate n! possible job assignments and for each such assignment, we compute its total cost and return the less expensive assignment. Since the solution is a permutation of the n jobs, its complexity is O(n!).

Solution 2: Hungarian Algorithm  

The optimal assignment can be found using the Hungarian algorithm. The Hungarian algorithm has worst case run-time complexity of O(n^3).

Solution 3: DFS/BFS on state space tree  

A state space tree is a N-ary tree with property that any path from root to leaf node holds one of many solutions to given problem. We can perform depth-first search on state space tree and but successive moves can take us away from the goal rather than bringing closer. The search of state space tree follows leftmost path from the root regardless of initial state. An answer node may never be found in this approach. We can also perform a Breadth-first search on state space tree. But no matter what the initial state is, the algorithm attempts the same sequence of moves like DFS.

Solution 4: Finding Optimal Solution using Branch and Bound  

The selection rule for the next node in BFS and DFS is “blind”. i.e. the selection rule does not give any preference to a node that has a very good chance of getting the search to an answer node quickly. The search for an optimal solution can often be speeded by using an “intelligent” ranking function, also called an approximate cost function to avoid searching in sub-trees that do not contain an optimal solution. It is similar to BFS-like search but with one major optimization. Instead of following FIFO order, we choose a live node with least cost. We may not get optimal solution by following node with least promising cost, but it will provide very good chance of getting the search to an answer node quickly.

There are two approaches to calculate the cost function:  

• For each worker, we choose job with minimum cost from list of unassigned jobs (take minimum entry from each row).
• For each job, we choose a worker with lowest cost for that job from list of unassigned workers (take minimum entry from each column).

In this article, the first approach is followed.

Let’s take below example and try to calculate promising cost when Job 2 is assigned to worker A. 

Since Job 2 is assigned to worker A (marked in green), cost becomes 2 and Job 2 and worker A becomes unavailable (marked in red). 

Now we assign job 3 to worker B as it has minimum cost from list of unassigned jobs. Cost becomes 2 + 3 = 5 and Job 3 and worker B also becomes unavailable. 

Finally, job 1 gets assigned to worker C as it has minimum cost among unassigned jobs and job 4 gets assigned to worker D as it is only Job left. Total cost becomes 2 + 3 + 5 + 4 = 14. 

Below diagram shows complete search space diagram showing optimal solution path in green. 

Complete Algorithm:  

Below is the implementation of the above approach:

Time Complexity: O(M*N). This is because the algorithm uses a double for loop to iterate through the M x N matrix.  Auxiliary Space: O(M+N). This is because it uses two arrays of size M and N to track the applicants and jobs.

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