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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:

• Carlos A. Alfaro   ORCID: orcid.org/0000-0001-9783-8587 1 ,
• Sergio L. Perez 2 ,
• Carlos E. Valencia 3 &
• Marcos C. Vargas 1  

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First, we give a detailed review of two algorithms that solve the minimization case of the assignment problem, the Bertsekas auction algorithm and the Goldberg & Kennedy algorithm. It was previously alluded that both algorithms are equivalent. We give a detailed proof that these algorithms are equivalent. Also, we perform experimental results comparing the performance of three algorithms for the assignment problem: the \(\epsilon \) - scaling auction algorithm , the Hungarian algorithm and the FlowAssign algorithm . The experiment shows that the auction algorithm still performs and scales better in practice than the other algorithms which are harder to implement and have better theoretical time complexity.

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Some results on an assignment problem variant

Integer Programming

A Full Description of Polytopes Related to the Index of the Lowest Nonzero Row of an Assignment Matrix

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Acknowledgements

This research was partially supported by SNI and CONACyT.

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Carlos A. Alfaro & Marcos C. Vargas

Mountain View, CA, 94043, USA

Sergio L. Perez

Departamento de Matemáticas, CINVESTAV del IPN, Apartado postal 14-740, 07000, Mexico City, Mexico

Carlos E. Valencia

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Alfaro, C.A., Perez, S.L., Valencia, C.E. et al. The assignment problem revisited. Optim Lett 16 , 1531–1548 (2022). https://doi.org/10.1007/s11590-021-01791-4

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Received : 26 March 2020

Accepted : 03 August 2021

Published : 16 August 2021

Issue Date : June 2022

DOI : https://doi.org/10.1007/s11590-021-01791-4

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WHAT IS ASSIGNMENT PROBLEM

Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons.

The assignment problem in the general form can be stated as follows:

“Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to assign each facility to one and only one job in such a way that the measure of effectiveness is optimised (Maximised or Minimised).”

Several problems of management has a structure identical with the assignment problem.

Example I A manager has four persons (i.e. facilities) available for four separate jobs (i.e. jobs) and the cost of assigning (i.e. effectiveness) each job to each ...

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Assignment Problem: Linear Programming

The assignment problem is a special type of transportation problem , where the objective is to minimize the cost or time of completing a number of jobs by a number of persons.

In other words, when the problem involves the allocation of n different facilities to n different tasks, it is often termed as an assignment problem.

The model's primary usefulness is for planning. The assignment problem also encompasses an important sub-class of so-called shortest- (or longest-) route models. The assignment model is useful in solving problems such as, assignment of machines to jobs, assignment of salesmen to sales territories, travelling salesman problem, etc.

It may be noted that with n facilities and n jobs, there are n! possible assignments. One way of finding an optimal assignment is to write all the n! possible arrangements, evaluate their total cost, and select the assignment with minimum cost. But, due to heavy computational burden this method is not suitable. This chapter concentrates on an efficient method for solving assignment problems that was developed by a Hungarian mathematician D.Konig.

"A mathematician is a device for turning coffee into theorems." -Paul Erdos

Formulation of an assignment problem

Suppose a company has n persons of different capacities available for performing each different job in the concern, and there are the same number of jobs of different types. One person can be given one and only one job. The objective of this assignment problem is to assign n persons to n jobs, so as to minimize the total assignment cost. The cost matrix for this problem is given below:

The structure of an assignment problem is identical to that of a transportation problem.

To formulate the assignment problem in mathematical programming terms , we define the activity variables as

for i = 1, 2, ..., n and j = 1, 2, ..., n

In the above table, c ij is the cost of performing jth job by ith worker.

Generalized Form of an Assignment Problem

The optimization model is

Minimize c 11 x 11 + c 12 x 12 + ------- + c nn x nn

subject to x i1 + x i2 +..........+ x in = 1          i = 1, 2,......., n x 1j + x 2j +..........+ x nj = 1          j = 1, 2,......., n

x ij = 0 or 1

In Σ Sigma notation

x ij = 0 or 1 for all i and j

An assignment problem can be solved by transportation methods, but due to high degree of degeneracy the usual computational techniques of a transportation problem become very inefficient. Therefore, a special method is available for solving such type of problems in a more efficient way.

Assumptions in Assignment Problem

• Number of jobs is equal to the number of machines or persons.
• Each man or machine is assigned only one job.
• Each man or machine is independently capable of handling any job to be done.
• Assigning criteria is clearly specified (minimizing cost or maximizing profit).

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Lesson 9. solution of assignment problem.

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Maximisation in an Assignment Problem: Optimizing Assignments for Maximum Benefit

Table of Contents

This blog will explain how to solve assignment problems using optimization techniques to achieve maximum results. Learn the basics, step-by-step approach, and real-world applications of maximizing assignment problems.

In an assignment problem, the goal is to assign n tasks to n agents in such a way that the overall cost or benefit is minimized or maximized. The maximization problem arises when the objective is to maximize the overall benefit rather than minimize the cost.

Understanding Maximisation in an Assignment Problem

The maximization problem can be solved using the Hungarian algorithm, which is a special case of the transportation problem. The algorithm involves converting the assignment problem into a matrix, finding the minimum value in each row, and subtracting it from all the elements in that row. Similarly, we find the minimum value in each column and subtract it from all the elements in that column. This is known as matrix reduction.

Next, we cover all zeros in the matrix using the minimum number of lines. A line covers a row or column that contains a zero. If the minimum number of lines is n, an optimal solution has been found. Otherwise, we continue with the next step.

In step three, we add the minimum uncovered value to each element covered by two lines, and subtract it from each uncovered element. Then, we return to step two and repeat until an optimal solution is found.

Solving Maximisation in an Assignment Problem

The above approach provides a step-by-step process to maximize an assignment problem. Here are the steps in summary:

• Convert the assignment problem into a matrix.
• Reduce the matrix by subtracting the minimum value in each row and column.
• Cover all zeros in the matrix with the minimum number of lines.
• Add the minimum uncovered value to each element covered by two lines and subtract it from each uncovered element.
• Repeat from step two until an optimal solution is found.

Real-World Applications

Maximization in an Assignment Problem has numerous real-world applications. For example, it can be used to optimize employee allocation to projects, to allocate tasks in a manufacturing process, or to assign jobs to machines for maximum productivity. By using optimization techniques to maximize the benefits of an assignment problem, businesses can save time, money, and resources.

This blog has provided an overview of Maximisation in an Assignment Problem, explained how to solve it using the Hungarian algorithm, and discussed real-world applications. By following the step-by-step approach, you can optimize your assignments for maximum benefit.

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