Assignment Problem: Meaning, Methods and Variations | Operations Research

assignment problem on operation 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:

assignment problem on operation research

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assignment problem on operation research

MBA Notes

How to Solve the Assignment Problem: A Complete Guide

Table of Contents

Assignment problem is a special type of linear programming problem that deals with assigning a number of resources to an equal number of tasks in the most efficient way. The goal is to minimize the total cost of assignments while ensuring that each task is assigned to only one resource and each resource is assigned to only one task. In this blog, we will discuss the solution of the assignment problem using the Hungarian method, which is a popular algorithm for solving the problem.

Understanding the Assignment Problem

Before we dive into the solution, it is important to understand the problem itself. In the assignment problem, we have a matrix of costs, where each row represents a resource and each column represents a task. The objective is to assign each resource to a task in such a way that the total cost of assignments is minimized. However, there are certain constraints that need to be satisfied – each resource can be assigned to only one task and each task can be assigned to only one resource.

Solving the Assignment Problem

There are various methods for solving the assignment problem, including the Hungarian method, the brute force method, and the auction algorithm. Here, we will focus on the steps involved in solving the assignment problem using the Hungarian method, which is the most commonly used and efficient method.

Step 1: Set up the cost matrix

The first step in solving the assignment problem is to set up the cost matrix, which represents the cost of assigning a task to an agent. The matrix should be square and have the same number of rows and columns as the number of tasks and agents, respectively.

Step 2: Subtract the smallest element from each row and column

To simplify the calculations, we need to reduce the size of the cost matrix by subtracting the smallest element from each row and column. This step is called matrix reduction.

Step 3: Cover all zeros with the minimum number of lines

The next step is to cover all zeros in the matrix with the minimum number of horizontal and vertical lines. This step is called matrix covering.

Step 4: Test for optimality and adjust the matrix

To test for optimality, we need to calculate the minimum number of lines required to cover all zeros in the matrix. If the number of lines equals the number of rows or columns, the solution is optimal. If not, we need to adjust the matrix and repeat steps 3 and 4 until we get an optimal solution.

Step 5: Assign the tasks to the agents

The final step is to assign the tasks to the agents based on the optimal solution obtained in step 4. This will give us the most cost-effective or profit-maximizing assignment.

Solution of the Assignment Problem using the Hungarian Method

The Hungarian method is an algorithm that uses a step-by-step approach to find the optimal assignment. The algorithm consists of the following steps:

  • Subtract the smallest entry in each row from all the entries of the row.
  • Subtract the smallest entry in each column from all the entries of the column.
  • Draw the minimum number of lines to cover all zeros in the matrix. If the number of lines drawn is equal to the number of rows, we have an optimal solution. If not, go to step 4.
  • Determine the smallest entry not covered by any line. Subtract it from all uncovered entries and add it to all entries covered by two lines. Go to step 3.

The above steps are repeated until an optimal solution is obtained. The optimal solution will have all zeros covered by the minimum number of lines. The assignments can be made by selecting the rows and columns with a single zero in the final matrix.

Applications of the Assignment Problem

The assignment problem has various applications in different fields, including computer science, economics, logistics, and management. In this section, we will provide some examples of how the assignment problem is used in real-life situations.

Applications in Computer Science

The assignment problem can be used in computer science to allocate resources to different tasks, such as allocating memory to processes or assigning threads to processors.

Applications in Economics

The assignment problem can be used in economics to allocate resources to different agents, such as allocating workers to jobs or assigning projects to contractors.

Applications in Logistics

The assignment problem can be used in logistics to allocate resources to different activities, such as allocating vehicles to routes or assigning warehouses to customers.

Applications in Management

The assignment problem can be used in management to allocate resources to different projects, such as allocating employees to tasks or assigning budgets to departments.

Let’s consider the following scenario: a manager needs to assign three employees to three different tasks. Each employee has different skills, and each task requires specific skills. The manager wants to minimize the total time it takes to complete all the tasks. The skills and the time required for each task are given in the table below:

The assignment problem is to determine which employee should be assigned to which task to minimize the total time required. To solve this problem, we can use the Hungarian method, which we discussed in the previous blog.

Using the Hungarian method, we first subtract the smallest entry in each row from all the entries of the row:

Next, we subtract the smallest entry in each column from all the entries of the column:

We draw the minimum number of lines to cover all the zeros in the matrix, which in this case is three:

Since the number of lines is equal to the number of rows, we have an optimal solution. The assignments can be made by selecting the rows and columns with a single zero in the final matrix. In this case, the optimal assignments are:

  • Emp 1 to Task 3
  • Emp 2 to Task 2
  • Emp 3 to Task 1

This assignment results in a total time of 9 units.

I hope this example helps you better understand the assignment problem and how to solve it using the Hungarian method.

Solving the assignment problem may seem daunting, but with the right approach, it can be a straightforward process. By following the steps outlined in this guide, you can confidently tackle any assignment problem that comes your way.

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Operations Research

1 Operations Research-An Overview

  • History of O.R.
  • Approach, Techniques and Tools
  • Phases and Processes of O.R. Study
  • Typical Applications of O.R
  • Limitations of Operations Research
  • Models in Operations Research
  • O.R. in real world

2 Linear Programming: Formulation and Graphical Method

  • General formulation of Linear Programming Problem
  • Optimisation Models
  • Basics of Graphic Method
  • Important steps to draw graph
  • Multiple, Unbounded Solution and Infeasible Problems
  • Solving Linear Programming Graphically Using Computer
  • Application of Linear Programming in Business and Industry

3 Linear Programming-Simplex Method

  • Principle of Simplex Method
  • Computational aspect of Simplex Method
  • Simplex Method with several Decision Variables
  • Two Phase and M-method
  • Multiple Solution, Unbounded Solution and Infeasible Problem
  • Sensitivity Analysis
  • Dual Linear Programming Problem

4 Transportation Problem

  • Basic Feasible Solution of a Transportation Problem
  • Modified Distribution Method
  • Stepping Stone Method
  • Unbalanced Transportation Problem
  • Degenerate Transportation Problem
  • Transhipment Problem
  • Maximisation in a Transportation Problem

5 Assignment Problem

  • Solution of the Assignment Problem
  • Unbalanced Assignment Problem
  • Problem with some Infeasible Assignments
  • Maximisation in an Assignment Problem
  • Crew Assignment Problem

6 Application of Excel Solver to Solve LPP

  • Building Excel model for solving LP: An Illustrative Example

7 Goal Programming

  • Concepts of goal programming
  • Goal programming model formulation
  • Graphical method of goal programming
  • The simplex method of goal programming
  • Using Excel Solver to Solve Goal Programming Models
  • Application areas of goal programming

8 Integer Programming

  • Some Integer Programming Formulation Techniques
  • Binary Representation of General Integer Variables
  • Unimodularity
  • Cutting Plane Method
  • Branch and Bound Method
  • Solver Solution

9 Dynamic Programming

  • Dynamic Programming Methodology: An Example
  • Definitions and Notations
  • Dynamic Programming Applications

10 Non-Linear Programming

  • Solution of a Non-linear Programming Problem
  • Convex and Concave Functions
  • Kuhn-Tucker Conditions for Constrained Optimisation
  • Quadratic Programming
  • Separable Programming
  • NLP Models with Solver

11 Introduction to game theory and its Applications

  • Important terms in Game Theory
  • Saddle points
  • Mixed strategies: Games without saddle points
  • 2 x n games
  • Exploiting an opponent’s mistakes

12 Monte Carlo Simulation

  • Reasons for using simulation
  • Monte Carlo simulation
  • Limitations of simulation
  • Steps in the simulation process
  • Some practical applications of simulation
  • Two typical examples of hand-computed simulation
  • Computer simulation

13 Queueing Models

  • Characteristics of a queueing model
  • Notations and Symbols
  • Statistical methods in queueing
  • The M/M/I System
  • The M/M/C System
  • The M/Ek/I System
  • Decision problems in queueing

Operations Research/Transportation and Assignment Problem

The Transportation and Assignment problems deal with assigning sources and jobs to destinations and machines. We will discuss the transportation problem first.

Suppose a company has m factories where it manufactures its product and n outlets from where the product is sold. Transporting the product from a factory to an outlet costs some money which depends on several factors and varies for each choice of factory and outlet. The total amount of the product a particular factory makes is fixed and so is the total amount a particular outlet can store. The problem is to decide how much of the product should be supplied from each factory to each outlet so that the total cost is minimum.

Let us consider an example.

Suppose an auto company has three plants in cities A, B and C and two major distribution centers in D and E. The capacities of the three plants during the next quarter are 1000, 1500 and 1200 cars. The quarterly demands of the two distribution centers are 2300 and 1400 cars. The transportation costs (which depend on the mileage, transport company etc) between the plants and the distribution centers is as follows:

Which plant should supply how many cars to which outlet so that the total cost is minimum?

The problem can be formulated as a LP model:

{\displaystyle x_{ij}}

The whole model is:

subject to,

{\displaystyle x_{11}+x_{12}=1000}

The problem can now be solved using the simplex method. A convenient procedure is discussed in the next section.

assignment problem on operation research

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  • Operations Research Problems

Statements and Solutions

  • Raúl Poler 0 ,
  • Josefa Mula 1 ,
  • Manuel Díaz-Madroñero 2

Research Centre on Production Management and Engineering, Polytechnic University of Valencia, Alcoy, Spain

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Escuela Politécnica Superior de Alcoy, Universidad Politécnica de Valencia, Alcoy, Spain

Universitat politècnica de valència, alcoy, spain.

Provides a valuable compendium of problems as a reference for undergraduate and graduate students, faculty, researchers and practitioners of operations research and management science

Identifies different operations management problems in order to improve the decision making process concerning readers

Addresses the following topics: Linear programming, integer programming, non-linear programming, network modeling, inventory theory, queue theory, tree decision, game theory, dynamic programming and markov processes

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Table of contents (10 chapters)

Front matter, linear programming.

  • Raúl Poler, Josefa Mula, Manuel Díaz-Madroñero
  • Integer Programming

Non-Linear Programming

Network modelling.

  • Inventory Theory

Queueing Theory

Decision theory, games theory.

  • Dynamic Programming
  • Markov Processes

Back Matter

  • Game Theory
  • Linear and Non-Linear Programming
  • Network Modeling
  • Queue Theory

From the reviews:

Josefa Mula

Manuel Díaz-Madroñero

Book Title : Operations Research Problems

Book Subtitle : Statements and Solutions

Authors : Raúl Poler, Josefa Mula, Manuel Díaz-Madroñero

DOI : https://doi.org/10.1007/978-1-4471-5577-5

Publisher : Springer London

eBook Packages : Engineering , Engineering (R0)

Copyright Information : Springer-Verlag London Ltd., part of Springer Nature 2014

Hardcover ISBN : 978-1-4471-5576-8 Published: 22 November 2013

Softcover ISBN : 978-1-4471-7190-4 Published: 23 August 2016

eBook ISBN : 978-1-4471-5577-5 Published: 08 November 2013

Edition Number : 1

Number of Pages : XV, 424

Number of Illustrations : 32 b/w illustrations, 55 illustrations in colour

Topics : Industrial and Production Engineering , Operations Research/Decision Theory , Game Theory, Economics, Social and Behav. Sciences

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Procedure, Example Solved Problem | Operations Research - Solution of assignment problems (Hungarian Method) | 12th Business Maths and Statistics : Chapter 10 : Operations Research

Chapter: 12th business maths and statistics : chapter 10 : operations research.

Solution of assignment problems (Hungarian Method)

First check whether the number of rows is equal to the numbers of columns, if it is so, the assignment problem is said to be balanced.

Step :1 Choose the least element in each row and subtract it from all the elements of that row.

Step :2 Choose the least element in each column and subtract it from all the elements of that column. Step 2 has to be performed from the table obtained in step 1.

Step:3 Check whether there is atleast one zero in each row and each column and make an assignment as follows.

assignment problem on operation research

Step :4 If each row and each column contains exactly one assignment, then the solution is optimal.

Example 10.7

Solve the following assignment problem. Cell values represent cost of assigning job A, B, C and D to the machines I, II, III and IV.

assignment problem on operation research

Here the number of rows and columns are equal.

∴ The given assignment problem is balanced. Now let us find the solution.

Step 1: Select a smallest element in each row and subtract this from all the elements in its row.

assignment problem on operation research

Look for atleast one zero in each row and each column.Otherwise go to step 2.

Step 2: Select the smallest element in each column and subtract this from all the elements in its column.

assignment problem on operation research

Since each row and column contains atleast one zero, assignments can be made.

Step 3 (Assignment):

assignment problem on operation research

Thus all the four assignments have been made. The optimal assignment schedule and total cost is

assignment problem on operation research

The optimal assignment (minimum) cost

Example 10.8

Consider the problem of assigning five jobs to five persons. The assignment costs are given as follows. Determine the optimum assignment schedule.

assignment problem on operation research

∴ The given assignment problem is balanced.

Now let us find the solution.

The cost matrix of the given assignment problem is

assignment problem on operation research

Column 3 contains no zero. Go to Step 2.

assignment problem on operation research

Thus all the five assignments have been made. The Optimal assignment schedule and total cost is

assignment problem on operation research

The optimal assignment (minimum) cost = ` 9

Example 10.9

Solve the following assignment problem.

assignment problem on operation research

Since the number of columns is less than the number of rows, given assignment problem is unbalanced one. To balance it , introduce a dummy column with all the entries zero. The revised assignment problem is

assignment problem on operation research

Here only 3 tasks can be assigned to 3 men.

Step 1: is not necessary, since each row contains zero entry. Go to Step 2.

assignment problem on operation research

Step 3 (Assignment) :

assignment problem on operation research

Since each row and each columncontains exactly one assignment,all the three men have been assigned a task. But task S is not assigned to any Man. The optimal assignment schedule and total cost is

assignment problem on operation research

The optimal assignment (minimum) cost = ₹ 35

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COMMENTS

  1. Assignment Problem: Meaning, Methods …

    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, …

  2. Assignment Problem: Meaning, Methods and Variations

    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 ...

  3. Operations Research with R

    Assignment Problem. The assignment problem is a special case of linear programming problem; it is one of the fundamental combinational optimization problems in the branch of optimization or operations research in mathematics. Its goal consists in assigning m resources (usually workers) to n tasks (usually jobs) one a one to one basis while ...

  4. Chapter 5: Assignment Problem

    5.1 INTRODUCTION. The assignment problem is one of the special type of transportation problem for which more efficient (less-time consuming) solution method has been devised by KUHN (1956) and FLOOD (1956). The justification of the steps leading to the solution is based on theorems proved by Hungarian mathematicians KONEIG (1950) and EGERVARY ...

  5. PDF ASSIGNMENT PROBLEM

    ASSIGNMENT PROBLEM Consider an assignment problem of assigning n jobs to n machines (one job to one machine). Let c ij be the unit cost of assigning ith machine to the jth job and,ith machine to jth job. Let x ij = 1 , if jth job is assigned to ith machine. x ij = 0 , if jth job is not assigned to ith machine. K.BHARATHI,SCSVMV. ASSIGNMENT ...

  6. ASSIGNMENT PROBLEM (OPERATIONS RESEARCH) USING PYTHON

    The Assignment Problem is a special type of Linear Programming Problem based on the following assumptions: However, solving this task for increasing number of jobs and/or resources calls for…

  7. How to Solve the Assignment Problem: A Complete Guide

    Step 1: Set up the cost matrix. The first step in solving the assignment problem is to set up the cost matrix, which represents the cost of assigning a task to an agent. The matrix should be square and have the same number of rows and columns as the number of tasks and agents, respectively.

  8. Operations Research

    The Assignment Problem is a classic optimization challenge in operations research, where the objective is to assign a set of tasks to a set of resources in a way that minimizes the total cost or ...

  9. Assignment problem

    The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment.

  10. Assignment Problem

    Title: "Cracking the Balanced Assignment Problem in Operations Research!"🔍 Uncover the secrets of the Balanced Assignment Problem in this quick guide to Ope...

  11. Operations Research/Transportation and Assignment Problem

    The Transportation and Assignment problems deal with assigning sources and jobs to destinations and machines. We will discuss the transportation problem first. Suppose a company has m factories where it manufactures its product and n outlets from where the product is sold. Transporting the product from a factory to an outlet costs some money ...

  12. Operations Research Problems: Statements and Solutions

    The objective of this book is to provide a valuable compendium of problems as a reference for undergraduate and graduate students, faculty, researchers and practitioners of operations research and management science. These problems can serve as a basis for the development or study of assignments and exams. Also, they can be useful as a guide ...

  13. Models

    Matrix model of the assignment problem. The network model is in Fig. 13. It is very similar to the transportation model except the external flows are all +1 or -1. The only relevant parameter for the assignment model is arc cost (not shown in the figure for clarity) ; all other parameters should be set to default values.

  14. Assignment problems: A golden anniversary survey

    Assignment problems involve optimally matching the elements of two or more sets, where the dimension of the problem refers to the number of sets of elements to be matched. ... of the assignment problem is discussed in almost every textbook for an introductory course in either management science/operations research or production and operations ...

  15. Solution of assignment problems (Hungarian Method)

    Solve the following assignment problem. Solution: Since the number of columns is less than the number of rows, given assignment problem is unbalanced one. To balance it , introduce a dummy column with all the entries zero. The revised assignment problem is. Here only 3 tasks can be assigned to 3 men.

  16. Operations Research (1): Models and Applications

    There are 6 modules in this course. Operations Research (OR) is a field in which people use mathematical and engineering methods to study optimization problems in Business and Management, Economics, Computer Science, Civil Engineering, Industrial Engineering, etc. This course introduces frameworks and ideas about various types of optimization ...

  17. Assignment problems: A golden anniversary survey

    Abstract. Having reached the 50th (golden) anniversary of the publication of Kuhn's seminal article on the solution of the classic assignment problem, it seems useful to take a look at the variety of models to which it has given birth. This paper is a limited survey of what appear to be the most useful of the variations of the assignment ...

  18. PDF Unit 4: ASSIGNMENT PROBLEM

    Problem 5 A typical assignment problem, presented in the classic manner, is shown in Fig. Here there are five machines to be assigned to five jobs. The numbers in the matrix indicate the cost of doing each job with each machine. Jobs with costs of M are disallowed assignments. The problem is to find the minimum cost matching of machines to jobs.

  19. PDF Introduction to Operations Research

    as the transportation problem, the assignment problem, the shortest path problem, the maximum flow problem, and the minimum cost flow problem. Very efficient algorithms exist which are many times more efficient than linear programming in the utilization of computer time and space resources. Introduction to Operations Research - p.6

  20. Revisiting the Evolution and Application of Assignment Problem ...

    'operation research', the ass ignment problem is very challenging and interesting that can represents many real-life problems. The optimal assignment problem is a classical combinatorial optimization problem. It entails optimally matching the elements of two or more sets, where the dimension of the problem refers to the number of sets of

  21. A Target-Assignment Problem

    Abstract. This paper is concerned with a target assignment model of a probabilistic and nonlinear nature, but nevertheless one which is closely related to the "personnel-assignment" problem. It is shown here that, despite the apparent nonlinearities, it is possible to devise a linear programming formulation that will ordinarily provide a ...

  22. Assignment problem

    ASSIGNMENT PROBLEM. 1 Prof Qureshi. An Automobiles dealer wishes to put five repairmen to five different jobs. The repairmen; have somewhat different kinds of skills and they exhibit different levels of efficiency from one job o another.

  23. Assignment Problems

    The assignment problem is classified into "Balanced assignment problem" and "Un-Balanced assignment problem". If the number of rows (jobs) is equal to the number of columns (Workers), then the problem is termed as a "Balanced Assignment Problem"; otherwise, an "Un-Balanced Assignment Problem".

  24. Transportation AND Assignment Problems

    Prof. Nagaraj operations research (ue18ie301) transportation and assignment problems contents sl. no. name of the topic formulation of transportation model, Skip to document. ... differences between Transportation problem and Assignment problem, Hungarian method-procedure and problems, Unbalanced Assignment problems. 10.

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    5 min read • June, 01 2023. Evidence-based practice in nursing involves providing holistic, quality care based on the most up-to-date research and knowledge rather than traditional methods, advice from colleagues, or personal beliefs. Nurses can expand their knowledge and improve their clinical practice experience by collecting, processing ...