## 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:

Task 1 | Task 2 | Task 3 | |
---|---|---|---|

Emp 1 | 5 | 7 | 6 |

Emp 2 | 6 | 4 | 5 |

Emp 3 | 8 | 5 | 3 |

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:

Task 1 | Task 2 | Task 3 | |
---|---|---|---|

Emp 1 | 0 | 2 | 1 |

Emp 2 | 2 | 0 | 1 |

Emp 3 | 5 | 2 | 0 |

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

Task 1 | Task 2 | Task 3 | |
---|---|---|---|

Emp 1 | 0 | 2 | 1 |

Emp 2 | 2 | 0 | 1 |

Emp 3 | 5 | 2 | 0 |

0 | 0 | 0 |

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
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- 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

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8 Integer Programming

- Some Integer Programming Formulation Techniques
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- Unimodularity
- Cutting Plane Method
- Branch and Bound Method
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9 Dynamic Programming

- Dynamic Programming Methodology: An Example
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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
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- 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

## 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:

## OPERATIONS RESEARCH

Lesson 9. solution of assignment problem.

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## Operations Research by P. Mariappan

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## 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 (1953), hence the method is named Hungarian.

## 5.2 GENERAL MODEL OF THE ASSIGNMENT PROBLEM

Consider n jobs and n persons. Assume that each job can be done only by one person and the time a person required for completing the i th job (i = 1,2,...n) by the j th person (j = 1,2,...n) is denoted by a real number C ij . On the whole this model deals with the assignment of n candidates to n jobs ...

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- Operations Research Problems

Statements and Solutions

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- 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
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## Applications and Mathematical Modeling in Operations Research

## From Operations Research to Dynamic Data Mining and Beyond

## Introduction to Stochastic Optimisation and Game Theory

Dynamic programming.

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## Integer Programming

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## Markov Processes

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## Table of contents (10 chapters)

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## Bibliographic Information

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

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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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Determine the optimum assignment schedule. Solution: 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. The cost matrix of the given assignment problem is. Column 3 contains no ...

Find step-by-step solutions and answers to Operations Research - 9781337798211, as well as thousands of textbooks so you can move forward with confidence. ... Assignment Problems. Section 7-6: Transshipment Problems. Page 408: Review Problems. Exercise 1. Exercise 2. Exercise 3. Exercise 4. Exercise 5. Exercise 6. Exercise 7. Exercise 8 ...

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.

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 ...

5.2 ASSIGNMENT PROBLEM AND ITS SOLUTION An assignment problem may be considered as a special type of transportation problem in which the number of sources and destinations are equal. The capacity of each source as well as the requirement of each destination is taken as 1. In the case of an assignment problem, the given matrix must necessarily

Step 2: next to examine the matrix for the best solutions to the assignment problem and first we try with value one (1) the cells having 1 are c12, c43, c45, using this cells we try for one sequence Let us try with assigning with c15 to c12 and c42 to c45. ∞. [1] 3. 6.

The assignment problem can be solved by the following four methods: a) Complete enumeration method. b) Simplex Method. c) Transportation method. d) Hungarian method. 9.2.1 Complete enumeration method. In this method, a list of all possible assignments among the given resources and activities is prepared.

Remember, consistent practice is key to mastering the art of operations research! Solution with PuLP: ... 11. Assignment Problem: A company needs to assign n workers to n tasks. Each worker has ...

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 ...

Operations Research I. Assignment problems. Ing. Lenka Skanderová, Ph.D. Description and definition. • Assignees are assigned to do the tasks • Asignee: • Employee, machine, vehicle, etc. The assignment problem satisfies these assumptions: 1. The number of assignees equals to the number of tasks 2. Each assignee can be assigned to do ...

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 ...

Solutions. 56:171 Operations Research Final Exam '98 page 11 of 14 For $20,000, Sue can hire a consultant who will predict the outcome of the trial, i.e., either he predicts a loss of the suit (event PL), or he predicts a win (event PW). The consultant predicts the correct outcome 80% of the time. 2.

The assignment problem represents a special case of linear programming problem used for allocating resources (mostly workforce) in an optimal way; it is a highly useful tool for operation and project managers for optimizing costs. The lpSolve R package allows us to solve LP assignment problems with just very few lines of code.

Operation Research Calculators ( examples ) 1.Assignment problem 1.1 Assignment problem (Using Hungarian method-2) 1.2 Assignment problem (Using Hungarian method-1) 2.1 Travelling salesman problem using hungarian method 2.2 Travelling salesman problem using branch and bound (penalty) method 2.3 Travelling salesman problem using branch and bound ...

OPERATIONS RESEARCH Chapter 2 Transportation and Assignment Problems Prof. Bibhas C. Giri Professor of Mathematics Jadavpur University ... The Hungarian method is an eﬃcient method for ﬁnding the optimal solution of an assignment problem. The method works on the principle of reducing the given cost matrix to a matrix of opportunity costs ...

Tags : Problem Questions with Answer, Solution , 12th Business Maths and Statistics : Chapter 10 : Operations Research Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

Problem Questions with Answer, Solution | Operations Research - Exercise 10.1: Transportation Problem | 12th Business Maths and Statistics : Chapter 10 : Operations Research. ... Solution of assignment problems (Hungarian Method) - Procedure, Example Solved Problem | Operations Research.

chapter 01: graphical solutions to linear operations research problems. chapter 02: linear programming(lp) - introduction. chapter 03: linear programming - the simplex method. chapter 04: linear programming-advanced methods. chapter 05: the transportation and assignment problems

OPERATIONS RESEARCH (UE18IE301) UNIT-3 TRANSPORTATION AND ASSIGNMENT PROBLEMS CONTENTS Sl. No. Name of the Topic 1. Formulation of Transportation model, Definition of Basic feasible solution

The transportation problem is a classic problem in operations research that involves finding the optimal way to move goods from one place to another. ... and transportation costs must be considered when choosing a solution method. 2.5. The assignment problem This problem is a combinatorial optimization problem that deals with allocating a set ...

Unit 8: Assignment Problem - Unbalanced. When an assignment problem has more than one solution, then it is Notes (a) Multiple Optimal solution (b) The problem is unbalanced (c) Maximization problem (d) Balanced problem. 8 Unbalanced Assignment Problem. If the given matrix is not a square matrix, the assignment problem is called an unbalanced ...

Ecole Polytechnique Problems and exercises in Operations Research Leo Liberti1 Last update: November 29, 2006 1 Some exercises have been proposed other authors, as detailed in the text. All the solutions, however, are the author, who takes full responsibility for their accuracy (or lack thereof).

Abstract. This note is concerned with two target assignment models. An optimum assignment is one which maximizes the expected value of targets destroyed. The first model, which admits an explicit solution, associates values only with the number of targets destroyed. An algorithm which enjoys a computational nicety is established when the values ...