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:
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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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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:
The whole model is:
subject to,
The problem can now be solved using the simplex method. A convenient procedure is discussed in the next section.
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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
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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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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 minimizing assignment costs.
Step 1: Develop the Cost Table from the given Problem: ADVERTISEMENTS: If the no of rows are not equal to the no of columns and vice versa, a dummy row or dummy column must be added. The assignment cost for dummy cells are always zero. Step 2: Find the Opportunity Cost Table:
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 method
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).
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.
Operations research. KEYWORDS online assignment problem, machine-learned advice, competitive analysis 1 INTRODUCTION The assignment problem has diverse applications in various fields. It is used by industries to assign jobs to workers, by transportation companies in assigning passengers to vehicles, and among others.
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…
4.8 (405 reviews) Beginner level Recommended experience Approx. 11 hours to complete 3 weeks at 3 hours a week Flexible schedule Learn at your own pace About Modules Recommendations Testimonials Reviews What you'll learn Formulate different types of mathematical models to tackle optimization problems with business applications.
Operations Research/Transportation and Assignment Problem < Operations Research The Transportation and Assignment problems deal with assigning sources and jobs to destinations and machines. We will discuss the transportation problem first.
This paper presents a new algorithm for solving the assignment problem. The algorithm is based on a scheme of relaxing the given problem into a series of simple network flow (transportation) problems for each of which an optimal solution can be easily obtained. The algorithm is thus seen to be able to take advantage of the nice properties in ...
Operations Research Problems Statements and Solutions Home Textbook Authors: Raúl Poler, Josefa Mula, Manuel Díaz-Madroñero Provides a valuable compendium of problems as a reference for undergraduate and graduate students, faculty, researchers and practitioners of operations research and management science
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.
Operations Research (OR) is a discipline that helps to make better decisions in complex scenarios by the application of a set of advanced analytical methods. It couples theories, results and theorems of mathematics, statistics and probability with its own theories and algorithms for problem solving. ... Unit 7: Assignment Problem. Unit 8 ...
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, Electrical Engineering, etc. The series of courses consists of three parts, we focus on deterministic optimization techniques, which is a major ...
The last phase, interpretation, encompasses making a decision and developing implementation plans. The paragraphs below explain the seven elements of the operations research problem solving process in greater detail. The activities that take place in each element are illustrated through some of the tools or methods commonly used.
The assignment problem is a standard topic discussed in operations research textbooks [8] and [10]. It is an important subject, put forward immediately after the transportation problem, is the assignment problem. This is particularly important in the theory of decision making. The assignment problem is one of the earliest
What is Operations Research? Optimization Problems and Applications Personal Examples What is Operations Reseach? Operations Research (OR) started just before World War II in Britain with the establishment of teams of scientists to study the strategic and tactical problems involved in military operations.
The assignment problem is a special type of transportation problem where the objec-tive is to minimize the cost of allocating a number of jobs to a number of persons or fa-cilities so that one person or facility is assigned to only one job.
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 ...
Computers & Operations Research. Volume 149, January 2023, 106047. ... Before the formal introduction of our problems, i.e., online Generalized Assignment Problem with Historical Information (online GAP-HI) and online d-GAP with Historical Information (online d-GAP-HI), we give the formulations of GAP and its online version at first. ...
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.
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 ...
Assignment Problem in Operational Research | PDF | Mathematical Optimization | Linear Programming. ASSIGNMENT PROBLEM IN OPERATIONAL RESEARCH - Read online for free.