define assignment problem in rmt

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 is a special type of linear programming problem which deals with the allocation of the various resources to the various activities on one to one basis. It does it in such a way that the cost or time involved in the process is minimum and profit or sale is maximum. Though there problems can be solved by simplex method or by transportation method but assignment model gives a simpler approach for these problems.

In a factory, a supervisor may have six workers available and six jobs to fire. He will have to take decision regarding which job should be given to which worker. Problem forms one to one basis. This is an assignment problem.

1. Assignment Model :

Suppose there are n facilitates and n jobs it is clear that in this case, there will be n assignments. Each facility or say worker can perform each job, one at a time. But there should be certain procedure by which assignment should be made so that the profit is maximized or the cost or time is minimized.

In the table, Co ij is defined as the cost when j th job is assigned to i th worker. It maybe noted here that this is a special case of transportation problem when the number of rows is equal to number of columns.

Mathematical Formulation:

Any basic feasible solution of an Assignment problem consists (2n – 1) variables of which the (n – 1) variables are zero, n is number of jobs or number of facilities. Due to this high degeneracy, if we solve the problem by usual transportation method, it will be a complex and time consuming work. Thus a separate technique is derived for it. Before going to the absolute method it is very important to formulate the problem.

Suppose x jj is a variable which is defined as

1 if the i th job is assigned to j th machine or facility

0 if the i th job is not assigned to j th machine or facility.

Now as the problem forms one to one basis or one job is to be assigned to one facility or machine.

The total assignment cost will be given by

The above definition can be developed into mathematical model as follows:

Determine x ij > 0 (i, j = 1,2, 3…n) in order to

Subjected to constraints

and x ij is either zero or one.

Method to solve Problem (Hungarian Technique):

Consider the objective function of minimization type. Following steps are involved in solving this Assignment problem,

1. Locate the smallest cost element in each row of the given cost table starting with the first row. Now, this smallest element is subtracted form each element of that row. So, we will be getting at least one zero in each row of this new table.

2. Having constructed the table (as by step-1) take the columns of the table. Starting from first column locate the smallest cost element in each column. Now subtract this smallest element from each element of that column. Having performed the step 1 and step 2, we will be getting at least one zero in each column in the reduced cost table.

3. Now, the assignments are made for the reduced table in following manner.

(i) Rows are examined successively, until the row with exactly single (one) zero is found. Assignment is made to this single zero by putting square □ around it and in the corresponding column, all other zeros are crossed out (x) because these will not be used to make any other assignment in this column. Step is conducted for each row.

(ii) Step 3 (i) in now performed on the columns as follow:- columns are examined successively till a column with exactly one zero is found. Now , assignment is made to this single zero by putting the square around it and at the same time, all other zeros in the corresponding rows are crossed out (x) step is conducted for each column.

(iii) Step 3, (i) and 3 (ii) are repeated till all the zeros are either marked or crossed out. Now, if the number of marked zeros or the assignments made are equal to number of rows or columns, optimum solution has been achieved. There will be exactly single assignment in each or columns without any assignment. In this case, we will go to step 4.

4. At this stage, draw the minimum number of lines (horizontal and vertical) necessary to cover all zeros in the matrix obtained in step 3, Following procedure is adopted:

(iii) Now tick mark all the rows that are not already marked and that have assignment in the marked columns.

(iv) All the steps i.e. (4(i), 4(ii), 4(iii) are repeated until no more rows or columns can be marked.

(v) Now draw straight lines which pass through all the un marked rows and marked columns. It can also be noticed that in an n x n matrix, always less than ‘n’ lines will cover all the zeros if there is no solution among them.

5. In step 4, if the number of lines drawn are equal to n or the number of rows, then it is the optimum solution if not, then go to step 6.

6. Select the smallest element among all the uncovered elements. Now, this element is subtracted from all the uncovered elements and added to the element which lies at the intersection of two lines. This is the matrix for fresh assignments.

7. Repeat the procedure from step (3) until the number of assignments becomes equal to the number of rows or number of columns.

Related Articles:

• Two Phase Methods of Problem Solving in Linear Programming: First and Second Phase
• Linear Programming: Applications, Definitions and Problems

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A Comparative Analysis of Assignment Problem

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• First Online: 06 June 2023
• Cite this conference paper

• 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  

Part of the book series: EAI/Springer Innovations in Communication and Computing ((EAISICC))

Included in the following conference series:

• International Conference on Big Data Innovation for Sustainable Cognitive Computing

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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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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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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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Operations Research Simplified Back Next

Goal programming Linear programming Simplex Method Transportation Problem

Unbalanced Assignment Problem: Definition, Formulation, and Solution Methods

Table of Contents

Are you familiar with the assignment problem in Operations Research (OR)? This problem deals with assigning tasks to workers in a way that minimizes the total cost or time needed to complete the tasks. But what if the number of tasks and workers is not equal? In this case, we face the Unbalanced Assignment Problem (UAP). This blog will help you understand what the UAP is, how to formulate it, and how to solve it.

What is the Unbalanced Assignment Problem?

The Unbalanced Assignment Problem is an extension of the Assignment Problem in OR, where the number of tasks and workers is not equal. In the UAP, some tasks may remain unassigned, while some workers may not be assigned any task. The objective is still to minimize the total cost or time required to complete the assigned tasks, but the UAP has additional constraints that make it more complex than the traditional assignment problem.

Formulation of the Unbalanced Assignment Problem

To formulate the UAP, we start with a matrix that represents the cost or time required to assign each task to each worker. If the matrix is square, we can use the Hungarian algorithm to solve the problem. But when the matrix is not square, we need to add dummy tasks or workers to balance the matrix. These dummy tasks or workers have zero costs and are used to make the matrix square.

Once we have a square matrix, we can apply the Hungarian algorithm to find the optimal assignment. However, we need to be careful in interpreting the results, as the assignment may include dummy tasks or workers that are not actually assigned to anything.

Solutions for the Unbalanced Assignment Problem

Besides the Hungarian algorithm, there are other methods to solve the UAP, such as the transportation algorithm and the auction algorithm. The transportation algorithm is based on transforming the UAP into a transportation problem, which can be solved with the transportation simplex method. The auction algorithm is an iterative method that simulates a bidding process between the tasks and workers to find the optimal assignment.

In summary, the Unbalanced Assignment Problem is a variant of the traditional Assignment Problem in OR that deals with assigning tasks to workers when the number of tasks and workers is not equal. To solve the UAP, we need to balance the matrix by adding dummy tasks or workers and then apply algorithms such as the Hungarian algorithm, the transportation algorithm, or the auction algorithm. Understanding the UAP can help businesses and organizations optimize their resource allocation and improve their operational efficiency.

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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

Assignment Model | Linear Programming Problem (LPP) | Introduction

What is assignment model.

→ Assignment model is a special application of Linear Programming Problem (LPP) , in which the main objective is to assign the work or task to a group of individuals such that;

i) There is only one assignment.

ii) All the assignments should be done in such a way that the overall cost is minimized (or profit is maximized, incase of maximization).

→ In assignment problem, the cost of performing each task by each individual is known. → It is desired to find out the best assignments, such that overall cost of assigning the work is minimized.

For example:

Suppose there are 'n' tasks, which are required to be performed using 'n' resources.

The cost of performing each task by each resource is also known (shown in cells of matrix)

• In the above asignment problem, we have to provide assignments such that there is one to one assignments and the overall cost is minimized.

How Assignment Problem is related to LPP? OR Write mathematical formulation of Assignment Model.

→ Assignment Model is a special application of Linear Programming (LP).

→ The mathematical formulation for Assignment Model is given below:

→ Let, C i j \text {C}_{ij} C ij ​ denotes the cost of resources 'i' to the task 'j' ; such that

→ Now assignment problems are of the Minimization type. So, our objective function is to minimize the overall cost.

→ Subjected to constraint;

(i) For all j t h j^{th} j t h task, only one i t h i^{th} i t h resource is possible:

(ii) For all i t h i^{th} i t h resource, there is only one j t h j^{th} j t h task possible;

(iii) x i j x_{ij} x ij ​ is '0' or '1'.

Types of Assignment Problem:

(i) balanced assignment problem.

• It consist of a suqare matrix (n x n).
• Number of rows = Number of columns

(ii) Unbalanced Assignment Problem

• It consist of a Non-square matrix.
• Number of rows ≠ \not=  = Number of columns

Methods to solve Assignment Model:

(i) integer programming method:.

In assignment problem, either allocation is done to the cell or not.

So this can be formulated using 0 or 1 integer.

While using this method, we will have n x n decision varables, and n+n equalities.

So even for 4 x 4 matrix problem, it will have 16 decision variables and 8 equalities.

So this method becomes very lengthy and difficult to solve.

(ii) Transportation Methods:

As assignment problem is a special case of transportation problem, it can also be solved using transportation methods.

In transportation methods ( NWCM , LCM & VAM), the total number of allocations will be (m+n-1) and the solution is known as non-degenerated. (For eg: for 3 x 3 matrix, there will be 3+3-1 = 5 allocations)

But, here in assignment problems, the matrix is a square matrix (m=n).

So total allocations should be (n+n-1), i.e. for 3 x 3 matrix, it should be (3+3-1) = 5

But, we know that in 3 x 3 assignment problem, maximum possible possible assignments are 3 only.

So, if are we will use transportation methods, then the solution will be degenerated as it does not satisfy the condition of (m+n-1) allocations.

So, the method becomes lengthy and time consuming.

(iii) Enumeration Method:

It is a simple trail and error type method.

Consider a 3 x 3 assignment problem. Here the assignments are done randomly and the total cost is found out.

For 3 x 3 matrix, the total possible trails are 3! So total 3! = 3 x 2 x 1 = 6 trails are possible.

The assignments which gives minimum cost is selected as optimal solution.

But, such trail and error becomes very difficult and lengthy.

If there are more number of rows and columns, ( For eg: For 6 x 6 matrix, there will be 6! trails. So 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 trails possible) then such methods can't be applied for solving assignments problems.

(iv) Hungarian Method:

It was developed by two mathematicians of Hungary. So, it is known as Hungarian Method.

It is also know as Reduced matrix method or Flood's technique.

There are two main conditions for applying Hungarian Method:

(1) Square Matrix (n x n). (2) Problem should be of minimization type.

Suggested Notes:

Modified Distribution Method (MODI) | Transportation Problem | Transportation Model

Stepping Stone | Transportation Problem | Transportation Model

Vogel’s Approximation Method (VAM) | Method to Solve Transportation Problem | Transportation Model

Transportation Model - Introduction

North West Corner Method | Method to Solve Transportation Problem | Transportation Model

Least Cost Method | Method to Solve Transportation Problem | Transportation Model

Tie in selecting row and column (Vogel's Approximation Method - VAM) | Numerical | Solving Transportation Problem | Transportation Model

Crashing Special Case - Multiple (Parallel) Critical Paths

Crashing Special Case - Indirect cost less than Crash Cost

Basics of Program Evaluation and Review Technique (PERT)

Numerical on PERT (Program Evaluation and Review Technique)

Network Analysis - Dealing with Network Construction Basics

Construct a project network with predecessor relationship | Operation Research | Numerical

Graphical Method | Methods to solve LPP | Linear Programming

Basics of Linear Programming

Linear Programming Problem (LPP) Formulation with Numericals

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How to formulate as an assignment problem?

How will I formulate this problem as an assignment problem?

I really don't know.

By definition of assignment problem, all the variables $x_{ij}$ are binary $(0$ or $1)$ and all the supply and demand are equal to $1.$

Will the 'cost' be represented by 'times in seconds' ?

Please someone help me how to solve this, how to translate this table into assignment problem?

I know how to solve the assignment problem, with the Hungarian method, but I did not see any examples like 8.3-4 in class..

• operations-research

• $\begingroup$ What are the decisions you are making? What can you assign to what? $\endgroup$ –  ConMan Commented Nov 29, 2018 at 23:39
• $\begingroup$ @ConMan To assign different swimmers to different strokes. $\endgroup$ –  user441848 Commented Nov 29, 2018 at 23:46
• $\begingroup$ So therefore your assignments are "Swimmer X is swimming stroke Y"? And if your objective is "minimise the sum of times", then the costs must therefore be times. $\endgroup$ –  ConMan Commented Nov 29, 2018 at 23:49
• $\begingroup$ Why the downvote.. 😔 $\endgroup$ –  user441848 Commented Dec 2, 2018 at 4:22

2 Answers 2

We can see that there are more swimmers than the number of swimming strokes.

We can introduce another dummy swimming stroke that cost time $0$ for everyone. Hence now we have a square table and we can perform Hungarian method to solve the problem.

\begin{array}{|c|c|c|c|c|}\hline 37.7 & 32.9 & 33.8 & 37 & 35.4 \\ \hline 43.4 & 33.1 & 42.2 & 34.7 & 41.8\\ \hline 33.3 & 28.5 & 38.9 & 30.4 & 33.6 \\ \hline 29.2 & 26.4 & 29.6 & 28.5 & 31.1 \\ \hline 0 & 0 & 0 & 0 & 0 \\ \hline \end{array}

Subtracting the minimum of each row from each row and then subtract the minimum of each column from each column and then cover up the zero (using color blue).

\begin{array}{|c|c|c|c|c|}\hline 4.8 & \color{blue}0 & 0.9 & 4.1 & 2.5 \\ \hline 10.3 & \color{blue}0 & 9.1 & 1.6 & 8.7\\ \hline 4.8 & \color{blue}0 & 10.4 & 1.9 & 5.1 \\ \hline 2.8 & \color{blue}0 & 3.2 & 2.1 & 4.7 \\ \hline \color{blue}0 & \color{blue}0 & \color{blue}0 & \color{blue}0 & \color{blue}0 \\ \hline \end{array}

We then shift the zero and repeat the procedure:

\begin{array}{|c|c|c|c|c|}\hline 3.9 & \color{blue}0 & \color{blue}0 & 3.2 & 1.6 \\ \hline 9.4 & \color{blue}0 & \color{blue}{8.2} & 0.7 & 7.8\\ \hline 3.9 & \color{blue}0 & \color{blue}{9.5} & 1 & 4.2\\ \hline 1.9 & \color{blue}0 & \color{blue}{2.3} & 1.2 & 3.8\\ \hline \color{blue}0 & \color{blue}{0.9} & \color{blue}0 & \color{blue}0 & \color{blue}0 \\ \hline \end{array}

\begin{array}{|c|c|c|c|c|}\hline 3.2 & \color{blue}0 & \color{blue}0 & \color{blue}{2.5} & 0.9 \\ \hline 8.7 & \color{blue}0 & \color{blue}{8.2} & \color{blue}{0} & 7.1\\ \hline 3.2 & \color{blue}0 & \color{blue}{9.5} & \color{blue}{0.3} & 3.5\\ \hline 1.2 & \color{blue}0 & \color{blue}{2.3} & \color{blue}{0.5} & 3.1\\ \hline \color{blue}0 & \color{blue}{1.6} & \color{blue}{0.7} & \color{blue}0 & \color{blue}0 \\ \hline \end{array}

\begin{array}{|c|c|c|c|c|}\hline \color{blue}{2.3} & \color{blue}0 & \color{blue}0 & \color{blue}{2.5} & \color{blue}0 \\ \hline \color{blue}{7.8} & \color{blue}0 & \color{blue}{8.2} & \color{blue}{0} & \color{blue}{6.2}\\ \hline 2.3 & \color{blue}0 & 9.5 & 0.3 & 2.6\\ \hline 0.3 & \color{blue}0 & 2.3 & 0.5 & 2.2\\ \hline \color{blue}0 & \color{blue}{2.5} & \color{blue}{1.6} & \color{blue}{0.9} & \color{blue}0 \\ \hline \end{array}

I indicate the selected position in red color.

\begin{array}{|c|c|c|c|c|}\hline 2.3 & 0.3 & \color{red}0 & 2.5 & 0 \\ \hline 7.8 & 0.3 & 8.2 & \color{red}{0} & 6.2\\ \hline 2 & \color{red}0 & 9.2 & 0 & 2.3\\ \hline \color{red}0 & 0 & 2 & 0.2 & 1.9\\ \hline 0 & 2.8 & 1.6 & 0.9 & \color{red}0 \\ \hline \end{array}

Hence Carl should do freestyle $(29.2)$ , Chris should do butterfly $(28.5)$ , David should do backstroke $(33.8)$ , Tony should do breaststroke $(34.7)$ . The total time by the team is $126.2$ .

Let's check our final answer by $R$ software:

• $\begingroup$ In the first part, should be more swimmers than swimming strokes? $\endgroup$ –  user441848 Commented Dec 7, 2018 at 18:32
• 1 $\begingroup$ oops, i made a mistake, yup, you are right. thanks. $\endgroup$ –  Siong Thye Goh Commented Dec 7, 2018 at 18:34

As you have said, the assignment is $x_{ij} = 1$ if swimmer $i$ is assigned to stroke $j$ , with $\forall i, j\sum_{j'} x_{ij'} = \sum_{i'} x_{i'j} = 1$ (since we want exactly one swimmer per stroke).

We are trying to get the minimum sum of times, meaning that our objective function is $\sum_{ij} x_{ij} t_{ij}$ where $t_{ij}$ is the time it takes for swimmer $i$ to swim stroke $j$ , hence the time is indeed our cost.

• $\begingroup$ In this solution, won't there be a square table ? $\endgroup$ –  user441848 Commented Nov 30, 2018 at 0:02
• $\begingroup$ Why did you write the ' here $\forall i, j\sum_{j'} x_{ij'} = \sum_{i'} x_{i'j} = 1$ ?, notice it's not defined $j'$ nor $i'$ $\endgroup$ –  user441848 Commented Nov 30, 2018 at 0:08
• 1 $\begingroup$ That's because they're summation variables. I mean that for any fixed $i$, the sum across the $j$s is 1 (i.e. each swimmer swims exactly one stroke). Since I wanted to squash both the sum over $i$ and the sum over $j$ into one statement, I couldn't use $i$ as the summation variable. $\endgroup$ –  ConMan Commented Nov 30, 2018 at 3:11
• $\begingroup$ And yes, there will be a square table of decisions. $\endgroup$ –  ConMan Commented Nov 30, 2018 at 3:11
• $\begingroup$ ok I understood the summation part. $\endgroup$ –  user441848 Commented Nov 30, 2018 at 3:55

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assignment problem

(classic problem)

Definition: The problem of finding a maximum (or minimum) weight matching in a weighted , bipartite graph .

Also known as marriage problem.

See also Munkres' assignment algorithm .

Note: From Algorithms and Theory of Computation Handbook, page 7-21, Copyright © 1999 by CRC Press LLC. Appearing in the Dictionary of Computer Science, Engineering and Technology, Copyright © 2000 CRC Press LLC.

Author: CRC-A

If you have suggestions, corrections, or comments, please get in touch with Paul Black .

Entry modified 16 May 2005. HTML page formatted Wed Mar 13 12:42:45 2019.

Cite this as: Algorithms and Theory of Computation Handbook, CRC Press LLC, 1999, "assignment problem", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed. 16 May 2005. (accessed TODAY) Available from: https://www.nist.gov/dads/HTML/assignment.html

Operations Research - Definition and formulation of Assignment Problem | 12th Business Maths and Statistics : Chapter 10 : Operations Research

Chapter: 12th business maths and statistics : chapter 10 : operations research, definition and formulation of assignment problem.

Definition and formulation

Consider the problem of assigning n jobs to n machines (one job to one machine). Let C ij be the cost of assigning i th job to the j th machine and x ij represents the assignment of i th job to the j th machine.

 x ij is missing in any cell means that no assignment is made between the pair of job and machine.( i.e ) x ij = 0.

x ij presents in any cell means that an assignment is made their.In such cases x ij = 1

The assignment model can be written in LPP as follows

Subject to the constrains

The optimum assignment schedule remains unaltered if we add or subtract a constant from all the elements of the row or column of the assignment cost matrix.

If for an assignment problem all C ij > 0 then an assignment schedule (x ij ) which satisfies ∑ C ij x ij   = 0 must be optimal.

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