## 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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## Solving the Rectangular assignment problem and applications

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- Published: 05 June 2010
- Volume 181 , pages 443–462, ( 2010 )

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- J. Bijsterbosch 1 &
- A. Volgenant 1

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The rectangular assignment problem is a generalization of the linear assignment problem (LAP): one wants to assign a number of persons to a smaller number of jobs, minimizing the total corresponding costs. Applications are, e.g., in the fields of object recognition and scheduling. Further, we show how it can be used to solve variants of the LAP, such as the k -cardinality LAP and the LAP with outsourcing by transformation. We introduce a transformation to solve the machine replacement LAP.

We describe improvements of a LAP-algorithm for the rectangular problem, in general and slightly adapted for these variants, based on the structure of the corresponding cost matrices. For these problem instances, we compared the best special codes from literature to our codes, which are more general and easier to implement. The improvements lead to more efficient and robust codes, making them competitive on all problem instances and often showing much shorter computing times.

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Operations Research Group, Faculty of Economics and Econometrics, University of Amsterdam, Roetersstraat 11, 1018 WB, Amsterdam, The Netherlands

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## About this article

Bijsterbosch, J., Volgenant, A. Solving the Rectangular assignment problem and applications. Ann Oper Res 181 , 443–462 (2010). https://doi.org/10.1007/s10479-010-0757-3

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Published : 05 June 2010

Issue Date : December 2010

DOI : https://doi.org/10.1007/s10479-010-0757-3

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## THE LITERATURE REVIEW FOR ASSIGNMENT AND TRANSPORTATION PROBLEMS.

Operations Research is a logical learning through interdisciplinary collaboration to determine the best usage of restricted assets. In this paper, the importance of Operations research is discussed and the literature of assignment and transportation problem is discussed in detail.

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Assignment problems deal with the question how to assign n objects to m other objects in an injective fashion in the best possible way. An assignment problem is completely specified by its two components the assignments, which represent the underlying combinatorial structure, and the objective function to be optimized, which models \\\\\\\"the best possible way\\\\\\\". The assignment problem refers to another special class of linear programming problem where the objective is to assign a number of resources to an equal number of activities on a one to one basis so as to minimize total costs of performing the tasks at hand or maximize total profit of allocation. In this paper we introduce a new technique to solve assignment problems namely, Divide Row Minima and Subtract Column Minima .For the validity and comparison study we consider an example and solved by using our technique and the existing Hungarian (HA) and matrix ones assignment method(MOA) and compare optimum result shown graphically.

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## COMMENTS

Problem 4. Job shop needs to assign 4 jobs to 4 workers. The cost of performing a job is a function of the skills of the workers. Table summarizes the cost of the assignments. Worker1 cannot do job3, and worker 3 cannot do job 4. Determine the optimal assignment using the Hungarian method. Job. Worker.

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 ...

e minimisation problem.3. The assignment problem wherein the number of rows is not equal to the number of columns is said t. be an unbalanced problem. Such a problem is handled by introducing dummy row(s) if the number of rows is less than the number of columns and dummy column(s) if the number of columns is le.

Tables 2, 3, 4, and 5 present the steps required to determine the appropriate job assignment to the machine. Step 1 By taking the minimum element and subtracting it from all the other elements in each row, the new table will be: Table 2 represents the matrix after completing the 1st step. Table 1 Initial table of a.

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

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 ...

problems such as the linear network flow and shortest path problems to take the form of an assignment problem. The assignment problem finds many applications; the most obvious being that of matching such as the matching of operators and machines or delivery vehicles and deliveries. There are however numerous other interesting applications.

minimizes the total cost or maximizes the total profit. The original version of the assignment problem is discussed in almost every textbook for an introductory course in either management science/operations research or production and operations management. Assignment problem is well structured linear programming problem,

book is on the applications of operations research in practice. Typically, a topic is introduced by means of a description of its applications, a model is formulated and ... 2.2.7 Transportation and Assignment Problems 43 Exercises 50 60 2.3.1 The Graphical Solution Method 60 2.3.2 Special Cases of Linear Programming Problems 70

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 ...

Assignment model is a powerful operations research techniques that can be used to solve assignment or allocation problem. This study applies the assignment model to the course allocation problem ...

The rectangular assignment problem is a generalization of the linear assignment problem (LAP): one wants to assign a number of persons to a smaller number of jobs, minimizing the total corresponding costs. Applications are, e.g., in the fields of object recognition and scheduling. Further, we show how it can be used to solve variants of the LAP, such as the k-cardinality LAP and the LAP with ...

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 ...

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

models the source is connected to one or more of destination. The most common. method to solve assignment models is the Hungarian metho d. In this paper. introduced another method to solve ...

Abstract. 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 ...

Assignment Problem - Notes - Free download as Word Doc (.doc / .docx), PDF File (.pdf), Text File (.txt) or read online for free. The document discusses the assignment problem and the Hungarian method for solving it. The assignment problem aims to allocate jobs to workers in a way that minimizes costs. The Hungarian method is an efficient algorithm for solving assignment problems by using a ...

1. Introduction. The Weapon-Target Assignment (WTA) problem is a fundamental problem arising in defense-related applications of operations research. The problem consists of optimally assigning weapons to the enemy-targets so that the total expected survival value of the targets after all the engagements is minimized.

speaking, an O.R. project comprises three steps: (1) building a model, (2) solving it, and. (3) implementing the results. The emphasis of this chapter is on the first and third steps. The second step typically involves specific methodologies or techniques, which could be.

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 ...

Operations Research is a logical learning through interdisciplinary collaboration to determine the best usage of restricted assets. In this paper, the importance of Operations research is discussed and the literature of assignment and transportation ... Refined Simplex Algorithm for the Classical Transportation Problem with Application to ...

This paper presents a review pertaining to assignment problem within the education domain, besides looking into the applications of the present research trend, developments, and publications ...

Limitations of Operations Research The Operations Research has number of applications, similarly it has certain limitations. These limitations are mostly related to model building and money & time factors problems that are involved in the application. Some of these are as given below: i) Distance between the O.R. specialist and Manager

The lack of a unique user equilibrium (UE) route flow in traffic assignment has posed a significant challenge to many transportation applications. The maximum-entropy principle, which advocates for the consistent selection of the most likely solution, is often used to address the challenge.