## 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:

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

• Open access
• Published: 05 June 2010
• Volume 181 , pages 443–462, ( 2010 )

• 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.

## On the rectangular knapsack problem: approximation of a specific quadratic knapsack problem

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Bertsekas, D. P., Castañon, D. A., & Tsaknakis, H. (1993). Reverse auction and the solution of asymmetric assignment problems. SIAM Journal on Optimization , 3 , 268–299.

Burkard, R., Dell’Amico, M., & Martello, S. (2009). Assignment Problems . Philadelphia: Society for Industrial and Applied Mathematics.

Caseau, Y., & Laburthe, F. (2000). Solving weighted matching problems with constraints. Constraints, an International Journal , 5 , 141–160.

Dell’Amico, M., & Martello, S. (1997). The k -cardinality assignment problem. Discrete Applied Mathematics , 76 , 103–121.

Dell’Amico, M., & Toth, P. (2000). Algorithms and codes for dense assignment problems: the state of the art. Discrete Applied Mathematics , 100 , 17–48.

Goldberg, A. V., & Kennnedy, J. R. (1995). An efficient cost scaling algorithm for the assignment problem. Mathematical Programming , 71 , 153–177.

Hsieh, A. J., Fan, K. C., & Fan, T. I. (1995). Bipartite weighted matching for on-line handwritten Chinese character recognition. Pattern Recognition , 28 , 143–151.

Jonker, R., & Volgenant, A. (1987). A shortest augmenting path algorithm for dense and sparse linear assignment problems. Computing , 38 , 325–340.

Knuth, D. E. (1981). The Art of Computer Programming , Vol. 2: Seminumerical Algorithms . Reading: Addison-Wesley.

Machol, R. E., & Wien, M. (1976). A hard assignment problem. Operations Research , 24 , 190–192.

Mosheiov, G., & Yovel, U. (2006). Minimizing weighted earliness-tardiness and due-date cost with unit processing-time jobs. European Journal of Operational Research , 172 , 528–544.

Pinedo, M. (2002). Scheduling Theory, Algorithms, and Systems (2nd edn.). Upper Saddle River: Prentice Hall.

Wiel Vander, R. J., & Sahinidis, N. V. (1997). The assignment problem with external interactions. Networks , 30 , 171–185.

Volgenant, A. (1996). Linear and semi-assignment problems, a core oriented approach. Computers & Operations Research , 10 , 917–932.

Volgenant, A. (2004). Solving the k -cardinality assignment problem by transformation. European Journal of Operational Research , 157 , 322–331.

## Author information

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

J. Bijsterbosch & A. Volgenant

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## Corresponding author

Correspondence to A. Volgenant .

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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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

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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The family of network optimization problems includes the following prototype models: assignment, critical path, max flow, shortest path, and transportation. Although it is long known that these problems can be modeled as linear programs (LP), this is generally not done. Due to the relative inefficiency and complexity of the simplex methods (primal, dual, and other variations) for network models, these problems are usually treated by one of over 100 specialized algorithms. This leads to several difficulties. The solution algorithms are not unified and each algorithm uses a different strategy to exploit the special structure of a specific problem. Furthermore, small variations in the problem, such as the introduction of side constraints, destroys the special structure and requires modifying andjor restarting the algorithm. Also, these algorithms obtain solution efficiency at the expense of managerial insight, as the final solutions from these algorithms do not have sufficient information to perform postoptimality analysis.Another approach is to adapt the simplex to network optimization problems through network simplex. This provides unification of the various problems but maintains all the inefficiencies of simplex, as well as, most of the network inflexibility to handle changes such as side constraints. Even ordinary sensitivity analysis (OSA), long available in the tabular simplex, has been only recently transferred to network simplex.This paper provides a single unified algorithm for all five network models. The proposed solution algorithm is a variant of the self-dual simplex with a warm start. This algorithm makes available the full power of LP perturbation analysis (PA) extended to handle optimal degeneracy. In contrast to OSA, the proposed PA provides ranges for which the current optimal strategy remains optimal, for simultaneous dependent or independent changes from the nominal values in costs, arc capacities, or suppliesJdemands. The proposed solution algorithm also facilitates incorporation of network structural changes and side constraints. It has the advantage of being computationally practical, easy for managers to understand and use, and provides useful PA information in all cases. Computer implementation issues are discussed and illustrative numerical examples are provided in the Appendix For teaching purposes you may try: Refined Simplex Algorithm for the Classical Transportation Problem with Application to Parametric Analysis, Mathematical and Computer Modelling, 12(8), 1035-1044, 1989. http://home.ubalt.edu/ntsbarsh/KahnRefine.pdf

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In a fast changing global market, a manager is concerned with cost uncertainties of the cost matrix in transportation problems (TP) and assignment problems (AP).A time lag between the development and application of the model could cause cost parameters to assume different values when an optimal assignment is implemented. The manager might wish to determine the responsiveness of the current optimal solution to such uncertainties. A desirable tool is to construct a perturbation set (PS) of cost coeffcients which ensures the stability of an optimal solution under such uncertainties. The widely-used methods of solving the TP and AP are the stepping-stone (SS) method and the Hungarian method, respectively. Both methods fail to provide direct information to construct the needed PS. An added difficulty is that these problems might be highly pivotal degenerate. Therefore, the sensitivity results obtained via the available linear programming (LP) software might be misleading. We propose a unified pivotal solution algorithm for both TP and AP. The algorithm is free of pivotal degeneracy, which may cause cycling, and does not require any extra variables such as slack, surplus, or artificial variables used in dual and primal simplex. The algorithm permits higher-order assignment problems and side-constraints. Computational results comparing the proposed algorithm to the closely-related pivotal solution algorithm, the simplex, via the widely-used pack-age Lindo, are provided. The proposed algorithm has the advantage of being computationally practical, being easy to understand, and providing useful information for managers. The results empower the manager to assess and monitor various types of cost uncertainties encountered in real-life situations. Some illustrative numerical examples are also presented."

IJAR Indexing

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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1. PDF Unit 4: ASSIGNMENT PROBLEM

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.

2. 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 ...

3. PDF UNIT 5 ASSIGNMENT PROBLEMS

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.

4. A Comparative Analysis of Assignment Problem

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.

5. PDF Solving The Assignment Problems Directly Without Any Iterations

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

6. 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 ...

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

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.

8. PDF A Brief Review on Classic Assignment Problem and its 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,

9. PDF Operations Research

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

10. 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 ...

11. (PDF) Program for Solving Assignment Problems and Its Application in

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 ...

12. Solving the Rectangular assignment problem and applications

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 ...

13. 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 ...

14. 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

15. (PDF) A New Method to Solve Assignment Models

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 ...

16. Solving the Assignment Problem by Relaxation

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 ...

17. Assignment Problem

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 ...

18. PDF MIT Sloan School of Management

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.

19. PDF Principles and Applications of Operations Research

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.

20. 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 ...

21. (Pdf) 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 ... Refined Simplex Algorithm for the Classical Transportation Problem with Application to ...

22. (PDF) An Assignment Problem and Its Application in Education Domain: A

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 ...

23. PDF Introduction to Operations Research

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

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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.