assignment model in operation research pdf

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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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.

Related papers

This study discusses the current scenario of Operations Research in the field of Logistics. Five sectors are considered in the study to form a brief understanding of how they use Operations Research techniques and why these techniques are used. Operation research technique help in reducing cost and improve decision making.

A transportation problem basically deals with the problem, which aims to find the best way to fulfil the demand of n demand points using the capacities of m supply points. Here we studied a new method for solving transportation problems with mixed constraints and described the algorithm to find an optimal more-for-less (MFL) solution. The optimal MFL solution procedure is illustrated with numerical example and also computer programming. Though maximum transportation problems in real life have mixed constraints, these problems are not be solved by using general method. The proposed method builds on the initial solution of the transportation problem which is very simple, easy to understand and apply.

Optimization, 1995

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

Journal of Applied Mathematics and Decision Sciences, 1998

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."

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.

The problem of finding the initial basic feasible solution of the Transportation Problem has long been studied and is well known to the research scholars of the field. So far three general methods for solving transportation methods are available in literature, namely Northwest, Least Cost and Vogel?s Approximation methods. These methods give only initial feasible solution. However here we discuss a new alternative method which gives Initial feasible solution as well as optimal or nearly optimal solution. In this paper we provide an alternate method to find IBFS (Initial Basic Feasible Solution) and compared the alternate method and the existing IBFS methods using a Graphical User Interface. It is also to be noticed that this method requires lesser number of iterations to reach optimality as compared to other known methods for solving the transportation problem and the solution obtained is as good as obtained by Vogel?s Approximation Method (VAM).

Omega-international Journal of Management Science, 1999

A new method called zero point method is proposed for finding an optimal solution for transportation problems with mixed constraints in a single stage. Using the zero point method, we propose a new method for finding an optimal more-for-less solution for transportation problems with mixed constraints. The optimal more-for-less solution procedure is illustrated with numerical examples. Mathematics Subject Classifications: 90C08 , 90C90

Handbooks in Operations Research and Management Science, 2005

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Int.J. Contemp.Math.Sciences, 2013

Journal of Supply Chain Management System, 2017

International Journal of Advanced Research in Computer Engineering & Technology (IJARCET), 2018

International Journal of Advance and Innovative Research Volume 6, Issue 1 (XXVII): January - March, 2019, 2019

Omega-international Journal of Management Science, 2000

International Journal on Advanced …, 2012

Omega-international Journal of Management Science, 2003

Operations Research Center …, 1988

Journal of Heuristics, 2005

Discrete Applied Mathematics, 1996

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• Operations Research

A Model-Based Approach

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• Carl-Louis Sandblom 1

Faculty of Business Administration, University of New Brunswick, Fredericton, Canada

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Dept. Industrial Engineering, Dalhousie University, Halifax, Canada

• Covers the standard operations research techniques
• Presents an approach to operations research that is heavily based on modeling and makes extensive use of sensitivity analyses
• Emphasis is on getting insight into problems, rather than computing solutions

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Introduction to Optimization

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Table of contents (13 chapters)

Front matter, introduction to operations research.

• H. A. Eiselt, C. -L. Sandblom

Linear Programming

Multiobjective programming, integer programming, network models, location models, project networks, machine scheduling, decision analysis, inventory models, stochastic processes and markov chains, waiting line models, back matter.

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H. A. Eiselt

Carl-Louis Sandblom

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Book Title : Operations Research

Book Subtitle : A Model-Based Approach

Authors : H. A. Eiselt, Carl-Louis Sandblom

DOI : https://doi.org/10.1007/978-3-642-10326-1

Publisher : Springer Berlin, Heidelberg

eBook Packages : Business and Economics , Business and Management (R0)

Copyright Information : Springer-Verlag Berlin Heidelberg 2010

eBook ISBN : 978-3-642-10326-1 Published: 17 May 2010

Edition Number : 1

Number of Pages : XII, 448

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Topics : Operations Research/Decision Theory , Optimization , Engineering, general , Operations Research, Management Science , Engineering Economics, Organization, Logistics, Marketing

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Home » Management Science » Transportation and Assignment Models in Operations Research

Transportation and Assignment Models in Operations Research

Transportation and assignment models are special purpose algorithms of the linear programming. The simplex method of Linear Programming Problems(LPP) proves to be inefficient is certain situations like determining optimum assignment of jobs to persons, supply of materials from several supply points to several destinations and the like. More effective solution models have been evolved and these are called assignment and transportation models.

The transportation model is concerned with selecting the routes between supply and demand points in order to minimize costs of transportation subject to constraints of supply at any supply point and demand at any demand point. Assume a company has 4 manufacturing plants with different capacity levels, and 5 regional distribution centres. 4 x 5 = 20 routes are possible. Given the transportation costs per load of each of 20 routes between the manufacturing (supply) plants and the regional distribution (demand) centres, and supply and demand constraints, how many loads can be transported through different routes so as to minimize transportation costs? The answer to this question is obtained easily through the transportation algorithm.

Similarly, how are we to assign different jobs to different persons/machines, given cost of job completion for each pair of job machine/person? The objective is minimizing total cost. This is best solved through assignment algorithm.

Uses of Transportation and Assignment Models in Decision Making

The broad purposes of Transportation and Assignment models in LPP are just mentioned above. Now we have just enumerated the different situations where we can make use of these models.

Transportation model is used in the following:

• To decide the transportation of new materials from various centres to different manufacturing plants. In the case of multi-plant company this is highly useful.
• To decide the transportation of finished goods from different manufacturing plants to the different distribution centres. For a multi-plant-multi-market company this is useful.
• To decide the transportation of finished goods from different manufacturing plants to the different distribution centres. For a multi-plant-multi-market company this is useful. These two are the uses of transportation model. The objective is minimizing transportation cost.

Assignment model is used in the following:

• To decide the assignment of jobs to persons/machines, the assignment model is used.
• To decide the route a traveling executive has to adopt (dealing with the order inn which he/she has to visit different places).
• To decide the order in which different activities performed on one and the same facility be taken up.

In the case of transportation model, the supply quantity may be less or more than the demand. Similarly the assignment model, the number of jobs may be equal to, less or more than the number of machines/persons available. In all these cases the simplex method of LPP can be adopted, but transportation and assignment models are more effective, less time consuming and easier than the LPP.

Related posts:

• Operations Research approach of problem solving
• Introduction to Transportation Problem
• Procedure for finding an optimum solution for transportation problem
• Initial Basic Feasible Solution of a Transportation Problem
• Introduction to Decision Models
• Transportation Cost Elements
• Modes of Transportation in Logistics
• Factors Affecting Transportation in Logistics

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• NOC:Introduction to Operations Research (Video) 
• Co-ordinated by : IIT Madras
• Available from : 2015-01-09
• Intro Video
• Product Mix problem and Notations
• Manpower and Production planning formulations
• Media selection problem and Bicycle problem
• Caterer problem
• Maximum flow and bin packing problems
• Graphical method (maximization)
• Graphical method (minimization)
• Algebraic method (maximization)
• Algebraic method (minimization)
• Comparing graphical and algebraic methods
• Algebraic form of simplex algorithm
• Tabular form of simplex (maximization)
• Tabular form (minimization)
• Unboundedness
• Infeasibility
• Motivation to the dual
• Writing the dual for a general LP
• Writing dual for a general LP (continued)
• Duality theorems
• Complimentary slackness theorem
• Dual solution using complimentary slackness
• Dual solution from simplex table; economic interpretation of dual
• Economic Interpretation of the dual; Dual Simplex algorithm
• Solving LPs with mixed type of constraints
• Matrix method for LP problems
• Introducing the transportation problem
• North West corner Rule and minimum cost method
• Penalty cost method
• Stepping stone method and Modified Distribution method
• MODI method; Dual of the transportation problem and the optimality of the MODI method
• Introducing the Assignment problem
• Solving the Assignment problem
• Hungarian algorithm; Alternate optimum
• Unequal number of rows and columns; Dual of the assignment problem
• Optimality of the Hungarian algorithm
• Setting up the problem and solving simple LP problems
• Unboundedness and infeasibility
• Solving other formulations
• Solving a transportation problem
• Solving an assignment problem
• Live Session 24-03-2021
• Watch on YouTube
• Assignments
• Transcripts