Ijraset Journal For Research in Applied Science and Engineering Technology

Abstract

Introduction

I. INTRODUCTION

It is observed that many real life problems, the cost of transporting a commodity from one place to another cannot be determined exactly. This may be attributed to many reasons such as variations in Crude oil ,fuel prices, Labour charges, power utilization, optimality utilization of man power, expenditure of utilities   etc. The inaccurate costs can be conveniently modeled by fuzzy numbers. Thus transportation problems with fuzzy costs are of a great importance and which are reliable too. 

An assignment problem with fuzzy costs has been considered by Lin and Wen [2004]. They have modeled the problem into a mixed integer programming problem, using the fuzzy decision of Bellman and Zadeh [1970] and then converted it into a linear fractional programming problem. They have developed a suitable labeling algorithm for the solution of the resultant linear fractional programming problem.

A definition of an optimal solution of a transportation problem with fuzzy cost coefficients and an algorithm for determining this solution was given by Chanas and Kuchta [1996].

Also, there are situations where the commodities supplied by the origins contain certain number of units of impurities and each destination has it’s own limitations on the number of units of impurities it can receive. For example, some industries receiving coal may fix some limits on the quantity of sulphur in the coal supplied. A method of solution of a multi-objective time transportation problem with impurity restrictions was developed by Singh and Saxena [2003].

The present chapter is aimed at developing a solution methodology for a transportation problem involving both the above aspects i.e costs are fuzzy and also impurity restrictions are imposed. The demand and supply values are assumed to be crisp numbers

This work ( chapter)  is divided into five sections. Section III.2 formulates the problem under consideration and also gives the basic notations and assumptions. Section III.3 develops the necessary theory and produces a linear fractional programming problem, which has the same optimal solution as of the problem under consideration. Section III.4 applies the developed method on a numerical example. Some concluding remarks are drawn in section III.5                           

Conclusion

The developed procedure for obtaining an optimal solution of a transportation problem with fuzzy costs and involving impurity restrictions is computationally efficient. The optimal values of xij’s can be obtained by the method by Kantiswarup [1965]. A computer program can be developed with ease in any programming language for this purpose. The optimal values of dij’s and hence the unit costs can be obtained using expressions given in the method. The fuzzy transportation problem considered in this work may not possess an optimal solution, in some cases. This may be attributed to the presence of the impurity restrictions. In such cases, a feasible solution and therefore an optimal solution may be obtained by relaxing the impurity restrictions of the destinations to some extent.

References

[1] Lin and Wen ; A GA Based fuzzy controller with slideing mode. [2] Bellman, R.E., Zadeh, L.A.: Decision making in a fuzzy environment, Management Science17 (1970) B141-B164. [3] Chanas and kuchta, A new approach to fuzzy goal programming, Proceedings of the 2nd International Conference in Fuzzy Logic and Technology, Leicester, United Kingdom, September 5-7, 2001 [4] Singh & Saxena , Analysis of intelligence techniques on sensor less speed control of Doubly fed — Induction machine. [5] Fuzzy Set Theory Fuzzy Logic and Their Applications Paperback – January 1, 2013 by A.K. Bhargava . [6] Fuzzy Set Theory—and Its Applications, Zimmermann, Hans-Jürgen [7] Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic-Theory and Applications, Prentice-Hall Inc. (1995). [8] Operations research , S.D.Sharma &V.K. Kapoor, Khanna Publications.

Copyright

Copyright © 2023 Dr. K. Sharath Babu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.