Sorting Out Fuzzy Transportation Problems via ECCT and Standard Deviation

Sorting Out Fuzzy Transportation Problems via ECCT and Standard Deviation

Krishna Prabha Sikkannan (PSNA College of Engineering, India) and Vimala Shanmugavel (Mother Teresa Women's University, India)
DOI: 10.4018/IJORIS.20210401.oa1
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Abstract

A well-organized arithmetical procedure entitled standard deviation is employed to find the optimum solution in this paper. This technique has been divided into two parts. The first methodology deals with constructing the entire contingency cost table, and the second deals with optimum allocation. In this work, the method of magnitude is used for converting fuzzy numbers into crisp numbers as this method is better than the existing methods. This technique gives a better optimal solution than other methods. A numerical example for the new method is explained, and the authors compared their method with existing methods such as north west corner method, least cost method, and Vogel's approximation method.
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1. Introduction

The study of perfect transportation and distribution of measures in mathematics and economics is named as transportation theory and the presumption was dignified by Gaspard Monge (1781). The notable collapse on the charge and the pricing of raw materials and goods is visibly due to Transportation cost. An most favorable allotment sketch for a single commodity is established by the transportation problem. Our aim is to reduce entire carrying outlay for the possessions transferring from resource to target. By balancing the total supply and demand we can divide the transportation problem as balanced or unbalanced. Dual simplex, Big M method and Interior Approach method can also be used to unravel the transportation problem. Techniques like North West corner rule, row minima, column minima, matrix minima, or the vogel’s approximation method are also applied to acquire the solution for transportation problems. Vogel’s method provides fairly accurate elucidation whereas MODI and Stepping Stone (SS) techniques are measured as a customary procedure for achieving the best possible elucidation.

The cost of transportation is managed by dealer and manufacturer. Excluding the conventional methods like North West corner method, row minima method, least cost method, column minima method, Vogel’s approximation method and modified distribution method etc, many researchers has endowed with new techniques to find a better initial basic feasible solution for the Transportation Problem.

The entire constraints of the transportation problems may not be identified specifically owing to unmanageable issues in real world applications. The limitations of the transportation problem are not forever accurately documented and unwavering. This ambiguity may pursue from the need of literal data, faltering in decision etc. All the parameters of the transportation problems may not be known precisely due to uncontrollable factors in real world applications. The parameters of the transportation problem are not always precisely identified and sure. This imprecision may go after the lack of exact information, uncertainty in judgment etc. To overcome this situation Zadeh (1965) introduced the notion of fuzziness that was reinforced by Bellman and Zadeh (1965). Chanas and Kuchta (1996) proposed the concept of optimal solution for the transportation problem with fuzzy coefficients which are expressed as fuzzy numbers. A fuzzy number is thus a special case of a convex, normalized fuzzy set of the real line. Calculations with fuzzy numbers allow the incorporation of uncertainty on parameters, properties, geometry, initial conditions, etc. Membership functions (MFs) are the building blocks of fuzzy set theory, i.e., fuzziness in a fuzzy set is determined by its MF. Accordingly, the shapes of MFs are important for a particular problem since they effect on a fuzzy inference system. They may have different shapes like triangular, trapezoidal, Gaussian, etc. Since triangular membership function are fewer complexes when splitting values (low, med and high MF) comparing other membership functions. Considering that parameters of transportation problem have uncertainties, this paper develops a generalized fuzzy transportation problem with fuzzy supply, demand and cost. For simplicity, these parameters are assumed to be triangular fuzzy numbers in this paper.

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