Fuzzy fractional stochastic transportation problem involving exponential distribution
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Fuzzy fractional stochastic transportation problem involving exponential distribution Prachi Agrawal1 · Talari Ganesh1 Accepted: 7 May 2020 © Operational Research Society of India 2020
Abstract This paper deals with the solution of the fuzzy fractional transportation problem in which the parameters of the transportation problem, supply, and demand, are stochastic in nature and considered as a fuzzy random variable that follows the exponential distribution with fuzzy mean and fuzzy variance. In a fuzzy fractional objective function, the costs are taken as a triangular fuzzy number. As the parameters are imprecise in nature, the obtained objective value should be a fuzzy number. To obtain the fuzzy objective value, one has to find out its lower and upper bounds, which represent the level of uncertainty. The mathematical form of bounds is expressed by applying dual formulation and variable substitution. Also, for converting the fuzzy constraints into deterministic, the chance-constrained and fuzzy programming was applied. The values of the bounds, which are calculated at different values of 𝜆 , the membership function of the objective value is approximated. A numerical example illustrates the considered methodology. Keywords Duality · Exponential distribution · Fractional transportation problem · Fuzzy programming · Stochastic programming
1 Introduction The transportation problem is the earliest and most important applications of linear programming problem [23]. It has wide applications in production planning, supply management, logistics systems, inventory control, etc. The lot of work has been done, by considering the parameters of standard transportation problem, cost, supply, and demand, were precisely known. But, in general, today’s market is highly * Prachi Agrawal [email protected] Talari Ganesh [email protected] 1
Department of Mathematics and Scientific Computing, National Institute of Technology Hamirpur, Hamirpur, Himachal Pradesh 177 005, India
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competitive, and the parameters may not be presented in a precise manner. The cost of the product may vary from time to time, or it may depend upon the production of the product. Also, supply and demand may be dubious in nature due to the unavailability of the information of the shipped product. Due to these factors, and to deal with imprecise information Zadeh [26] introduced the concept of fuzziness. When the ratio of two or more function being optimized, the problem is known as a fractional programming problem. It has been widely used in investment problems, game theory, production planning, etc. [23]. In 1956, linear fractional programming was first identified by Isbell and Marlow [13]. They developed an algorithm that generates a sequence of linear programs whose solution converges to the solution of the fractional program. The several methods for finding out the solution given by many researchers (Gilmore and Gomory [11], Martos and Whinston [17], Charnes and Cooper [6]). In the next few years, transport
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