Transportation Problem with Multi-choice Cost and Demand and Stochastic Supply
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Transportation Problem with Multi-choice Cost and Demand and Stochastic Supply Sankar Kumar Roy1
Received: 10 July 2015 / Revised: 9 January 2016 / Accepted: 14 March 2016 © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2016
Abstract This paper analyzes the multi-choice stochastic transportation problem where the cost coefficients of the objective function and the demand parameters of the constraints follow multi-choice parameters. Assume that the supply parameters of the constraints in a transportation problem (TP) follow logistic distribution. The main objective of this paper is to select an appropriate choice from the multi-choices for the cost coefficients of the objective function and the demand of the constraints in the TP by introducing Lagrange’s interpolating polynomial in such a way that the total cost is minimized and satisfies the required demand. Using stochastic programming, the stochastic supply constraints of the TP are transformed into deterministic constraints. Finally, a non-linear deterministic model is formulated. Using Lingo software, the optimal solution of the proposed problem is derived. To illustrate the methodology, a real-life problem on the TP is considered. Keywords Transportation problem · Multi-choice programming · Lagrange’s interpolating polynomial · Stochastic programming Mathematics Subject Classification
90B06 · 65K05 · 90C15
1 Introduction Transportation problem (TP) is a special type linear programming problem (LPP) in which the objective is to transport various quantities of a single homogeneous com-
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Sankar Kumar Roy [email protected] Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, West Bengal 721102, India
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modity, which are initially stored at various origins to different destinations, in such a way that the total transportation cost is minimal. The transportation problem mainly contains three parameters namely cost coefficient, supply, and demand parameters. Due to some unpredictable factors, these parameters are not always fixed exactly. This imprecision follows from the lack of exact information. Stochastic programming deals with situations where some or all of the parameters of the optimization problem are described by random variables rather than by deterministic quantity. In today’s highly competitive market, assume that the supply parameters ai (i = 1, 2, · · · , m) (sources) follow random variables. In this paper, the supply parameters follow logistic distribution. The logistic distribution is symmetric about the location parameter α, and it can be used as a substitute for normal distribution. It is also used to analyze data related to stocks. Logistic distribution has many applications such as in nuclear medicine, in soil-water retention, to study citrus rust damage on oranges, etc. Many real-life decision making problems of practical importance are designed with multi-choice parameters.
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