# Solving Solid Transportation Problems with Multi-Choice Cost and Stochastic Supply and Demand

Sankar Kumar Roy (Vidyasagar University, India) and Deshabrata Roy Mahapatra (Vidyasagar University, India)
DOI: 10.4018/978-1-4666-7272-7.ch023
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## Abstract

In this chapter, the authors propose a new approach to analyze the Solid Transportation Problem (STP). This new approach considers the multi-choice programming into the cost coefficients of objective function and stochastic programming, which is incorporated in three constraints, namely sources, destinations, and capacities constraints, followed by Cauchy's distribution for solid transportation problem. The multi-choice programming and stochastic programming are combined into a solid transportation problem, and this new problem is called Multi-Choice Stochastic Solid Transportation Problem (MCSSTP). The solution concepts behind the MCSSTP are based on a new transformation technique that will select an appropriate choice from a set of multi-choice, which optimize the objective function. The stochastic constraints of STP converts into deterministic constraints by stochastic programming approach. Finally, the authors construct a non-linear programming problem for MCSSTP, and by solving it, they derive an optimal solution of the specified problem. A realistic example on STP is considered to illustrate the methodology.
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## Introduction

The classical transportation problem (TP) can be described as a special case of linear programming problem. Its model is applied to determine an optimal solution of the TP of how many units of commodity to be shipped from each origin to various destinations, satisfying source availability and destination demand and minimizing the total cost of transportation. The amounts of available goods at the supply points and the amounts required at the demand points are the parameters of the TP, and these parameters are not always exactly known and stable. This imprecision may follow from the lack of exact information. The STP is an important extension of the traditional TP. The traditional TP is a well known optimization problem in operations research, in which two kinds of constraints are taken into consideration, i.e., source constraints and destination constraints. But in the real system, we always deal with other constraints besides of source constraints and destination constraints, such as product type constraints or transportation mode constraints. For such cases, the traditional TP turns into STP. The STP is a generalization of the well-known TP in which three items (they are source, destination and conveyance) are taken into account in the constraint set instead of two (source and destination). In many industrial problems, a homogeneous product is delivered from an origin to destination by means of different modes of transport called conveyances, such as trucks, cargo flights, goods trains, ships, etc. In reality, due to changes in market supply and demand, weather conditions, road conditions and other unpredictable factors, the STP is important for both theoretical and practical significance.

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. The random variables are defined as sources, destinations and conveyances which are depending on the nature and the type of problem. Decision making problems of stochastic optimization arise when certain coefficients of the optimization model are not fixed or known. In such cases, the quantities are random. In recent years, methods of multi-objective stochastic optimization have become increasingly important in the field of economics, industry, transportation, military purpose and technology. The Cauchy’s distribution is a good example of a continuous stable distribution and this distribution is a source of counter examples, having little connection with statistical practice. The Cauchy’s distribution is an old and very important problem in the statistical literature. Particularly, in the recent years it has gained more importance. The Cauchy’s distribution is empirical as many large data sets exhibit on heavy tails and skewness. The strong empirical evidence for these features combined with the generalized central limit theorem is used for many types of physical and economic systems.

In the recent past normal, log-normal, extreme value distribution and other random variables have been considered in the stochastic programming model. Cauchy’s distribution has also two parameters namely location parameter and scale parameter. It has some similarities with normal distribution. Assuming Cauchy’s distribution is a suitable one for stochastic programming model, we consider the source, the availability and the conveyance parameters of the transportation problem as Cauchy’s distribution. Here, we consider the probability density function of Cauchy’s distribution with location parameter and scale parameter as,

(1.1)

## Key Terms in this Chapter

Stochastic Programming: Stochastic programming is a framework for modeling optimization problems that involve uncertainty.

Solid Transportation Problem: The STP is an important extension of the traditional TP with constraints besides of source constraints and destination constraints, such as product type constraints or transportation mode constraints.

Mixed-Integer Programming: Mixed integer linear programming (MILP) involves problems in which only some of the variables are constrained to be integers, while other variables are allowed to be non-integers.

Transportation Problem: It is a model that is applied to determine how many units of commodity to be shipped from each origin to various destinations, satisfying source availability and destination demand and minimizing the total cost of transportation.

Cauchy’s Distribution: It is the distribution of a random variable that is the ratio of two independent standard normal variables.

Transformation Technique: Selects only one choice from a set of multi-choice for each coefficient of objective function and provides an optimal solution to the model.

Multi-Choice Programming: The Multi-choice programming (MCP) is a mathematical programming technique in which the decision maker collects multiple information exists for all parameters.

Binary Variable: Variable which takes only two values.

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