Pareto optimal solutions of a cost–time trade-off bulk transportation problem

Abstract

A cost–time trade-off bulk transportation problem with the objectives to minimize the total cost and duration of bulk transportation without according priorities to them is considered. The entire requirement of each destination is to be met from one source only; however a source can supply to any number of destinations subject to the availability of the commodity at it. Two new algorithms are provided to obtain the set of Pareto optimal solutions of this problem. This work extends and generalizes the work related to single-objective and prioritized two-objective bulk transportation problems done in the past while providing flexibility in decision making.

Introduction

The normal transportation problem wherein the requirement of each destination can be met from one or more than one source with the single objective to minimize the total cost of transportation has long been studied and is well known. In the recent past, this very transportation problem with a different single objective to minimize the duration of transportation has been studied by many researchers – Hammer, 1969, Hammer, 1971, Garfinkel and Rao, 1971, Szwarc, 1971, Bhatia et al., 1977, Ramakrishnan, 1977, Sharma and Swarup, 1977, Seshan and Tikekar, 1980, Prakash, 1982. Prakash (1981) has solved the transportation problem with two objectives after having accorded first and second priorities to the minimization of the total cost and duration of transportation respectively. Purushotam et al. (1984) have dealt with this problem with the reversed order of priorities. Bhatia et al. (1976) have developed a method to find the set of Pareto optimal solutions of the normal transportation problem with the objectives to minimize the total cost and cumulative time of transportation. Later on, Glickman and Berger, 1977, Prakash et al., 1988 have dealt with the just aforementioned two-objective transportation problem after replacing the objective of the cumulative time of transportation by that of the duration of transportation.

There is another version of the normal transportation problem called the bulk transportation problem which differs from the normal transportation problem in that it stipulates that the requirement of each destination has to be met from one source only; however a source can supply to any number of destinations subject to the availability of the commodity at it. Maio and Roveda (1971) were the first to formulate a bulk transportation problem with the single objective to minimize the total cost of bulk transportation and devise a solution procedure based on zero–one implicit enumeration. Srinivasan and Thompson (1973) presented an algorithm consisting of two phases to solve this problem. In the first phase, the problem is solved ignoring the constraint which stipulates that the requirement of each destination has to be met from one source only. In the second phase, a branch and bound technique is used to obtain a row unique solution. Murthy (1976) proposed a method based on the principle of lexicographic minimum to solve this problem. Prakash and Ram (1995) have considered a bulk transportation problem with the minimization of the total cost and duration of bulk transportation as primary and secondary objectives respectively.

Sometimes it happens that the decision maker is not able to assign priorities to his/her objectives. In such a situation, help can be provided to the decision maker by presenting him/her with a set of Pareto optimal solutions. He/she can pick up that solution out of this set which suits him/her most with regard to his/her objectives and liking. In the present paper, an attempt is made to deal with this type of situation in the context of bulk transportation. For this purpose, Prakash and Ram’s problem is considered with the alteration that the two objectives are now not assigned priorities in contrast to the earlier problem wherein they were assigned priorities. Two algorithms are developed for obtaining the set of Pareto optimal solutions of this altered problem. The newly developed algorithms are quite different from the previous ones employed to solve the single-objective bulk transportation problem. Both the new algorithms modify and extend the previous algorithm employed to solve the cost–time trade-off bulk transportation problem with prioritized objectives, for obtaining the set of Pareto optimal solutions of the cost–time trade-off bulk transportation problem without according priorities to the two objectives. As the set of Pareto optimal solutions of the altered problem contains the optimal solution of Prakash and Ram’s problem, the altered problem extends and generalizes their work while providing flexibility in decision making.

A solution of a multi-objective optimization problem is Pareto optimal if no solution of the multi-objective problem exists, which is superior to it with respect to at least one objective function but is not inferior to it with respect to any objective function. The terms efficient solution and non-dominated solution are also used in the literature for Pareto optimal solution. Specifically, for the cost–time trade-off bulk transportation problem with the total cost C and duration T of bulk transportation as the two minimizing objective functions, a solution $\overline{y}$ is Pareto optimal if there exists no solution $\overline{x}$ of the problem satisfying the conditions (i) $C(\overline{y})⩾C(\overline{x})$ and (ii) $T(\overline{y})⩾T(\overline{x})$ with inequality sign strictly holding in at least one of the conditions out of (i) and (ii). Here $C(\overline{x})$ and $C(\overline{y})$ refer to the total costs and $T(\overline{x})$ and $T(\overline{y})$ refer to the durations of bulk transportation for the solutions $\overline{x}$ and $\overline{y}$ respectively. Here $\overline{x}=(x_{\mathit{ij}}:i=1\text{,}\dots \text{,}m\text{;}j=1\text{,}\dots \text{,}n)$. The variables x_ij’s and other notations are explained hereinafter. A detailed discussion about Pareto optimal solution can be found in the works of Ignizio, 1982, Steuer, 1986.