Continuous Optimization
Solving fuzzy transportation problems based on extension principle

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Abstract

Transportation models play an important role in logistics and supply chain management for reducing cost and improving service. This paper develops a procedure to derive the fuzzy objective value of the fuzzy transportation problem, in that the cost coefficients and the supply and demand quantities are fuzzy numbers. The idea is based on the extension principle. A pair of mathematical programs is formulated to calculate the lower and upper bounds of the fuzzy total transportation cost at possibility level α. From different values of α, the membership function of the objective value is constructed. Two different types of the fuzzy transportation problem are discussed: one with inequality constraints and the other with equality constraints. It is found that the membership function of the objective value of the equality problem is contained in that of the inequality problem. Since the objective value is expressed by a membership function rather than by a crisp value, more information is provided for making decisions.

Introduction

In today’s highly competitive market the pressure on organizations to find better ways to create and deliver value to customers becomes stronger. How and when to send the products to the customers in the quantities they want in a cost-effective manner become more challenging. Transportation models provide a powerful framework to meet this challenge. They ensure the efficient movement and timely availability of raw materials and finished goods.

Transportation problem is a linear programming problem stemmed from a network structure consisting of a finite number of nodes and arcs attached to them. Efficient algorithms have been developed for solving the transportation problem when the cost coefficients and the supply and demand quantities are known exactly. However, there are cases that these parameters may not be presented in a precise manner. For example, the unit shipping cost may vary in a time frame. The supplies and demands may be uncertain due to some uncontrollable factors. To deal quantitatively with imprecise information in making decisions, Bellman and Zadeh [1] and Zadeh [16] introduce the notion of fuzziness.

Since the transportation problem is essentially a linear program, one straightforward idea is to apply the existing fuzzy linear programming techniques [2], [3], [6], [7], [9], [10], [12], [13] to the fuzzy transportation problem. Unfortunately, most of the existing techniques [2], [3], [6], [9], [12], [13] only provide crisp solutions. The method of Julien [7] and Parra et al. [10] is able to find the possibility distribution of the objective value provided all the inequality constraints are of “⩾” type or “⩽” type. However, due to the structure of the transportation problem, in some cases their method requires the refinement of the problem parameters to be able to derive the bounds of the objective value. There are also studies discussing the fuzzy transportation problem. Chanas et al. [5] investigate the transportation problem with fuzzy supplies and demands and solve them via the parametric programming technique in terms of the Bellman–Zadeh criterion. Their method is to derive the solution which simultaneously satisfies the constraints and the goal to a maximal degree. Chanas and Kuchta [4] discuss the type of transportation problems with fuzzy cost coefficients and transform the problem to a bicriterial transportation problem with crisp objective function. Their method is able to determine the efficient solutions of the transformed problem; nevertheless, only crisp solutions are provided. Verma et al. [14] apply the fuzzy programming technique with hyperbolic and exponential membership functions to solve a multi-objective transportation problem, the solution derived is a compromise solution. Similar to the method of Chanas and Kuchta [4], only crisp solutions are provided.

Obviously, when the cost coefficients or the supply and demand quantities are fuzzy numbers, the total transportation cost will be fuzzy as well. In this paper, we develop a solution procedure that is able to calculate the fuzzy objective value of the fuzzy transportation problem, where at least one of the parameters are fuzzy numbers. The idea is to apply Zadeh’s extension principle [15], [16], [17]. A pair of two-level mathematical programs is formulated to calculate the lower and upper bounds of the α-level cut of the objective value. The membership function of the fuzzy objective value is derived numerically by enumerating different values of α.

In the sections that follow, firstly, the fuzzy transportation problem is briefly stated. Then a pair of mathematical programs for calculating the α-cuts of the fuzzy total transportation cost is formulated based on the extension principle. We use an example to illustrate the difference between inequality-constraint and equality-constraint problems. Finally, some conclusions are drawn from the discussions.

Section snippets

Fuzzy transportation problem

Consider a transportation problem with m supply nodes and n demand nodes, in that sj>0 units are supplied by supply node i and dj>0 units are required by demand node j. Associated with each link (i,j) from supply node i to demand node j, there is a unit shipping cost cij for transportation. The problem is to determine a feasible way of shipping the available amount to satisfy the demand that minimizes the total transportation cost.

Let xij denote the number of units to be transported from Supply

The solution procedure

Denote the α-cuts of Cij, Si and Dj as:(Cij)α=[(Cij)Lα,(Cij)Uα]=[mincij{cij∈S(Cij)|μC̃ij(cij)⩾α},maxcij{cij∈S(Cij)|μC̃ij(cij)⩾α}](Si)α=[(Si)Lα,(Si)Uα]=[minsi{si∈S(Si)|μS̃i(si)⩾α},maxsi{si∈S(Si)|μS̃i(si)⩾α}](Dj)α=[(Dj)Lα,(Dj)Uα]=[mindj{dj∈S(Dj)|μD̃j(dj)⩾α},maxdj{dj∈S(Dj)|μD̃j(dj)⩾α}]These intervals indicate where the unit shipping cost, supply, and demand lie at possibility level α. Suppose we are interested in deriving the membership function of the total transportation cost Z. The major

Equality constraints

In the preceding section, the transportation model being considered has inequality constraints. The total supply must be greater than or equal to the total demand to assure feasibility. In this section, we discuss the transportation model with equality constraints:Z=mini=1mj=1ncijxijs.t.j=1nxij=si,i=1,…,m,i=1mxij=dj,j=1,…,n,xij⩾0,∀i,j.This model is feasible if and only if ∑i=1msi=∑j=1ndj.

When the shipping costs, supplies, and demands are not known exactly, we have the following fuzzy

Conclusion

Transportation models have wide applications in logistics and supply chain for reducing the cost. Some previous studies have devised solution procedures for fuzzy transportation problems. The objective values derived from those studies are crisp values, rather than fuzzy numbers. This paper develops a method to find the membership function of the fuzzy total transportation cost when the unit shipping costs, the supply quantities, and the demand quantities are fuzzy numbers. The idea is based on

Acknowledgements

This research is supported by the National Science Council of Republic of China under Contract NSC89-2416-H-006-020.

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