Multi-Index Bi-Criterion Transportation Problem: A Fuzzy Approach

This paper represents a non linear bi-criterion generalized multi-index transportation problem (BGMTP) is considered. The generalized transportation problem (GTP) arises in many real-life applications. It has the form of a classical transportation problem, with the additional assumption that the quantities of goods change during the transportation process. Here the fuzzy constraints are used in the demand and in the budget. An efficient new solution procedure is developed keeping the budget as the first priority. All efficient time-cost trade-off pairs are obtained. D1-distance is calculated to each trade-off pair from the ideal solution. Finally optimum solution is reached by using D1-distance.


INTRODUCTION
The cost minimizing classical multi-index transportation problems play important rule in practical problems. The cost minimizing classical multi-index transportation problems have been studied by several authors [14,15,16,17] etc. Some times there may exist emergency situation eg police services, time services, hospital management etc. where time of transportation is of greater importance than cost of transportation. In this situation, it is to be noted that the cost as well as time play prominent roles to obtain the best decision. Here the two aspects (ie cost and time) are conflicting in nature. In general one can not simultaneously minimize both of them. Bi-criterion transportation problem have been studied by several authors [3,4,8,17,11] etc.
There are many business problems, industrial problems, machine assignment problems, routing problems, etc. that have the characteristic in common with generalized transportation problem that have been studied by several authors [1,2,4,5,9,10,14 ] etc.
In real world situation, most of the intimations are imprecise in nature involving vagueness or to say fuzziness. Precise mathematical model are not enough to tackle all practical problems. Fuzzy set theory was developed for solving the imprecise problems in the field of artificial intelligence. To tackle this situation fuzzy set theory are used. In this field area pioneer work came from Bellman and Zadeh [6]. Fuzzy transportation problem have been studied by several authors [12,18,19,20,21,23,24] etc.
The importance of fuzzy generalized multi-index transportation problem is increasing in a great deal but the method for finding time-cost trade-off pair in a bicriterion fuzzy generalized multi-index transportation problem has been paid less attention. In this paper, we have developed a new algorithm to find time-cost trade-off pair of bi-criterion fuzzy generalized multi-index transportation problem. Thereafter an optimum time-cost trade-off pair has been obtained.

Problem Formulation:
Let there be m-origins, n-destinations and qproducts in a bi-criterion generalized multi-index fuzzy transportation problem. Let, xijk = the amount of the k-th type of product transported from the i-th origin to the j-th destination, tijk = the time of transporting the k-th type of product from the i-th origin to the j-th destination which is independent of amount of commodity transported so long as xijk > 0, rijk = the cost involved in transporting per unit of the k-th type of product from the i-th origin to the j-th destination, Some times there may arise emergency situation, eg, hospital managements, fire services, police services etc., where the time of transportation is of greater importance than that of cost. Then time minimizing transportation problem arises. The time minimizing transportation problem can be written as: Subject to the constraints (1). Combining the problem P1 and P 1 , the fuzzy BGMTP appears as: subject to the constraints (1).

Difference between Classical Multi-index Transportation Problem (MTP) and Generalized Multiindex Transportati on Problem (GMTP):
There are several important differences between classical MTP and GMTP which are given below: (i) The rank of the co-efficient matrix [xijk]m × n × q is in general m + n + q rather than m + n + q -2, ie, all the constraints are in general independent.
(ii) In GMTP the value of xijk may not be integer, though it is integer in classical MTP. (iii) The activity vector in GMTP is (iv) In GMTP it need not be true that cells corresponding to a basic solution form a tree. Or in other words vectors in the loop are linearly independent. But in classical MTP vectors in the loop are linearly dependent. The problem consists of two parts, P1 : the problem of solving the fuzzy GMTP P 1 : the problem of minimizing the time.
To solve the problem P, the following technique is used.
The triangular membership function for the fuzzy demand constraints are The linear membership function of the fuzzy budget goal can be written as: Where Z* is the upper tolerance limit of the budget goal and

II.
SOLUTION PROCEDURE The fuzzy programming model of problem P1 is equivalent to the following linear programming problem as:

The Algorithm:
Step -1: Set b = 1, where b is the number of iteration.

Numerical Examples:
A manufacturing company produces three types of products at two factories. They supply their products at four destinations. The corresponding data are given in Table -1.

Time -Cost Graph
So the optimum time-cost trade-off pair is (1275, 40).