A Weighted Least Cost Matrix Approach in Transportation Problem

Abstract

Transportation models are of multidisciplinary fields of interest. Mainly in the business arena, it is common to encounter the problem of transportation of some products/goods from source/origin to sink/destination so that cost of transportation is minimal and also satisfy the constraints related to demand and the supply. A huge number of physical problems are modeled as Transportation problem (TP) which includes inventory problem, assignment problem, traffic problem and so on. TPs are required for analyzing and formulating such models. In order to minimize the transportation cost satisfying all constraint, transportation model first provides the Initial Basic Feasible Solution (IBFS) and then IBFS be optimized by some related optimization algorithm if IBFS is not optimized. So the primary objective of transportation model is to find out a good IBFS of TPs. In classical transportation approaches, the flow of allocation is controlled by the cost entries such as West Corner Method (WCM), Least Cost Method (LCM) etc. and/or manipulation of cost entries, so called Distribution Indicator (DI) or Total Opportunity Cost (TOC) like Vogel’s Approximation Method (VAM) and its variations. In LCM, the flow of allocation is directly controlled by the cost entries i.e. lowest cost prefers first. On the other hand, for examples, on VAM, its variants and some other methods, the flow of allocations is controlled by the DI or TOC tables. But these DI or TOC tables are formulated by the manipulation of cost entries only. None of them considers demand and/or supply entry to formulate the DI/ TOC table. In this thesis, we have first developed a new procedure of control of allocation named Weighted Opportunity Cost (WOC) matrix by incorporating supply/demand entries. At first, weight factors are formulated by using demand and supply entries which is off-course statistically valid. Then virtual weighted cost entries are formulated by manipulation of cost entries along with weight factors. Finally WOC matrix is formulated in which supply/demand entries acts as weight factor upon corresponding cost entries. Several examples are provided to demonstrate the concept of WOC matrix. After successfully development of WOC, our intension is go upon the development of an algorithm to find out IBFS of TPs. It is known that, in Least Cost Matrix method, the flows of allocations are controlled by the cost entries only. The flows of allocations are predefined according to the cost entries i.e the ascending order of cell cost and the whenever identical costs are encountered whatever be the structure of demand/supply entries. So, the algorithm does not need to update allocation direction in subsequent steps. On the other hand in VAM, the flow of allocation is controlled by the DI table rather than directly cost matrix. By incorporating these two ideas, we have proposed a Weighted Opportunity Cost based on LCM (WOC-LCM) approach. In this proposed approach, the flow of allocation is controlled by the WOC matrix rather than cost matrix as in LCM approach. But WOC matrix is invariant through all over the allocation procedures like cost matrix in LCM method whereas DI table is updated after each step of allocations. Some experiments have been carried out to justify the validity and the effectiveness of the proposed WOC-LCM approach. Experimental results have shown that the WOC-LCM approach outperforms LCM. Moreover, sometime this approach is able to find out optimal solution too.