In many distribution problems, the transportation cost consists of a fixed cost, independent of the amount transported and a variable cost, proportional to the amount shipped. In such fixed-charge transportation problems, is it possible to find a solution with less (or equal) cost than the optimal solution by shipping more units, under the condition that at least the same amount is shipped from each supply point and to each market? This question has not received any attention in the literature, and no algorithm (either analytical or heuristic) is known to address this problem. The more-for-less analysis could be useful for managers in decisions such as increasing warehouse/plant capacity, analyzing company acquisitions, mergers, consolidations or downsizing. In this paper we develop a quick sufficient condition to identify candidate markets and supply points to ship more for less in fixed-charge transportation problems.

On the fixed-charge transportation problem

This paper presents a variant of Vogel's approximation method (VAM) for transportation problems. The importance of determining efficient solutions for large sized transportation problems is borne out by many practical problems in industries, the military, etc. With this motivation, a few variants of VAM incorporating the total opportunity cost (TOC) concept were investigated to obtain fast and efficient solutions. Computational experiments were carried out to evaluate these variants of VAM. The quality of solutions indicates that the basic version of the VAM coupled with total opportunity cost (called the VAM–TOC) yields a very efficient initial solution. In these experiments, on an average, about 20% of the time the VAM–TOC approach yielded the optimal solution and about 80% of the time it yielded a solution very close to optimal (0.5% loss of optimality). The CPU time required for the problem instances tested was very small (on an average, less than 10 s on a 200 MHz Pentium machine with 64 MB RAM).

http://www.angelfire.com/ak6/invo_escom/lectura17_transporte_vogel.pdf

Experimental Analysis of some Variants of Vogel's Approximation Method

The more-for-less (MFL) phenomenon in distribution problems occurs when it is possible to ship more total goods for less (or equal) total cost, while shipping the same quantity or more from each origin and to each destination. This paradox occurs often in fixed-charge transportation problems (FCTPs), and further analysis could bring significant reduction in costs. The MFL phenomenon for FCTPs has received minimal attention in the literature despite the fact that existing analytical algorithms, such as branch and bound, are limited to small problems due to excessive computational effort. In this paper, we develop a simple heuristic algorithm to identify the demand destinations and the supply points to ship MFL in FCTPs. The proposed method builds upon any existing basic feasible solution. It is easy to implement and can serve as an effective tool for managers for solving the more-for-less paradox for large distribution problems.

More-for-less algorithm for fixed-charge transportation problems

We develop here a new approach to postoptimality analyses of the transportation problem (TP) through the use of perturbation analysis. In contrast to the usual sensitivity analysis the proposed methods allow for simultaneous, independent (or dependent) perturbation of the supplies, demands and transportation rates cost from their nominal values whilst preserving the current optimal basis. The essential parts of the calculation involve the same manipulations as the solution algorithm used in solving the parametric perturbed supply-demand version of the nominal TP. Furthermore, the coefficient of the perturbed parameters in the optimal parametric cost function is used to indicate the existence of a cheaper (or equivalent cost) solution than the current optimal one by shipping more total goods. The proposed approach permits computation of the maximal allowable additional units and distribution of these extra units in a systematic manner. The proposed unified procedures are demonstrated by numerical examples.

Postoptimality analyses of the transportation problem

This paper presents a two stage cost minimizing fuzzy transportation problem with multi-objective constraints. A new approach is used to conclude the optimal solution of a multi-objective two stage fuzzy transportation problem. Numerical example is given to ensure the strength of the proposed method.

Optimal Solution for Multi-Objective Two Stage Fuzzy Transportation Problem

This paper re-examines use of the linear programming (LP) formulation to solve the transportation problem (TP). The proposed method is a general-purpose algorithm which uses only one operation, the Gauss Jordan pivoting used in the simplex method. The final tableau can be used for post-optimality analysis of TP. This algorithm appears to be faster than simplex, more general than stepping-stone and simpler than both in solving general TP. A numerical example illustrates the methodology. It is assumed the reader is familiar with simplex terminology.

A Simplex-Type Algorithm for General Transportation Problems: An Alternative to Stepping-Stone

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A Simplex-Type Algorithm for General Transportation Problems: An Alternative to Stepping-Stone

Postoptimality analyses of the transportation problem