Optimization Concepts and Applications in Engineering

Introduction

Transportation, assignment, and network problems are of great significance in engineering industry, agriculture, global economy, and commerce. These problems can be solved by the conventional simplex method of linear programming. However, because of their simpler structures, they are more efficiently solved by special methods. In most cases, the calculations involved are simple additions and subtractions and the methods are effective and robust. Many problems in totally different situations may be closely related to these models and the methods may be used for their solution. We consider the transportation, assignment, and network flow problems with a view to develop the formulation of the problems, the solution algorithms, and their implementation in simple computer codes.

Transportation Problem

There are m supply points where quantities s1, s2,…, sm, respectively, are produced or stocked. There are n delivery points or destinations where respective quantities of d1, d2,…, dn, are in demand. Cost is incurred on an item shipped from origination point i to the destination j. The problem is to determine the amounts (≥ 0) originating at i and delivered at j so that the supply and demand constraints are satisfied and the total cost is a minimum.