On the Location of Supply Points to Minimize Transportation Costs

of locating supply points optimally with respect to transportation costs i s given. Although the algorithm may fail to converge to an optimal solution, repeated application with judicious selections of alternative starting values will assure a good, i f not optimal, solution. The algorithm has been tested and some sample results are included. The location of factories, warehouses-and supply points in general, to serve customers distributed over a network of cities-is often influenced by transportation costs. If transportation costs are uniform and linear with respect to distance, the total transportation cost is proportional to the sum of the distances from the supply points to the cities served, each weighted by the volume of shipments. To illustrate notation and terminology, a network is shown in Figure 1. The cities in a network will be labelled with subscripted ~i~~~~ 1 p's and referred to as points or nodes. The set of p's will be denoted by P. The distance along a path from a node pi to pi not passing through intervening nodes will be denoted by d i , and, of course d i. j = d. 1. I. and di. = 0. The shortest path from one node to another may pass through one or more intervening nodes-as in Figure 1 where the shortest path from p l to p3 passes through p,. If a direct path exists between a pair of nodes, it will be assumed to be unique, but there may be no direct path between nodes, and in such cases we will put d i , i = m for computational purposes. However, it will be a.ssumed that some path exists connecting every pair of nodes in the network. The demand or volume of shipments to be made to node pi ~ will be denoted by an associated weight wi. A node a t which a supply point is located will be called a source; a node having a demand to be satisfied will be referred to as a sink. We may wish .I to consider locating a source a t a city having no demand, and it will be convenient to assign a weight of zero to such cities in order that they may be treated formally as sinks rather than on an exception basis. Thus, in mathematical terms, we have the following problem. where D,.i is the minimal path length from p i to pi. …