Extensions to the planar p-median problem

Abstract

In this paper we propose three models for locating multiple facilities anywhere in the plane. The facilities serve demand points and require raw materials from a list of available sources. Problem characteristic originally proposed in 1909 by Weber for manufacturing systems. Weber argued that optimal locations involve minimizing total transport cost which was comprised of the costs of transporting the raw materials and the delivery cost of the final product when plant production and location costs were invariant across the plane. Both the parameters of raw material sources and demand points affect the best locations for the facilities. In this paper, a special algorithm is designed to heuristically solve these three models. The algorithm exploits the special structure of the models. Problems with up to 2000 demand points and 20 facilities were tested. The results are compared with applying available non-linear solvers in a multi-start approach. The special algorithm performed better in most instances especially for a large number of facilities and a large number of demand points.

Appendix

The first solution method for the Weber problem was proposed by Weiszfeld (1937). It is described and improvements suggested in Drezner (1992, 1996) among others. One improvement was proposed by Ostresh (1978) and tested in Drezner (1992) is as follows. Let $(x',y')$ be the current iteration, and (x", y") be the next iteration by the Weiszfeld algorithm. A simple and effective suggestion is to use a multiplier $\lambda$ and apply $(x^*, y^*)=[x'+\lambda (x"-x'),~y'+\lambda (y"-y')]$ for the next iteration. In the original Weiszfeld algorithm $\lambda =1$. Ostresh (1978) proved convergence for $1\le \lambda \le 2$, and Drezner (1992, 1996) showed by simulation that $\lambda =1.8$ is the most efficient value. Even though convergence is proved for $1\le \lambda \le 2$, slow convergence is observed when the current iteration is close to a demand point because zero distance to a demand point leads to a $\frac{0}{0}$ expression. We therefore set $\lambda =1$ if the current iteration is close to a demand point.

Drezner and Simchi-Levi (1992) showed that the probability that the optimal solution, for a problem with equal weights, is at a demand point is about $\frac{1}{n}$. Therefore, such convergence issues are more likely for small values of n. A condition that the optimal solution is located at demand point k is (Drezner & Simchi-Levi, 1992; Francis et al., 1992):

$$\begin{aligned} \left[ \sum \limits _{i\ne k}\frac{w_i(x_k-x_i)}{\sqrt{(x_i-x_k)^2+(y_i-y_k)^2}}\right] ^2~+~\left[ \sum \limits _{i\ne k}\frac{w_i(y_k-y_i)}{\sqrt{(x_i-x_k)^2+(y_i-y_k)^2}}\right] ^2~\le ~w_k^2 \end{aligned}$$

(10)

This condition can be checked for a demand point next to the current iteration, and stop the iterations if an optimal solution is verified. However, checking this condition for the closest demand point each iteration is time consuming and its usefulness is doubtful. Recall that the Weber problem is convex. The modified Weiszfeld procedure code in MATLAB, without checking (10), is depicted in Fig. 2.