Chapter 2 Basic Models for the Location Problem

Outline 11.3 Techniques for Discrete Space Location Problems 11.3.1 Qualitative Analysis 11.3.2 Quantitative Analysis 11.3.3 Hybrid Analysis

Outline Cont... 11.4 Techniques for Continuous Space Location Problems 11.4.1 Median Method 11.4.2 Contour Line Method 11.4.3 Gravity Method 11.4.4 Weiszfeld Method

11.4.4 Weiszfeld Method

Weiszfeld Method: The objective function for the single facility location problem with Euclidean distance can be written as: As before, substituting wi=cifi and taking the derivative of TC with respect to x and y yields

Weiszfeld Method:

Weiszfeld Method:

Weiszfeld Method:

Weiszfeld Method:

Weiszfeld Method: Step 0: Set iteration counter k = 1;

Weiszfeld Method: Step 1: Set Step 2: If xk+1 = xk and yk+1 = yk, Stop. Otherwise, set k = k + 1 and go to Step 1

Example 8: Consider Example 6. Assuming the distance metric to be used is Euclidean, determine the optimal location of the new facility using the Weiszfeld method. Data for this problem is shown in Table 11.

Table 11.17 Coordinates and weights for 4 departments

Table 11.17: Departments # xi yi wi 1 10 2 6 2 10 10 20 3 8 6 8 1 10 2 6 2 10 10 20 3 8 6 8 4 12 5 4

Solution: Using the gravity method, the initial seed can be shown to be (9.8, 7.4). With this as the starting solution, we can apply Step 1 of the Weiszfeld method repeatedly until we find that two consecutive x, y values are equal.

Summary: Methods for Single-Facility, Continuous Space Location Problems Rectilinear Squared Euclidean Euclidean Method Median Gravity Weiszfeld