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

11.4.3 Single-facility Location Problem with Squared Euclidean Distances

Gravity Method: The cost function is As before, we substitute wi = fi ci, i = 1, 2, ..., m and rewrite the objective function as

Gravity Method (Cont) Since the objective function can be shown to be convex, partially differentiating TC with respect to x and y, setting the resulting two equations to 0 and solving for x, y provides the optimal location of the new facility

Gravity Method (Cont) Similarly, Thus, the optimal locations x and y are simply the weighted averages of the x and y coordinates of the existing facilities

Example 7: Consider Example 5. Suppose the distance metric to be used is squared Euclidean. Determine the optimal location of the new facility using the gravity method.

Solution - Table 11.16 Department i xi yi wi wixi wiyi 1 10 2 6 60 12 1 10 2 6 60 12 2 10 10 10 100 100 3 8 6 8 64 48 4 12 5 4 48 20 Total 28 272 180

Example 6. Cont... If this location is not feasible, we only need to find another point which has the nearest Euclidean distance to (9.7, 6.4) and is a feasible location for the new facility and locate the copiers there