1. Chapter 2 Basic Models for the Location Problem

2. Outline 11.3 Techniques for Discrete Space Location Problems
Qualitative Analysis
Quantitative Analysis
Hybrid Analysis

3. Outline Cont... 11.4 Techniques for Continuous Space Location Problems
Median Method
Contour Line Method
Gravity Method
Weiszfeld Method

4. 11.4.3 Single-facility Location Problem with Squared Euclidean Distances

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

6. 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

7. 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

8. 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.

9. Solution - Table 11.16 Department i xi yi wi wixi wiyi 1 10 2 6 60 12
Total

10. 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