1 Chapter 2 Basic Models for the Location Problem

Techniques for Discrete Space Location Problems 11.3 Techniques for Discrete Space Location Problems Qualitative Analysis Quantitative Analysis Hybrid Analysis OutlineOutline

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

Techniques For Continuous Space Location Problems

Model for Rectilinear Metric Problem Consider the following notation: f i = Traffic flow between new facility and existing facility i c i = Cost of transportation between new facility and existing facility i per unit x i, y i = Coordinate points of existing facility i

6 Model for Rectilinear Metric Problem (Cont) Where TC is the total distribution cost The median location model is then to minimize:

7 Model for Rectilinear Metric Problem (Cont) Since the c i f i product is known for each facility, it can be thought of as a weight w i corresponding to facility i.

8 Median Method: Step 1: List the existing facilities in non- decreasing order of the x coordinates. Step 2: Find the j th x coordinate in the list at which the cumulative weight equals or exceeds half the total weight for the first time, i.e.,

9 Median Method (Cont) Step 3: List the existing facilities in non- decreasing order of the y coordinates. Step 4: Find the k th y coordinate in the list (created in Step 3) at which the cumulative weight equals or exceeds half the total weight for the first time, i.e.,

10 Median Method (Cont) Step 4: Cont... The optimal location of the new facility is given by the j th x coordinate and the k th y coordinate identified in Steps 2 and 4, respectively.

11 NotesNotes 1. It can be shown that any other x or y coordinate will not be that of the optimal location’s coordinates 2. The algorithm determines the x and y coordinates of the facility’s optimal location separately 3. These coordinates could coincide with the x and y coordinates of two different existing facilities or possibly one existing facility

12 Example 5: Two high speed copiers are to be located in the fifth floor of an office complex which houses four departments of the Social Security Administration. Coordinates of the centroid of each department as well as the average number of trips made per day between each department and the copiers’ yet-to-be-determined location are known and given in Table 9 below. Assume that travel originates and ends at the centroid of each department. Determine the optimal location, i.e., x, y coordinates, for the copiers.

13 Table Centroid Coordinates and Average Number of Trips to Copiers

14 Table Dept.Coordinates Average number of #xy daily trips to copiers

15 Solution:Solution: Using the median method, we obtain the following solution: Step 1: Dept.x coordinates inWeightsCumulative #non-decreasing orderWeights

16 Solution:Solution: Step 2: Since the second x coordinate, namely 10, in the above list is where the cumulative weight equals half the total weight of 28/2 = 14, the optimal x coordinate is 10.

17 Solution:Solution: Step 3: Dept.y coordinates inWeightsCumulative #non-decreasing orderWeights

18 Solution:Solution: Step 4: Since the third y coordinates in the above list is where the cumulative weight exceeds half the total weight of 28/2 = 14, the optimal y coordinate is 6. Thus, the optimal coordinates of the new facility are (10, 6).

19 Equivalent Linear Model for the Rectilinear Distance, Single- Facility Location Problem Parameters f i = Traffic flow between new facility and existing facility i f i = Traffic flow between new facility and existing facility i c i = Unit transportation cost between new facility and existing facility i c i = Unit transportation cost between new facility and existing facility i x i, y i = Coordinate points of existing facility i x i, y i = Coordinate points of existing facility i Decision Variables x, y = Optimal coordinates of the new facility x, y = Optimal coordinates of the new facility TC = Total distribution cost TC = Total distribution cost

20 The median location model is then to Equivalent Linear Model for the Rectilinear Distance, Single- Facility Location Problem

21 Since the c i f i product is known for each facility, it can be thought of as a weight w i corresponding to facility i. The previous equation can now be rewritten as follows Equivalent Linear Model for the Rectilinear Distance, Single- Facility Location Problem

22 Equivalent Linear Model for the Rectilinear Distance, Single- Facility Location Problem

23 Equivalent Linear Model for the Rectilinear Distance, Single- Facility Location Problem

24 Equivalent Linear Model for the Rectilinear Distance, Single- Facility Location Problem