Example 7.6 Facility Location Models

| 7.2 | 7.3 | 7.4 | 7.5 | 7.7 | 7.8 | 7.9 | 7.10 | Background Information n The Lafferty Company wants to locate a warehouse from which it will ship products to four customers. n The location (in the x-y plane) of the four customers and the number of shipments per year needed by each customer are given in this table. Data for Lafferty Example xyShipments per Year Customer Customer Customer Customer

| 7.2 | 7.3 | 7.4 | 7.5 | 7.7 | 7.8 | 7.9 | 7.10 | Background Information -- continued n All locations are in miles, relative to the point x=0 and y=0. n A single warehouse must be used to service all of the customers. n Lafferty wants to determine the location of the warehouse that minimizes the total distance traveled from the warehouse to the customers.

| 7.2 | 7.3 | 7.4 | 7.5 | 7.7 | 7.8 | 7.9 | 7.10 | Solution n For the spreadsheet model we need to keep track of the following: –The x and y coordinates of the warehouse and each customer –The distance between the warehouse and each customer –The total annual distance traveled from the warehouse to customers.

| 7.2 | 7.3 | 7.4 | 7.5 | 7.7 | 7.8 | 7.9 | 7.10 | LAFFERTY.XLS n The model appears on the next slide. n This file can be used to build the model.

| 7.2 | 7.3 | 7.4 | 7.5 | 7.7 | 7.8 | 7.9 | 7.10 |

| 7.2 | 7.3 | 7.4 | 7.5 | 7.7 | 7.8 | 7.9 | 7.10 | Developing the Model n The model can be formed as follows. –Inputs. Enter the given customer data in the shaded ranges. –Coordinates of warehouse. Enter any trial values in Location range for the x- and y-coordinates of the warehouse. –Distances from warehouse to customers. Calculate the distances from the warehouse to the customers in the Distances range. To do so, recall that the (straight line) distance between the two points ( a,b ) and (c,d ) is Therefore, enter the formula =SQRT((B5-$B$11)^2+(C5-$C$11)^2) in cell B14 and copy it to the rest of the Distances range.

| 7.2 | 7.3 | 7.4 | 7.5 | 7.7 | 7.8 | 7.9 | 7.10 | Developing the Model -- continued –Total annual distance. The total annual distance traveled from the warehouse to meet the demands of all customers is where n i is the number of trips per year for customer i and d i is the distance from the warehouse to customer i. Therefore, calculate the total annual distance traveled from the customers to the warehouse in the TotDistance cell with the formula =SUMPRODUCT(Shipments,Distances).

| 7.2 | 7.3 | 7.4 | 7.5 | 7.7 | 7.8 | 7.9 | 7.10 | Using the Solver n This model requires the “leanest” Solver setup so far. n All we need to specify is that TotDistance should be minimized and the Location range contains the changing cells. n There are no constraints, not even nonnegativity constraints. n Also, because of the squares in the straight-line distance formula, this model is nonlinear, so the Assume Linear Model box should not be checked.

| 7.2 | 7.3 | 7.4 | 7.5 | 7.7 | 7.8 | 7.9 | 7.10 | Solution n The warehouse should be located at x=9.31 and y=5.03. Each year a total of miles will be traveled annually from the warehouse to the customers. n This solution represents a compromise. On the other hand, Lafferty would like to position the facility near customer 4 because customer 4 is fairly far from the other customers, the warehouse is located in a more central position.

| 7.2 | 7.3 | 7.4 | 7.5 | 7.7 | 7.8 | 7.9 | 7.10 | Sensitivity Analysis n One possible sensitivity analysis is to see how the optimal location of the warehouse changes as the annual number of shipments to any particular customer increases. n We do this for customer 4 in the table on the following slide. n We run SolverTable with the number of shipments to customer 4 as the single input cell, allowing it to vary from 300 to 700 in increments of 50, and we keep track of the total annual distance and the warehouse location coordinates.

| 7.2 | 7.3 | 7.4 | 7.5 | 7.7 | 7.8 | 7.9 | 7.10 | Sensitivity Analysis -- continued n As expected, the total annual distance increases as the annual shipments to customer 4 increase. n Also, the warehouse gradually gets closer to customer 4. In fact, when the annual shipments to customer 4 are 600 or above, the optimal location for the warehouse is at customer 4.

| 7.2 | 7.3 | 7.4 | 7.5 | 7.7 | 7.8 | 7.9 | 7.10 | Is the Solver Solution Optimal? n The Lafferty model has no constraints. n An NLP with no constraints is called an unconstrained NLP. Therefore, we know that the Solver will find an optimal solution if the objective is a convex function of the x- and y-coordinates of the warehouse. n It can be shown that the annual distance traveled is indeed a convex function of the coordinates of the warehouse.Therefore, we know that the Solver solution is optimal.

| 7.2 | 7.3 | 7.4 | 7.5 | 7.7 | 7.8 | 7.9 | 7.10 | Is the Solver Solution Optimal? - - continued n However, what if you do not know whether the objective is a convex function? n Then the best strategy is to try different starting solutions in the Locations range, run the Solver on each of them, and see whether they all take you to the same solution. n In fact, we have made this easy for you in the LAFFERTY.XLS file. LAFFERTY.XLS

| 7.2 | 7.3 | 7.4 | 7.5 | 7.7 | 7.8 | 7.9 | 7.10 | Is the Solver Solution Optimal? - - continued n We have written two short macros that are automated by clicking on buttons. n Click on the left buttons to randomly generate a new starting location. n Then click on the right button to run Solver. n You should find that they always take you to the same solution.