Question 10.15: Solving a total cover problem

A major retail firm is considering a major change in its distribution system. Specifically, in the past, it shipped merchandise from a few distribution centers (DC_{s}) to its stores on a

Table 10.4 Cover Coefficients for Example 10.14

weekly basis. Consumer surveys indicate that sales are being lost because of out-of-stock situations occurring. The competition seems to be able to maintain its inventory in its stores without incurring excessive inventory costs. Among the options being considered is locating distribution centers such that all retail stores can be replenished within 24 hours of a request
for merchandise. To evaluate the feasibility of such an approach, a region of the country with numerous stores is to serve as a test region. Shown in Table 10.4 is a matrix of cover coefficients for 10 stores and 10 potential sites for distribution centers. If store i can be replenished within 24 hours from site j, then a_{ij} = 1; otherwise, a_{ij} = 0.
The total covering problem is solved using SOLVER, as shown in Figure 10.22. The cover coefficients and SOLVER parameters are displayed in (a); the solution is given in (b),

row 13 contains the values of the decision variables; the entry in the target cell (L13) is the sum of the values in column L for rows 3 through 13. The problem is set up to minimize the target cell value by changing the values of the decision variables. The constraints consist of the binary constraint for the decision variables and one constraint for each retail store to be replenished. (Excel’s® SUMPRODUCT function was used to generate the values in column L1.)
In the example, we desire to minimize the number of distribution centers required to have at least one DC within a 24 hour replenishment window. Consequently, a constraint was added for each customer, requiring that the sumproduct of the decision variables and cover coefficients be at least equal to 1. If redundant coverage is desired, then the sumproduct constraint could be set to be greater than or equal to, say, 2. As shown in (c), a solution was obtained, satisfying all the constraints and requiring only two distribution centers: DCs are to be located at sites 1 and 6. In this example, there
are multiple optimum solutions: locating DCs at sites 2 and 7 will also provide the required coverage. (We stumbled across the alternate solution by trying a variety of initial values for the decision variables [e.g., odd-numbered sites having a value of 1 and even-numbered sites having a value of 0, and vice versa].)
As noted previously, a non-optimum solution can be obtained when using SOLVER.
For the example, if the SOLVER search is initiated by letting x_{1} = x_{5} = x_{6} = 1 and x_{2} = x_{3} =x_{4} = x_{7}= x_{8}= x_{9}= x_{10} = 0, then a solution requiring three distribution centers is obtained.
On the other hand, letting all decision variables equal 0 or letting all decision variables equal 1 produced the optimum solution of having two DCs located at sites 1 and 6.