Question

The U.S. Cable Company uses a distribution system with five distribution centers and eight customer zones. Each customer zone is assigned a sole source supplier; each customer zone receives all of its cable products from the same distribution center. In an effort to balance demand and workload at the distribution centers, the company’s vice president of logistics specified that distribution centers may not be assigned more than three customer zones. The following table shows the five distribution centers and cost of supplying each customer zone (in thousands of dollars):

a. Determine the assignment of customer zones to distribution centers that will minimize cost.

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b. Which distribution centers, if any, are not used?

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c. Suppose that each distribution center is limited to a maximum of two customer zones. How does this constraint change the assignment and the cost of supplying customer zones?

Answer 1

(a) A linear program model is a mathematical model with a linear objective function, a set of linear constraints, and nonnegative variables. Let represent the customer zone assigned distribution center . For the customer zones, let 1, 2, 3, 4, 5, 6, 7, and 8 represent Los Angeles, Chicago, Columbus, Atlanta, Newark, Kansas City, Denver, and Dallas, respectively. For the distribution centers, let 1, 2, 3, 4, and 5 represent Plano, Nashville, Flagstaff, Springfield, and Boulder, respectively.

Use EXCEL to solve for the optimal solution report using the following steps.

1. Select Solver from the Analysis Group in the Data Ribbon

2. Enter the proper values for the constraints and the objective function

3. Select Simplex LP

4. Click the checkbox for Make Unconstrained Variables Non-negative

5. Click Solve

The optimal solution is to assign Kansas City and Dallas to the Plano distribution center, Los Angeles to the flagstaff distribution center, Chicago, Columbus, and Atlanta to the Springfield distribution center, and Newark and Denver to the Boulder distribution center. The total cost is $216,000.

(b) None of the customer zones are assigned to the Nashville distribution center.

(c) If distribution centers are limited to only two customer zones, the right hand-side values for the first five constraints is now 2. Resolve the linear program.

Use EXCEL to solve for the optimal solution report using the following steps.

1. Select Solver from the Analysis Group in the Data Ribbon

2. Enter the proper values for the constraints and the objective function

3. Select Simplex LP

4. Click the checkbox for Make Unconstrained Variables Non-negative

5. Click Solve

The optimal solution is to assign Kansas City and Dallas to the Plano distribution center, Columbus to the Nashville distribution center, Los Angeles to the flagstaff distribution center, Chicago, and Atlanta to the Springfield distribution center, and Newark and Denver to the Boulder distribution center. The revised total cost is $227,000. Limiting distribution centers to two customer zones would increase costs by $11,000.