15. Sandford Tile Company makes ceramic and porcelain tile for residential and commercial use. They produce three different grades of tile (for walls, residential flooring, and commercial flooring), each of which requires different amounts of materials and production time, and generates different contributions to profit. The following information shows the percentage of materials needed for each grade and the profit per square foot.

Grade I Grade II Grade III Profit/square foot $2.50 $4.00 $5.00 Clay 50% 30% 25%

Silica 5% 15% 10%

Sand 20% 15% 15%

Feldspar 25% 40% 50%

Each week, Sanford Tile receives raw-material shipments, and the operations manager must schedule the plant to efficiently use the materials to maximize profitability.

Currently, inventory consists of 6,000 pounds of clay, 3,000 pounds of silica, 5,000 pounds of sand, and 8,000 pounds of feldspar. Because demand varies for the different grades, marketing estimates that at most 8,000 square feet of Grade III tile should be produced, and that at least 1,500 square feet of Grade I tiles are required. Each square foot of tile weighs approximately two pounds.

a. Develop a linear optimization model to determine how many of each grade of tile the company should make next week to maximize profit contribution.

b. Implement your model on a spreadsheet and find an optimal solution.

c. Explain the sensitivity information for the objective coefficients. What happens if the profit on Grade I is increased by $0.05?

d. If an additional 500 pounds of feldspar is available, how will the optimal solution be affected?

e. Suppose that 1,000 pounds of clay are found to be of inferior quality. What should the company do?

f. Use the auxiliary variable cells technique to handle the bound constraints and generate all shadow prices.