Question

University Ceramics manufactures plates, mugs, and steins that include the campus name and logo for sale in campus bookstores. The time required for each item to go through the two stages of production (molding and finishing), the material required (clay), and the corresponding unit profits are given in the following table, along with the amount of each resource available in the upcoming production period. Plates Mugs Steins Available Molding (minutes) 4 6 3 2400 Finishing (minutes) 8 14 12 7200 Clay (ounces) 5 4 3 3000 Unit Profit $3.10 $4.75 $4.00 A linear programming model has been formulated in a spreadsheet to determine the production levels that would maximize profit. The solved spreadsheet model and corresponding sensitivity report are given. Answer the following questions based off of the sensitivity report. Each question is independent (i.e., any change made in one part does not apply to any other parts). a. Suppose the profit per plate decreases from $3.10 to $2.80. Will this change the optimal production quantities? What can be said about the change in total profit? b. Suppose a worker in the molding department is sick. Now eight fewer hours are available that day in the molding department. How much would this affect total profit? Would this change the optimal production quantities? GRBU 503 he allowable decrease in the mugs’ objective coefficient and the shadow price for the available clay constraint are both missing from the sensitivity report. What numbers should be there? Explain how you were able to deduce each number.

Answer 1

a)   The decrease is within the allowable decrease, so the optimal production quantities stay the same. Total profit will decrease by ($0.30)(300) = $90 to $2440.

b)   8 hours, or 480 minutes, is within the allowable decrease for molding, so the shadow price is valid. The change in total profit is therefore ∆Profit = ($0.22)(-480) = -$105.60. The optimal production quantities will change.

c)   300. The shadow price is 0 because there is slack in this constraint. The shadow price will remain 0 so long as there is slack. There will remain slack so long as the right-hand side decreases no more than 300 minutes.