Problem 4 (70 points) Chris is coordinating a bake sale for a nonprofit organization. From donations he has acquired 90 cups of butter, 150 cups of flour, 200 cups of sugar and 65 cups of cocoa. He can make brownies and cookies. The ingredients needed for both recipes are butter, flour, sugar, and cocoa. The following table shows the amounts and costs of ingredients used per batch of each baked good. Each batch of brownies can be sold for $6.50 and each batch of cookies can be sold for $7.10. For this part of the problem assume that any noninteger batch of brownies or cookies can be made. Ingredients Butter (cups) Flour (cups) Sugar (cups) Cocoa (cups) Profit per batch Brownies 0.60 1.50 1.40 1.00 $6.50 Cookies 0.70 1.20 1.80 0.35 $7.10

a) (15 points) How can Chris best use the donated ingredients to raise the largest amount of money? Mathematically formulate a linear optimization problem. To get full credit you need to identify the decision variables, write the objective function and the constraints in mathematical formulation.

b) (15 points) Implement the linear optimization model in Excel and use Solver to provide the Solver Answer report. You need to use the linear optimization format we used in class that includes the Math Formulation, the Data, and the Model sections.

c) (5 points) Identify which constraints are binding.

d) (3 points) Produce the Sensitivity report.

e) (6 points) Using only the information in the sensitivity report, answer the following: If the brownies are sold for $7.10, how will the optimal solution and profit change? Explain.

f) (6 points) Using only the information in the sensitivity report, answer the following: If 20 extra cups of flour were available, how will the optimal solution and profit change? Explain.

g) (20 points) Suppose that we restrict the batch of cookies and the batch of brownies to be integers. On a new spreadsheet tab, solve the problem and include the answer report. How much difference is there between the optimal integer solution objective function and the linear optimization solution objective function? Would rounding the continuous linear programming solution have provided the optimal integer solution? Explain.