Problem 3 (35 points): Consider the following linear programming model. Let and be the two decision variables. The objective of the model is maximization, and the objective function is 2 + . In addition to 0 and 0, there are two functional constraints: + 5 and 3 + 2 10. The whole model is given below: Note: An Excel file will have to be created for question 3a (two tabs: one for the model and one for the sensitivity report).

2 + + 5 3 + 2 10 0, 0

3a) (8 points) Formulate a spreadsheet model for the above model and solve it using Excel Solver, and then generate the sensitivity report. 3b) (8 points, 2 points each) Answer the following questions based on the sensitivity report generated in part a and explain your answers (2-4 sentences are sufficient). Do not resolve the model using Excel Solver. The answers will be solely based on the sensitivity report. i. If the objective function coefficient for X increases from 2 to 3, the optimum solution will not change. True or False? Explain why briefly. ii. If the objective function coefficient for Y increases from 1 to 3, the optimum solution will not change. True of False? Explain why briefly. iii. If the first constraints right-hand-side decreases from 5 to 3, can you tell how much the optimum objective function value will decrease? If yes, calculate the amount of decrease referring to the values in the table above. If no, explain why. iv. If the second constraints right-hand-side increases from 10 to 12, can you tell how much the optimum objective function value will increase? If yes, calculate the amount of increase referring to the values in the table above. If no, explain why. 3c) (8 points) Mathematically formulate the dual of the linear programming model given above (the primal model). To do so, clearly define the dual variables and which constraints of the primal model they correspond to. 3d) (8 points) Given the optimal primal solution from your solution in part a, determine the optimum dual solution to the dual model you formulated in part c using complementary slackness. Clearly show how you use complementary slackness conditions. 3e) (3 points) Looking at the Constraints section of the above sensitivity report, what is the optimal solution of the dual problem? That is, give the optimal values for each dual variable from the Constraints section of the report.