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1. (a) What is an infeasible linear optimization problem? How do…
1. (a) What is an infeasible linear optimization problem? How do we find if a given linear optimization problem is infeasible? Give a real world example of a linear optimization problem where infeasibility may occur.
(b) Describe the concept and process of spreadsheet modeling and analysis. Give a real world example where spreadsheet modeling and analysis is useful. Answer in at least eight sentences.
(c) What is a marketing problem in applications of linear optimization? Briefly discuss the decision variables, the objective function, and the constraint requirements in a marketing problem. Give a real world example of a marketing problem. Answer in at least eight sentences.
(d) Explain how the simulation process is used in business analytics models. What are the advantages of using simulation? What are its limitations? How can a simulation model be verified? Give a real world example where using simulation is appropriate.
2. Given the following linear optimization problem
Maximize 60x + 90y
Subject to
x + y < 120 3x + y < 180 x + y > 40
x, y > 0
(a) Graph the constraints and determine the feasible region.
(b) Find the coordinates of each corner point of the feasible region.
(c) Determine the optimal solution and optimal objective function value.
Answer Questions 3 and 4 are based on the following linear optimization problem.
Maximize 20X1 + 25X2 + 10X3 + 15X4 Total Profit
Subject to X1 + X2 + X3 + X4 > 150 At least a total of 150 units of all four products are needed
X1 + 3X2 + 2X3 + 2X4 = 400 Resource 1
2X1 + X2 + 2X3 + X4 = 280 Resource 2
And X1, X2, X3, X4 = 0
Where X1, X2, X3 and X4 represent the number of units of Product 1, Product 2, Product 3 and Product 4 to be manufactured.
The Excel Solver output for this problem is given below.
3. (a) Determine the optimal solution and the optimal value for this problem and interpret their meanings.
(b) Determine the slack (or surplus) value for each constraint in this problem and interpret its meaning.
4. (a) What are the ranges of optimality for the profit of Product 1, Product 2, Product 3, and Product 4?
(b) Find the shadow prices of the three constraints and interpret their meanings. What are the ranges in which each of these shadow prices is valid?
(c) If the profit contribution of Product 1 changes from $20 per unit to $30 per unit, what will be the optimal solution? What will be the new total profit? (Note: Answer this question by using the sensitivity results given above. Do not solve the problem again).
(d) Which resource should be obtained in larger quantity to increase the profit most? (Note: Answer this question using the sensitivity results given above. Do not solve the problem again).
5. An oil company wants to decide how to allocate its budget. The government grants certain tax breaks if the company invests funds in research concerned with energy conservation. However, the government stipulates that at least 35 percent of the funds must be funneled into research for automobile efficiency (methanol fuel research and emission reduction). The company has a budget of $4 million for investment. The research proposal data are shown in the following table.
Maximum Investment Annual Return
Project Allowed on Investment
______________________________________________________________
Methanol fuel research $900,000 4.5%
Emission reduction $485,000 4.0%
Solar cells $800,000 4.0%
Windmills $650,000 3.8%
______________________________________________________________
The company wants to receive the government tax break. How much money should be invested in each project if the company wants to maximize total annual return on its investments?
Formulate a linear optimization model for the above situation.
(a) Define the decision variables for this problem.
(b) Determine the objective function for this problem. What does it represent?
(c) Determine all the constraints for this problem. Briefly describe what each constraint represents.
Note: Do NOT solve the problem after formulating.
6. A consulting firm has four projects to consider. Each project will require time (in days) in the next three months according to the table below.
Project Time in first month Time in second month Time in third month Revenue
A 6 8 6 22000
B 5 6 8 18000
C 8 7 6 16000
D 5 5 4 15000
Revenue from each project is also shown. There is 20 days’ time available in the first month, 22 days’ time available in the second month, and 22 days’ time available in the third month to do these projects. The management wants to select at most 3 projects. If project B is selected, then project C must also be selected. The objective of the firm is to maximize the total revenue.
Formulate an integer optimization model for this problem by defining the decision variables, the objective function and all the constraints. Briefly describe what the objective function and each constraint represent. What type of integer optimization model is this?
Note: Do NOT solve the problem after formulating.
