# What are the two different types of sensitivity ranges?

All questions are required.
Question 1: 24 Points, Question 3: 20 Points, Questions 2, 4, 5, and 6: 14 Points each

1. (a) What are the two different types of sensitivity ranges? Describe each type briefly and give a real world example for each type.

(b) What is a marketing problem in applications of linear programming? Briefly discuss the decision variables, the objective function and constraint requirements in a marketing problem. Give a real world example of a marketing problem.

(c) What is the required format of a linear programming problem to be solved by QM for Windows? What results are available from QM for Windows after solving a linear programming problem? Discuss briefly.

(d) What is a transportation problem? Briefly discuss the decision variables, the objective function and constraint requirements in a transportation problem. Give a real world example of the transportation problem.

Answer Questions 2 and 3 based on the following LP problem.

Let P1 = number of Product 1 to be produced
P2 = number of Product 2 to be produced
P3 = number of Product 3 to be produced
P4 = number of Product 4 to be produced

Maximize 80P1 + 100P2 + 120P3 + 70P4 Total profit
Subject to
8P1 + 12P2 + 10P3 + 8P4 ? 6000 Production budget constraint
4P1 + 3P2 + 2P3 + 3P4 ? 2000 Labor hours constraint
P1 > 200 Minimum quantity needed for Product 1 constraint
P2 > 100 Minimum quantity needed for Product 2 constraint
And P1, P2, P3, P4 ? 0 Non-negativity constraints

The QM for Windows output for this problem is given below.

Linear Programming Results:
Variable Status Value
P1 Basic 200
P2 Basic 100
P3 Basic 320
P4 NONBasic 0
slack 1 NONBasic 0
slack 2 Basic 260
surplus 3 NONBasic 0
surplus 4 NONBasic 0
Optimal Value (Z) 64400

P1 P2 P3 P4 RHS Dual
Maximize 80 100 120 70
Constraint 1 8 12 10 8 <= 6000 12
Constraint 2 4 3 2 3 <= 2000 0
Constraint 3 1 0 0 0 >= 200 -16
Constraint 4 0 1 0 0 >= 100 -44
Solution-> 200 100 320 0 Optimal Z-> 64400

Ranging Results:
Variable Value Reduced Cost Original Val Lower Bound Upper Bound
P1 200 0 80 -Infinity 96
P2 100 0 100 -Infinity 144
P3 320 0 120 100 Infinity
P4 0 26 70 -Infinity 96

Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound
Constraint 1 12 0 6000 2800 7300
Constraint 2 0 260 2000 1740 Infinity
Constraint 3 -16 0 200 0 308.3333
Constraint 4 -44 0 100 0 366.6667

2. (a) Determine the optimal solution and optimal value and interpret their meanings.
(b) Determine the slack (or surplus) value for each constraint and interpret its meaning.

3. (a) What are the ranges of optimality for the profit of Product 1, Product 2, Product 3, and Product 4?
(b) Find the dual prices of the four constraints and interpret their meanings. What are the ranges in which each of these dual prices is valid?
(c) If the profit contribution of Product 2 changes from \$100 per unit to \$125 per unit, what will be the optimal solution? What will be the new total profit? (Note: Answer this question by using the ranging results given above).
(d) Which resource should be obtained in larger quantity to increase the profit most? (Note: Answer this question using the ranging results given above.).

4. A professional football player is retiring, and he is thinking about going into the insurance business. He plans to sell three types of policies— homeowner’s insurance, auto insurance and life insurance. The average amount of profit returned per year by each type of insurance policy is as follows:

Policy Yearly Profit/Policy
Homeowner’s \$35
Auto 24
Life 60

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