1. (a) What are the differences between QM for Windows and Excel when solving a linear programming problem? Which one you like better? Why?
(b) What is a diet problem? Briefly discuss the objective function and constraint requirements in a diet problem. Give a real world example of a diet problem.
(c) What do you understand by sensitivity analysis? Why is it important? How can you use the sensitivity ranges to decide whether your recommendations based on the optimal solution of a linear programming problem are robust or not?
(d) What is an assignment problem? Briefly discuss the decision variables, the objective function and constraint requirements in an assignment problem. Give a real world example of the assignment 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 15P1 + 20P2 + 24P3 + 15P4 Total profit
Subject to
8P1 + 12P2 + 10P3 + 8P4 ≤ 3000 Material requirement constraint
4P1 + 3P2 + 2P3 + 3P4 ≤ 1000 Labor hours constraint
P2 > 120 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 NONBasic 0
P2 Basic 120
P3 Basic 156
P4 NONBasic 0
slack 1 NONBasic 0
slack 2 Basic 328
surplus 3 NONBasic 0
Optimal Value (Z) 6144
Ranging Results:
Variable Value Reduced Cost Original Val Lower Bound Upper Bound
P1 0 4.2 15 -Infinity 19.2
P2 120 0 20 -Infinity 28.8
P3 156 0 24 18.75 Infinity
P4 0 4.2 15 -Infinity 19.2
Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound
Material Constraint 2.4 0 3000 1440 4640
Labor Constraint 0 328 1000 672 Infinity
Product 2 Constraint -8.8 0 120 0 250
2. (a) Determine the optimal solution and the 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 three 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 3 changes from $24 per unit to $50 per unit, what will be the optimal solution? What will be the new total profit? (Note: Answer this question by using the ranging results. 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 ranging results given above. Do not solve the problem again).
4. A brokerage firm has just been instructed by one of its clients to invest $500,000 of her money. The analysts at the brokerage firm are considering the following options for investment:
Investment Option Projected Rate of Return (%)
Municipal bonds 2.5
Company A stocks 11.0
Company B stocks 9.2
Company C stocks 8.2
The client has specified the following guidelines:
- Municipal bonds should constitute at least 25% of the money invested.
- At least 60% of the funds available should be placed in a combination of A, B, and C stocks.
- No more than 40% of the amount invested in municipal bonds should be invested in stock C.
The client's goal is to maximize total projected return on investments.
Formulate a linear programming model for this investment problem.
(a) Define the decision variables.
(b) Determine the objective function. What does it represent?
(c) Determine all the constraints. Briefly describe what each constraint represents.
Note: Do NOT solve the problem after formulating.
5. A congressman is running for reelection. His campaign manager is planning the marketing campaign. She has selected four ways to advertise: television ads, radio ads, billboards, and newspaper ads. The cost of each type of ad, the audience reached by each type of ad, and the maximum number of ads available in each medium is shown in the following table:
MEDIUM COST PER AD AUDIENCE REACHED/AD MAXIMUM NUMBER OF ADS AVAILABLE
TV $5000 22,000 25
Radio $4400 18,000 14
Billboards $5200 15,000 10
Newspapers $1800 10,000 10
The campaign manager has decided that there should be at least a total of 20 ads in all four media combined. The number of ads on radio must be at least as many as the number of ads on TV. The budget for advertising has been set at $100,000. How many ads of each type should be placed to maximize the total audience reached?
Formulate a linear programming model by determining
(a) Define the decision variables.
(b) Determine the objective function. What does it represent?
(c) Determine all the constraints. Briefly describe what each constraint represents.
Note: Do NOT solve the problem after formulating.
6. A company freezes fresh vegetables at three plants and then ships them to three retailers. The cost (in cents) of shipping one pound of frozen corn from a plant to a retailer, the capacity at each plant (in pounds), and the demand at each retailer (in pounds), are summarized below:
Cost
Retailer A Retailer B Retailer C Capacity __________________________________________________________________
Plant 1 7 10 8 15000
Plant 2 6 11 12 17000
Plant 3 6 7 10 12500 _________________________________________________________________
Demand 12500 16000 10000
Due to some road repair problems, the corn cannot be shipped from plant 3 to retailer B. The management of the company has decided to ship at least 8000 pounds of frozen corn from plant 1 and plant 2 to retailer C. Formulate a linear programming model to minimize the total cost of shipping for this transportation problem.
(a) Define the decision variables.
(b) Determine the objective function. What does it represent?
(c) Determine all the constraints. Briefly describe what each constraint represents.
Note: Do NOT solve the problem after formulating.