## Linear Programming $25

You can use a calculator to do numerical calculations. No graphing calculator is allowed. Please DO NOT USE ANY COMPUTER SOFTWARE to solve the problems.

1. (a) 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.

(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 are the differences between QM for Windows and Excel when solving a linear programming problem? Which one you like better? Why?

(d) What are the dual prices? In what range are they valid? Why are they useful in making recommendations to the decision maker? Give a real world example.

**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

10P1 + 12P2 + 10P3 + 8P4 ≤ 3200 Production budget constraint

4P1 + 3P2 + 2P3 + 3P4 ≤ 1000 Labor hours constraint

5P1 + 4P2 + 3P3 + 3P4 ≤ 1200 Material constraint

P1 > 100 Minimum quantity needed for Product 1 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 100

P2 NONBasic 0

P3 Basic 220

P4 NONBasic 0

slack 1NONBasic 0

slack 2Basic 160

slack 3Basic 40

surplus 4 NONBasic 0

Optimal Value (Z) 34400

Original problem w/answers:

P1 P2 P3 P4 RHS Dual

Maximize 80 100 120 70

Constraint 1 10 12 10 8 <= 3200 12

Constraint 2 4 3 2 3 <= 1000 0

Constraint 3 5 4 3 3 <= 1200 0

Constraint 4 1 0 0 0 >= 100 -40

Solution-> 100 0 220 0 Optimal Z-> 34400

Ranging Results:

Variable Value Reduced Cost Original Val Lower Bound Upper Bound

P1 100 0 80 -Infinity 120

P2 0 44 100 -Infinity 144

P3 220 0 120 87.5 Infinity

P4 0 26 70 -Infinity 96

Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound

Constraint 1 12 0 3200 1000 3333.333

Constraint 2 0 160 1000 840 Infinity

Constraint 3 0 40 1200 1160 Infinity

Constraint 4 -40 0 100 0 120

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 $130 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. The Portfolio Manager of Charm City Pension Planners, Inc., has been asked to invest $1,000,000 of a large pension fund. The management of the company has identified five mutual funds as possible investment options. The details of these five mutual funds are given below:

## Mutual Fund

1 2 3 4 5

## Annual return (in dollars) 12% 10% 8.5% 10% 11%

### Risk amount (in dollars) 9.8% 8% 7.2% 7.1% 7.3%

To control the risk, the management of the company has specified that the total risk amount cannot exceed $200,000. In addition, the management wants to invest at least $150,000 in mutual fund 2 and at least $125,000 in mutual fund 3.

With these restrictions, how much money should the portfolio manager of the company invest in each mutual fund so as to maximize the total annual return?

(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.

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**Note: Do NOT solve the problem after formulating.**

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5. A charity wants to contact people to collect donations. A person can be contacted morning or evening, by phone, or door-to-door. The average donation resulting from each type of contact is given below:

Phone Door-to-Door

____________________________________

Morning $35 $60

Evening $40 $70

The Charity has 150 volunteer hours in the morning and 120 volunteer hours in the evening. Each phone contact takes 6 minutes and each door-to-door contact takes 15 minutes to conduct. The Charity wants to have at least 550 phone and at least 400 door-to-door contacts.

Formulate a linear programming model that meets these restrictions and maximizes the total average donations by determining

(a) The decision variables.

(b) Determine the objective function. What does it represent?

(c) Determine all the constraints. Briefly describe what each constraint represents.

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**Note: Do NOT solve the problem after formulating**

** **

6. The Charm City Truck Rental Inc. has accumulated extra trucks at three of its truck leasing outlets, as shown in the following table:

**Leasing Outlet Extra Trucks**

1. Atlanta 70

2. St. Louis 115

3. Greensboro 60

The firm also has three outlets with shortages of rental trucks, as follows:

**Leasing Outlet Truck Shortage**

A. New Orleans 80

B. Cincinnati 50

C. Baltimore 45

The firm wants to transfer trucks from those outlets with extras to those with shortages at the minimum total cost. The following costs of transporting these trucks from city to city have been determined:

**To (cost in dollars) **

**From A B C**

1 75 80 45

2 115 50 55

3 100 60 40

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.

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**Note:** **Do NOT solve the problem after formulating.**