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Homework 5: OPRE 315

Please answer the following questions giving all details.

1. The Charming City Engineering Company is considering five projects for the next year. The

number of engineers and the number of support personnel required for each project, and the

expected profit for each project are given below.

Project

1

2

3

4

5

_________________________________________________

Engineers Required

28 26 36 31 45

Support Personnel Required

26 28 38 46 36

Profit (in thousands of dollars) 250 320 250 210 350

Formulate a capital budgeting problem that maximizes the company’s total profit subject to the

following constraints:

– Use no more than 120 engineers

– Use no more than 150 support personnel

– If project 1 is done, then project 2 must be done

– At most four projects can be selected

(a) Define the decision variables.

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

(c) Determine all the constraints. What does each constraint represent?

Note: Do NOT solve the problem after formulating.

2. The director of career advising at the Charm City Community College wants to use decision

analysis to provide information to help students decide which 2-year degree program they should

pursue. The director has set up the following payoff table for four of the most popular and

successful degree programs at the college that shows the estimated 5-year gross income (in

thousands of dollars) from each degree for three future economic conditions:

Degree Program

Graphic design

Nursing

Real estate

Medical technology

Economic Condition

Recession

Average

Good

165

185

185

145

205

225

135

205

245

175

175

175

Determine which 2-year degree program to pursue and the profit associated with it by finding the

optimal decision using the following decision criteria:

a. Maximax

b. Maximin

c. Equal likelihood

d. Minimax regret

1

3. For the problem given in Question 2, the probabilities for the economic conditions are given by

P(Recession) = 0.2, P(Average) = 0.4, and P(Good) = 0.4.

a. Compute the expected value for each decision and select the best one.

b. Compute the expected regret value for each decision and select the best one.

c. Calculate and interpret the expected value of perfect information.

4. Solve the following integer programming model by using a computer software:

Maximize Z = 5×1 + 6×2

Subject to

3×1 + 4×2 < 10
4x1 + 2x2 < 15
x1, x2 > 0 and integer

5. Juan Hernandez, a Cuban athlete who visits the United States and Europe frequently, is

allowed to return with a limited number of consumer items not generally available in Cuba. The

items, which are carried in a duffel bag, cannot exceed a weight of 8 pounds. Once Juan is in

Cuba, he sells the items at highly inflated prices. The three most popular items in Cuba are denim

jeans, CD players, and CDs of U.S. rock groups. The weight and profit (in U.S. dollars) of each

item are as follows:

Item

Weight (lb.)

Profit

Denim jeans

2.2

$75

CD players

3.1

110

Compact discs

1.2

20

Juan wants to determine the combination of items he should pack in his duffel bag to maximize

his profit. This problem is an example of a type of integer programming problem known as a

“knapsack” problem. Formulate this problem by defining the decision variables, objective

function, and all the constraints. Do NOT solve after formulating.

6. Consider a capital budgeting example with five projects from which to select. Let Xi = 1 if

project i is selected, 0 if not, for i = 1, 2, 3, 4, 5. Write the appropriate constraint(s) for each of

the following conditions. Conditions are independent of one another.

(a) Select at least four projects.

(b) If project 2 is selected, then project 4 must also be selected.

(c) Projects cost 160, 200, 100, 225, and 175 dollars respectively. The budget is 500 dollars.

(d) No more than three of the five projects can be selected.

7. The Charm City Consulting has five projects to consider. Each will require employee time

(weeks) in the next two months according to the table below. Profit from each project is also

shown in the table below.

Project Weeks required in Month 1

Weeks required in Month 2

Profit in dollars

_______________________________________________________________________

1

5

10

20000

2

7

9

21000

3

6

8

10000

4

6

7

23000

5

4

7

12500

There is 25 weeks employee time available in the first month and 30 weeks in the second

month.

2

Formulate a zero-one capital budgeting integer programming problem to maximize total profit for

this situation by defining the decision variables, objective function, and all the constraints.

Do NOT solve after formulating.

3

…

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