Questions are attachedInstructions: 1. No graphing calculator is allowed. Please DO NOT USE EXCEL OR ANY COMPUTER SOFTWARE OR WEBSITE to solve the problems or to draw graphs. If you answer any question using a graphing calculator, software or a website, you will not receive any credit for it. 2. Show all work. Just writing the final answer will not get you full credit. 3.Make sure you have answered all six questions. 4. Please submit the exam in Microsoft Word. 1. (a) 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. (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 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.

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Business Applications of Decision Sciences

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OPRE 315

Business Applications of Decision Sciences

Instructions:

1. No graphing calculator is allowed. Please DO NOT USE EXCEL OR ANY COMPUTER

SOFTWARE OR WEBSITE to solve the problems or to draw graphs. If you answer any question

using a graphing calculator, software or a website, you will not receive any credit for it.

2. Show all work. Just writing the final answer will not get you full credit.

3. Make sure you have answered all six questions.

4. Please submit the exam in Microsoft Word.

1. (a) 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.

(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 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.

1

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

Maximize 8C + 10R + 7T

Subject to

7C + 10R + 5T = 2000

2C + 3R + 2T = 660

C = 200

R = 300

T = 150

And C, R, T = 0

Total profit

Production budget constraint

Labor hours constraint

Maximum demand for clocks constraint

Maximum demand for radios constraint

Maximum demand for toasters constraint

Non-negativity constraints

Where C = number of clocks to be produced

R = number of radios to be produced

T = number of toasters to be produced

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

Solution List:

Variable

Status

C

Basic

R

NONBasic

T

Basic

slack 1

NONBasic

slack 2

Basic

slack 3

Basic

slack 4

Basic

slack 5

NONBasic

Optimal Value (Z)

Value

178.57

0

150

0

2.86

21.43

300

0

2478.57

2

Linear Programming Results:

C

Maximize

8

Constraint 1

7

Constraint 2

2

Constraint 3

1

Constraint 4

0

Constraint 5

0

Solution

178.57

R

10

10

3

0

1

0

0

T

7

5

2

0

0

1

150

Ranging Results:

Variable

Value

C

178.57

R

0

T

150

Reduced Cost

0

1.43

0

Slack/Surplus

0

2.86

21.43

300

0

Constraint 1

Constraint 2

Constraint 3

Constraint 4

Constraint 5

Dual Value

1.14

0

0

0

1.29

RHS

Dual

<=
<=
<=
<=
<=
2000
660
200
300
150
2478.57
1.14
0
0
0
1.29
Original Val
8
10
7
Lower Bound
7
-Infinity
5.71
Upper Bound
9.8
11.43
Infinity
Original Val
2000
660
200
300
150
Lower Bound
750
657.14
178.57
0
120
Upper Bound
2010
Infinity
Infinity
Infinity
155
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 a clock, a radio and a toaster?
(b) Find the dual prices of the five constraints and interpret their meanings. Determine the ranges in which
each of these dual prices is valid.
(c) If the profit contribution of a clock changes from $8 to $9, 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 campus dietitian for a college wants to formulate a nutritious meal plan for students. She wants to satisfy
the following requirements:
- at least 2400 calories
- at most 2600 calories
- no more than 60 grams of fat
- at least 28 grams of protein
- no more than 100 grams of carbohydrates.
On a particular day, her food stock includes five items that can be prepared and served to meet these
requirements. The cost per pound for each food item and the contribution to each of the four nutritional
requirements are given in the table below.
3
Food
Beef
Chicken
Beans
Arugula
Potatoes
Calories
/pound
400
360
130
100
320
Fat
grams/pound
12.3
10.5
1.1
1.0
0.8
Protein
grams/pound
70
65
14
12
6
Carbohydrates
grams/pound
0
0
40
22
75
Cost
Dollars/pound
2.20
2.50
0.85
2.40
0.50
Formulate a linear programming model that meets these restrictions and minimizes the total cost of the meal
by determining
(a) The decision variables. (Hint: There are 5 decision variables.)
(b) The objective function. What does it represent?
(c) All the constraints. Briefly describe what each constraint represents. (Hint: There are 5 constraints in
addition to non-negativity constraints.)
Note: Do NOT solve the problem after formulating.
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 $2.5 million for investment. The research
proposal data are shown in the following table.
Maximum Investment
Annual Return
Project
Allowed
on Investment
______________________________________________________________
Methanol fuel research
$800,000
2.3%
Emission reduction
$500,000
2.2%
Solar cells
$700,000
2.8%
Windmills
$620,000
2.6%
______________________________________________________________
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 programming model for the above situation.
(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.
4
6. A software development company wants to assign 5 software engineers to four projects. The estimated time
of completion (in days) of each project by each software engineer is given in the following table.
Project 1 Project 2 Project 3
Project 4
___________________________________________________________
Software Engineer A
30
25
17
14
Software Engineer B
40
30
22
32
Software Engineer C
38
19
22
Software Engineer D
32
25
15
19
Software Engineer E
35
24
15
20
___________________________________________________________
Software engineer C cannot be assigned to project 2 because she does not have enough training to do that
project. The company wants project 4 to be assigned to software engineer A or software engineer B. The
objective is to minimize the total time of completion of all four projects.
For this assignment 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
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