```MGMT 306 Fall 2009

Final Exam: Tuesday, December 15th, from 8:00 – 10:00 AM in Lambert F101. The
seating assignment will be announced very soon.
 You may bring 3 sheets of "crib" notes (double-sided). Standard normal distribution table
will be provided, if needed.
 Bring your student ID, because we will check your ID during the final exam.
 Topics included in the Final Exam are:
1. Conceptual Understanding of LP.
2. LP Sensitivity Analysis and Excel Solver Output Interpretation.
3. LP Formulation
4. IP Formulation
5. Nonlinear Programming
6. Simulation modeling using @Risk.
7. Project Management
8. Decision Analysis including PrecisionTree.
9. Multi-criteria Decision Modeling and Solution
 The Graphical Solution method will not be on the exam.
Question 1:
A textile manufacturer produces two types of cotton cloth: corduroy and denim. Corduroy is a
heavier grade of cotton cloth and, as such, requires 7.5 pounds of raw cotton to produce one
yard, whereas denim requires 5 pounds of raw cotton per yard. A yard of corduroy requires 3.2
hours of processing time, and a yard of denim requires 3 hours. Although the demand for denim
is practically unlimited, the maximum demand for corduroy is 510 yards per month. The textile
manufacturer has a maximum of 6500 pounds of cotton available each month due to limited
storage capacity and 3000 hours of processing time available each month.
The processing time each month is a sunk cost. The cost of cotton is 10 cents per pound, and it
is considered a variable cost accounting for only the amount of cotton used.
The
manufacturer sells the cloth for \$3.85 per yard for corduroy and \$2.75 per yard for denim. The
manufacturer wants to know how many yards of each type of cloth to produce each month to
maximize profit.
a. Clearly define the decision variables you would use to solve this problem.
b. Formulate a linear program for the above problem using only the variables defined in a.
c. Due to a new process each type of cloth must undergo after being manufactured, at most 40%
of the total production can be corduroy. Make the necessary changes in the original
formulation to represent this restriction. Define any additional variables if you need them.
d. An agreement has been reached with the cotton supplier that enables the textile manufacturer
to use more than 6500 pounds of cotton in a month. According to the agreement, cotton in
excess of 6500 pounds can be stored in the supplier's depot during the month. The supplier
will charge 5 cents per pound for storage for the month and will store up to 4000 pounds each
month. Make the necessary changes in the original formulation to represent this situation.
Define any additional variables if you need them.
Question 2:
1
The Tots Toys Company is trying to schedule production of two very popular toys for the next
three months: a rocking horse and a scooter. Information about both toys is given below.
Toy
Rocking Horse
Scooter
Summer Schedule
June
July
August
Beginning
Inventory on
June 1st
25
55
Required
Plastic
Per Toy
5
4
Required
Time
Per Toy
2
3
Production
Cost
Per Toy
12
14
Plastic
Available
3500
5000
4800
Time
Available
2100
3000
2500
Monthly Demand
Horse
220
350
600
Inventory
Cost
Per Toy
1
1.2
Monthly Demand
Scooter
450
700
520
Develop a model that would tell the company how many of each toy to produce during each
month. You are to minimize total cost. Inventory cost will be levied on any items in inventory
on June 30, July 31, or August 31 after demand for the month has been satisfied. The company
wants to end the summer with 150 rocking horses and 60 scooters as beginning inventory for
Sept. 1. Don't forget to define your decision variables.
2
Question 3:
No-name B-School is trying to decide how many of each type of computer to buy for its
computer labs. The Appease (A) model can only be used for spreadsheets and word processing
and costs \$1400. The Basic (B) model can be used for spreadsheets, word processing and
databases and costs \$2000. The Capture (C) model can only be used for multimedia and costs
\$2500. Finally, the Doitall (D) model can do any task and costs \$3500. Due to class
enrollments, the school must have at least 100 machines capable of doing spreadsheets and word
processing, at least 50 machines capable of doing databases, and at least 30 machines capable of
doing multimedia. The dealer the school is buying from has only 45 of model A available. The
school wishes to minimize their total purchase cost. The following is an LP formulation for the
problem with solution and sensitivity analysis.
Let A=number of Appease model purchased
Let C=number of Capture model purchased
Minimize 1400A+2000B+2500C+3500D
Subject to
A +B
+ D  100
B
+ D  50
C
+ D  30
A
 45
A, B, C  0
Let B=number of Basic model purchased
Let D=number of Doitall model purchased
(constraint for database course)
(constraint for multimedia course)
(number of model A’s available)
Target Cell (Min)
Cell
Name
\$G\$3 Obj. (Min)
Original Value Final Value
0
218000
Cell
Name
\$B\$2 A
\$C\$2 B
\$D\$2 C
\$E\$2 D
Original Value Final Value
0
45
0
25
0
0
0
30
Constraints
Cell
Name
\$G\$4 Sp/wp
\$G\$5 Dbase
\$G\$6 Multimedia
\$G\$7 Model A
Cell Value
Formula
100\$G\$4>=\$I\$4
55\$G\$5>=\$I\$5
30\$G\$6>=\$I\$6
45\$G\$7<=\$I\$7
Status
Slack
Binding
0
Not Binding
5
Binding
0
Binding
0
3
Sensitivity Report
Cell
\$B\$2
\$C\$2
\$D\$2
\$E\$2
Name
A
B
C
D
Final Reduced Objective Allowable Allowable
Value
Cost
Coefficient Increase Decrease
45
0
1400
600
1E+30
25
0
2000
1500
600
0
1000
2500
1E+30
1000
30
0
3500
1000
1500
Constraints
Cell
\$G\$4
\$G\$5
\$G\$6
\$G\$7
Final Shadow Constraint Allowable Allowable
Name
Value Price
R.H. Side Increase Decrease
Sp/wp
100
2000
100
1E+30
5
Dbase
55
0
50
5
1E+30
Multimedia
30
1500
30
25
30
Model A
45
-600
45
5
45
a. If enrollment in the database course is limited to 45 students instead of the original 50, what
would be the new total purchase costs for all the computers?
b. If the dealer could supply three additional model A’s, what would be the new total purchase
costs for all the computers?
c. If the dealer agrees to lower the price for Model C computers, at what new lower price would
the B-school consider buying any Model C computers?
d. The dealer has warned that the price of Model D computers may actually increase before the
purchase order goes through. At what higher price would the school consider changing the
order?
e. A local community college would like to offer a 60 person multimedia course in the Bschool’s labs, so the school will need 60 instead of 30 multimedia computers. The
community college is willing to make a one-time payment of \$30,000 to do this. Assuming
the B-school gets no other advantages than this one time payment, should the school take the
offer? Why or why not?
4
Question 4:
A product manager for a soap manufacturer must decide whether or not to offer a new,
biodegradable laundry detergent. The projected profit from a successful detergent is \$2 million,
whereas failure of the product would result in a loss of \$1 million. The manager currently thinks
there is a 40% chance that the product will be successful. Not offering the product would not
change profits.
a. Construct a decision tree for this decision.
 What is the optimal strategy?
 What is the EMV of this strategy?
 What is the risk profile?
 What is the EVPI?
The manager also has the opportunity to test the product before taking it to market. At a cost of
\$100,000, the product can be tested. Consumer testing can be favorable, a 50% chance, or
unfavorable. Given a favorable test result, the chance of product success is judged to be 80%.
However, for an unfavorable test result, the chance of product success is judged to be only 30%.
b. Construct a decision tree in PrecisionTree for this problem.
 What is the optimal strategy and its expected value?
 What is the risk profile for this strategy?
Question 5:
Remington Manufacturing is planning its next production cycle. The company can produce
three products, each of which must undergo machining, grinding and assembly operations.
The following table summarizes the hours of machining, grinding and assembly required by
each unit of each product, and the total hours of capacity available for each operation.
Operation
Machining
Grinding
Assembly
Hours Required by
Product 1
Product 2
2
6
5
Product 3
3
3
6
6
4
2
Total hours
Available
600
300
400
The per unit profit from each of the products and the setup costs are listed in the table below:
Profit per unit
Setup cost
Product 1
\$48
\$1000
Product 2
\$55
\$800
Product 3
\$50
\$900
Since there is a heavy demand for the products, the marketing department believes that all the
products produced within the available capacities can be sold. The management of
Remington wants to determine the most profitable mix of products to produce. Formulate an
integer-programming model on behalf of Remington. Define any variables you use and
clearly specify the objective function and constraints.
5
Question 6:
Building a backyard swimming pool consists of six major activities. According to the knowledge
of activities and their immediate predecessors, the project network is drawn below:
A
C
E
START
FINISH
B
D
F
Assume that the activity time estimates (in days) for the swimming pool construction project are
Activity
A
B
C
D
E
F
Optimistic
hidden
hidden
hidden
hidden
1
hidden
Most Probable
Hidden
hidden
hidden
hidden
2
hidden
Pessimistic
hidden
hidden
hidden
hidden
3
hidden
Expected Time
5
4
6
9
8
Variance
0.5
0.6
0.1
0.3
0.2
(a) What are the expected time and variance for activity E? (fill them into the above table)
(b) What are the critical activities?
(c) What is the expected time to complete the project?
(d) What is the probability that the project can be completed in 21 or fewer days?
6
Question 7:
Office Automation, Inc. has developed a proposal for introducing a new computerized office
system that will improve word processing and interoffice communications for a particular
company. Contained in the proposal is a list of activities that must be accomplished to complete
the new office system project. The following information about the activities is given.
Activity
A
B
C
D
E
F
Immediate
Description
Predecessor
Plan needs
Order equipment
A
Install equipment
B
Set up training lab
A
Conduct training lab
D
Test system
C, E
Time (weeks)
Normal
Crash
10
8
8
6
10
7
7
6
10
8
3
3
Cost (\$1000s)
Normal Crash
30
70
120
150
100
160
40
50
50
75
60
-
The length of the critical path for the above project is 31 weeks. However, the company wants to
complete the project in 26 weeks. Formulate a linear programming model that could be used in
making the crashing decisions.
Question 8:
EasyRent just started its DVD rental business. It charges \$3 for overnight rental of new movies
and guarantees that every new DVD they have is never out of stock. If a customer is not able to
rent a DVD, EasyRent offers three free rentals as compensation. The estimated cost of this
compensation is \$8. Not every customer asks for the free rentals for compensation. EasyRent
estimates that 80% of customers will ask for the free rental offer if the movie is not on the shelf.
Recently, a new action movie is just about to be launched on DVD. According to historical data
about action movies, the daily demand during the first month of release is a normal distribution
with mean 130 and standard deviation 15. EasyRent has to pay \$60 per copy of DVD for right to
rent out the movie during its first month of release. Considering the stock levels of 120, 130,
140, 150, 160, and 170 use simulation approach to decide which stock level EasyRent should
have in order to maximize its profit during the first month (30 days.)
a. Create an @RISK model for the first month of rentals to simulate net monthly
income for the DVD for each stock level.
b. How many copies of the DVD should they stock?
7
Multiple Choice Questions:
1. Which of the following statements is correct?
a. Sensitivity analysis of a linear program investigates how the optimal solution changes
when the value of a decision variable changes.
b. When the objective function coefficient of a variable changes, the current optimal
solution may become infeasible.
c. If a constraint is not binding at the current optimal solution, then its shadow price is zero.
d. As long as the objective function coefficient of a variable is in its optimality range, the
optimal objective function value remains the same.
2. The earliest start time for an activity is:
a. Based on the length of the critical path
b. Determined by the maximum of the earliest finish times of its immediate predecessors
c. The same as the latest start time of its immediate predecessor
d. None of the above
3. The project management strategy of injecting additional resources in order to reduce the length of
the project is called:
a. Rushing
b. Panicking
c. Crashing
d. None of the above
4. Which of the following are ways of crashing an activity?
a. Using overtime
b. Hiring more workers
c. Using specialized equipment
d. All of the above
5. Which of the following is not part of PERT?
a. Uncertain activity durations.
b. A list of activities and precedence relations.
c. Activities overlapping with their immediate predecessors.
d. A calculation of expected project completion time.
6. Which of the following statements related to decision trees are correct?
I The branches emanating from each decision node represent the precedence relationships
between decision alternatives.
II The sum of probabilities of each state of nature (outcome) branch emanating from a given
event node must be equal to one.
III Nodes of a decision tree can be ordered arbitrarily.
a.
b.
c.
d.
Only I
Only II
I and II
I, II and III
8
Consider the following for Questions 7 - 11.
The Big Ten Company (BTC) is considering expanding production to meet increases in demand. BTC has
three alternatives, and the payoffs for each alternative under various market situations are presented in the
table below:
Payoff Table (profit in \$1000s)
Market Situations
A
B
C
Alternative 1
100
120
180
Alternative 2
200
100
60
Alternative 3
120
140
120
7. If BTC takes a conservative approach, the choice will be:
a. alternative 1 and/or 2.
b. alternative 1.
c. alternative 2.
d. alternative 3.
8. Which one of the following is correct?
a. The regret for alternative 3 under state A is 100.
b. The regret for alternative 3 under state B is 40.
c. The maximum regret for alternative 3 is 140.
d. The maximum regret for alternative 3 is 80.
9. Under Minimax regret approach, BTC should select:
a. alternative 1.
b. alternative 2.
c. alternative 3.
d. alternative 2 and/or alternative 3.
For questions 10 and 11, refer to the above payoff matrix, and suppose the probabilities for the three
market situations A, B, and C are 0.3, 0.5, and 0.2, respectively.
10. To maximize the expected profit, BTC should select:
a. alternative 1.
b. alternative 2.
c. alternative 3.
d. alternative 2 and/or alternative 3.
11. Suppose BTC could hire a consultant who could predict the future with 100% accuracy. Which
one of the following is correct?
a. With such perfect information, BTC’s expected payoff would be 130.
b. BTC would be willing to pay at most \$44 to the consultant.
c. The value of the perfect information provided by the consultant is 36.
d. The information provided by the consultant will not increase BTC’s expected profit.
9
12. Which one of the following is correct about goal programming models?
a. In a goal-programming model with preemptive priorities, we always permit tradeoffs
between higher and lower level goals.
b. There can only be one goal at each priority level.
c. Deviation variables represent the difference between the target value for the goal and the
level achieved.
d. The priority levels of the goals are represented in the goal equations.
13. The Royal Seas Company wants to run TV ads promoting its Caribbean cruises to high-income
men, high-income women, and retirees. The company is considering four ad campaigns, namely A, B,
C, D, and would like to select a campaign with the objectives of minimizing the cost of the campaign,
and maximizing estimated exposures to the target audience.
Campaign
A
B
C
D
Total Cost (\$)
130,000
120,000
80,000
90,000
Estimated exposures
to the target audience
7 million
8 million
2 million
1 million
The Royal Seas Company can eliminate from consideration campaigns:
a.
A and B
b.
B and C
c.
C and D
d.
A and D
10
Consider the following for Questions 14, 15 and 16.
The Lofton Company, a distributor of exercise equipment, wants to decide on how many units to
order from two of the most popular models. The company has developed the following linear
program to maximize its profits, but found out that it is infeasible.
x = Number of units to be ordered from model 1
y = Number of units to be ordered from model 2
Max
s.t.
Storage constraint
Budget constraint
Demand constraint
5x
+
3y
2x
2x
x
+
+
+
x,
y
3y
y
y




10
24
16
0
In revision, Lofton drops the original objective and establishes the following three goals in order of
importance:
Priority 1, Goal 1: Don’t exceed 10 in the storage constraint.
Priority 2, Goal 2: Don’t exceed 24 in the budget constraint.
Priority 3, Goal 3: Don’t fall short of 16 in the demand constraint.
Let di+ be the over-achievement of goal i, and let di- be the under-achievement of goal i.
14. Converting the inequality 2x + y  10 into a goal equation will result in the following:
a. 2x + y + d1+ - d1- = 10
b. 2x + y + d1- - d1+ = 10
c. 2x + y + d1- - d1+  10
d. 2x + y + d1+ - d1-  10
15. Which one of the following is the goal equation for goal 3?
a. x + y + d3+ - d3- = 16
b. x + y + d3- - d3+ = 16
c. x + y + d3+ = 16
d. x + y - d3- = 16
16. What will be the new objective function?
a. Min P1 (d1-) + P2 (d2-) + P3 (d3+ )
b. Max P1 (d1-) + P2 (d2-) + P3 (d3+ )
c. Min P1 (d1+) + P2 (d2+) + P3 (d3- )
d. Min P1 (d1+) + P1 (d2+) + P2 (d3- )
11
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