BA 452
Quantitative Analysis
Final Exam
This is a 150-minute exam (2hr. 30 min.). There are 6 questions (25
minutes per question). To avoid the temptation to cheat, you must abide by
these rules then sign below after you understand the rules and agree to
them:
 Turn off your cell phones.
 You cannot leave the room during the exam, not even to use the
restroom.
 The only things you can have in your possession are pens or pencils
and a simple non-graphing, non-programmable, non-text calculator.
 All other possessions (including phones, computers, or papers) are
prohibited and must be placed in the designated corner of the room.
 Possession of any prohibited item (including phones, computers, or
papers) during the exam (even if you don’t use them but keep them
in your pocket) earns you a zero on this exam, and you will be
reported to the Academic Integrity Committee for further action.
Print name here:______________________________________________
Sign name here:______________________________________________
Each individual question on the following exam is graded on a 4-point
scale. After all individual questions are graded, I sum the individual scores,
and then compute that total as a percentage of the total of all points
possible. I then apply a standard grading scale to determine your letter
grade: 90-100% A; 80-89% B; 70-79% C; 60-70% D; 0-59% F
Finally, curving points may be added to letter grades for the entire class (at
my discretion), and the resulting curved letter grade will be recorded on a
standard 4-point numerical scale.
Tip: Explain your answers. And pace yourself. When there is only ½ hour
left, spend at least 5 minutes outlining an answer to each remaining
question.
For all of the exam, you may only use blank or graph paper, pencils,
and a calculator. You may not use a computer or notes.
1
BA 452
Quantitative Analysis
Final Exam
Manipulating Alternatives to No Agreement
Question 1. Consider negotiations over wages for
adjunct Management Science Professors at the
Business Administration Division of Seaver College of
Pepperdine University. Pepperdine seeks a professor
for the next 5 semesters. The only candidate (Charlie) is willing to work for
as little as $5,000 per semester and Pepperdine is willing to pay up to
$35,000 per semester. Before the first semester, Pepperdine confronts the
candidate Charlie over wages. Pepperdine presents its wage offer.
Charlie either accepts it or rejects it and returns the next semester with a
counteroffer. Offers alternate thereafter. Employment only occurs in those
semesters after an agreement is reached.
What wage per semester should Pepperdine offer to Charlie? Should
Charlie accept that initial offer?
Now suppose candidate Charlie finds an alternative employment offer by
UCLA, who makes Charlie a take-it-or-leave-it offer of $15,000 for each of
the same 5 semesters. Suppose the candidate cannot accept the job at
both UCLA and Pepperdine University. Re-compute Pepperdine’s initial
wage offer to Charlie.
Finally suppose, in addition to the alternative employment offer to
candidate Charlie by UCLA described above, the problem changes
because Pepperdine finds an alternative candidate Chris who makes
Pepperdine a a take-it-or-leave-it offer to work for $25,000 per semester for
each of the same 5 semesters. Re-compute Pepperdine’s initial wage offer
to Charlie.
Answer to Question:
2
BA 452
Quantitative Analysis
Final Exam
Answer to Question:
To consider all possible wage offers (offers by Pepperdine to buy labor by
the candidate to sell labor), consider a bargaining payoff table. The game
ends if an offer is accepted or if the semesters end without an accepted
offer, and gains are measured as a percentage of the gain from the
candidate accepting Pepperdine’s offer before the first semester.
Rounds t o
End of
Game
Offer by
T ot al Gain
t o Divide
P's Gain
Offered
C's Gain
Offered
1
P
20
20
0
2
C
40
20
20
3
P
60
40
20
4
C
80
40
40
5
P
100
60
40
Without the alternative offer, the candidate’s opportunity cost of Pepperdine
employment is $5,000 per semester, and so the gain from Pepperdine
employment is $30,000=$35,000-$5,000 per semester. For Pepperdine to
receive 60% of those gains, Pepperdine gains $18,000 per semester,
which means offer wages $17,000=$35,000-$18,000 per semester, or
$85,000 for all five semesters. And Charlie should accept that initial offer.
With the alternative employment offer to candidate Charlie, the candidate’s
opportunity cost of Pepperdine employment is $15,000 per semester, and
so the gain from Pepperdine employment is $20,000=$35,000-$15,000 per
semester. For Pepperdine to receive 60% of those gains, Pepperdine
gains $12,000 per semester, which means offer wages $23,000=$35,000$12,000 per semester, or $115,000 for all five semesters. And Charlie
should accept that initial offer.
Comment: Charlie finding alternative employment raised candidate
Charlie’s Pepperdine wages.
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BA 452
Quantitative Analysis
Final Exam
With the alternative employment offer to candidate Charlie and the
alternative employment offer from candidate Chris, candidate Charlie’s
opportunity cost of Pepperdine employment is $15,000 per semester and
Pepperdine’s value to Charlie’s employment is $25,000 per semester. So,
the gain from Pepperdine employment is $10,000=$25,000-$15,000 per
semester. For Pepperdine to receive 60% of those gains, Pepperdine
gains $6,000 per semester, which means offer wages $19,000=$25,000$6,000 per semester, or $95,000 for all five semesters. And Charlie
should accept that initial offer.
Comment: Pepperdine finding an alternative candidate lowered candidate
Charlie’s Pepperdine wages.
4
BA 452
Quantitative Analysis
Final Exam
Sensitivity to Constants
Question 2. The Ford Motor Company makes cars
and trucks on an assembly line. Each car and truck
requires labor and needs to visit three stations:
engine installation, hood installation, and wheel
installation. The hours of labor and hours required at each station are as
follows:
Vehicle
Car
Truck
Labor
4
3
Engine
Installation
6
8
Hood
Installation
4
5
Wheel
Installation
3
4
There are available 12 hours of labor, 48 hours of engine installation, 20
hours of hood installation, and 12 hours of wheel installation. Labor costs
$20 per hour, and each installation station costs $10 per hour to operate.
Cars sell for $710 each, and trucks sell for $530 each.
a. Formulate and graphically solve for the recommended production
quantities. Do not require production units to be integers.
b. How much should Ford be willing to pay to another potential worker
(Joe) to supply one more unit of labor. (Joe is not part of the 12
hours of labor mentioned above.)
Answer to Question:
5
BA 452
Quantitative Analysis
Final Exam
Answer to Question:
a. The profit contributions are $500 for each car, and $300 for each
truck.
Let C = cars produced.
Let T = trucks produced.
Max 500C + 300T
s.t. 4C + 3T < 12
6C + 8T < 48
4C + 5T < 20
3C + 4T < 12
C, T  0
(labor)
(engine)
(hood)
(wheel)
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
A graph of the feasible set shows the engine and hood constraints are
redundant, and the iso-value lines show the optimal solution occurs where
the labor constraint and the non-negativity of T bind (C = 3 and T = 0).
6
BA 452
Quantitative Analysis
Final Exam
b. How much should Ford be willing to pay to change the problem by
having Joe supply one more unit of labor.
Repeat analysis after changing the right hand side of the first constraint
from 12 to 13:
Max 500C + 300T
s.t. 4C + 3T < 13
6C + 8T < 48
4C + 5T < 20
3C + 4T < 12
C, T  0
(labor)
(engine)
(hood)
(wheel)
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
The new graph of the feasible set shows the engine and hood constraints
are redundant, and the iso-value lines show the optimal solution occurs
where the labor constraint and the non-negativity of T bind (C = 13/4 = 3.25
and T = 0). That new solution yields profit 500C + 300T = 1625, which is
125 higher than the previous profit of 1500. So, Ford should be willing to
pay up to 145 = 20+125 dollars to Joe.
7
BA 452
Quantitative Analysis
Final Exam
Make or Buy with Fixed Costs
Question 3. Danchuk Manufacturing produces a variety of classic
automobiles, including a 1955 Chevy and a 1955 Thunderbird. Each car
consists of three components that can be manufactured by Danchuk: a
body, an interior, and an engine. Both cars use the same engine, but
different bodies and different interiors.
Danchuk’s sales forecast indicates that 300 Chevys and 500 Thunderbirds
will be needed to satisfy demand during the next year. Because only 2000
hours of in-house manufacturing time is available, Danchuk is considering
purchasing some, or all, of the components from outside suppliers. If
Denchuk manufactures a component in-house, it incurs a fixed setup cost
as well as a variable manufacturing cost. The following table shows the
setup cost, the manufacturing time per component, the manufacturing cost
per component, and the cost to purchase each of the components from an
outside supplier:
Manufacturing
Purchase
Setup Cost Manufacturing Cost per Unit Cost per Unit
(thousands Time per Unit (thousands of (thousands
Component
of dollars)
(hours)
dollars)
of dollars)
Chevy Body
100
3
4
5
Thunderbird
90
2
2
3
Body
Chevy
10
1
2
3
Interior
Thunderbird
20
2
3
4
Interior
Engine
20
2
4
5
Formulate a linear program for Denchuk to minimize total cost to meet the
sales forecasts. But you need not compute an optimum.
Tip: Your written answer should define the decision variables, carefully
state which variables are continuous and which are binary, and formulate
the objective and constraints.
Answer to Question:
8
BA 452
Quantitative Analysis
Final Exam
Answer to Question:
Continuous (or integer) variables are used for the number of units made:
Let CB = the number of Chevy Bodies to make
Let TB = the number of Thunderbird Bodies to make
Let CI = the number of Chevy Interiors to make
Let TI = the number of Thunderbird Interiors to make
Let E = the number of Engines to make
Continuous (or integer) variables are also used for the number of units
purchased:
Let CBP = the number of Chevy Bodies to purchase
Let TBP = the number of Thunderbird Bodies to purchase
Let CIP = the number of Chevy Interiors to purchase
Let TIP = the number of Thunderbird Interiors to purchase
Let EP = the number of Engines to purchase
Binary variables are used to indicate whether production is positive, and
setup costs are incurred:
Let CBS = 1 if Chevy Bodies are produced; 0, if not.
Let TBS = if Thunderbird Bodies are produced; 0, if not.
Let CIS = 1 if Chevy Interiors are produced; 0, if not.
Let TIS = if Thunderbird Interiors are produced; 0, if not.
Let ES = if Engines are produced; 0, if not.
Objective: Minimize Total Cost
4CB + 2TB + 2CI + 3TI + 4E (production costs)
+5CBP + 3TBP + 3CIP + 4TIP + 5EP (purchase costs)
+100CBS + 90TBS + 10CIS + 20TIS + 20ES (setup costs)
Input Constraints:
3CB + 2TB + 1CI + 2TI + 2E < 2000 (In-house hours)
9
BA 452
Quantitative Analysis
Final Exam
Sales Constraints:
CB + CBP = 300 (manufactured or purchased bodies used for Chevys)
TB + TBP = 500 (manufactured or purchased bodies used for
Thunderbirds)
CI + CIP = 300 (manufactured or purchased interiors used for Chevys)
TI + TIP = 500 (manufactured or purchased interiors used for
Thunderbirds)
E + EP = 800 (manufactured or purchased engines used for both Chevys
and Thunderbirds)
Setup Constraints (given sales constraints imply CB < 300, TB < 500, CI <
300, TI < 500, and E < 800):
CB < 300CBS
TB < 500TBS
CI < 300CIS
TI < 500TIS
E < 800ES
10
BA 452
Quantitative Analysis
Final Exam
Economic Analysis with Teamwork
Question 4. Food on Foot operates a weekly meal
program every Sunday in Hollywood. They are
considering three alternative distribution systems.
Arrivals to the distribution stand follow the Poisson
distribution with an average of one homeless person arriving every 12
minutes. The cost for a homeless person waiting is $2 per hour.
The first system involves one volunteer (who talks to the homeless and
serves food and serves drinks) and costs $10 per hour to operate (the
opportunity cost of the volunteer’s time), and it has an exponential service
rate, and it can serve an average of one homeless person every 6 minutes.
The second system involves two volunteers working as a team (one talks to
the homeless and serves food, and the other serves drinks) and costs $20
per hour to operate (the opportunity cost of the 2 volunteer’s time), and it
has an exponential service rate, and it can serve an average of one
homeless person every 3 minutes.
The third system involves three volunteers working as a team (one talks to
the homeless, one serves food, and the third serves drinks) and costs $30
per hour to operate (the opportunity cost of the 3 volunteer’s time), and it
has an exponential service rate, and it can serve an average of one
homeless person every 1 minute.
Which system should be chosen? Re-compute your answer if the
customers value their time at $20 per hour, rather than $2 per hour.
You may use any or all of the following analytical formulas for an M/M/1
system to compute your answer:
The probability of no units in the system: P0 = 1-
The average number of units in the waiting line: Lq = 
The average number of units in the system: L = Lq + 
The average time a unit spends in the waiting line: Wq = Lq/
The average time a unit spends in the system: W = 1/
The probability that an arriving unit has to wait: Pw = 
Probability of n units in the system: Pn = ()nP0
11
BA 452
Quantitative Analysis
Final Exam
Answer to Question:
First system: k = 1 channel,  = 5 per hour, and  = 10 per hour.
The average number of units in the waiting line:
Lq =  = 5*5/(10(5)) = 1/2
The average number of units in the system: L = Lq +  = 1/2 + 1/2 = 1
Cost to operate = $10 + $2*L = $12 per hour.
Second system: k = 1 channel,  = 5 per hour, and  = 20 per hour.
The average number of units in the waiting line:
Lq =  = 5*5/(20(15)) = 1/12
The average number of units in the system: L = Lq +  = 1/12 + 1/4 = 1/3
= 0.3333
Cost to operate = $20 + $2*L = $20.66 per hour.
Third system: k = 1 channel,  = 5 per hour, and  = 60 per hour.
The average number of units in the waiting line:
Lq =  = 5*5/(60(55)) = 1/132
The average number of units in the system: L = Lq +  = 1/132 + 1/12 =
0.0909
Cost to operate = $30 + $2*L = $30.18 per hour.
Choose the first system. (Going beyond the basic answer, ask any other
volunteers beyond two to put in extra time at work and donate their income.
The homeless people will benefit more from the extra money than from the
faster service.)
At $20 per hour for customers,
First system cost to operate = $10 + $20*(1) = $30 per hour.
Second system cost to operate = $20 + $20*(0.333) = $26.66 per hour.
Third system cost to operate = $30 + $20*(0.090) = $31.80 per hour.
Choose the second system.
12
BA 452
Quantitative Analysis
Final Exam
Backward Induction
Question 5. Use backward induction (and draw a
decision tree, if needed) to help make the following
sequential decisions, where later decisions depend on
earlier decisions:
1. Should the Bayer Biological Products Group spend $20 million to
begin preclinical development of a new blood-clot-busting drug?
a. The probability of successful testing on humans is 0.7
b. If the testing is not successful, the project is terminated
c. If the testing is successful, the project requires $2 million to file
a license with to the FDA. The probability of successful filing
0.9
d. If the filing is successful,, demand and revenues are uncertain:
i. With probability 0.4, demand is low and revenue is $60
million.
ii. With probability 0.1, demand is medium and revenue is
$80 million.
iii. With probability 0.5, demand is high and revenue is $100
million.
2. If the testing on humans were successful, should the Bayer Biological
Products Group sell its rights in the project for $70 million?
Compute expected profit for the Bayer Biological Products Group.
Answer to Question:
13
BA 452
Quantitative Analysis
Final Exam
Answer to Question:
Step 1: Replace uncertain payoffs with expected value:
1. Should the Bayer Biological Products Group spend $20 million to
begin preclinical development of a new blood-clot-busting drug?
a. The probability of successful testing on humans is 0.7
b. If the testing is not successful, the project is terminated
c. If the testing is successful, the project requires $2 million to file
a license with to the FDA. The probability of successful filing
0.9
d. If the filing is successful,, expected revenue =
0.4(60)+0.1(80)+0.5(100)=$82 million demand and revenues
are uncertain:
i. With probability 0.4, demand is high and revenue is $60
million.
ii. With probability 0.1, demand is medium and revenue is
$80 million.
iii. With probability 0.5, demand is low and revenue is $100
million.
2. If the testing on humans were successful, should the Bayer Biological
Products Group sell its rights in the project for $70 million?
14
BA 452
Quantitative Analysis
Final Exam
Step 2: Replace uncertain payoffs with expected value:
1. Should the Bayer Biological Products Group spend $20 million to
begin preclinical development of a new blood-clot-busting drug?
a. The probability of successful testing on humans is 0.7
b. If the testing is not successful, the project is terminated
c. If the testing is successful, the project requires $2 million to file
a license with to the FDA, and earn expected revenue = 0.9(82)
+ 0(0) = $73.8 million. The probability of successful filing 0.9
d. If the filing is successful,, expected revenue =
0.4(60)+0.1(80)+0.5(100)=$82million demand and revenues are
uncertain:
i. With probability 0.4, demand is high and revenue is $60
million.
ii. With probability 0.1, demand is medium and revenue is
$80 million.
iii. With probability 0.5, demand is low and revenue is $100
million.
2. If the testing on humans were successful, should the Bayer Biological
Products Group sell its rights in the project for $70 million?
15
BA 452
Quantitative Analysis
Final Exam
Step 3: Make the second decision:
1. Should the Bayer Biological Products Group spend $20 million to
begin preclinical development of a new blood-clot-busting drug?
a. The probability of successful testing on humans is 0.7
b. If the testing is not successful, the project is terminated
c. If the testing is successful, the project requires $2 million to file
a license with to the FDA, and earn expected revenue = 0.9(82)
+ 0(0) = $73.8 million. The probability of successful filing 0.9
d. If the filing is successful,, expected revenue =
0.4(60)+0.1(80)+0.5(100)=$82million demand and revenues are
uncertain:
i. With probability 0.4, demand is high and revenue is $60
million.
ii. With probability 0.1, demand is medium and revenue is
$80 million.
iii. With probability 0.5, demand is low and revenue is $100
million.
2. If the testing on humans were successful, should the Bayer Biological
Products Group sell its rights in the project for $70 million?
Answer: Assume testing on humans were successful. Selling rights in the
project earns $(70-20) = $50 million profit. Keeping rights earns $(73.8-220) = $51.8 million profit. So keep the rights.
16
BA 452
Quantitative Analysis
Final Exam
Step 4: Replace uncertain payoffs with expected value, and make the first
decision:
1. Should the Bayer Biological Products Group spend $20 million to
begin preclinical development of a new blood-clot-busting drug?
a. The probability of successful testing on humans is 0.7
b. If the testing is not successful, profit = -$20 million. If the
testing is successful, expected profit = $51.8 million.
So, preclinical development yields expected profit = 0.3(-20) + 0.7(51.8) =
$30.26 million, which being positive indicates that, yes, the Bayer
Biological Products Group should begin preclinical development. And from
previous calculations, if the project is successful, they should not sell their
rights.
17
BA 452
Quantitative Analysis
Final Exam
Expected Value of Sample Information
Question 6. Ryoei Saito has received from the
Siegfried Kramarsky family a take-it-or-leave-it offer to
sell to Ryoei Saito the painting Portrait of Dr. Gachet,
by Vincent van Gogh for $150 million. Ryoei Saito is
an expected-value maximizer. Ryoei Saito has determined probability
estimates of the painting's worth to other buyers, based on economic
outcomes, as:
P($120 million) = .3, P($140 million) = .4,
P($160 million) = .1, and P($180 million) = .2
a.
Should Ryoei Saito buy the painting? Explain your answer.
b.
What is the expected value of perfect information?
Ryoei Saito can have an economic forecast performed by Christie’s auction
house. The forecast indicates either G = Good resale conditions or B =
Bad resale conditions. Conditional probabilities of the indicators conditional
on the collection's worth are
P(G$120 million) = .1, P(G$140 million) = .3,
P(G$160 million) = .2, and P(G | $180 million) = .4
c.
Should Ryoei Saito purchase that forecast if the forecast cost $1
million? What is the maximum amount Ryoei Saito should be willing to pay
for that forecast?
Answer to Question:
18
BA 452
Quantitative Analysis
Final Exam
Answer to Question: There are four states of nature corresponding to
the asset’s worth.
S1 (state 1) = $120 million, S2 = $140 million, S3 = $160 million, S4 = $180
million.
And there are two decision choices:
D1 = buy for $150 million, D2 = don’t buy
The payoffs from decisions are
S1
S2
S3
S4
D1
-30
-10
10
30
D2
0
0
0
0
a. Given priors
Priors
P(Si)
S1
S2
S3
0.3
0.4
0.1
S4
0.2
the expected value from decisions is
S1
S2
S3
S4
EV
D1
-30
-10
10
30
-6
D2
0
0
0
0
0
So the optimum choice is D2 (don’t buy), and value is 0.
19
BA 452
Quantitative Analysis
Final Exam
b. EVPI (expected value of perfect information) is based expected value if
you know the state before making the decision, and so you always
achieve maximum value.
S1
S2
S3
S4
D1
-30
-10
10
30
D2
0
0
0
0
Max
0
0
10
30
Given prior probabilities, the expected value from decisions is
S1
S2
S3
S4
EV
D1
-30
-10
10
30
-6
D2
0
0
0
0
0
Max
0
0
10
30
7
Putting it all together, EVPI is the increase in value from 0 to 7, EVPI = 7
million dollars.
20
BA 452
Quantitative Analysis
Final Exam
c. To evaluate the forecast, compute posterior probabilities under the two
possible forecasts: G = Good conditions or B = Bad conditions.
First, consider the forecast G.
S1
S2
S3
S4
Priors
P(Si)
0.3
0.4
0.1
0.2
Conditionals
P(G|Si)
0.1
0.3
0.2
0.4
Joint
P(G&Si)
0.03
0.12
0.02
0.08
P(G) =
0.25
Posterior
P(Si|G)
0.120
0.480
0.080
0.320
Given those posterior probabilities, the expected value from decisions is
S1
S2
S3
S4
EV
D1
-30
-10
10
30
2
D2
0
0
0
0
0
So the optimum choice is D1, and value is 2.
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BA 452
Quantitative Analysis
Final Exam
Next, consider the forecast B.
S1
S2
S3
S4
Priors
P(Si)
0.3
0.4
0.1
0.2
Conditionals
P(B|Si)
0.9
0.7
0.8
0.6
Joint
P(B&Si)
0.27
0.28
0.08
0.12
P(B) =
0.75
Posterior
P(Si|B)
0.360
0.373
0.107
0.160
Given those posterior probabilities, the expected value from decisions is
S1
S2
S3
S4
EV
D1
-30
-10
10
30
-8.6666667
D2
0
0
0
0
0
So the optimum choice is D2, and value is 0.
Putting it all together, P(G) = .25 of a Good report and expected value =2,
and P(B) = .75 of a Bad report and expected value = 0. Overall, expected
value is .25x2 + .75x0 = 0.50. Therefore, EVSI = 0.50 million dollars, and
you would pay up to 0.50 million dollars for the economic forecast.
22