Quantitative Analysis BA 452 Exam A Questions
BA 452
Dr. Jon Burke
Exam A
This is a 100-minute exam (1hr. 40 min.). There are 5 questions (20
minutes per question). One of those questions involves solving a problem
with a computer. You must turn in a USB drive with a copy of the program
files you used for your solution to have the chance to earn full credit; if you
do not turn in a USB drive with the files then you automatically loose 2 of
the 4 points possible for that question. The rest of the questions are either
solved without a computer or are formulated only, without requiring a
solution.
To avoid the temptation to cheat on Exam A, you must agree to the
following rules before taking your exam:
• 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,
graph paper and a ruler, and a simple non-graphing, nonprogrammable, 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.
Tip: Pace yourself. When there is only ½ hour left, spend at least 5
minutes outlining an answer to each remaining question.
1
Quantitative Analysis BA 452 Exam A Questions
Part 1. You may only use blank or graph paper,
pencils, a ruler, and a calculator. You may not
use a computer or notes. After you finish all
questions in Part 1, turn in your answers, then
you may use the school computer to run the
Management Scientist to complete Part 2 of the
exam.
2
Quantitative Analysis BA 452 Exam A Questions
Non-Unique Optimal Solutions
Question 1: Wilson Sporting Goods manufactures a
standard-size racket and an oversize racket. The
firm’s rackets are extremely light due to the use of a
magnesium-graphite alloy that was invented by the
firm’s founder. Each standard-size racket uses 0.1 kilograms of the alloy
and each oversize racket uses 0.4 kilograms; over the next two-week
production period only 80 kilograms of the alloy are available. Each
standard-size racket uses 10 minutes of manufacturing time an each
oversize racket uses 12 minutes. The profit contribution are $5 for each
standard-size racket and $6 for each oversize racket, and 40 hours of
manufacturing time are available each week. Management specified that at
least 30% of the total production of rackets must be the standard-size
racket. Assume that because of the unique nature of their products, Wilson
can sell as many rackets as they can produce.
Formulate the problem of how many rackets of each type to manufacture
over the next two weeks to maximize the total profit contribution. Compute
all optimal solutions without a computer.
Answer to Question:
3
Quantitative Analysis BA 452 Exam A Questions
Answer to Question 1:
Let S = number of standard size rackets produced every 2 weeks
V = number of oversize size rackets produced every 2 weeks
Max
5S
+
6V
0.7S
-
0.3V
>
0
10S
+
12V
<
4800
0.1S
+
0.4V
<
80
S, V
>
0
s.t.
% standard
Time (min./2 weeks)
Alloy
(The time constraint in hours, rather than minutes, reads, 0.167S + 0.2V <
80.) Graphing the constraints and isovalue lines indicates there are
multiple optimal solutions. That consists of all points on a line segment.
One end of the segment is where the second and third constraints bind
(10S+12V= 4800 and 0.1S+0.4V= 80), so (S,V) = (342.86,114.29). The
other end of the segment is where the second constraint and the nonnegativity of V bind (10S+12V=4800 and V= 0), so (S,V) = (480,0).
600
Standard S on horizontal axis
500
Oversize V on vertical axis
First constraint, through (0,0) and (300,700)
400
Second constraint, through (480,0) and (0,400)
300
Third constraint, through (800,0) and (0,200)
200
Feasible Region
100
0
0
100
200
300
400
500
4
600
700
800
Quantitative Analysis BA 452 Exam A Questions
Relative Patience
Question 2. Consider negotiations over the rental
price for air compressors between Best Buy in
Thousand Oaks and Radkat Compressed Air Co. of
Simi Valley. Best Buy seeks a compressor for the 5
days of Spring Break Week (Monday, Tuesday, Wednesday, Thursday,
Friday). The only company currently known to have the right kind of
compressor (Radkat) is willing to rent their only compressor for as little as
$200 per day. Best Buy and Radkat hire quantitative analysts who
determine that Best Buy using the compressor adds $500 to profits each
day (that is, the dual price is $500).
• On Sunday afternoon at 1pm, Radkat confronts Best Buy over the
rental price for the 5 days of Spring Break Week. Radkat presents
their offer of their price to rent.
• Best Buy either accepts it or rejects it and returns at 2pm with a
counteroffer for the 5 days of Spring Break Week.
• Radkat either accepts it or rejects it and returns at 3pm with a
counteroffer for the 5 days of Spring Break Week.
• Best Buy either accepts it or rejects it and returns at 4pm with a
counteroffer for the 5 days of Spring Break Week.
• Radkat either accepts it or rejects it and there will be no rental.
Suppose Best Buy discounts 10% between each hour of negotiation, and
Radkat discounts 20% between each hour of negotiation.
What initial rental rate should Radkat offer to Best Buy? Should Best Buy
accept that initial offer?
Now suppose that, before negotiations begin between Best Buy and
Radkat, Best Buy is given a take-it-or-leave-it offer to be supplied a
compressor by Compressed Air Supply Co. for $400 per day for each of the
5 days of the Spring Break Week (Monday, Tuesday, Wednesday,
Thursday, Friday). Re-compute Radkat’s rental rate offer to Best Buy.
5
Quantitative Analysis BA 452 Exam A Questions
Finally suppose, in addition to the alternative rental offer to Best Buy from
Compressed Air Supply Co. described above, the problem changes
because, before negotiations begin between Best Buy and Radkat, Radkat
finds an alternative rental customer who makes a take-it-or-leave-it offer to
rent their only compressor for $300 per day for each of the 5 days of the
Spring Break Week (Monday, Tuesday, Wednesday, Thursday, Friday).
Re-compute Radkat’s rental rate offer to Best Buy.
Answer to Question:
6
Quantitative Analysis BA 452 Exam A Questions
Answer to Question:
There are 4 bargaining rounds. Here is a bargaining payoff table for Best
Buy (Bargainer B) and Radkat (Bargainer R), with Bargainer R making the
first offer, with R discounting 10% between each bargaining round, and with
B discounting 20% between each bargaining round.
Rounds t o
End of
Game
Offer by
T ot al Gain
t o Divide
B's Gain
Offered
(10% dis.)
R's Gain
Offered
(20% dis.)
1
B
100
100.00
0.00
2
R
100
90.00
10.00
3
B
100
92.00
8.00
4
R
100
82.80
17.20
Part 1: Without the alternative offers, the gain from Best Buy’s rental from
Radkat is $300=$500-$200 per day. For Radkat to receive 17.2% of those
gains, Radkat should offer to keep $51.60 gain per day, or offer rental
price $251.60 per day, or $1258 for all five days. And Best Buy should
accept that initial offer.
With the alternative supply offer to Best Buy, Best Buy’s maximum
willingness to pay drops from $500 per day to $400 per day, and so the
gain from Best Buy’s rental from Radkat drops to $200=$400-$200 per day.
For Radkat to receive 17.2% of those gains, Radkat should offer to keep
$34.40 gain per day, or offer rental price $234.40 per day, or $1172 for
all five days. And Best Buy should accept that initial offer.
Comment: Best Buy finding an alternative supplier lowered the rental price
they pay Radkat.
With the alternative supply offer to Best Buy and the alternative demand
offer to Radkat, Best Buy’s maximum willingness to pay is $400 per day
and Radkat’s willingness to sell raises from $200 per day to $300 per day,
and so the gain from Best Buy’s rental from Radkat drops to $100=$300$200 per day. For Radkat to receive 17.2% of those gains, Radkat should
7
Quantitative Analysis BA 452 Exam A Questions
offer to keep $17.20 gain per day, or offer rental price $317.20 per day, or
$1586 for all five days. And Best Buy should accept that initial offer.
Comment: Radkat finding an alternative renter raised the rental price they
receive from Best Buy.
8
Quantitative Analysis BA 452 Exam A Questions
Sensitivity to Constants
Question 3. Blue Ridge Hot Tubs produces two
types of hot tubs: Aqua-Spas and Hydro-Luxes.
To make the tubs, there are 4 pumps available at the
cost of $4 each, 18 hours of labor available at a cost
of $1 per hour, and 16 feet of tubing available at a cost of $0.50 per foot.
Here are the input requirements, sales prices, and unit profits:
Aqua-Spa
Hydro-Lux
Pumps
1
1
Labor
3 hours
6 hours
Tubing
6 feet
4 feet
Sales Price
$13
$19
Unit Profit
$3
$7
Formulate Blue Ridge’s problem as a linear program.
Solve graphically for the optimum.
How much should Blue Ridge be willing to pay for one more pump? for
one more hour of labor? for one more foot of tubing?
Tip: Your written answer should define the decision variables, formulate the
objective and constraints, and solve the problem.
Answer to Question:
9
Quantitative Analysis BA 452 Exam A Questions
Answer to Question:
Let A = number of Aqua-Spa’s and H = number of Hydro-Lux’s produced.
Max 3 A + 7 H
s.t.
A+H<4
3A + 6H < 18
6A + 4H < 16
(pump constraint)
(labor constraint)
(tubing constraint)
Graphing the constraints and isovalue lines indicates the non-negativity of
A and the labor constraint are binding at the optimum (so A = 0 and 3A +
6H = 18), with the pump constraint and the tubing constraint both slack.
Therefore, the solution is to produce 0 Aqua-Spa’s and 3 Hydro-Lux’s.
Blue Ridge should be willing to pay just the embedded input prices of $4 for
one more pump, $0.50 for one more foot of tubing?
To find the dual price of labor, recompute the optimum for 19 hours
available: A = 0 and 3A + 6H = 19. So, 0 Aqua-Spa’s and 3 1/6 HydroLux’s. That increases profit $7x(1/6) = $1.17. Hence, Blue Ridge should
be willing to pay $1 + 1.17 = $2.17 for one more hour of labor.
Aqua Spa’s on horizontal axis
Hydro-Lux’s on vertical axis
Pump-constraint intercepts A=5 and H=5
Labor-constraint intercepts A=6 and H=3
Tubing-constraint intercepts A=2.7 and H=4
Sample isovalue line (dashed)
6
5
4
3
2
1
0
0
1
2
3
4
5
6
10
7
Quantitative Analysis BA 452 Exam A Questions
Revenue Management
Question 4. Avis Car Rental in Austin, TX has 50
high-performance Shelby-H Mustangs in its rental
fleet. These cars will be in greater demand than
usual during the last weekend in July when the
Central Texas Mustang Club holds its annual rally in
Austin. At times like this, Avis uses a revenue management system to
determine the optimal number of reservations to have available for the
Shelby-H cars.
Avis has agreed to have at least 60% of its Shelby-H Mustangs available
for rally attendees at a special rate. Although many of the rally attendees
will request a Saturday and Sunday two-day package, some attendees may
select a Saturday only or a Sunday only reservation. Customers not
attending the rally may also request a Saturday and Sunday two-day
package, or make a Saturday only or Sunday only reservation. Thus, six
types of reservations are possible. The cost for each type of reservation is
shown here.
Two-Day Saturday
Sunday
Package
Only
Only
Rally
$125
$75
$65
Regular
$150
$85
$75
The anticipated demand for each type of reservation is as follows:
Two-Day Saturday
Sunday
Package
Only
Only
Rally
20
10
15
Regular
10
20
25
Avis Car Rental would like to determine how many Shelby-H Mustangs to
make available for each type of reservation in order to maximize total
revenue.
Formulate a linear-programming model for this revenue-management
application.
Tip: Your written answer should define the decision variables, and
formulate the objective and constraints.
Answer to Question:
11
Quantitative Analysis BA 452 Exam A Questions
Answer to Question 4:
RATW = number of reservations available for rally attendees for two days
RGTW = number of reservations available for regular customers for two
days
RASA = number of reservations available for rally attendees for Saturday
only
RGSA = number of reservations available for regular customers for
Saturday only
RASU = number of reservations available for rally attendees for Sunday
only
RGSU = number of reservations available for regular customers for Sunday
only
Max 125RATW + 150RGTW + 75RASA + 85RGSA + 65RASU + 75RGSU
s.t.
RATW + RGTW + RASA + RGSA < 50 (Shelby-H Saturday Capacity)
RATW + RGTW + RASU + RGSU < 50 (Shelby-H Sunday Capacity)
RATW + RASA > 30
(60% Saturday Capacity for Rally
attendees)
RATW + RASU > 30
(60% Sunday Capacity for Rally
attendees)
RATW < 20
(Two-Day Package Rally Demand)
RGTW < 10
(Two-Day Package Regular Demand)
RASA < 10
(Saturday-Only Rally Demand)
RGSA < 20
(Saturday-Only Regular Demand)
RASU < 15
(Sunday-Only Rally Demand)
RGSU < 25
(Sunday-Only Regular Demand)
RATW,RGTW,RASA,RGSA,RASU,RGSU > 0 (Non-negativity)
12
Quantitative Analysis BA 452 Exam A Questions
Part 2. To answer the remaining question, you
may now use the school computer to run the
Management Scientist. After you finish the
question, reattach your answers to the rest of
your exam.
13
Quantitative Analysis BA 452 Exam A Questions
14
Quantitative Analysis BA 452 Exam A Questions
A.11 Data Envelopment Analysis
Question 5: The Health Care Administrators
Association is trying to determine the relative
efficiency of the top three hospitals for Diabetes and
Endocrinology. In particular, it wants to evaluate the
Mayo Clinic. The administrators are evaluating
performance output number of patients served, bed productivity, and the
average turnover interval as a function of the inputs of the total salary for
doctors, and the total salary for nurses.
Cleveland Clinic
Massachusetts
General Hospital
Mayo Clinic
6
4
5
4
2
6
800
900
720
6
5
4
6
5
7
Inputs:
Total salary of
doctors
($1,000,000’s)
Total salary of
nurses
($1,000,000’s)
Outputs:
Number of
patients served
Bed productivity
Average
turnover interval
Use that data to formulate a linear program for a Data Envelopment
Analysis to determine whether the Mayo Clinic is efficient, or whether it can
be improved. Solve with the computer.
Tip: Your written answer should define the decision variables and formulate
the objective and constraints, then solve using Management Scientist.
Explain the meaning of your solution.
Answer to Question:
15
Quantitative Analysis BA 452 Exam A Questions
Answer to Question 5:
Define the decision variables
E = Fraction of Mayo Clinic 's input resources required by the
composite hospital
w1 = Weight applied to Cleveland Clinic's input/output resources by
the composite high school
w2 = Weight applied to Massachusetts General’s input/output
resources by the composite high school
w3 = Weight applied to Mayo Clinic's input/output resources by the
composite high school
Define the objective function. Minimize the fraction of Mayo Clinic 's input
resources required by the composite hospital: Min E
Constrain the sum of the weights to one: (1) w1 + w2 + w3 = 1
Constrain each output of the composite hospital to be at least Mayo
Clinic’s:
(2) 800w1 + 900w2 + 720w3 > 720 (Patients served)
(3) 6w1 + 5w2 + 4w3 > 4 (Bed productivity)
(4) 6w1 + 5w2 + 7w3 > 7 (Turnover interval)
Constrain the inputs used by the composite hospital to be no more than the
multiple, E, of the inputs available to Mayo Clinic:
(5) 6w1 + 4w2 + 5w3 < 5E (Doctor salary)
(6) 4w1 + 2w2 + 6w3 < 6E (Nurse salary)
16
Quantitative Analysis BA 452 Exam A Questions
Note: In that formulation, the input constraints involving E are subtracted
from the right-hand side and moved to the left.
Management Scientist reveals optimal E = 1, so Mayo Clinic is efficient in
the sense that no composite hospital can match outputs while reducing
inputs.
17