ADM2302 Section M, N, P and Q
Solutions Assignment 3
Assignment # 3 Solutions and Marking Scheme
Decision Analysis and Project Scheduling
ADM2302 students are reminded that submitted assignments must be typed (i.e. can NOT be
hand written), neat, readable, and well-organized. However, it is ok to plot Decision Tree and
Project Network by hand and to SCAN/INCLUDE them within the Word document file as long as
they are placed well within the Word document (proper rotation, readable quality, and well
organized). Assignment marks will be adjusted for sloppiness, poor grammar, spelling mistakes,
technical errors as well as wrong formats such as PDF files. Submitted assignment solutions (if
applicable) must include “managerial statements” that communicate the results of the analysis in
plain language.
The assignment is to be submitted electronically as a single Word Document file via Brightspace
by Sunday April 11th prior to 23:59. The front page of the Word document has to include the
title of the assignment, the course code and section, and the student name and number. The second
page is the SIGNED Statement of Integrity.
Note: Each student must provide an individual original submission of completed Assignment #3.
Please also note: Assignment #3 copies that are submitted jointly (i.e., by more than one author)
will not be graded.
E-mail questions related to the assignment should be sent to the Teaching Assistant or posted on
the Brightspace course website “Discussion page” (viewed by all).
Section M: Parisa Keshavarz (pkesh064@uottawa.ca)
Section N: Scott Pasiechnyk (spasi018@uottawa.ca)
Section P: Afshin Kamyabniya (akamy007@uottawa.ca)
Section Q: Josianne Absi (jabsi022@uottawa.ca)
Total: 100 points
General marking rules
- If the student forgot to sign the statement of integrity, please send him/her an e-mail
requesting the student to send it to you. Then you can copy/paste the signed statement of
integrity into the student assignment 3 document before uploading the feedback.
- Don’t penalize twice for an error that occurs at the start and it does affect the results that
follows.
- Provide a brief explanation that would allow the students to understand were the error was
committed. Also indicate how many points were deducted next to each error.
- Please provide me with the most common mistakes, so I can send the feedback to the
students via e-mail.
Winter 2021
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ADM2302 Section M, N, P and Q
Solutions Assignment 3
Problem 1 (28 points)
A major imaging center is not able to meet the increased demand from patients for MRIs. The
administration is willing to explore different possibilities by evaluating such alternatives as
adding one or two additional units or outsourcing to other image centers and earning a
commission of $30.00 per MRI. A feasibility analysis showed that three major demand levels
could occur in the future, summarized as 500, 750 and 1000 additional MRI cases. The financial
analysis of the potential decisions summarizes profits/losses under additional MRI demand levels
in a payoff table shown in the Table below (given in thousands of dollars).
Alternatives
500 Cases
750 Cases
1000 Cases
Buy One MRI Unit
-$15
$200
$300
Buy Two MRI Units
-$150
$100
$725
Outsource
$15
$22.5
$40
a) Provide an analysis that would satisfy an optimistic approach and another that would satisfy a
pessimistic approach. Show your work/calculations. (8 points)
b) What is the best option that minimizes regrets? Show your work/calculations. (8 points)
c) What would be the optimal choice if market research suggested that the likelihood of 500
cases was 0.2, 750 cases was 0.6 and 1000 cases was 0.2? Use the EMV/EP criterion. Show
your work/calculations. (4 points)
d) What is the expected value of perfect information (EVPI) under the probabilities given in c)?
Show your work/calculations. Verbally communicate the result. (8 points)
Answer:
P.S: please note that the payoff Table had a typos mistake in the value for
outsource with 1000 Cases. It is supposed to be $30 (i.e. $30/MRI *1000 MRI =
$30,000). If the student adjusts in the table the value to $30 instead of $40, also
consider as correct.
Winter 2021
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ADM2302 Section M, N, P and Q
Solutions Assignment 3
Question 1 a) (8 points)
Alternatives
500 Cases
750 Cases
1000 Cases Best
(3 points)
Worst
(3 points)
Buy One
MRI Unit
-15
200
300
300
-15
Buy Two
MRI Units
-150
100
725
725
-150
Outsource
15
22.5
40
40
15
Thus the optimist would choose to buy two MRI (1 point) units and the pessimist/conservative
would choose to outsource (1 point).
Question 1 b) (8 points)
4 points for constructing and providing the regret table.
Alternatives
500 Cases
750 Cases
1000 Cases
Worst
(2 points)
Buy One MRI
Unit
15 – (-15) = 30
0
725-300 = 425
425
Buy Two MRI
Units
15 – (-150) =
165
200-100
= 100
0
165
Outsource
0
200-22.5
=177.5
725-40
= 685
685
Thus the minimum regret is achieved by buy two MRIs. (2 points)
Winter 2021
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ADM2302 Section M, N, P and Q
Solutions Assignment 3
Question 1c) (4 points)
3 points: Computing the EP (Expected Time)
1 point: Thus the option of buying one MRI maximizes the expected value ($177,000) and so
becomes the optimal action.
Probability
.2
.6
.2
Expected
Value
Alternatives
500 Cases
750 Cases
1000 Cases
Buy One MRI Unit
-15
200
300
177
Buy Two MRI Units
-150
100
725
175
Outsource
15
22.5
40
24.5
Question 1d) (8 points)
EVPI = EPC – EP = EVUC - EMV
EVUC = EPC = sum_j p_j (Best of O_ij given S_j)
EVUC = EPC = .2*15000+.6*200000+.2*725000) = 268000 (3 points)
EMV = EP = 177000
EVP1 = 268000 – 177000 = 91000 (2 points)
Alternative calculation:
EPC = 15*0.2 + 200*0.6 + 725*0.2
EPC = 268
EMV = 177 (calculated in part c)
EVPI = 268 – 177 = 91 in thousands of dollars.
3 points: The maximum willingness to spend on perfect information is $91,000.
If the students say $91 then (– 2 points)
Problem 2 (35 points)
David is a film producer and he is currently evaluating a new horror movie script. David estimates
that the probability of a new movie being a success is 0.10 and the probability that it will fail at
the box office is 0.90. The studio’s accounting department estimates that if this new movie is a hit,
it will make $25 million in profit, whereas if it fails, it will lose $8 million.
David would also like to consider the service of a noted film critic, Alison, and ask her to assess
the movie’s chance of success. Alison is able to correctly predict a successful film 70% of the
time. She is also able to correctly predict an unsuccessful film 80% of the time.
Winter 2021
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ADM2302 Section M, N, P and Q
Solutions Assignment 3
a) Use the likelihoods/conditional and prior probabilities, and calculate all the revised
probabilities. Show your work/calculations. (6 points)
b) Draw a decision tree for this problem to help David make the best decision. Solve the decision
tree and determine the decision strategy David should follow and its expected profit. (Show
your work/calculations). Verbally communicate the decision strategy (20 points)
c) What is the expected value of sample information provided by Alison (EVSI)? Show your
work/calculations. Verbally communicate the result. (3 points)
d) Calculate the expected value of perfect information (EVPI) and find out the efficiency of the
information provided by Alison. Show your work/calculations. (6 points)
Answer:
(a)
Summary of the information provided:
P(s) = 0.1
P(f) = 0.9
G = Good review
B = Bad review
P(G|s) = 0.7
P(B|s) = 0.3
P(G|f) = 0.2
P(B|f) = 0.8
Use Bayes theorem to revise probabilities:
(2 Points): P(s|G) = 0.28 P(f|G) = 0.72
(2 Points): P(s|B) = 0.04 P(f|B) = 0.96
Therefore
(2 points): P(G) = 0.25 and P(B) = 0.75
State of nature
Success (s)
Flop (f)
State of nature
Hit
Flop
Winter 2021
Good Review (G)
Conditional Porb.Prior Prob Joint
0.7
0.1
0.07
0.2
0.9
0.18
P(G) =
0.25
Prior PROB
Bad review (B)
Conditional Porb.Prior Prob Joint
0.3
0.1
0.03
0.8
0.9
0.72
P(B) =
0.75
Revised Probability
0.280 P(s/G)
0.720 P(f/G)
Revised Probability
0.04 P(s/B)
0.96 P(f/B)
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ADM2302 Section M, N, P and Q
Solutions Assignment 3
if student mix up the conditional probabilities with the prior probabilities or enter the wrong
value of conditional probabilities and get the answer below – 4 points.
This example below is wrong because of entering wrong value for conditional probabilities.
State of nature
Success (s)
Flop (f)
State of nature
Hit
Flop
Good Review (G)
Conditional Porb.Prior Prob Joint
0.7
0.1
0.07
0.3
0.9
0.27
P(G) =
0.34
Prior PROB
Bad review (B)
Conditional Porb.Prior Prob Joint
0.2
0.1
0.02
0.8
0.9
0.72
P(B) =
0.74
Revised Probability
0.206 P(s/G)
0.794 P(f/G)
Revised Probability
0.03 P(s/B)
0.97 P(f/B)
Message to the TA: OR any other mistakes of mixing the values, then use the Excel file that I
sent you to figure out if other mistakes were committed and if so deduct further marks.
(b)
Tree structure with arc labels: 6 points
If a decision (square) and probability (circle) node are not represented as such then (-1 point) for
the decision node and (– 1 point) for the probability node. For a maximum of (-2 points).
For major mistake that impact the decision tree (-2 points).
Inputting the probability on the tree correctly 3 points
Each mistake in assignment of probabilities (-1 point)
Cost flows/payoffs: 3 points (deduct 1 point per mistake)
Computing the EP on the TREE (PRUNE THE TREE CORRECTLY): 5 points (Deduct two
points per mistake for a maximum deduction of 5 points. As such not using the proper formulas
to calculate the EP at a state of nature node, not choosing the maximum value decision at a
decision node, etc. …). It is ok if the student doesn’t put the value on top of the node but instead
label the nodes with number and show the value below the tree.
-2 points for each logical mistake: Choosing for a probability node instead of adding.
-2 points for each logical mistake: Averaging or adding at a decision node instead of choosing.
-1 point for a calculation mistake and NOT a logical mistake in the value for the EP.
3 points for stating the decision strategy: Hire Alison, if good review produce, if bad review do
not produce. The expected payoff of such decision strategy is $310,000.
-2 points if INCOMPLETE or NOT decision strategy is verbally given.
Winter 2021
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ADM2302 Section M, N, P and Q
Solutions Assignment 3
(c)
EVSI = EVwSI – EVwOI = $0.31M – (0) = $0.31M (2 points)
The maximum amount that David is willing to pay Allison is $310,000 (1 point)
(d)
EVPI = EVwPI – EVwOI = 25(0.1) + 0(0.9) – (0) = 2.5 M
2 points for the correct value of EPC/EVwPI = 25(0.1) + 0(0.9) = 2.5 millions
2 points EVPI = EPC – MAX(EP) = 2.5 M – 0 = 2.5 M.
Payoffs in Millions of dollars
Success Flop
0.1
0.9
produce
25
-8
Don't Produce
0
0
Best outcome
25
0
EP
Choice
-4.70
0
Best
Expected Value WITH Perfect Information(EPC) =
Best Expected Payoff (EP) =
Expected Value OF Perfect Information (EVPI) =
2.50
0.00
2.50
Millions
Millions
Millions
Efficiency = EVSI/EVPI = 0.31/2.5= 0.124 (2 points)
Or 12.4%
Winter 2021
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ADM2302 Section M, N, P and Q
Solutions Assignment 3
Problem 3 (37 points)
Russia is hosting the 2018 World Cup, a quadrennial international soccer tournament. A new
stadium must be constructed in the City of Rostov-on-Don, Russia. The new stadium will have a
capacity of 45,000 seats and host five matches during the tournament. The following activities
would have to be undertaken before construction can begin on the stadium.
Activity
Description
Immediate Predecessor Time (weeks)
A
Survey building site
6
B
Develop initial design
8
C
Obtain board approval
A, B
12
D
Select architect
C
4
E
Establish budget
C
6
F
Finalize design
D, E
15
G
Obtain financing
E
12
H
Hire contractor
F, G
8
Construct a network for this problem. (4 points).
Determine ES, EF, LS, and LF time, and slack for each activity. (8 points)
Determine the critical path and the expected project completion time. (4 points)
Based on the information thus far, would a delay of 1 week in activity D (select architect)
delay the completion time of the project? Justify. (3 points)
Now assume that the duration of some activities is not known with certainty. The estimates of
these activities are shown below. Assuming that the duration of the other activities remain
unchanged, answer the following questions.
a.
b.
c.
d.
Activity Optimistic Most Likely Pessimistic Expected Time
e.
f.
g.
h.
Winter 2021
A
4
6
8
B
3
8
13
E
5
6
7
F
13
15
17
Variance
What is the project’s expected completion time and variance given this new
information about uncertainty in duration of some activities? (7 points)
The project must be completed within 52 weeks or else there are penalties. What is the
probability of the organizing committee incurring these penalties? Show your
work/calculations. (4 points)
What time should the organizing committee report such that they are 85% sure that the
project will be completed by that time? Show your work/calculations. (4 points)
The organizing committee wants to reduce the duration of the project by 1 week. They
have looked at several different options to bring in more resources to expedite an
activity. The options are listed in the following table. Which activity should be crashed
in order to reduce the overall duration of the project by 1 week? Justify.
(3 points)
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ADM2302 Section M, N, P and Q
Solutions Assignment 3
Activity
Description
Crashing Cost (1 week)
B
Develop initial design
$2000
D
Select architect
$900
E
Establish budget
$1300
G
Obtain financing
$1500
Answer:
a) (4 points)
b) No need to have the End node, as H is the only activity with no successor.
Deduct 1 point if forgot start node.
Deduct 1 point if student don’t put the arrows on the arcs.
Deduct 1 point of any other mistake.
c) (8 points). Reduce 2 points per mistake
If they carry their mistake through no further deductions (ie. If they make ES for C = 6, then ES
for D and E will be 18…)
EST
EFT
LST
LFT
Slack
Start
0
0
0
0
0
A
0
6
2
8
2
B
0
8
0
8
0
C
8
20
8
20
0
D
20
24
22
26
2
E
20
26
20
26
0
F
26
41
26
41
0
G
26
38
29
41
3
H
41
49
41
49
0
End
49
49
49
49
0
Winter 2021
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ADM2302 Section M, N, P and Q
Solutions Assignment 3
c) The critical path is Start  B  C  E  F  H  End (2 points)
The expected project length is 49 weeks (2 points)
d) No, because D is not a critical activity or because it has a slack. (2 points for the justification.)
e) (7 points)
If the expected time and variances are calculated correctly for some (see below), then give 2 points
Activity Optimistic Most Likely Pessimistic Expected Time Variance
A
4
6
8
6
0.44
B
3
8
13
8
2.77
E
5
6
7
6
0.11
F
13
15
17
15
0.44
The project’s expected completion time is 49 weeks and the variance is 3.32
Expected completion time = 8 + 12 + 6 + 15 + 8 = 49 (2 points)
The duration time to finish task C and H are known with certainty. As such no variability on the
time to complete those tasks. So the variances of these activities are Zero.
Variance = 2.77 + 0+0.11 + 0.44 + 0 = 3.32 (3 points)
f) Solve for 1 - P(T ≤ 52)
Z = (52 – 49) / √(3.32) = 1.648 (2 points)
Z = 1.648
P(T ≤ 52) = 95.05%
1 - P(T ≤ 52) = 1 – 0.9505 = 0.0495
The probability of the organizing committee incurring the penalty fee is 0.0495 or
approximately 5%
(2 points)
g) Find Z value for 0.85  Z = 1.04 or if the student choose a value between 1.03 and 1.04
(inclusive) it is still correct. (2 points)
(T – 49) / √(3.32) = 1.04
T = 50.89 weeks (2 points)
The organizing committee is 85% sure that the project will be completed by 50.89 weeks
IF THE STUDENT PROVIDES AN EXPECTED DURATION < 49 weeks (- 4 POINTS)
because the 49 weeks has a 50% percent chance of occurring.
h) Activity E should be crashed because it is a critical activity and has the lowest crashing cost of
the critical activities that can be crashed.
-2 points if not justification or incorrect justification.
Winter 2021
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