Chapter 05S – Decision Theory
5S–31
Find the intersection between A & B:
20 + 100P = 40 + 20P
100P – 20P = 40 – 20
80P = 20
P = 20/80
P = 0.2500
Find the intersection between B & C:
40 + 20P = 110 – 100P
20P + 100P = 110 – 40
Chapter 05S – Decision Theory
5S–32
16. a. Determine the range over which each alternative would be best in terms of the
value of P(High).
Plot each alternative relative to P(#2). Plot the payoff value for #1 on the left side of the
graph and the payoff value for #2 on the right side of the graph.
State of Nature
Alternative
#1
#2
A
$20
$140
B
$120
$80
C
$100
$40
A
C
Payoff
#2
140
1.0
120
Payoff
#1
P(#2)
B
Chapter 05S – Decision Theory
b.
Alternative C is lower than Alternative B for all values of P(#2), so it would never be
appropriate in terms of maximizing profits.
c.
For low and intermediate values of P(#2), Alternative B is best because it has the
highest expected value. For higher values of P(#2), Alternative A is best.
To find the exact values of the ranges, we must determine where the upper parts of the
lines intersect. For each line, b is the slope of the line and x = P(#2). The slope of each
line = Right-hand value – Left-hand value.
Equations:
A: 20 +120P (slope = 140 – 20)
B: 120 – 40P (slope = 80 – 120)
Find the intersection between A & B:
20 +120P = 120 – 40P
120P + 40P = 120 – 20
160P = 100
P = 100/160
P = .6250
Conclusion: Select Alternative A if P(#2) is greater than .6250.
d.
Conclusion: For P(#1), choose Alternative A if P(#1) is less than .3750 (i.e., 1.00 –
.6250).
17.
a. [Refer to the diagram in the solution for Problem 16]
b.
Alternative B is higher than Alternative C for all values of P(#2), so Alternative B
would never be appropriate in terms of minimizing costs.
c.
For low values of P(#2), Alternative A is best. For intermediate and higher values of
P(#2), Alternative C is best.
Equations:
A: 20 +120P (slope = 140 – 20)
C: 100 – 60P (slope = 40 – 100)
Find the intersection between A & C:
20 +120P = 100 – 60P
120P + 60P = 100 – 20
180P = 80
P = 80/180
P = .4444
Conclusion: Select Alternative A for P(#2) less than .4444 and choose Alternative C
for P(#2) greater than .4444.
d.
Conclusion: In terms of P(#1), choose Alternative A for P(#1) greater than .5556 (1.00
– .4444) and choose Alternative C for P(#1) less than .5556.
Chapter 05S – Decision Theory
5S-35
18. Given: Payoffs are provided in the table below.
Plot each alternative relative to P(No Contract). Plot the payoff value for Receive Contract on
the left side of the graph and the payoff value for No Contract on the right side of the graph.
State of Nature
Alternative
Receive
Contract
No
Contract
#1
10
-2
#2
8
3
#3
5
5
#4
0
7
Alternative 1 is best for the lowest range of P(No Contract), followed by Alternative 2 for the
next range, then Alternative 3 for the range after that, and then Alternative 4 for the highest
range.
Equations:
1: 10 – 12P (slope = –2 – 10)
2: 8 – 5P (slope = 3 – 8)
10
8
#3
Payoff for
No Contract
7
5
3
5
0
-2
1.0
#1
#2
#4
Payoff for
Contract
P(No Contract)
Chapter 05S – Decision Theory
5S–36
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of
McGraw-Hill Education.
Find the intersection between 1 & 2:
10 – 12P = 8 – 5P
-12P – (-5P) = 8 – 10
-7P = –2
P = -2/-7
P = .2857
Find the intersection between 2& 3:
8 – 5P = 5 + 0P
8 – 5P = 5
-5P = 5 – 8
5 = 7P
P = 5/7
P = .7143
Optimal ranges:
#1: P(No Contract) = 0 to < .2857
Chapter 05S – Decision Theory
Enrichment Model: Advanced Decision Tree Problems
In this section two additional decision tree problems are presented
1. Space engineers have three alternative designs for the configuration of a component for an
unmanned space shuttle. The space vehicle is likely to encounter one of four different
conditions, which have probabilities of occurrence as listed in the following payoff table
(costs in $00) with the payoffs for each combination of design and state of nature. Additional
5S–38
2. Demand for movie rentals at a video store on Saturdays during summer months is related to
the weather. If it is raining, or if the chance of rain is greater than 50%, demand tends to
follow one distribution, whereas if it is not raining and the chance of rain does not exceed
50%, demand follows a different distribution. This is important to the video store because the
manager must decide early on Saturday how many employees to schedule for Saturday
afternoon and early evening.
The two distributions are:
P(Rain) > 50% P(Rain) 50%
Demand
Probability
Demand
Probability
Low
.10
Low
.60
Moderate
.20
Moderate
.30
High
.70
High
.10
The regular staff can handle Low demand. Moderate demand requires two additional
employees, and High demand requires another two employees. The payoff table (profits in
$000) is shown below:
Demand
Low
Moderate
High
0
2
3
4
Extra Staff
2
1
4
5
4
0
3
6
a. Construct a tree diagram showing the payoffs for this situation.
b. Determine the number of additional staff needed for a rainy Saturday.
c. Determine the number of staff needed when the chance of rain is 20% for a Saturday.
Chapter 05S – Decision Theory
5S–39
Solution to Problem 1
1.
A
B
C
D
Design
.3
.4
.2
.1
Expected
Value
001
20
10
10
0
12
002
15
10
0
40
12.5
003
10
20
30
30
20
Design 001 minimizes the expected cost. Expected payoff under risk = $12 ($1,200).
Find the best payoff under each state of nature:
A: Best (minimum cost) = 10
B: Best (minimum cost) =10
C: Best (minimum cost) = 0
D: Best (minimum cost) = 0
Develop the opportunity loss table:
Design
A
B
C
D
001
10
0
10
0
002
5
0
0
40
003
0
10
30
30
Expected regret for each alternative:
001: .3(10) + .4(0) + .2(10) + .1(0) = $5
Chapter 05S – Decision Theory
Solution to Problem 2
2. a.
4
5
Medium demand
High demand
2
No additional staff
4
1
High demand (.7)
Low demand (.1)
1
b.
EV1 = (.1)(2) + (.2)(3) + (.7)(4) = 3.6
c.
1
4
5
0
3
No additional staff
Four additional staff
2
3
4
1
0
3
6
Medium demand
High demand
Low demand
Low demand
Medium demand
High demand
Low demand
1
3
2
4
5
0
3
6
Low demand (.1)
High demand (.7)
Medium demand (.2)
High demand (.7)
Low demand (.1)
2
3
No additional staff
2
3
4
Medium demand (.3)
High demand (.1)
Low demand (.6)
Low demand (.6)
1
Four additional staff