Chapter 05S – Decision Theory
5S–11
6. Given: Management must decide whether to renew the lease for another 10 years or to
relocate near the site of a proposed motel. The net present values of the two alternatives under
each state of nature are given in the table below.
States of Nature
Alternative
Motel Approved
Motel Rejected
Renew
$500,000
$4,000,000
Relocate
$5,000,000
$100,000
a. Maximax: Determine best possible payoff for each alternative and choose the alternative
that has the “best.”
States of Nature
Alternative
Motel Approved
Best Payoff
Renew
$500,000
$4,000,000
Relocate
$5,000,000
$5,000,000
Best of
the Best
Conclusion: Select Relocate alternative with a payoff of $5,000,000.
b. Maximin: Determine the worst possible payoff for each alternative and choose the
alternative that has the “best worst.”
States of Nature
Alternative
Motel Approved
Motel Rejected
Worst
Payoff
Renew
$500,000
$4,000,000
$500,000
Best of
the
Worst
Relocate
$5,000,000
$100,000
$100,000
Chapter 05S – Decision Theory
5S–12
c. Laplace: Determine the average payoff for each alternative and choose the alternative with
the best average.
States of Nature
Alternative
Motel Approved
Motel Rejected
Average
Payoff
Renew
$500,000
$4,000,000
$2,250,000
Relocate
$5,000,000
$100,000
$2,550,000
Best
Conclusion: Select Relocate alternative with an average payoff of $2,550,000.
d. Minimax Regret: Prepare a table of regrets (opportunity losses)—for each column,
subtract every payoff from the best payoff in that column. Identify the worst regret for
each alternative. Select the alternative with the “best worst.”
Regrets
Alternative
Motel Approved
Motel Rejected
Worst Regret
Renew
$4,500,000
($5,000,000-
$500,000)
$0
($4,000,000-
$4,000,000)
$4,500,000
Relocate
$0
$3,900,000
$3,900,000
Best of the
Chapter 05S – Decision Theory
5S–13
7. Given: Probability that the motel’s application will be approved = .35.
The probability that the motel will be rejected = 1.00 – .35 = .65.
Alternative
Expected Value
a.
Renew
(.35)500,000 + (.65)4,000,000 = $2,775,000
Relocate
(.35)5,000,000 + (.65)100,000 = $1,815,000
Conclusion:
Renew lease.
b.
Reject (.65)
$5,000,000
$100,000
$1,815,000
c. Expected value of perfect information (EVPI) = Expected payoff under certainty –
Expected payoff under risk
Find the best payoff under each state of nature:
Motel Approved: Best Payoff = $5,000,000
Motel Rejected: Best Payoff = $4,000,000
$500,000
$4,000,000
Approve (.35)
Reject (.65)
Approve (.35)
$2,775,000
E.V.
Renew
Chapter 05S – Decision Theory
5S–14
8. a. Determine the range over which each alternative would be best in terms of the value of
P(Approved).
Plot each alternative relative to P(Approved). Plot the payoff value for Rejected on the left
side of the graph and the payoff value for Approved on the right side of the graph.
Relocate
value.
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(Approved). The slope of each line
= Right-hand value – Left-hand value.
Equations:
-8,400,000P = –3,900,000
P = -3,900,000/-8,400,000
P = .4643
Alternative
Rejected
Approved
Renewal better than
(millions)
Relocation
should be to renew the 10-year
lease because a. P(Approved) of
.35 < .4643 & b. P(Approved)
of .45 < .4643
Exp.
Value
100,000 + 4,900,000x
5
3
2
1
Relocate
x = .4643
4,000,000 – 3,500,000x
.5
5
3
2
1
Chapter 05S – Decision Theory
Optimal ranges:
b. Conclusion: Still should renew because P = .45 is less than P = .4643.
c.
Given:
Renew = .35(500,000) + .65(4,000,000)
Relocate = .35(5,000,000) + .65(100,000) = $1,815,000
Plug in x for 4,000,000 in Renew equation:
Renew = .35(500,000) + .65x
Relocate = .35(5,000,000) + .65(100,000) = $1,815,000
Set the two equations equal and solve for x:
.35(500,000) + .65x = $1,815,000
175,000 + .65x = 1,815,000
.65x = 1,815,000 – 175,000
.65x = 1,640,000
x = 1,640,000/.65
x = 2,523,077
Relocate.
Conclusion:
The decision to Renew remains the same for the range of x >
$2,523,077.
At x = $2,523,077, we would be indifferent between Renew and
Chapter 05S – Decision Theory
5S–16
Expand Greatly
48
.8(48)
1
.2 Low
Expand
53.6
Analyze the decisions from right to left:
1) Determine which alternative would be selected for each possible second decision
(marked as “2”).
2) Determine the product of the chance probabilities and their respective payoffs for the
remaining branches.
Small
Low demand: .2($42) = $8.4
High demand: .8($48) = $38.4
Medium
.8 High
.2 Low
2
.2 Low
2
$ 42
42
22
46
72
46.8
44.4
53.6
.2(42)
+
.2(22)
+
+
.8(72)
Medium
9. a.
46.8
48
44.4
50
53.6
Chapter 05S – Decision Theory
5S–17
3) Determine the expected value of each initial alternative.
b. Maximin Alternative: Determine the worst possible payoff for each alternative and choose
the alternative that has the “best worst.”
Next Year’s
Demand
Alternative
Low
High
Worst Payoff
Small
$42
$48
$42
Best of the Worst
Medium
$22
$50
$22
Large
-$20
$72
-$20
c. Expected value of perfect information (EVPI) = Expected payoff under certainty –
Expected payoff under risk
Find the best payoff under each state of nature:
Low Demand: Best Payoff = $42
High Demand: Best Payoff = $72
Chapter 05S – Decision Theory
5S–18
d. Determine the range over which each alternative would be best in terms of the value of
P(High).
Next Year’s
Demand
Alternative
Low
High
Small
$42
$48
Medium
$22
$50
Large
-$20
$72
The graph above shows the range of values of P(High) over which each alternative is optimal.
For low values of P(High), Small is best because it has the highest expected value.
For interemediate and higher values of P(High), Large is best.
Equations:
Small: 42 + 6P (slope = 48 – 42)
Large: -20 + 92P (slope = 72 – (-20))
72
50
48
42
22
Small
Medium
Large
0 .7209 1.0
P (High)
–20
Low
High
Chapter 05S – Decision Theory
5S–19
Find the intersection between Small & Large:
42 + 6P = -20 + 92P
6P – 92P = -20 – 42
-86P = –62
5S–20
10. Decision Tree
buy 2
100
75
.30 low
Analyze the decisions from right to left:
1) Determine which alternative would be selected for each possible second decision.
Buy 1 with High Demand: Select Subcontract. Payoff = $110. Place a double dash
through Do Nothing and Buy 2nd.
2) Determine the product of the chance probabilities and their respective payoffs for the
Low demand: .30($75) = $22.5
Medium demand: .70($130) = $91
3) Determine the expected value of each initial alternative.
Buy 1 = $27+ $77 = $104
buy 1
$90
90
110
130
.30 low
subcontract
do nothing
.70 high
.7 high