Answer to Assignments, Chapter 12, Decision Theory, BSNS2120
Problem #7, Page 574
(a) MaxiMax and (b) MaxiMin
Investment
Material Costs
(a)
(b)
Alternatives
Stable
Increase
Row Maximums
Row Minimums
Houses
70,000
30,000
70,000
30,000
Shopping Center
105,000
20,000
20,000
105,000
Leasing
40,000
40,000
40,000
40,000
Column Max
(for regret
105,000
40,000
105,000
40,000
calculations)
(a) The decision choice by MaxiMax payoff is Shopping Center, $105,000.
(b) The decision choice by MaxiMin payoff is Leasing, $40,000
(c) Regret Table and MiniMax Regret
Investment
Material Costs
(c)
Alternatives
Stable
Increase
Row Maximums
Houses
35,000
10,000
35,000
Shopping Center
0
20,000
20,000
Leasing
65,000
0
65,000
(c) The decision choice by MiniMax Regret is Shopping Center with regret value $20,000.
(d) Hurwicz and (e) Equally Likely
Material Costs
(d)
(e)
Hurwicz Values with =0.2
Investment
Equally likely
Stable Surplus
Alternatives
(Plain average)
(Best*+Worst*(1-))
Houses
70,000 30,000 70,000*0.2 + 30,000*0.8 = 38,000
(70,000+30,000) / 2 = 50,000
Shopping Center
105,000 20,000 105,000*0.2 + 20,000*0.8 = 37,000
(105,000+20,000) / 2 = 62,500
Leasing
40,000 40,000 40,000*0.2 + 40,000*0.8 = 40,000
(40,000+40,000) / 2 = 40,000
(d) The maximum Hurvica value is 40,000. The decision choice is Leasing.
(e) The maximum equally likely value is 62,500. The decision choice is Shopping Center.
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Problem #8, Page 575
(a) MaxiMax and (b) MaxiMin
Investment
Gasoline Availability
(a)
Alternatives
Shortage
Stable
Surplus
Row Maximums
Motel
15,000
20,000
20,000
8,000
Restaurant
2,000
8,000
6,000
8,000
Theater
6,000
6,000
5,000
6,000
Column Max
(for regret
6,000
15,000
20,000
20,000
calculations)
(a) The MaxiMax payoff is 20,000. So the decision choice is Motel.
(b) The MaxiMin payoff is 5,000. So the decision choice is Theater.
(b)
Row Minimums
8,000
2,000
5,000
5,000
(c) Regret Table and MiniMax Regret
Investment
Gasoline Availability
(c)
Alternatives
Shortage
Stable
Surplus
Row Maximums
Motel
14,000
0
0
~
Restaurant
4,000
7,000
14,000
14,000
Theater
0
9,000
15,000
15,000
(c) The MiniMax Regret value is 14,000. So the decision choice would be either Motel or Restaurant.
(d) Hurwicz and (e) Equally Likely
Gasoline Availability
(d)
Hurwicz Values with =0.4
Investment
Shortage
Stable
Surplus
Alternatives
(Best*+Worst*(1-))
Motel
15,000
20,000
8,000
20,000*0.4 + (8,000)*0.6 = 3,200
Restaurant
2,000
8,000
6,000 8,000*0.4 + 2,000*0.6 = 4,400
Theater
6,000
6,000
5,000 6,000*0.4 + 5,000*0.6 = 5,400
(d) The maximum Hurwicz value is 5,400. So, the decision choice is Theater.
(e) The maximum equally likely value is 9,000. So, the decision choice is Motel.
(e)
Equally likely
(Plain average)
(8,000+15,000+20,000) / 3 = 9,000
(2,000+8,000+6,000) / 3 = 5,333.3
(6,000+6,000+5,000) / 3 = 5,666.7
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Problem #18, Page 578
(a) Payoff table and Maximum expected payoff
Market Conditions
Favorable
Stable
Unfavorable
Investment
Alternatives
0.2
0.7
0.1
Expected Values
Widget
120,000
70,000
30,000 120,000*0.2+70,000*0.7+(30,000)*0.1 = 70,000
Hummer
60,000
40,000
20,000 60,000*0.2+40,000*0.7+20,000*0.1 = 42,000
Nimnot
35,000
30,000
30,000 35,000*0.2+30,000*0.7+30,000*0.1 = 31,000
Column Max
(for EVPI and
120,000*0.2+70,000*0.7+30,000*0.1 = 76.000
120,000
70,000
30,000
opportunity loss
(This is expected value with perfect information, EVwPI)
calculations)
The decision choice is Widget, which has the maximum expected payoff 70,000.
(b) Opportunity loss table and minimum EOL
Market Conditions
Favorable
Stable
Unfavorable
Product
alternatives
0.2
0.7
0.1
Expected Values (EOL’s)
Widget
0
0
60,000 0*0.2+0*0.7+60,000*0.1 = 6,000
Hummer
60,000
30,000
10,000 60,000*0.2+30,000*0.7+10,000*0.1 = 34,000
Nimnot
85,000
40,000
0 85,000*0.2+40,000*0.7+0*0.1 = 45,000
The minimum expected opportunity loss (Min EOL) is 6,000. So the decision choice is Widget.
(c)
Expected payoff without perfect information (EVw/oPI) = 70,000 (as calculated in (a))
Expected payoff with perfect information (EVwPI) = 120,000*0.2+70,000*0.7+30,000*0.1 = 76.000 (see the payoff table in (a)).
Expected value of perfect information (EVPI) = EVwPI – EVw/oPI = 76,000  70,000 = 6,000. (It is always the same as Min EOL).
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Problem #27, page 581.
Cost per case = $10/case
Regular selling price = $12/case
Selling price for unsold ones = $2/case
Cost of shortage = $4/case
States of nature: Possible weekly demand, 15, 16, 17, or 18 cases of milk.
Decision alternatives: To stock 15, 16, 17, or 18 cases of milk each week.
Payoff: Total net profit each week.
Total net profit = total revenue – total cost
= $ in - $ out
= (Rev. from selling at regular price + Rev. from selling the unsold ones)
– (Cost to order from supplier + Cost of shortage)
For example:
If to stock 17 cases and weekly demand turns out to be 15 cases, then
total net profit = ($12*15 + $2*2) – ($10*17 + $4*0) = $184 - $170 = $14.
If to stock 16 cases and weekly demand turns out to be 17 cases, then
total net profit = ($12*16 + $2*0) – ($10*16 + $4*1) = $192 - $164 = $28.
Payoff Table and expected payoff of each alternative, (a) and (b)
Demands (states of nature) and Probabilities
To Stock
15 cases
16 cases
17 cases
18 cases
(decision
alternatives)
0.2
0.25
0.4
0.15
Expected Payoff
15 cases
$30
$26
$22
$18
$24
16 cases
$22
$32
$28
$24
$27.2
17 cases
$14
$24
$34
$30
$26.9
18 cases
$6
$16
$26
$36
$21
Column Max $30
$32
$34
$36
$33
The best decision alternative is to stock 16 cases of milk, which has the maximum
expected weekly payoff $27.2..
(c) The opportunity loss table
Demands (states of nature) and Probabilities
To Stock
15 cases
16 cases
17 cases
18 cases
(decision
alternatives)
0.2
0.25
0.4
0.15
Expected value (EOL)
15 cases
0
6
12
18
9
16 cases
8
0
6
12
5.8
17 cases
16
8
0
6
6.1
18 cases
24
16
8
0
12
The minimum EOL is $5.8, which is associated with decision alternative of
stocking 16 cases. Therefore, stocking 16 cases of milk each week is the best decision.
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(d) EVPI = minimum EOL = $5.8.
Or
Expected payoff without perfect information (EVw/oPI) = $27.2, as calculated in (b).
Expected payoff with perfect information (EVwPI) = 30*0.2+32*0.25+34*0.4+36*0.15 =
$33. Also see the last row in the payoff table in (a) and (b).
Expected value of perfect information (EVPI) = EVwPI – EVw/oPI = $33 – $27.2 = $5.8.
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Problem #29, page 582.
Cost of stocking = $2/dozen
Regular selling price = $3/dozen
Selling price for unsold ones = $0.75/dozen
Cost of shortage = $1/dozen
States of nature: Possible daily demand, 20, 22, 24, 26, 28, or 30 dozens of carnations.
Decision alternatives: To stock 20, 22, 24, 26, 28, or 30 dozens of carnations each day.
Payoff: Total net profit each day.
Total net profit = total revenue – total cost
= $ in - $ out
= (Rev. from selling at regular price + Rev. from selling the unsold ones)
– (Cost of stocking + Cost of shortage)
For example:
If to stock 20 dozens and demand turns out to be 28 dozens, then
total net profit = ($3*20 + $0.75*0) – ($2*20 + $1*8) = $60 - $48 = $12.
If to stock 30 dozens and demand turns out to be 24 dozens, then
total net profit = ($3*24 + $0.75*6) – ($2*30 + $1*0) = $76.5 - $60 = $16.5.
(a) Payoff table and Expected values:
Stocking
(decision
alternatives)
20 dozens
22 dozens
24 dozens
26 dozens
28 dozens
30 dozens
20
0.05
$20
$17.5
$15
$12.5
$10
$7.5
Demand (dozens) & Probability
22
24
26
28
0.1
0.25
0.3
0.2
$18
$22
$19.5
$17
$14.5
$12
$16
$20
$24
$21.5
$19
$16.5
$14
$18
$22
$26
$23.5
$21
$12
$16
$20
$24
$28
$25.5
30
0.1
Expected Payoffs
$10
$14
$18
$22
$26
$30
$14.4
$18.075
$21.1
$22.5
$21.95
$20.1
(b) The expected payoff of each decision alternative is shown in the table above.
The largest expected daily payoff is $22.5 which is associated with alternative “stocking
26 dozens”. So, the best decision is stocking 26 dozens of carnations.
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