Chapter 22
22.1 a
1
a
2
1
2
4
22.2
1
2
22.4
22.5 a
1
a
2
a
3
s
1
0 15 21
2
3
1
2
3
1
22.7a Produce
Demand a
0
a
1
a
2
a
3
s
0
0 -3.00 –6.00 -9.00
1
2
3
b Produce
Demand a
0
a
1
a
2
a
3
s
0
0 3.00 6.00 9.00
1
2
3
c
22.8a EMV(a
0
) = 0
EMV(a
1
) = .25(-3.00) + .25(5.00) + .25(5.00) + .25(5.00) = 3.00
2
EMV decision is a
2
(bake 2 cakes)
b EOL(a
0
) = .25(0) + .25(5.00) + .25(10.00) + .25(15.00) = 7.50
1
2
2
2
22.9 a
1
(flat fee) a
2
Pay per snowfall
s
0
-40,000 0
1
2
3
4
22.10 EMV(a
1
) = -40,000
2
1
22.11a Payoff Table
a
100
a
200
a
300
s
100
12(100)-10(100) 12(100)-9(200)+6(100) 12(100)-8.50(300)+6(200)
= 200 = 0 = –150
150
200
250
b Opportunity Loss Table
a
100
a
200
a
300
100
150
200
250
c
22.12
EMV(a
100
) = 200
200
300
22.13 P(s
0
) = .607, P(s
1
) = .303, P(s
2
) = .076, P(s
3
) = .012, P(s
4
) = .002
Payoff Table
a
0
a
1
a
2
a
3
s
0
0 -6,000 –12,000 -18,000
1
2
3
Opportunity Loss Table
a
0
a
1
a
2
a
3
s
0
0 6,000 12,000 18,000
1
2
3
22.14a EMV(Small) = .15(-220) + .55(-330) + .30(-440) = -346.5
EMV(Medium) = .15(-300) + .55(-320) + .30(-390) = -338.0
EMV(Large) = .15(-350) + .55(-350) + .30(-350) =-350.0
EMV decision: build a medium size plant; EMV*= -338.0
b Opportunity Loss Table
Small Medium Large
Low 0 80 130
22.15a P(s
10
) = 9/90 = .10, P(s
11
) = 18/90 = .20, P(s
12
) = 36/90 = .40, P(s
13
) = 27/90 = .30
Payoff Table
a
10
a
11
a
12
a
13
s
10
30 10(5)- 11(2)+2 10(5)-12(2)+3.50 10(5)-13(2)+4.50
11
12
13
b EMV(a
10
) = 30
EMV(a
11
) = .10(30) + .20(33) + .40(33) + .30(33) = 32.70
12
13
22.16 Payoff Table
Decision
Produce Don’t produce
Market share
5% -28 million 0
22.17 EPPI = .10(110) + .25(150) + .50(220) + .15(250) = 196
22.18 Opportunity Loss Table
a
1
a
2
a
3
s
1
50 0 35
2
3
4
1
2
3
22.19 EPPI = .5(65) + .5(110) = 87.5
EMV(a
1
) = .5(65) + .5(70) = 67.5
2
3
4
22.20 a EPPI = .75(65) + .25(110) = 76.25
EMV(a
1
) = .75(65) + .25(70) = 66.25
2
3
4
b EPPI = .95(65) + .05(110) = 67.25
EMV(a
1
) = .95(65) + .05(70) = 65.25
2
3
4
22.21 As the difference between the two prior probabilities increases EVPI decreases.
22.22 Posterior Probabilities for I
1
s
j
P(s
j
) P(I
1
|s
j
) P(s
j
and I
1
) P(s
j
| I
1
)
__________________________________________________________________________
s
1
.25 .40 (.25)(.40) = .10 .12/.20 = .500
1
Posterior Probabilities for I
2
s
j
P(s
j
) P(I
2
|s
j
) P(s
j
and I
2
) P(s
j
| I
2
)
__________________________________________________________________________
s
1
.25 .30 (.25)(.30) = .075 .075/.28 = .268
P(I
2
) = .28
Posterior Probabilities for I
3
s
j
P(s
j
) P(I
3
|s
j
) P(s
j
and I
3
) P(s
j
| I
3
)
__________________________________________________________________________
s
1
.25 .20 (.25)(.20) = .05 .05/.29 = .172
3
4
j
j
4
j
j
4
j
4
1
4
22.23 Posterior Probabilities for I
1
s
j
P(s
j
) P(I
1
|s
j
) P(s
j
and I
1
) P(s
j
| I
1
)
1
1
Posterior Probabilities for I
2
s
j
P(s
j
) P(I
2
|s
j
) P(s
j
and I
2
) P(s
j
| I
2
)
__________________________________________________________________________
1
2
22.24a Prior probabilities: EMV(a
1
) = .5(10) + .5(22) = 16
2
3
I
1
: EMV(a
1
) = .951(10) + .049(22) = 10.588
EMV(a
2
) = .951(18) + .049(19) = 18.049
EMV(a
3
) = .951(23) + .049(15) = 22.608
Optimal act: a
3
2
1
2
3
1
22.25 Prior probabilities: EMV(a
1
) = .333(60) + .333(90) + .333(150) = 100
EMV(a
2
) = 90
EMV* = 100
Posterior Probabilities for I
1
s
j
P(s
j
) P(I
1
|s
j
) P(s
j
and I
1
) P(s
j
| I
1
)
__________________________________________________________________________
1
3
1
Posterior Probabilities for I
2
s
j
P(s
j
) P(I
2
|s
j
) P(s
j
and I
2
) P(s
j
| I
2
)
__________________________________________________________________________
s
1
.333 .3 (.333)(.3) = .100 .100/.534 = .187
2
I
1
: EMV(a
1
) = .499(60) + .358(90) + .143(150) = 83.61
EMV(a
2
) = 90
2
1
2
22.26 Prior probabilities: EMV(a
1
) = .5(60) + .4(90) + .1(150) = 81
EMV(a
2
) = 90
EMV* = 90
Posterior Probabilities for I
1
s
j
P(s
j
) P(I
1
|s
j
) P(s
j
and I
1
) P(s
j
| I
1
)
__________________________________________________________________________
s
1
.5 .7 (.5)(.7) = .35 .35/.57 = .614
1
Posterior Probabilities for I
2
s
j
P(s
j
) P(I
2
|s
j
) P(s
j
and I
2
) P(s
j
| I
2
)
__________________________________________________________________________
1
1
1
1
EMV(a
2
) = 90
I2: EMV(a
1
) = .349(60) + .465(90) + .186(150) = 90.69
EVSI = EMV` – EMV* = 90.30 – 90 = .30
22.27 Prior probabilities: EMV(a
1
) = .90(60) + .05(90) + .05(150) = 66
2
1
s
j
P(s
j
) P(I
1
|s
j
) P(s
j
and I
1
) P(s
j
| I
1
)
__________________________________________________________________________
s
1
1
2
j
j
j
j
2
j
2
1
.90 .7 (.90)(.7) = .63 .63/.665 = .947
1
I
1
: EMV(a
1
) = .947(60) + .038(90) + .015(150) = 62.49
EMV(a
2
) = 90
I
2
: EMV(a
1
) = .806(60) + .075(90) + .119(150) = 72.96
EMV(a
2
) = 90
EMV` = .665(90) + .335(90) = 90
EVSI = EMV` – EMV* = 90 – 90 = 0
22.28 As the prior probabilities become more diverse EVSI decreases.
22.29
Payoff Table
Demand Purchase lot Don’t purchase lot