Solution to Sample Questions
#1
Let
xS  acres of soybeans to plant
xC = acres of corn to plant
xO = acres of oats to plant
C = number of cows to purchase
H = number of hens to purchase
T = number of hours to work in town
Maximize 600x S  900x C  450xO  1000C  5H  6T
subject to
land:
xS  xC  x O  1.5C  125 acres
funds:
1200C  9H  $40,000
labor:
70x S  105xC  50xO  100C  0.9H  T  7500 hours
barn:
C  32
chicken house:
H  3000
and
xS , xC , xO ,C, H,T  0.
#2 a.
b.
c.
d.
Pro = 1000, College = 200, High School = 0. Profit = $4200.
The solution will change (outside of allowable decrease). Profit will decrease.
The solution will not change (inside of allowable increase). Profit increases $2000.
∆Z = (Shadow Price)(∆RHS) = ($0.40)(+1000) = +$400 (Note: shadow price valid
because inside of allowable increase). This is less than the leasing cost, so they should not
lease the sewing machine.
e. ∆Z = (Shadow Price)(∆RHS) = (–$3)(–100) = +$300 (Note: shadow price valid because
inside of allowable decrease). Thus, profit increases by $300. The solution will change to
generate this extra profit.
f. The allowable decrease should be 4000. The shadow price is zero because there is slack
in this constraint. The shadow price will remain zero so long as there is slack. The righthand-side can be reduced up to 4000 before there is no longer slack.
g. The change is outside of the allowable decrease, so the shadow price is not valid, and we
can not determine exactly how much this will cost them. On the other hand, the shadow
price is valid for the first 3000 minutes of decrease. For just these 3000 minutes lost, the
cost will be ∆Z = (Shadow Price)(∆RHS) = ($0.40)(–3000) = –$1200. Thus, they will
lose at least $1200. Since they will lose at least $1200, they would clearly be better off to
spend $1000 to replace the sewing machines.
#3 a) They should purchase the 21-Day advance ticket, with an expected cost of $695.
Full Fare
900
900
900
0.5
Can be reissued
2
0.4
Meeting changes
695
575
75
0
987.5
575
0.5
Can not be reissued
21-Day Advance
1400
900
500
1400
695
0.6
Meeting doesn’t change
500
0
500
b) 20% of the time, the schedule will change. Of these, half can be reissued, so a total of
10% will need to be reissued, and a total of 10% will need to be repurchased (full fare).
The expected cost of reissue/repurchase = (0.1)($75) + (0.1)($900) = $97.50. Let T be the
cost of the ticket. The total expected cost will be T + $97.50. Since T+$97.50<$695 when
T<$597.50, this is the most you should be willing to pay.
#4 a) They should go to high capacity at the start of year 1. Expected payoff = $512,000.
0.8
Y2 Demand Low (20,000)
320000
Stay Low Capacity (20,000)
0
320000
0.4
Y1 Demand Low (20,000)
200000
320000
0.2
Y2 Demand High (40,000)
320000
200000
320000
1
200000
320000
0.8
Y2 Demand Low (20,000)
240000
Expand Capacity (40,000)
-80000
280000
200000
240000
0.2
Y2 Demand High (40,000)
440000
Low Capacity (20,000)
-80000
400000
368000
440000
0.2
Y2 Demand Low (20,000)
320000
Stay Low Capacity (20,000)
0
320000
0.6
Y1 Demand High (40,000)
200000
320000
0.8
Y2 Demand High (40,000)
320000
200000
320000
2
200000
400000
0.2
Y2 Demand Low (20,000)
240000
Expand Capacity (40,000)
200000
240000
2
512000
-80000
400000
0.8
Y2 Demand High (40,000)
440000
400000
440000
0.8
Y2 Demand Low (20,000)
0.4
Y1 Demand Low (20,000)
200000
320000
280000
200000
280000
0.2
Y2 Demand High (40,000)
480000
High Capacity (40,000)
-120000
400000
512000
480000
0.2
Y2 Demand Low (20,000)
0.6
Y1 Demand High (40,000)
400000
640000
480000
200000
480000
0.8
Y2 Demand High (40,000)
680000
400000
680000
#4 b) Assign the highest payoff ($680,000) a utility value of 1.
Assign the lowest payoff ($240,000) a utility value of 0.
$6 80,0 00 (U=1 )
p
$2 40,0 00 (U=0 )
1-p
x
For each other final payoff in tree (from part a), what does p need to be so that I’m
indifferent between that payoff (x) and the above lottery with probability p of $680,000
(and probability 1-p of $240,000). Then U(x) = p.
For me: when x = $280,000, p must be 0.15, so U($280,000) = 0.15.
When x = $320,000, p must be 0.3, so U($320,000) = 0.3.
When x = $440,000, p must be 0.6, so U($440,000) = 0.6.
When x = $480,000, p must be 0.7, so U($480,000) = 0.7.
(These answers will vary.)
This leads to the following utility function graph, which is risk-averse.
Utility
1.0
0.8
0.6
0.4
0.2
0
240
280
320
360
400
440
480
Payoff ($000)
520
560
600
640
680
#4 c)
0.8
Y2 Deman d Lo w (20,0 00)
0.3
Sta y Low Capa city (20,0 00)
0
20 0000
0.3
0.4
Y1 Deman d Lo w (20,0 00)
0.3
0.2
Y2 Deman d High (40,000)
0.3
20 0000
0.3
1
20 0000
0.3
0.8
Y2 Deman d Lo w (20,0 00)
0
Expan d Capa city (40,0 00)
-80 000
20 0000
0.12
0
0.2
Y2 Deman d High (40,000)
0.6
Lo w Ca pacity (20 ,000 )
-80 000
40 0000
0.408
0.6
0.2
Y2 Deman d Lo w (20,0 00)
0.3
Sta y Low Capa city (20,0 00)
0
20 0000
0.3
0.6
Y1 Deman d High (40,000)
0.3
0.8
Y2 Deman d High (40,000)
0.3
20 0000
0.3
2
20 0000
0.48
0.2
Y2 Deman d Lo w (20,0 00)
0
Expan d Capa city (40,0 00)
20 0000
0
2
0.668
-80 000
0.48
0.8
Y2 Deman d High (40,000)
0.6
40 0000
0.6
0.8
Y2 Deman d Lo w (20,0 00)
0.4
Y1 Deman d Lo w (20,0 00)
20 0000
0.26
0.15
20 0000
0.15
0.2
Y2 Deman d High (40,000)
0.7
High Capacity (4 0,00 0)
-12 0000
40 0000
0.668
0.7
0.2
Y2 Deman d Lo w (20,0 00)
0.6
Y1 Deman d High (40,000)
40 0000
0.94
0.7
20 0000
0.7
0.8
Y2 Deman d High (40,000)
1
40 0000
1
#5 a) Jerry should open a large shop, with an expected payoff of $25,000.
0.5
Market Favorable
80
Large Shop
80
0
25
80
0.5
Market Unfavorable
-30
-30
-30
0.5
Market Favorable
50
1
Small Shop
50
50
25
0
20
0.5
Market Unfavorable
-10
-10
-10
No Shop
15
15
15
b) With the information, Jerry would open a large shop with a favorable prediction, and
open no shop with a negative prediction, with an expected payoff of $34,250.
EVSI = $34,250 – $25,000 = $9250 (This is the most Jerry should pay for the
information).
0.727273
Market Favorable
80
Large Shop
0
80
50
80
0.272727
Market Unfavorable
-30
-30
0.727273
Market Favorable
0.55
Predict Favorable
50
1
0
-30
Small Shop
50
50
50
0
33.636
0.272727
Market Unfavorable
-10
-10
-10
No Shop
15
15
15
0.222222
Market Favorable
34.25
80
Large Shop
0
80
-5.556
80
0.777778
Market Unfavorable
-30
-30
0.222222
Market Favorable
0.45
Predict Unfavorable
50
3
0
-30
Small Shop
50
50
15
0
3.3333
0.777778
Market Unfavorable
-10
-10
-10
No Shop
15
15
15
c) EVPI = 0.5($80,000) + (0.5)($15,000) – $25,000 = $22,500.