3-27 Farm Grown, Inc., produces cases of perishable food products.

Each case contains an

assortment of vegetables and other farm products. Each case costs $5 and sells for $15. If
there are any cases not sold by the end of the day, they are sold to a large food processing
company for $3 a case. The probability that daily demand will be 100 cases is 0.3, the
probability that the demand will be 200 cases is 0.4, and the probability that daily demand
will be 300 cases is 0.3. Farm Grown has a policy of always satisfying customer demands. If
its own supply of cases is less than the demand, it buys the necessary vegetables from a
competitor. The estimated cost of doing this is $16 per case.
a) Draw the decision table for this problem
100 200 300 EMV
100 (100 x 15) - [(100 x 15) – (100 [(100 x 15) - (100 0.3(1000) +
(100 x 5) x 5)] – [(100 x 16) x 5)] – [(200 x 16) 0.4 (900) +
= 1000 – (100 x 15)] – (200 x 15)] 0.3 (800)
=900 =800 = 900
200 [(100 x 15) - (200 x 15) - (200 x [(200 x 15) - (200 0.3(800) + 0.4
(200 x 5)] + 5) x 5)] – [(100 x 16) (2000) + 0.3
(100 x 3) = 2000 – (100 x 15)] (1900)
= 800 = 1900 = 1610
300 [(100 x 15) - [(200 x 15) - (300 (300 x 15) - (300 x 0.3(600) + 0.4
(300 x 5)] + x 5)] + (100 x 3) 5) (1800) + 0.3
(200 x 3) = 1800 = 3000 (3000)
= 600 = 1800
Probabilit 0.3 0.4 0.3 Produce 300
y cases

b) What do you recommend

 Produce only 300 cases each day because it yields the highest EMV among all
 Proceed with selling to the large food processor to minimize loss and to gain some
profit
 Buy from competitors when demand is high to avoid loss of customer and to retain
loyal customer

3-28 In Problem 3-27, Farm Grown, Inc. has reason to believe the probabilities may not be
reliable due to changing conditions. If these probabilities are ignored, what decision would
be made using the optimistic criterion? What decision would be made using the pessimistic
criterion?
100 200 300 EMV Optimistic Pessimistic
100 100 900 800 900 1000 800
0
200 800 200 190 1610 2000 800
0 0
300 600 180 300 1800 3000 600

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 0 0
Probabilit 0.3 0.4 0.3 Produce 300 Produce 300 Produce 100
y cases cases cases

 Optimistic - Farm Grown. Inc. must produce 300 cases each day to gain maximum
profit of $3000.
 Pessimistic - Farm Grown. Inc. can choose to produce 100 cases and get maximum
profit of $800 and need to continue borrowing from the competitors to meet the
demand in case out of stock.

3-29 Mick Karra is the manager of MCZ Drilling Products, which produces a variety of
specialty Valves for oil equipment. Recent activity in the oil fields has caused demand to
increase drastically, and a decision has been made to open a new manufacturing facility.
Three locations are being considered and the size of the facility would not be the same in
each location. Thus, overtime might be necessary at times. The following table gives the
total monthly cost (in $1,000s) for each possible location under each demand possibility. The
probabilities for the demand levels have been determined to be 20% for low demand, 30%
for medium demand, and 50% for high demand.

Demand is low Demand is medium Demand is High

Ardmore, Ok 85 110 150
Sweetwater, Tx 90 100 140
lake Charles, LA 110 120 130

(a) Which location would be selected based on the optimistic criterion?

Low Medium Hig Optimistic

h
Ardmore, 85 110 150 85
Ok
Sweetwater, 90 100 140 90
Tx
Lake 110 120 130 110
Charles, LA
Probability 0.20 0.30 0.5

(b) Which location would be selected based on the pessimistic criterion?

Low Medium Hig Optimistic Pessimistic

h
Ardmore, 85 110 150 85 150
Ok
Sweetwater, 90 100 140 90 140

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 Tx
Lake 110 120 130 110 130
Charles, LA
Probability 0.20 0.30 0.5

(c) Which location would be selected based on the minimax regret criterion?

Opportunity loss table

Low Medium High Maximum in

row
Ardmore, Ok 85 - 85 100- 110 130 – 150 20
=0 = 10 = 20
Sweetwater, 85- 90 100 – 100 130 – 140 10
Tx =5 =0 = 10
lake Charles, 85 – 110 100 – 120 130 – 130 25
LA = 25 =20 =0

(d) Which location should be selected to minimize the expected cost of operation?

Low Medium Hig Optimistic Pessimistic EMV

h
Ardmore, 85 110 150 85 150 0.2(85) + 0.3
Ok (110) + 0.5 (150)
=125
Sweetwater, 90 100 140 90 140 0.2(90) + 0.3
Tx (100) + 0.5 (140)
=118
Lake 110 120 130 110 130 0.2(110) + 0.3
Charles, LA (120) + 0.5 (130)
=123
Probability 0.20 0.30 0.5

(e) How much is a perfect forecast of the demand worth?

(f) Which location would minimize the expected opportunity loss?

Low Mediu High Maximum in EOL

m row
Ardmore, Ok 0 10 20 20 0.2(0) + 0.3
(10) + 0.5 (20)
=13

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 Sweetwater, Tx 5 0 10 10 0.2(5) + 0.3
(0) + 0.5 (10)
=6
Lake Charles, 25 20 0 25 0.2(25) + 0.3
LA (20) + 0.5 (0)
=11
Probability 0.20 0.30 0.5

(g) What is the expected value of perfect information in this situation?

EVPI = EV wPI – maximum EMV

= [ 0.2 (85) + 0.3 (100) + 0.5 (130) ] – 118

= 112-118

= -6

3-30 Even though independent gasoline stations have been having a difficult time, Susan
Solomon has been thinking about starting her own independent gas station. Susan’s
problem is to decide how large her station should be. The annual returns will depend on
both the size of the station and a number of marketing factors related to oil industry and
demand for gasoline. After careful analysis, Susan developed the following table

Size of first station Good market Fair market Poor market

Small 50 000 20 000 -10 000
Medium 80 000 30 000 -20 000
Large 100 000 30 000 -40 000
Very large 300 000 25 000 -160 000

For Example, if Susan constructs a small station and the market is good, she will realize a
profit of $50 000.

a. Develop a decision table for this decision.

Size of first station Good market Fair market Poor market

Small 50 000 20 000 -10 000
Medium 80 000 30 000 -20 000
Large 100 000 30 000 -40 000
Very large 300 000 25 000 -160 000

b. What is the Maximax decision?

Size of first station Good market Fair market Poor market Optimistic

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 Small 50 000 20 000 -10 000 50 000
Medium 80 000 30 000 -20 000 80 000
Large 100 000 30 000 -40 000 100 000
Very large 300 000 25 000 -160 000 300 000

c. What is the Maximin decision?

Size of first station Good market Fair market Poor market Pessimistic
Small 50 000 20 000 -10 000 -10 000
Medium 80 000 30 000 -20 000 -20 000
Large 100 000 30 000 -40 000 -40 000
Very large 300 000 25 000 -160 000 -160 000

d. What is the equally likely decision

Size of first station Good market Fair market Poor market Equally likely decision
Small 50 000 20 000 -10 000 20 000
Medium 80 000 30 000 -20 000 30 000
Large 100 000 30 000 -40 000 30 000
Very large 300 000 25 000 -160 000 55 000

e. What is the criterion of realism decision? Use an α value of 0.8

Size of first station Good market Fair market Poor market Criterion of realism
decision
Small 50 000 20 000 -10 000 0.8 ( 50 000) + 0.2 (-10
000)
= 38 000
Medium 80 000 30 000 -20 000 0.8 ( 80 000) + 0.2 (-20
000)
= 60 000
Large 100 000 30 000 -40 000 0.8 ( 100 000) + 0.2 (-
40 000)
= 72 000
Very large 300 000 25 000 -160 000 0.8 300 000) + 0.2 (-
160 000)
= 208 000

f. Develop an Opportunity Loss Table

Size of first station Good market Fair market Poor market

Small 300 000 -50 000 30 000 - 20 000 -10 000 – (-10 000)
= 250 000 = 10 000 =0
Medium 300 000 - 80 000 30 000 - 30 000 -10 000 – (-20 000)

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 = 220 000 =0 = 10 000
Large 300 00 - 100 000 30 000- 30 000 -10 000 – (-40 000)
= 200 000 =0 = 30 000
Very large 300 000 – 300 25 000 – 30 000 -10 000 – (-160
000 = 5 000 000)
=0 = 150 000

g. What is the Minimax Regret Decision?

Size of first Good market Fair market Poor market Maximum

station in row
Small 300 000 -50 000 30 000 - 20 000 -10 000 – (-10 000) 250 000
= 250 000 = 10 000 =0
Medium 300 000 - 80 000 30 000 - 30 000 -10 000 – (-20 000) 220 000
= 220 000 =0 = 10 000
Large 300 00 - 100 000 30 000- 30 000 -10 000 – (-40 000) 200 000
= 200 000 =0 = 30 000
Very large 300 000 – 300 25 000 – 30 000 -10 000 – (-160 150 000
000 = 5 000 000)
=0 = 150 000

3-31 Beverly Mills has decided to lease a hybrid car to save on gasoline expenses and to do
her part to help keep the environment clean. The car she selected is available from only one
dealer in the local area, but that dealer has several leasing options to accommodate a
variety of driving patterns. All the leases are for 3 years and require no money at the time of
signing the lease. The first option has a monthly cost of $330, a total mileage allowance of
36,000 miles (an average of 12,000 miles per year), and a cost of $0.35 per mile for any miles
over 36,000. The following table summarizes each of the three lease options:

3-YEAR LEASE MONTHLY MILEAGE COST PER

COST ALLOWANCE EXCESS MILE
Option 1 $330 36 000 $0.35
Option 2 $380 45 000 $0.25
Option 3 $430 54 000 $0.15

Beverly has estimated that, during the 3 years of the lease, there is a 40% chance she will
drive an average of 12,000 miles per year, a 30% chance she will drive an average of 15,000
miles per year, and a 30% chance that she will drive 18,000 miles per year. In evaluating
these lease options, Beverly would like to keep her costs as low as possible.

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 (a) Develop a payoff (cost) table for this situation.

12 000 15 000 18 000

Option 1 36 x 330 (36 x 330) + 0.35 [(3x 15000) (36 x 330) + 0.35 [(3x
36 000 = 11 880 – 36 000)] 18000) – 36 000)]
= 15 030 = 18 180
Option 2 36 x 380 (36 x 380) + 0.25 [(3x 15000) (36 x 380) + 0.25 [(3x
45 000 = 13 680 – 45 000)] 18000) – 45 000)]
= 13 680 = 15 930
Option 3 36 x 430 36 x 430 (36 x 430) + 0.15 [(3x
54 000 = 15 480 = 15 480 18000) – 54 000)]
= 15 480
Probabilit 0.4 0.3 0.3
y

(b) What decision would Beverly make if she were optimistic?

12 000 15 000 18 000 Optimistic

Option 1 11 880 15 030 18 180 11 880
36 000
Option 2 13 680 13 680 15 930 13 680
45 000
Option 3 15 480 15 480 15 480 15 480
54 000
Probabilit 0.4 0.3 0.3
y

(c) What decision would Beverly make if she were pessimistic?

12 000 15 000 18 000 Pessimistic

Option 1 11 880 15 030 18 180 18 180
36 000
Option 2 13 680 13 680 15 930 15 930
45 000
Option 3 15 480 15 480 15 480 15 480
54 000
Probability 0.4 0.3 0.3

(d) What decision would Beverly make if she wanted to minimize her expected cost
(monetary value)?

12 000 15 000 18 000 EMV

Option 1 11 880 15 030 18 180 0.4 ( 11 880 ) + 0.3 ( 15 030) +
36 000 0.3 ( 18 180)

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 = 14 715
Option 2 13 680 13 680 15 930 0.4 (13 680) + 0.3 (13 680) + 0.3
45 000 (15 930)
= 14 355
Option 3 15 480 15 480 15 480 0.4 (15 480 ) + 0.3 (15 480 ) +
54 000 0.3 (15 480)
= 15 480
Probabilit 0.4 0.3 0.3
y

(f) Calculate the expected value of perfect information for this problem.

EVPI = EV wPI – maximum EMV

= [ 0.4 (11 880) + 0.3 (13 680) + 0.3 (15 480) ] – 14 355

= 13 500 - 14 355

= -855

Note this problem is based on costs, so the minimum values are the best.1. For a 3-year
lease, there are 36 months of payments.

Option 1 total monthly payments: 36($330) = $11,880

Option 2 total monthly payments: 36($380) = $13,680

Option 3 total monthly payments: 36($430) = $15,480

Excess mileage costs based on 36,000 mileage allowance for Option 1, 45,000 for Option 2,
and 54,000 for option 3.

Option 1 excess mileage cost if 45,000 miles are driven = (45000 – 36000)(0.35) = 3150

Option 1 excess mileage cost if 54,000 miles are driven = (54000 – 36000)(0.35) = 6300

Option 2 excess mileage cost if 54,000 miles are driven = (54000 – 45000)(0.25) = 2250

The total cost for each option in each state of nature is obtained by adding the total monthly
payment cost to the excess mileage cost.

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