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Supplement A
Decision Making
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A - 01
Decision Making Tools
• Break-even analysis
– Analysis to compare processes by finding the volume at which
two processes have equal total costs.
• Preference matrix
– Table that allows managers to rate alternatives based on several
performance criteria.
• Decision theory
– Approach when outcomes associated with alternatives are
in doubt.
• Decision Tree
– Model to compare alternatives and their possible
consequences.
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Break-even analysis notation
• Variable cost (c)– The portion of the total cost that varies directly with
volume of output.
• Fixed cost (F) –
– The portion of the total cost that remains constant
regardless of changes in levels of output.
• Quantity (Q) –
– The number of customers served or units produced per
year.
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Break-Even Analysis
Total cost = F + cQ
Total revenue = pQ
By setting revenue equal to total cost
pQ = F + cQ
F
Q=
p-c
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Example A.1
A hospital is considering a new procedure to be offered at
$200 per patient. The fixed cost per year would be $100,000
with total variable costs of $100 per patient. What is the
break-even quantity for this service? Use both algebraic and
graphic approaches to get the answer.
The formula for the break-even quantity yields
100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100
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Example A.1
The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients)
(Q)
Total Annual Cost ($)
(100,000 + 100Q)
Total Annual Revenue ($)
(200Q)
0
100,000
0
2,000
300,000
400,000
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Example A.1
Dollars (in thousands)
400 –
300 –
(2000, 400)
Profits
Total annual revenues
(2000, 300)
Total annual costs
Break-even quantity
200 –
100 –
Loss
0–
Fixed costs
|
500
|
|
1000
1500
Patients (Q)
|
2000
 The two lines
intersect at
1,000
patients, the
break-even
quantity
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Application A.1
The Denver Zoo must decide whether to move twin polar bears to Sea
World or build a special exhibit for them and the zoo. The expected
increase in attendance is 200,000 patrons. The data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food
$2,400,000
$220,000
$30,000
Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Is the predicted
increase in
attendance
sufficient to
break even?
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Application A.1
TR = pQ
$0
$6,000,000
TC = F + cQ
$2,650,000
$5,400,000
7–
Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11
6–
Cost and revenue
(millions of dollars)
Q
0
250,000
5–
4–
3–
2–
1–
0 |–
|
|
|
|
|
50
100
150
200
250
Q (thousands of patrons)
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Application A.1
Q
TR = pQ
TC = F + cQ
0
250,000
$0
$6,000,000
$2,650,000
$5,400,000
Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11
Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846
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Example A.2
If the most pessimistic sales forecast for the proposed
service from Example 1 was 1,500 patients, what would be
the procedure’s total contribution to profit and overhead per
year?
pQ – (F + cQ) =
200(1,500) – [100,000 + 100(1,500)]
= $50,000
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Make-or-buy decision notation
• Fb
– The fixed cost (per year) of the buy option
• Fm
– The fixed cost of the make option
• cb
– The variable cost (per unit) of the buy option
• cm
– The variable cost of the make option
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Make-or-buy decision
• Total cost to buy
Fb + cbQ
• Total cost to make
Fm + cmQ
Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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Example A.3
• A fast-food restaurant featuring hamburgers is adding
salads to the menu
• The price to the customer will be the same
• Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
• Preassembled salads could be purchased from a local
supplier at $2.00 per salad
• Preassembled salads would require additional
refrigeration with an annual fixed cost of $2,400
• Expected demand is 25,000 salads per year
• What is the break-even quantity?
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Example A.3
The formula for the break-even quantity yields the
following:
Fm – Fb
Q= c –c
b
m
=
12,000 – 2,400
= 19,200 salads
2.0 – 1.5
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Application A.2
• At what volume should the Denver Zoo be
indifferent between buying special sweatshirts from
a supplier or have zoo employees make them?
Fixed costs
Buy
$0
Make
$300,000
Variable costs
$9
$7
Fm – Fb
Q= c –c
b
m
300,000 – 0
Q= 9–7
Q = 150,000
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Preference Matrix
• A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria.
– The criteria can be scored on any scale as long as the same
scale is applied to all the alternatives being compared.
• Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100.
– The total score is the sum of the weighted scores (weight ×
score) for all the criteria and compared against scores for
alternatives.
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Example A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion
Weight (A) Score (B) Weighted Score (A  B)
Market potential
30
8
240
Unit profit margin
20
10
200
Operations compatibility
20
6
120
Competitive advantage
15
10
150
Investment requirements
10
2
20
5
4
20
Project risk
Weighted score =
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750
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Example A.4
Because the sum of the weighted scores is 750, it falls short
of the score of 800 for another product. This result is
confirmed by the output from OM Explorer’s Preference
Matrix Solver below
Total
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750
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Application A.3
The following table shows the performance criteria, weights, and
scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion
Weight (A) Score (B) Weighted Score (A  B)
Market potential
10
5
50
Unit profit margin
30
8
240
Operations compatibility
20
10
Competitive advantage
25
7
Investment
requirements
10
3
30
5
4
20
Project risk
No.
Because
715 >800
Weighted score =
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200
175
715
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Decision Theory Steps
• List a reasonable number of feasible alternatives
• List the events (states of nature)
• Calculate the payoff table showing the payoff for
each alternative in each event
• Estimate the probability of occurrence for each
event
• Select the decision rule to evaluate the alternatives
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Example A.5
• A manager is deciding whether to build a small or a large
facility
• Much depends on the future demand
• Demand may be small or large
• Payoffs for each alternative are known with certainty
• What is the best choice if future demand will be low?
Alternative
Small facility
Large facility
Do nothing
Possible Future Demand
Low
High
200
270
160
800
0
0
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Example A.5
• The best choice is the one with the highest payoff
• For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
• Under these conditions, the larger facility has a payoff of
only $160,000
Alternative
Small facility
Large facility
Do nothing
Possible Future Demand
Low
High
200
270
160
800
0
0
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Decision Making under Uncertainty
• Maximin
• Maximax
• Laplace
• Minimax Regret
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Example A.6
Reconsider the payoff matrix in Example 5. What is the best
alternative for each decision rule?
a. Maximin. An alternative’s worst payoff is the lowest
number in its row of the payoff matrix, because the
payoffs are profits. The worst payoffs ($000) are
Alternative
Small facility
Large facility
Worst Payoff
200
160
The best of these worst numbers is $200,000, so the
pessimist would build a small facility.
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Example A.6
b. Maximax. An alternative’s best payoff ($000) is the
highest number in its row of the payoff matrix, or
Alternative
Best Payoff
Small facility
270
Large facility
800
The best of these best numbers is $800,000, so the
optimist would build a large facility.
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Example A.6
c. Laplace. With two events, we assign each a probability
of 0.5. Thus, the weighted payoffs ($000) are
Alternative
Small facility
Large facility
Weighted Payoff
0.5(200) + 0.5(270) = 235
0.5(160) + 0.5(800) = 480
The best of these weighted payoffs is $480,000, so
the realist would build a large facility.
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Example A.6
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to
be low, the regret is 40 (or 200 – 160).
Regret
Alternative
Low Demand
High Demand
Small facility
200 – 200 = 0
800 – 270 =530
Large facility
200 – 160 = 40
800 – 800 = 0
Maximum
Regret
530
40
The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a
large facility. The biggest regret is associated with having
only a small facility and high demand.
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Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability
of these policy “events.” The following data are available:
Payoffs (Profits)
Alternative
Land Routes
No treaty
Land Routes
Treaty
Sea Routes
Only
Arrows
$840,000
$440,000
$190,000
Barrels
$370,000
$220,000
$670,000
Wagons
$25,000
$1,150,000
($25,000)
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Application A.4
• Which product would be favored by Fletcher (realist)?
– Fletcher (realist – Laplace) would choose arrows
• Which product would be favored by Cooper (pessimist)?
– Cooper (pessimist – Maximin) would choose barrels
• Which product would be favored by Wainwright (optimist)?
– Wainwright (optimist – Maximax) would choose wagons
• What is the minimax regret solution?
– The Minimax Regret solution is arrows
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Decision Making Under Risk
• Use the expected value rule
• Weigh each payoff with associated probability
and add the weighted payoff scores.
• Choose the alternative with the best expected
value.
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Example A.7
Reconsider the payoff matrix in Example 5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
The expected value for each
alternative is as follows:
Alternative
Possible Future
Demand
Alternative
Small
Large
Small facility
200
270
Large facility
160
800
Expected Value
Small facility
0.4(200) + 0.6(270) = 242
Large facility
0.4(160) + 0.6(800) = 544
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The large
facility is
the best
alternative.
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Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
What alternative has the best expected results?
Land routes,
No Treaty
(0.50)
Land Routes,
Treaty Only
(0.30)
Sea routes,
Only (0.20)
Arrows
840,000
440,000
190,000
Barrels
370,000
220,000
670,000
Wagons
25,000
1,150,000
-25,000
Alternative
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Application A.5
Land routes, No
Treaty
(0.50)
Land Routes,
Treaty Only
(0.30)
Arrows
(.50) * 840,000` +
(.30)* 440,000 + (.20) * 190,000 590,000
Barrels
(.50) * 370,000` +
(.30)* 220,000 + (.20) * 670,000 385,000
Wagons
(.50) * 25,000` +
Alternative
(.30)* 1,150,000 +
Sea routes
Only (0.20)
Expected Value
(.20) * -25,000 352,500
Arrows is the
best alternative.
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Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability
Payoff 1
Payoff 2
Payoff 3
Alternative 3
1
2
1st
decision
Alternative 5
Possible
2nd decision
E2 & Probability
= Event node
Alternative 4
E3 & Probability
Payoff 1
Payoff 2
Payoff 3
Payoff 1
Payoff 2
= Decision node
Ei = Event i
P(Ei) = Probability of event i
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Example A.8
• A retailer will build a small or a large facility at a new location
• Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
• For a small facility and high demand, not expanding will have a
payoff of $223,000 and a payoff of $270,000 with expansion
• For a small facility and low demand the payoff is $200,000
• For a large facility and low demand, doing nothing has a payoff
of $40,000
• The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7, respectively
• For a modest response the payoff is $20,000 and $220,000 if the
response is sizable
• For a large facility and high demand the payoff is $800,000
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Example A.8
Low demand [0.4]
Don’t expand
2
1
Expand
$200
$223
$270
Do nothing
$40
3
Modest response [0.3]
Advertise
Sizable response [0.7]
$20
$220
High demand [0.6]
$800
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Example A.8
Low demand [0.4]
Don’t expand
2
1
Expand
$200
$223
$270
0.3 x $20 = $6
Do nothing
$40
3
Modest response [0.3]
$20
Advertise
$6 + $154 = $160
Sizable response [0.7]
$220
0.7 x $220 = $154
High demand [0.6]
$800
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A - 38
Example A.8
Low demand [0.4]
Don’t expand
2
Expand
1
$200
$223
$270
Do nothing
$40
3
Modest response [0.3]
Advertise
$160
$160
Sizable response [0.7]
$20
$220
High demand [0.6]
$800
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A - 39
Example A.8
Low demand [0.4]
Don’t expand
2
Expand
$270
1
$200
$223
$270
Do nothing
$40
3
Modest response [0.3]
Advertise
$160
$160
Sizable response [0.7]
$20
$220
High demand [0.6]
$800
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A - 40
Example A.8
Low demand [0.4]
$200
x 0.4 = $80
$80 + $162 = $242
Don’t expand
2
Expand
$270
1
$223
$270
x 0.6 = $162
Do nothing
$40
3
Modest response [0.3]
Advertise
$160
$160
Sizable response [0.7]
$20
$220
High demand [0.6]
$800
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A - 41
Example A.8
Low demand [0.4]
$200
$242
Don’t expand
2
Expand
$270
1
$223
$270
Do nothing
$40
3
Advertise
$160
0.4 x $160 = $64
$544
Modest response [0.3]
$160
High demand [0.6]
$800
Sizable response [0.7]
$20
$220
x 0.6 = $480
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A - 42
Example A.8
Low demand [0.4]
$200
$242
Don’t expand
2
Expand
$270
1
$223
$270
Do nothing
$40
$544
3
Advertise
$160
$160
$544
Modest response [0.3]
Sizable response [0.7]
$20
$220
High demand [0.6]
$800
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A - 43
Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application 5
b. What is the expected payoff for the best alternative
in the decision tree below?
Land routes,
No Treaty
(0.50)
Land Routes,
Treaty Only
(0.30)
Arrows
840,000
440,000
190,000
Barrels
370,000
220,000
670,000
Wagons
25,000
1,150,000
-25,000
Alternative
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Sea routes, Only
(0.20)
A - 44
Application A.6
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A - 45
Solved Problem 1
• A small manufacturing business has patented a new
device for washing dishes and cleaning dirty kitchen sinks
• The owner wants reasonable assurance of success
• Variable costs are estimated at $7 per unit produced and
sold
• Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic and
graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?
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Solved Problem 1
a. Beginning with the algebraic approach, we get
Q=
56,000
F
=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first draw
two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A - 47
Solved Problem 1
250 –
Dollars (in thousands)
200 –
Total revenues
150 –
Break-even
quantity
100 –
$77.7
Total costs
50 –
3.1
0–
|
|
|
|
|
|
|
|
1
2
3
4
5
6
7
8
Units (in thousands)
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A - 48
Solved Problem 1
b. Total profit contribution
= Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000
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A - 49
Solved Problem 2
Herron Company is screening three new product idea: A, B, and C.
Resource constraints allow only one of them to be commercialized. The
performance criteria and ratings, on a scale of 1 (worst) to 10 (best),
are shown in the following table. The Herron managers give equal
weights to the performance criteria. Which is the best alternative, as
indicated by the preference matrix method?
Performance Criteria
1. Demand uncertainty and project risk
2. Similarity to present products
3. Expected return on investment (ROI)
4. Compatibility with current
manufacturing process
5. Competitive Strategy
Product A
3
7
10
4
Rating
Product B
9
8
4
7
Product C
2
6
8
6
4
6
5
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A - 50
Solved Problem 2
Each of the five criteria receives a weight of
1/5 or 0.20
Product
Calculation
Total Score
A
(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)
= 5.6
B
(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)
= 6.8
C
(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)
= 5.4
The best choice is product B as Products A and C are well behind in
terms of total weighted
score
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Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret
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A - 52
Solved Problem 3
The payoff table for this problem is
Demand for Red Roses
Low
(25 dozen)
Alternative
Medium
(60 dozen)
High
(130 dozen)
Order 25 dozen
$625
$625
$625
Order 60 dozen
$100
$1,500
$1,500
($950)
$450
$3,250
$0
$0
$0
Order 130 dozen
Do nothing
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Solved Problem 3
a. Under the Maximin criteria, Weiss should order 25 dozen, because
if demand is low, Weiss’s profits are $625, the best of the worst
payoffs.
b. Under the Maximax criteria, Weiss should order 130 dozen. The
greatest possible payoff, $3,250, is associated with the largest
order.
c. Under the Laplace criteria, Weiss should order 60 dozen. Equally
weighted payoffs for ordering 25, 60, and 130 dozen are about
$625, $1,033, and $917, respectively.
d. Under the Minimax regret criteria, Weiss should order 130 dozen.
The maximum regret of ordering 25 dozen occurs if demand is
high: $3,250 – $625 = $2,625. The maximum regret of ordering 60
dozen occurs if demand is high: $3,250 – $1,500 = $1,750. The
maximum regret of ordering 130 dozen occurs if demand is low:
$625 – (–$950) = $1,575.
A - 54
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Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will
be necessary. Each lift can accommodate 250 people per day
and skiing occurs 7 days per week in the 14-week season and
lift tickets cost $20 per customer per day. The table below
shows all the costs and probabilities for each alternative and
condition. Should the resort purchase one lift or two?
Alternatives
Conditions
One lift
Two lifts
Utilization
Installation
Operation
Bad times (0.3)
0.9
$50,000
$200,000
Normal times (0.5)
1.0
$50,000
$200,000
Good times (0.2)
1.0
$50,000
$200,000
Bad times (0.3)
0.9
$90,000
$200,000
Normal times (0.5)
1.5
$90,000
$400,000
Good times (0.2)
1.9
$90,000
$400,000
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Solved Problem 4
The decision tree is shown on the following slide. The payoff
($000) for each alternative-event branch is shown in the
following table. The total revenues from one lift operating at
100 percent capacity are $490,000 (or 250 customers × 98 days
× $20/customer-day).
Alternatives
Economic Conditions
One lift
Bad times
0.9(490) – (50 + 200) = 191
Normal times
1.0(490) – (50 + 200) = 240
Good times
1.0(490) – (50 + 200) = 240
Bad times
0.9(490) – (90 + 200) = 151
Normal times
1.5(490) – (90 + 400) = 245
Good times
1.9(490) – (90 + 400) = 441
Two lifts
Payoff Calculation (Revenue – Cost)
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A - 56
Solved Problem 4
Bad times [0.3]
0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]
$191
$240
One lift
$225.3
$256.0
Good times [0.2]
Bad times [0.3]
Two lifts
Normal times [0.5]
$256.0
0.3(151) + 0.5(245) +
0.2(441) = 256.0
Good times [0.2]
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
$240
$151
$245
$441
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