supplement a decision making copyright ©2013 pearson education, inc. publishing as prentice hall a...

Supplement ADecision Making

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 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.


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.


Break-Even Analysis

By setting revenue equal to total cost

pQ = F + cQ

Q =F

p - c

Total cost = F + cQ

Total revenue = pQ


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

Q = Fp - c

= 1,000 patients= 100,000200 – 100


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


Example A.1

Total annual costs

Fixed costs

Break-even quantity

Profits

Loss

Patients (Q)

Dol

lars

(in

thou

sand

s)

400 –

300 –

200 –

100 –

0 –

| | | |

500 1000 1500 2000

(2000, 300)

Total annual revenues

The two lines intersect at 1,000 patients, the break-even quantity

(2000, 400)


Application A.1The 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 ExhibitGate receipts $4Concessions $5Licensed apparel $15

Estimated Fixed CostsExhibit construction $2,400,000Salaries $220,000Food $30,000

Estimated Variable Costs per PersonConcessions $2Licensed apparel $9

Is the predicted increase in attendance sufficient to break even?

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A -08

Application A.1Q TR = pQ TC = F + cQ0 $0 $2,650,000

250,000 $6,000,000 $5,400,000

7 –

6 –

5 –

4 –

3 –

2 –

1 –

0 –| | | | | |

50 100 150 200 250

Cost

and

reve

nue

(mill

ions

of d

olla

rs)

Q (thousands of patrons)

Total Cost

Total Revenue

Wherep = 4 + 5 + 15 = $24F = 2,400,000 + 220,000 + 30,000

= $2,650,000c = 2 + 9 = $11


Application A.1

Q TR = pQ TC = F + cQ

0 $0 $2,650,000250,000 $6,000,000 $5,400,000

Wherep = 4 + 5 + 15 = $24F = 2,400,000 + 220,000 + 30,000

= $2,650,000c = 2 + 9 = $11

Algebraic solution of Denver Zoo problempQ = F + cQ

24Q = 2,650,000 + 11Q13Q = 2,650,000

Q = 203,846


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?

200(1,500) – [100,000 + 100(1,500)]pQ – (F + cQ) =

= $50,000


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


Make-or-buy decision

• Total cost to buy Fb + cbQ

• Total cost to make Fm + cmQ


Fb + cbQ = Fm + cmQ

Q =Fm – Fb

cb – cm

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?


The formula for the break-even quantity yields the following:

Q =Fm – Fb

cb – cm

= 19,200 salads= 12,000 – 2,4002.0 – 1.5

Example A.3


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?


Buy Make

Fixed costs $0 $300,000

Variable costs $9 $7

Q =Fm – Fb

cb – cm

Q =300,000 – 0

9 – 7 Q = 150,000

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.


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?

Example A.4

Performance Criterion Weight (A) Score (B) Weighted Score (A B)Market potential

30 8240

Unit profit margin20 10

200

Operations compatibility20 6

120

Competitive advantage15 10

150

Investment requirements10 2

20

Project risk 54

20

Weighted score = 750


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

Example A.4

Total 750


Application A.3The 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

10 550

Unit profit margin30 8

240

Operations compatibility20 10

200

Competitive advantage25 7

175

Investment requirements 10 3

30

Project risk 54

20

Weighted score = 715

No. Because 715 >800

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


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?

Possible Future DemandAlternative Low HighSmall facility 200 270Large facility 160 800Do nothing 0 0


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

Possible Future DemandAlternative Low HighSmall facility 200 270Large facility 160 800Do nothing 0 0


Decision Making under Uncertainty

• Maximin

• Maximax

• Laplace

• Minimax Regret


Example A.6Reconsider 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 Worst PayoffSmall facility 200Large facility 160

The best of these worst numbers is $200,000, so the pessimist would build a small facility.


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.


Example A.6

c. Laplace. With two events, we assign each a probability of 0.5. Thus, the weighted payoffs ($000) are

The best of these weighted payoffs is $480,000, so the realist would build a large facility.

0.5(200) + 0.5(270) = 235

0.5(160) + 0.5(800) = 480

Alternative Weighted PayoffSmall facility

Large facility


Example A.6d. 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 Maximum Regret

Small facility 200 – 200 = 0 800 – 270 =530 530

Large facility 200 – 160 = 40 800 – 800 = 0 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.


Application A.4Fletcher (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 RoutesNo treaty

Land RoutesTreaty

Sea RoutesOnly

Arrows $840,000 $440,000 $190,000

Barrels $370,000 $220,000 $670,000

Wagons $25,000 $1,150,000 ($25,000)

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

A - 30Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

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.


Example A.7Reconsider 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:

Possible Future Demand

Alternative Small Large

Small facility 200 270

Large facility 160 800

0.4(200) + 0.6(270) = 242

0.4(160) + 0.6(800) = 544

Alternative Expected Value

Small facility

Large facility


The large facility is the best

alternative.

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?


AlternativeLand 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

Application A.5

Application A.5

A - 34

AlternativeLand routes, No

Treaty(0.50)


(0.30)Sea routes Only (0.20) Expected Value

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` + (.30)* 1,150,000 + (.20) * -25,000 352,500

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Arrows is the best alternative.

Payoff 1

Payoff 2

Payoff 3

Alternative 3

Alternative 4

Alternative 5

Payoff 1

Payoff 2

Payoff 3

E1 & Probability

E2 & Probability

E3 & Probability

Altern

ative 1

Alternative 2

E2 & Probability

E3 & Probability

E 1 &

Probabilit

y

Payoff 1

Payoff 2

1stdecision

1

Possible2nd decision

2

Decision Trees

= Event node

= Decision node

Ei = Event i

P(Ei) = Probability of event i


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


Example A.8$200

$223

$270

$40

$800

$20

$220

Don’t expand

Expand

Low demand [0.4]

High demand [0.6]

2

Low demand

[0.4]

High demand [0.6]

3

Do nothing

Advertise

Modest response [0.3]

Sizable response [0.7]

Small facil

ity

Large facility

1


Example A.8$200

$223

$270

$40

$800

$20

$220

Don’t expand

Expand

Low demand [0.4]

High demand [0.6]

2

Low demand

[0.4]

High demand [0.6]

3

Do nothing

Advertise



Small facil

ity

Large facility

1 0.3 x $20 = $6

0.7 x $220 = $154

$6 + $154 = $160


Example A.8$200

$223

$270

$40

$800

$20

$220

Don’t expand

Expand

Low demand [0.4]

High demand [0.6]

2

Low demand

[0.4]

High demand [0.6]

3

Do nothing

Advertise



Small facil

ity

Large facility

1

$160$160


Example A.8$200

$223

$270

$40

$800

$20

$220

Don’t expand

Expand

Low demand [0.4]

High demand [0.6]

2

Low demand

[0.4]

High demand [0.6]

3

Do nothing

Advertise



Small facil

ity

Large facility

1

$160$160

$270


Example A.8$200

$223

$270

$40

$800

$20

$220

Don’t expand

Expand

Low demand [0.4]

High demand [0.6]

2

Low demand

[0.4]

High demand [0.6]

3

Do nothing

Advertise



Small facil

ity

Large facility

1

$160$160

$270

x 0.4 = $80

x 0.6 = $162

$80 + $162 = $242


Example A.8$200

$223

$270

$40

$800

$20

$220

Don’t expand

Expand

Low demand [0.4]

High demand [0.6]

2

Low demand

[0.4]

High demand [0.6]

3

Do nothing

Advertise



Small facil

ity

Large facility

1

$160$160

$270

$242

x 0.6 = $480

0.4 x $160 = $64

$544


Example A.8$200

$223

$270

$40

$800

$20

$220

Don’t expand

Expand

Low demand [0.4]

High demand [0.6]

2

Low demand

[0.4]

High demand [0.6]

3

Do nothing

Advertise



Small facil

ity

Large facility

1

$160$160

$270

$242

$544

$544


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?

AlternativeLand routes,

No Treaty(0.50)


(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


Application A.6


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?


Solved Problem 1

a. Beginning with the algebraic approach, we get

Q =F

p – c

= 3,111 units

= 56,00025 – 7

Using the graphic approach, shown in Figure A.6, we first draw two lines:

The two lines intersect at Q = 3,111 units, the break-even quantity

Total revenue =Total cost =

25Q56,000 + 7Q


Total costs

Break-evenquantity

250 –

200 –

150 –

100 –

50 –

0 –

Units (in thousands)

Dol

lars

(in

thou

sand

s)

| | | | | | | |

1 2 3 4 5 6 7 8

Total revenues

3.1

$77.7

Solved Problem 1


Solved Problem 1

b. Total profit contribution = Total revenue – Total cost= pQ – (F + cQ)

= 15(10,000) – [56,000 + 7(10,000)]= $24,000


Solved Problem 2Herron 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?

RatingPerformance Criteria Product A Product B Product C1. Demand uncertainty and project risk

39 2

2. Similarity to present products7

8 6

3. Expected return on investment (ROI)10

4 8

4. Compatibility with current manufacturing process 4

7 6

5. Competitive Strategy4

6 5Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 50

Solved Problem 2

Each of the five criteria receives a weight of 1/5 or 0.20

The best choice is product B as Products A and C are well behind in terms of total weighted score

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) + (0.20 × 4) + (0.20 × 4) = 5.6

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) + (0.20 × 7) + (0.20 × 6) = 6.8

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) + (0.20 × 6) + (0.20 × 5) = 5.4

Product Calculation Total Score

A

B

C


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. Maximinb. Maximaxc. Laplaced. Minimax regret


Solved Problem 3

The payoff table for this problem is

Demand for Red Roses

Alternative Low(25 dozen)

Medium(60 dozen)

High(130 dozen)

Order 25 dozen $625 $625 $625

Order 60 dozen $100 $1,500 $1,500

Order 130 dozen ($950) $450 $3,250Do nothing $0 $0 $0


Solved Problem 3a. 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.


Solved Problem 4White 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 Utilization Installation OperationOne lift Bad times (0.3) 0.9 $50,000 $200,000

Normal times (0.5) 1.0 $50,000 $200,000Good times (0.2) 1.0 $50,000 $200,000

Two lifts Bad times (0.3) 0.9 $90,000 $200,000Normal times (0.5) 1.5 $90,000 $400,000Good times (0.2) 1.9 $90,000 $400,000


Solved Problem 4The 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).

0.9(490) – (50 + 200) = 191

1.0(490) – (50 + 200) = 240

1.0(490) – (50 + 200) = 240

0.9(490) – (90 + 200) = 151

1.5(490) – (90 + 400) = 245

1.9(490) – (90 + 400) = 441

Alternatives Economic Conditions Payoff Calculation (Revenue – Cost)

One lift Bad times

Normal times

Good times

Two lifts Bad times

Normal times

Good times


Bad times [0.3]

Normal times [0.5]

Good times [0.2]

$191

$240

$240

Bad times [0.3]

Normal times [0.5]

Good times [0.2]

$151

$245

$441

One lift

Two lifts

$256.0

$225.3

$256.0

Solved Problem 4

0.3(191) + 0.5(240) + 0.2(240) = 225.3

0.3(151) + 0.5(245) + 0.2(441) = 256.0



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