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Lecture No. 41

Chapter 12

Contemporary Engineering Economics

Copyright 2010

Contemporary Engineering Economics, 5th edition, 2010


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Decision Tree Analysis A graphical tool for

describing

(1) the actions available tothe decision-maker,

(2) the events that canoccur, and

(3) the relationshipbetween the actions and

events.



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Constructing a

Decision TreeA Company is considering marketing a

new product. Once the product is

introduced, there is a 70% chance of

encountering a competitive product. Two

options are available each situation.

Option 1 (with competitive product):

Raise your price and see how your

competitor responds. If the competitor

raises price, your profit will be $60. If

they lower the price, you will lose $20.

Option 2 (without competitive

product): You still two options: raise yourprice or lower your price.

The conditional profits associated with

each event along with the likelihood of

each event is shown in the decision tree.


Conditional

Profit

Decision Points

Events

( ) Probability

Do not market

Market

Competitive

Product (0.7)

No Competitive

Product (0.3)

High

Low

High

Low

High

Low

High

Low

$60

-$20

$40

$10

$100

$30

(0.5)

(0.5)

(0.2)

(0.8)

$0

Our Price

Competitors price

First Decision Point Second Decision Point


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Rollback Procedure To analyze a decision tree, we begin at the end of the

tree and work backward.

For each chance node, we calculate the expected

monetary value (EMV), and place it in the node toindicate that it is the expected value calculated over allbranches emanating from that node.

For each decision node, we select the one with the

highest EMV (or minimum cost). Then those decisionalternatives not selected are eliminated from furtherconsideration.



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Making Sequential Investment Decisions


Do not market

Market

Competitive

Product (0.7)

No Competitive

Product (0.3)

Set High Price

Low

Set High Price

Low

High

Low

High

Low

$60

-$20

$40

$10

$100

$30

(0.5)

(0.5)

(0.2)

(0.8)

$0

$20

$16

$100

$20

$44

$44


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Decision RulesMarket the new product.

Whether or not you encounter a competitive

product, raise your price.

The expected monetary value associated

with marketing the new product is $44.



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Practice Problem A company is considering the purchase of a new labor-

saving machine.

The machines cost will turn out to be $55 per day. Each

hour of labor that is saved reduces costs by $5.However, there is some uncertainty over the number ofhours that actually will be saved.

It is judged that the hours of labor saved per day will be

10, 11, or 12, with probabilities of 0.10, 0.60, 0.30,respectively.

Let us define profit as the excess of labor-cost savingsover the machine cost.



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Construct a Decision Tree


Invest

Do not invest

-$5

0

$5

0

0.10

0.60

0.30

10

11

12

$1$1.0

EMV = $1.0

Decision: Purchase the equipment


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Expected Value of Perfect Information (EVPI)

What is EVPI? This is equivalent to asking yourself how much you canimprove your decision if you had perfect information.

Mathematical Relationship:

EVPI = EPPI EMV = EOL

where EPPI (Expected profit with perfect information) is the expectedprofit you could obtain if you had perfect information, and EMV(Expected monetary value) is the expected profit you could obtainbased on your own judgment. This is equivalent to expected

opportunity loss (EOL).



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Expected Value of Perfect Information (EVPI)


State of

Nature

Best

Strategy

Maximum

Payoff

Probability

the State of

Nature

Occurs

Expected

Payoff or

each State

10

11

12

Dont Buy

Indifferent

Buy

0

0

5

0.10

0.60

0.30

0

0

1.5

Expected Profit with Perfect Information (EPPI):

(0.10)(0) + (0.60)(0) + (0.30)(5) = $1.5

Expected Value of Perfect Information (EVPI) = EPPI EMV

$1.5 - $1 = $0.5


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Bills Decision Problem $50,000 to Invest

Decision Problem:

Buying a highly

speculative stock (d1) with

three potential levels of

return High (50%),

Medium (9%), and Low (-

30%).

Buying a very safe U.S.Treasury bond (d2) with a

guaranteed 7.5% return.

Seek advice from an expert?

Seek professional advice

before making the decision

Do not seek professional

advice do on his own.



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Decision Tree for Bills Investment Problem



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(c) 2001 Contemporary Engineering Economics 13

EMV = $898Or, prior optimaldecision is Option 2

Option 1:1) Period 0: (-$50,000 - $100) = -$50,100

Period 1: (+$75,000 - $100) - 0.20($24,800) =$69,940

PW(5%)=-$50,100 + $69,940 (P/F, 5%, 1) = $16,510

2) Period 0: (-$50,000 - $100)= -$50,100

Period 1: (+$54,500 - $100)- (0.20)($4,300) = $53,540PW(5%) = -$50,100 + $53,540 (P/F, 5%, 1) = $890

3) Period 0: (-$50,000 - $100) = -$50,100

Period 1: (+$35,000 - $100) (0.20)(-$14,800) = $37,940

PW(5%)= - $50,100 + $37,940 (P/F, 5%, 1) = -$13,967

Option 2:

Period 0: (- $50,000 - $150) = -$50,150

Period 1: (+$53,750 - $150) = $53,600

PW (5%)= -$50,150 + $53,600 (P/F, 5%, 1) = $898

Evaluating Options in Bills Investment Problem


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Expected Value of Perfect Information


Potential

Return Level Probability

Decision OptionOptimal

Choice with

Perfect

Information

OpportunityLoss

Associated

with Investing

in BondsOption1:

Invest in

Stock

(Prior

Optimal)

Option 2:

Invest inBonds

High (A) 0.25 $16,510 $898 Stock $15,612

Medium (B) 0.40 890 898 Bond 0

Low(C) 0.35 -13,967 898 Bond 0

EMV -$405 $898 $3,903

EPPI = (0.25)($16,510) + (0.40)($898)

+ (0.35)($898) = $4,801

EVPI = EPPI EV

= $4,801 - $898

= $3,903

EOL = (0.25)($15,612)

+ (0.40)(0) + (0.35)(0)

= $3,903


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Bills Investment Problem with an Option of Getting

Professional AdviceUpdating Conditional Profit (or Loss) after

Paying a Fee to the Expert (Fee = $200)Revised Decision Tree



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Conditional Probabilities of the Experts Prediction,

Given a Potential Return on the Stock


Given Level of Stock Performance

What the Report

Will Say

High

(A)

Medium

(B)

Low

(C)

Favorable (F) 0.80 0.65 0.20

Unfavorable (UF) 0.20 0.35 0.80


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Natures Tree: Conditional Probabilities and

Joint ProbabilitiesNatures Tree Joint & Marginal Probabilities


P(A,F) = P(F|A)P(A) = (0.80)(0.25) = 0.20

P(A,UF|A)P(A) = (0.20)(0.25) = 0.05

P(B,F) = P(F|B)P(B) = (0.65)(0.40) = 0.26

P(B,UF) = P(UF|B)P(B) = (0.35)(0.40) = 0.14

P(F) = 0.20 + 0.26 + 0.07 = 0.53

P(UF) = 1 P(F) = 1 0.53 =

0.47


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Joint and Marginal Probabilities


What the Report Will Say

Joint Probabilities

When Potential

Level of Return is

Given

Favorable (F) Unfavorable (UF)

Marginal

Probabilities of

Return Level

High (A) 0.20 0.05 0.25

Medium (B) 0.26 0.14 0.40

Low (C) 0.07 0.28 0.35

Marginal

Probabilities

0.53 0.47 1.00


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Determining Revised Probabilities


P(A|F) = P(A,F)/P(F) = 0.20/0.53 = 0.38

P(B|F) = P(B,F)/P(F) = 0.26/0.53 = 0.49

P(C|F) = P(C,F)/P(F) = 0.07/0.53 = 0.13

P(A|UF) = P(A,UF)/P(UF) = 0.05/0.47 = 0.11

P(B|UF) + P(B,UF)/P(UF) = 0.14/0.47 = 0.30

P(C|UF) = P(C,UF)/P(UF) = 0.28/0.47 = 0.59


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Decision Making after Having Imperfect Information


- $6,319

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