Decision-Tree Analysis

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Lecture No. 41

Chapter 12

Contemporary Engineering Economics

Copyright © 2010

Contemporary Engineering Economics, 5th edition, © 2010

Decision Tree Analysis

 A graphical tool for describing

(1) the actions available to the decision-maker,

(2) the events that can occur, and

(3) the relationship between the actions and events.

Contemporary Engineering Economics, 5th edition, © 2010

Constructing a

Decision Tree

A 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.

Decision Points

Events

( ) Probability

Market

 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 your price or lower your price.

Do not market

First Decision Point

$0

Our Price

High

Competitor’s price

High

(0.5)

Conditional

Profit

$60

(0.5)

Low

-$20

Competitive

Product (0.7)

High

(0.2)

$40

Low

No Competitive

Product (0.3) Low

(0.8)

$10

High

$100

Low

$30

Second Decision Point

The conditional profits associated with each event along with the likelihood of each event is shown in the decision tree.

Contemporary Engineering Economics, 5th edition, © 2010

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 to indicate that it is the expected value calculated over all branches emanating from that node.

 For each decision node, we select the one with the highest EMV (or minimum cost). Then those decision alternatives not selected are eliminated from further consideration.

Contemporary Engineering Economics, 5th edition, © 2010

Making Sequential Investment Decisions

$20

Set High Price

High

(0.5)

(0.5)

Low

$60

-$20

$44

Market

$44

Competitive

Product (0.7)

$20

Do not market

No Competitive

Product (0.3)

Low

High

(0.2)

$40

$16

Set High Price

Low

(0.8)

$10

$100

$100

Low

$0

$30

Contemporary Engineering Economics, 5th edition, © 2010

Decision Rules

 Market 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.

Contemporary Engineering Economics, 5th edition, © 2010

Practice Problem

 A company is considering the purchase of a new laborsaving machine.

 The machine’s 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 of hours 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 savings over the machine cost.

Contemporary Engineering Economics, 5th edition, © 2010

Construct a Decision Tree

-$5

0.10

$1.0

10

$1

Invest

0.60

11 0

12

0.30

Do not invest

$5

EMV = $1.0

Decision: Purchase the equipment

Contemporary Engineering Economics, 5th edition, © 2010

0

Expected Value of Perfect Information (EVPI)

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

Mathematical Relationship:

EVPI = EPPI – EMV = EOL where EPPI (Expected profit with perfect information) is the expected profit you could obtain if you had perfect information, and EMV

(Expected monetary value) is the expected profit you could obtain based on your own judgment. This is equivalent to expected opportunity loss ( EOL ).

Contemporary Engineering Economics, 5th edition, © 2010

Expected Value of Perfect Information (EVPI)

State of

Nature

10

11

12

Best

Strategy

Don’t Buy

Indifferent

Buy

Maximum

Payoff

0

0

5

Probability the State of

Nature

Occurs

0.10

0.60

0.30

Expected

Payoff or each State

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

Contemporary Engineering Economics, 5th edition, © 2010

Bill’s Decision Problem – $50,000 to Invest

 Decision Problem:

 Buying a highly speculative stock (d

1

) with three potential levels of return – High (50%),

Medium (9%), and Low (-

30%).

 Buying a very safe U.S.

Treasury bond (d

2

) 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.

Contemporary Engineering Economics, 5th edition, © 2010

Decision Tree for Bill’s Investment Problem

Contemporary Engineering Economics, 5th edition, © 2010

Evaluating Options in Bill’s Investment Problem

• 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) = $53,540

PW(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

EMV = $898

Or, prior optimal decision is Option 2

(c) 2001 Contemporary Engineering Economics 13

Expected Value of Perfect Information

Decision Option

Potential

Return Level

High (A)

Medium (B)

Probability Option1:

Invest in

Stock

0.25

$16,510

0.40

890

(Prior

Optimal)

Option 2:

Invest in

Bonds

$898

898

Low(C) 0.35

-13,967 898

EMV -$405 $898

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

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

EVPI = EPPI – EV

= $4,801 - $898

= $ 3,903

Optimal

Choice with

Perfect

Information

Stock

Bond

Opportunity

Loss

Associated with Investing in Bonds

$15,612

0

Bond

$3,903

EOL = (0.25)($15,612)

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

= $ 3,903

0

Contemporary Engineering Economics, 5th edition, © 2010

Bill’s Investment Problem with an Option of Getting

Professional Advice

Updating Conditional Profit (or Loss) after

Paying a Fee to the Expert (Fee = $200)

Revised Decision Tree

Contemporary Engineering Economics, 5th edition, © 2010

Conditional Probabilities of the Expert’s Prediction,

Given a Potential Return on the Stock

What the Report

Will Say

Favorable (F)

Unfavorable (UF)

Given Level of Stock Performance

High

(A)

0.80

0.20

Medium

(B)

0.65

0.35

Low

(C)

0.20

0.80

Contemporary Engineering Economics, 5th edition, © 2010

Nature’s Tree: Conditional Probabilities and

Joint Probabilities

Nature’s 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

Contemporary Engineering Economics, 5th edition, © 2010

Joint and Marginal Probabilities

What the Report Will Say

Joint Probabilities

When Potential

Level of Return is

Given

High (A)

Favorable (F)

0.20

Unfavorable (UF)

0.05

Medium (B)

Low (C)

Marginal

Probabilities

0.26

0.07

0.53

0.14

0.28

0.47

Contemporary Engineering Economics, 5th edition, © 2010

Marginal

Probabilities of

Return Level

0.25

0.40

0.35

1.00

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

Contemporary Engineering Economics, 5th edition, © 2010

Decision Making after Having Imperfect Information

- $6,319

Contemporary Engineering Economics, 5th edition, © 2010

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