3-1 Quantitative Analysis for Management Chapter 3 Fundamentals of Decision Theory Models

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Quantitative Analysis Quantitative Analysis for Managementfor Management

Chapter 3Chapter 3Fundamentals of Fundamentals of

Decision Theory ModelsDecision Theory Models

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Chapter OutlineChapter Outline3.1 Introduction

3.2 The Six Steps in Decision Theory

3.3 Types of Decision-Making Environments

3.4 Decision Making Under Risk

3.5 Decision Making Under Uncertainty

3.6 Marginal Analysis with a Large Number of Alternatives and States of Nature

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Learning ObjectivesLearning ObjectivesStudents will be able to:

List the steps of the decision-making processDescribe the types of decision-making

environmentsUse probability values to make decisions under

riskMake decisions under uncertainty where there is

risk but probability values are not knownUse computers to solve basic decision-making

problems

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IntroductionIntroduction Decision theory is an analytical and

systematic way to tackle problems

A good decision is based on logic.

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The Six Steps in Decision The Six Steps in Decision TheoryTheory

Clearly define the problem at hand List the possible alternatives Identify the possible outcomes List the payoff or profit of each combination

of alternatives and outcomes Select one of the mathematical decision

theory models Apply the model and make your decision

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Decision Table Decision Table for Thompson Lumberfor Thompson Lumber

State of NatureAlternative

200,000 -180,000

100,000 -20,000

0 0

Construct alarge plantConstruct a small plant

Do nothing

Favorable Market ($)

Unfavorable Market ($)

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Types of Decision-Making Types of Decision-Making EnvironmentsEnvironments

Type 1: Decision-making under certaintydecision-maker knows with certaintyknows with certainty the

consequences of every alternative or decision choice

Type 2: Decision-making under riskThe decision-maker knowsknows the probabilities of

the various outcomes Decision-making under uncertainty

The decision-maker does not knowdoes not know the probabilities of the various outcomes

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Decision-Making Under RiskDecision-Making Under Risk

n nature, of states ofnumber the to 1 j where

))P(S* (Payoff i) ativeEMV(Alternn

1jjSj

Expected Monetary Value:Expected Monetary Value:

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Decision Table Decision Table for Thompson Lumberfor Thompson Lumber


Probabilities

200,000 -180,000

100,000 -20,000

0 0


Do nothing



0.50 0.50

EMV

10,000

40,000

0

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Expected Value of Perfect Expected Value of Perfect Information (Information (EVPI))

EVPIEVPI places an upper bound on what one would pay for additional information

EVPIEVPI is the expected value with perfect information minus the maximum EMV

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Expected Value With Perfect Expected Value With Perfect Information (Information (EV|PI))

n nature, of states ofnumber the to 1 j

)P(S*j) nature of statefor outcomebest j

where

(PI|EVn

j

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Expected Value of Perfect Expected Value of Perfect InformationInformation

EVPIEVPI = EV|PIEV|PI - maximum EMVEMV

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Probabilities

200,000

0


Do nothing



0.50 0.50

EMV

40,000

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EVPIEVPI = expected value with perfect

information - max(EMVEMV)

= $200,000*0.50 + 0*0.50 - $40,000

= $60,000

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Expected Opportunity LossExpected Opportunity Loss EOLEOL is the cost of not picking the best

solution EOLEOL = Expected Regret

We want to maximize EMV or

minimize EOL

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Computing EOL - The Computing EOL - The Opportunity Loss TableOpportunity Loss Table

State of Nature

Alternative Favorable Market($)

UnfavorableMarket ($)

Large Plant 200,000 - 200,000 0 - (-180,000)

Small Plant 200,000 - 100,000 0 -(-20,000)

Do Nothing 200,000 - 0 0-0

Probabilities 0.50 0.50

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The Opportunity Loss Table The Opportunity Loss Table continuedcontinued

State of Nature

Alternative Favorable Market($)

UnfavorableMarket ($)

Large Plant 0 180,000

Small Plant 100,000 20,000

Do Nothing 200,000 0

Probabilities 0.50 0.50

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The Opportunity Loss Table The Opportunity Loss Table continuedcontinued

Alternative EOL

Large Plant (0.50)*$0 +(0.50)*($180,000)

$90,000

Small Plant (0.50)*($100,000)+ (0.50)(*$20,000)

$60,000

Do Nothing (0.50)*($200,000)+ (0.50)*($0)

$100,000

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Sensitivity AnalysisSensitivity Analysis

EMV(Large Plant) = $200,000PP - (1-P1-P)$180,000

EMV(Small Plant) = $100,000PP - $20,000(1-P1-P)

EMV(Do Nothing) = $0PP + 0(1-P1-P)

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Sensitivity Analysis - Sensitivity Analysis - continuedcontinued

-200000

-150000

-100000

-50000

0

50000

100000

150000

200000

250000

0 0.2 0.4 0.6 0.8 1

Values of P

EMV

Val

ues

Point 1 Point 2

EMV (Small Plant)

EMV(Large Plant)

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Decision MakingDecision Making Under Uncertainty Under Uncertainty

Maximax

Maximin

Equally likely (Laplace)

Criterion of Realism

Minimax

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Maximax - Choose the alternative with the maximum output


Probabilities

200,000 -180,000

100,000 -20,000

0 0


Do nothing



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Maximin - Choose the alternative with the maximum minimum output


Probabilities

200,000 -180,000

100,000 -20,000

0 0


Do nothing



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Equally likely (Laplace) - Assume all states of nature to be equally likely, choose maximum EMV


Probabilities

200,000 -180,000

100,000 -20,000

0 0


Do nothing



0.50 0.50

EMV

10,000

40,000

0

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Criterion of Realism (Hurwicz):CR = *(row max) + (1-)*(row min)


Probabilities

200,000 -180,000

100,000 -20,000

0 0


Do nothing



0.50 0.50

CR

124,000

76,000

0

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Minimax - choose the alternative with the minimum maximum Opportunity Loss


Probabilities

0 180,000

100,000 20,000

200,000 0

Construct alarge plantConstruct a small plantDo nothing



0.50 0.50

Max in row

180,000

100,000

200,000

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Marginal AnalysisMarginal Analysis P P = probability that demand is greater than

or equal to a given supply 1-P1-P = probability that demand will be less

than supply MPMP = marginal profit MLML = marginal loss Optimal decision rule is: P*MP P*MP (1-P)*ML (1-P)*ML or

MLMPML

P

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Marginal Analysis -Marginal Analysis -Discrete DistributionsDiscrete Distributions

Steps using Discrete Distributions:Determine the value for PP

Construct a probability table and add a

cumulative probability column

Keep ordering inventory as long as the

probability of selling at least one additional unit

is greater than PP

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Café du Donut ExampleCafé du Donut Example

Daily Sales(Cartons)

Probability of Salesat this Level

Probability that SalesWill Be at this Level

or Greater4 0.05 1.00

5 0.15 0.95

6 0.15 0. 80

7 0.20 0.65

8 0.25 0.45

9 0.10 0.20

10 0.10 0.10

1.00

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Café du Donut Example Café du Donut Example continuedcontinued

Marginal profit = selling price - cost

= $6 - $4 = $2 Marginal loss = cost Therefore:

.

MPMLML

P

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Café du Donut Example Café du Donut Example continuedcontinued

DailySales

(Cartons)

Probability ofSales at this Level

Probability thatSales Will Be at this

Level or Greater4 0.05 1.00 0.665 0.15 0.95 0.666 0.15 0. 80 0.667 0.20 0.65

8 0.25 0.45

9 0.10 0.20

10 0.10 0.10

1.00

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Marginal AnalysisMarginal AnalysisNormal DistributionNormal Distribution

= average or mean sales

= standard deviation of sales

MPMP = marginal profit

MLML = Marginal loss

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Marginal Analysis -Marginal Analysis -Discrete DistributionsDiscrete Distributions

Steps using Normal Distributions:Determine the value for P.

Locate P on the normal distribution. For a given area under the curve, we find Z from the standard Normal table.

Using we can now solve for X*

MPMLML

P

*XZ

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Joe’s Newsstand Example AJoe’s Newsstand Example A MLML = 4

MPMP = 6

= Average demand = 50 papers per day

= Standard deviation of demand = 10

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Joe’s Newsstand Example A Joe’s Newsstand Example A continuedcontinued

Step 1:

Step 2: Look on the Normal table for

PP = 0.6 (i.e., 1 - .04) ZZ = 0.25,

and

or:

.

MPMLML

P

*X.

newspapersor ..*X *

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Joe’s Newsstand Example A Joe’s Newsstand Example A continuedcontinued

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Joe’s Newsstand Example BJoe’s Newsstand Example B MLML = 8

MPMP = 2

= Average demand = 100 papers per day

= Standard deviation of demand = 10

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Joe’s Newsstand Example B Joe’s Newsstand Example B continuedcontinued

Step 1:

Step 2:

ZZ = -0.84 for an area of 0.80

and

or:

.

MPMLML

P

*X.

newspapersor ..X *

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Joe’s Newsstand Example B Joe’s Newsstand Example B continuedcontinued

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3-1 Quantitative Analysis for Management Chapter 3 Fundamentals of Decision Theory Models