Chapter 8 Decision Analysis

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Chapter 8Decision Analysis

MGS3100Julie Liggett De Jong

Terminology

States of nature

Payoffs / payoff table

Probability

The payoff tablepayoff table is a fundamental component in decision analysis models

State of Nature1 2 … mDecision

d1 r11 r12 … r1m

d2 r21 r22 … r2m

dn rn1 rn2 … rnm

… … … … …

Table 1, p81

Terminology

Expected Return

Regret

EVPI

EVSI

Three Classes of Decision Models

Decisions under:

certainty

risk

uncertainty

Decisions under certainty

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If I know for sure that it will be raining when I leave work this afternoon, should I take my umbrella to work today?

If I know for sure that it will be raining when I leave work this afternoon, should I take my umbrella to work today?

Rain

Take Umbrella 0Do Not -7.00

Table 2, p82

Decisions under risk Multiple states of nature

We size up the likelihood of each state of nature happening

Historical frequencies

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Historical frequencies

Subjective estimates

We calculate Expected ReturnsExpected Returns

E(X) = Σpixi

ERi* = maximum overall i of ERi

We choose the alternative that yields the maximum expected return. In other words, i* is the optimal decision where

c) Calculate expected values: expected shortage (S) & expected excess (E) inventory

1.70

0.18*1=0.18

0.18*1=0.180.14*3=0.420.18*1=0.180.16*2=0.32

0.14*3=0.42

Exp(s)

0.240.180.240.240.180.140.180.160.080.14

Prob.

2

E

0.16Expected

1

1312

3

S

42434242434543444045

Demand

.51

.58

.29

.50

.55

.96

.66

.80

.02

.97

RN

109876543

0.08*2=0.1621

Exp(E)Week

All-Ways-Open Market The Newsvendor ModelThe Newsvendor ModelThe Newsvendor Model

Selling Price: $ .75Purchase Price: $ .40Goodwill cost: $ .50

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A B C D E1 Selling Price 752 Purchase Cost 403 Goodwill Cost 5045 States of Nature6 Decision 0 1 2 37 0 0 -50 -100 -1508 1 -40 35 -15 -659 2 -80 -5 70 20

10 3 -120 -45 30 105

The Newsvendor ModelThe Newsvendor ModelThe Newsvendor Model

Table 4, p84


Selling Price: $ .75Purchase Price: $ .40Goodwill cost: $ .50

Demand distribution:

P0 = Prob(demand = 0) = 0.1P1 = Prob(demand = 1) = 0.3P2 = Prob(demand = 2) = 0.4P3 = Prob(demand = 3) = 0.2

What is the Expected Return?

A B C D E F1 Selling Price 752 Purchase Cost 403 Goodwill Cost 5045 States of Nature6 Decision 0 1 2 3 Expected Return7 0 0 -50 -100 -150 -858 1 -40 35 -15 -65 -12.59 2 -80 -5 70 20 22.5

10 3 -120 -45 30 105 7.51112 Probabilities 0.1 0.3 0.4 0.2

The Newsvendor ModelThe Newsvendor ModelThe Newsvendor ModelDecisions under uncertainty

Multiple states of nature

Don’t know what state of nature will occur

Decisions under uncertainty

LaplaceMaximinMaximaxMinimax regret

Laplace

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Laplace

Assume all Assume all states of nature states of nature are equally likely are equally likely to occurto occur

What is the Expected Return?


10 3 -120 -45 30 105 7.51112 Probabilities 0.1 0.3 0.4 0.2


Maximin

extremely conservative or pessimistic approach to making decisions

Maximin

Evaluate minimumminimumpossible return associated with each decision.

Maximin

Select decision yielding maxmaximum value of minminimum returns.

Table 1, p81

Maximin

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Different criterion yields different decisions. Consider the decision table below:

• Under the Maximin criterion, you would choose decision 1.

• Under the Maximax criterion, you would choose decision 2.

Maximax

optimistic approach to making decisions

Maximax

Evaluate maximummaximumpossible return associated with each decision

Maximax

Select decision yielding maxmaximum of these maxmaximum returns.

Maximax Different criterion yields different decisions.

• Under the Maximin criterion, you would choose decision 1.

• Under the Maximax criterion, you would choose decision 2.

Which is the best choice?

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Minimax regret Regret measures the desirability of an outcome.

Choose the decision that minimizes the regret for making that choice.

a)Find the maximum value in columncolumn 1

b)Subtract every value in column 1 from this value

c)Repeat for each column







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After regret table is built:

d)Choose the maximum value in each row

e)choose the smallest

After regret table is built:

d)Choose the maximum value in each row

e)Choose the smallest (minimum of the maximum)

A B C D E F1 Selling Price 752 Purchase Cost 403 Goodwill Cost 5045 States of Nature6 Decision 0 1 2 37 0 0 -50 -100 -1508 1 -40 35 -15 -659 2 -80 -5 70 20

10 3 -120 -45 30 10511 0 35 70 1051213 Regret MinMax Regret14 0 0 85 170 255 25515 1 40 0 85 170 17016 2 80 40 0 85 8517 3 120 80 40 0 1201819 85

Minimax Regret Each method yields different decisions regarding the newsvendor data:

CriteriaCriteria DecisionLaPlace Cash Flow Order 2 papers

Maximax Cash Flow Order 3 papers

Minimax Regret Order 2 papers

Maximin Cash Flow Order 1 paper

How much would you be willing to pay for perfect information?


10 3 -120 -45 30 105 7.51112 Probabilities 0.1 0.3 0.4 0.2

What is the most money the newsvendor should be willing to pay for perfect information?

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EVPI = -expected return

with perfectinformation

maximum possibleexpected returnwithout sample

information


10 3 -120 -45 30 105 7.51112 Probabilities 0.1 0.3 0.4 0.2

Decision Trees

Graphical tool used to analyze decisions under risk

Useful to analyze sequences of decisions

TreePlanAn add-in used to draw decision trees in Excel.

Bayes’ TheoremAllows us to incorporate new information into the process.

SonoraloCellular Phones

3 strategies

Aggressive

Major commitment

Major capital expenditure

Large inventory

Major global marketing campaign

Basic

Move productionto existing facility

Modify current line

Maintain inventory for popular items

Local/regional advertising

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Cautious

Use excess capacity

Minimize retooling

Produce enough to satisfy demand

Advertise at discretion of local dealer

States of Nature

Strong Demand (S)

Weak Demand (W)

Sonoralo Cellular PhonesPayoff table

A square nodesquare node represents a point at which a decision must be made.

Each line (branch) leading from a square represents a possible decision.

A square nodesquare node represents a point at which a decision must be made.

A circular nodecircular node represents an event (a situation when the outcome is not certain).

Each line (branch) leading from a circle represents a possible outcome.

• Insert the CD into the CD-ROM drive. • Select Run... from the Windows Start

menu. • Type d:\html\TreePlan\Treeplan.xla &

select "OK". • TreePlan will launch in Microsoft Excel

as an add-in to the Tools menu. • In the Microsoft Excel dialog box,

select Enable Macros. • For additional assistance go to Help.

TREE PLAN

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The Completed Decision Tree

Decision Trees: Incorporating New Information

Before implementing the Basic strategy, the corporate marketing research group performs a marketing study and reports on whether the study is encouraging (E) or discouraging (D).

We will consider the new information before we make a decision.

Prior Probabilities

Conditional Probabilities / Reliabilities

Joint & Marginal ProbabilitiesJoint & Marginal Probabilities

Posterior ProbabilitiesPosterior Probabilities

Terminology

Prior Probabilities:

Initial estimates, such as P(S) and P(W).

A MARKET RESEARCH STUDY FOR CELLULAR PHONES

Sonorola has estimated the prior probabilities as P(S) = 0.45 and P(W) = 0.55.

Conditional Probabilities / Reliabilities:

For two events A and B, the conditional probability [P(A|B)], is the probability of event A occurs givengiven that event B will occur.


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For example, P(E|S) is the conditional probability that marketing gives an encouraging report given given that the market is in fact going to be strong.



If marketing were perfectly reliable, P(E|S) = 1.


Marketing has the following “track record” in predicting the market:

P(E|S) = 0.6P(E|S) = 0.6

P(D|S) = 1 P(D|S) = 1 -- P(E|S) = 0.4P(E|S) = 0.4

P(D|W) = 0.7P(D|W) = 0.7

P(E|W) = 1 P(E|W) = 1 -- P(D|W) = 0.3P(D|W) = 0.3

Posterior Probabilities:

Conditional probabilities, such as P(S|E).


We’ll use Bayes’ Theorem to calculate the posterior probabilities.

Calculating Posterior Probabilities:

1. Enter given Reliabilities (conditional probabilities).

2. Calculate Joint Probabilities by multiplying Reliabilities by Prior Probabilities.

3. Compute Marginal Probabilities by summing the entries in each row.

4. Generate Posterior Probabilities by dividing each row entry of joint probability table by its row sum.

P(E|W)P(D|W)

P(S) P(W)

P(E&S)

P(W|E)P(W|D)

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A new decision tree!

I

II

IV

V

VI

III

VII

VIII

IX

E

D

AB

C

AB

C

SW

S

WS

W

S

WS

WS

W

P(E)

P(D)

P(W|E)P(S|E)

P(S|E)

P(S|E)

P(S|D)

P(W|E)

P(W|E)

P(W|D)

P(W|D)

P(W|D)

P(S|D)

P(S|D)

30

-820

75

1530

-820

75

15

How much should we be willing to spend on sample information?

Sonoralo Cellular PhonesPayoff table (w/out sample info) EVSI = -

maximum possibleexpected return

with sampleinformation


information

EVSI is the upper bound of how much one would be willing to pay for this particular sample information.

THE EXPECTED VALUE OF SAMPLE INFORMATION

EVSI = 13.46 – 12.85 = $0.61 million.


with perfect information

maximum possibleexpected returnwithout perfect

information

THE EXPECTED VALUE OF PERFECT INFORMATION

EVPI is the maximum possible increase in the expected return that can be obtained from new information. ERPI = 30(0.45) + 15(0.55) = 21.75

THE EXPECTED RETURN WITH PERFECT INFORMATION

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with perfect information


information

THE EXPECTED VALUE OF PERFECT INFORMATION

EVPI = 21.75 – 12.85 = $8.90 million

EVPI is the maximum possible increase in the expected return that can be obtained from new information.

EVSI = -maximum possible

expected returnwith sampleinformation


information


with perfectinformation


information

Sequential Decisions: Sequential Decisions: To Test or Not to TestTo Test or Not to Test

The value in performing the market research test depends on how Sonorola uses the information generated by the test.

The value of an initial decision depends on a sequencesequence of decisions and uncertain events that will follow the initial decision. This is called a sequential sequential decision modeldecision model.

Chapter 8 Decision Analysis

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