Chapter 8 Decision Analysis

Chapter 8 Decision Analysis

Terminology States of nature Payoffs / payoff table Probability

MGS3100 Julie Liggett De Jong

The payoff table is a fundamental component in decision analysis models State of Nature 2 …

m

r11

r12

…

r1m

d2

r21

r22

…

r2m

…

…

…

…

…

dn

rn1

rn2

…

rnm

Decision

1

d1

Terminology Expected Return Regret EVPI EVSI

Table 1, p81

Three Classes of Decision Models Decisions under:

Decisions under certainty

certainty risk uncertainty

1

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 Do Not

0 -7.00

Table 2, p82

Decisions under risk

We size up the likelihood of each state of nature happening

Multiple states of nature

Historical frequencies

2

Historical frequencies

We calculate Expected Returns

Subjective estimates

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

E(X) = Σpixi

ERi* = maximum overall i of ERi

The Newsvendor Model

All-Ways-Open Market c) Calculate expected values: expected shortage (S) & expected excess (E) inventory Week

RN

Demand

Prob.

S

1

.97

45

0.14

3

E

Exp(s)

2

.02

40

0.08

3

.80

44

0.16

2

0.16*2=0.32

4

.66

43

0.18

1

0.18*1=0.18

5

.96

45

0.14

3

0.14*3=0.42

6

.55

43

0.18

1

0.18*1=0.18

7

.50

42

0.24

8

.29

42

0.24

9

.58

43

0.18

1

0.18*1=0.18

10

.51

42

0.24

Exp(E)

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

0.14*3=0.42 2

Expected

0.08*2=0.16

1.70

0.16

3

The Newsvendor Model 1 2 3 4 5 6 7 8 9 10

A Selling Price Purchase Cost Goodwill Cost

B

C

D

The Newsvendor Model Selling Price: $ .75 Purchase Price: $ .40 Goodwill cost: $ .50

E

75 40 50

Decision

0 0 -40 -80 -120

0 1 2 3

States of Nature 1 2 -50 -100 35 -15 -5 70 -45 30

3 -150 -65 20 105

Demand distribution: P0 = Prob(demand = 0) = 0.1 P1 = Prob(demand = 1) = 0.3 P2 = Prob(demand = 2) = 0.4 P3 = Prob(demand = 3) = 0.2 Table 4, p84

The Newsvendor Model 1 2 3 4 5 6 7 8 9 10 11 12

A Selling Price Purchase Cost Goodwill Cost

Decision 0 1 2 3 Probabilities

B

C

D

E

F

75 40 50

0 0 -40 -80 -120 0.1

States of Nature 1 2 -50 -100 35 -15 -5 70 -45 30 0.3

0.4

3 Expected Return -150 -85 -65 -12.5 20 22.5 105 7.5 0.2

What is the Expected Return?

Decisions under uncertainty

Decisions under uncertainty Multiple states of nature Don’t know what state of nature will occur

Laplace

Laplace Maximin Maximax Minimax regret

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The Newsvendor Model

Laplace Assume all states of nature are equally likely to occur

1 2 3 4 5 6 7 8 9 10 11 12

A Selling Price Purchase Cost Goodwill Cost

Decision 0 1 2 3 Probabilities

B

C

D

E

F

75 40 50

0 0 -40 -80 -120 0.1

States of Nature 1 2 -50 -100 35 -15 -5 70 -45 30 0.3

0.4

3 Expected Return -150 -85 -65 -12.5 20 22.5 105 7.5 0.2

What is the Expected Return?

Maximin

Maximin

extremely conservative or pessimistic approach to making decisions

Evaluate minimum possible return associated with each decision.

Maximin

Maximin

Select decision yielding maximum max value of minimum min returns. Table 1, p81

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

Maximax optimistic approach to making decisions

• Under the Maximin criterion, you would choose decision 1. • Under the Maximax criterion, you would choose decision 2.

Maximax

Maximax

Evaluate maximum possible return associated with each decision

Select decision yielding maximum of these max maximum returns. max

Maximax

Different criterion yields different decisions.

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• 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

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

Regret measures the desirability of an outcome.

a) Find the maximum value in column 1 b)Subtract every value in column 1 from this value c) Repeat for each column

a) Find the maximum value in column 1

a) Find the maximum value in column 1

b)Subtract every value in column 1 from this value

b)Subtract every value in column 1 from this value

c) Repeat for each column

c) Repeat for each column

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After regret table is built:

After regret table is built:

d)Choose the maximum value in each row

d)Choose the maximum value in each row

e) choose the smallest

e) Choose the smallest (minimum of the maximum)

Each method yields different decisions regarding the newsvendor data:

Minimax Regret 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

A Selling Price Purchase Cost Goodwill Cost

Decision

B

C

D

E

F

75 40 50

0 1 2 3

0 0 -40 -80 -120 0

0 1 2 3

0 40 80 120

States of Nature 1 2 -50 -100 35 -15 -5 70 -45 30 35 70

3 -150 -65 20 105 105

Regret 85 0 40 80

170 85 0 40

255 170 85 0

MinMax Regret 255 170 85 120

Criteria LaPlace Cash Flow

Order 2 papers

Decision

Maximin Cash Flow

Order 1 paper

Maximax Cash Flow

Order 3 papers

Minimax Regret

Order 2 papers

85

1 2 3 4 5 6 7 8 9 10 11 12

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

A Selling Price Purchase Cost Goodwill Cost

Decision 0 1 2 3 Probabilities

B

C

D

E

F

75 40 50

0 0 -40 -80 -120 0.1

States of Nature 1 2 -50 -100 35 -15 -5 70 -45 30 0.3

0.4

3 Expected Return -150 -85 -65 -12.5 20 22.5 105 7.5 0.2

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

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1 2 3 4 5 6 7 8 9 10 11 12

A Selling Price Purchase Cost Goodwill Cost

Decision 0 1 2 3 Probabilities

EVPI =

B

C

D

E

F

75 40 50

0 0 -40 -80 -120

States of Nature 1 2 -50 -100 35 -15 -5 70 -45 30

0.1

0.3

expected return with perfect information

0.4

3 Expected Return -150 -85 -65 -12.5 20 22.5 105 7.5

Decision Trees Graphical tool used to analyze decisions under risk

0.2

maximum possible expected return without sample information

TreePlan An add-in used to draw decision trees in Excel.

Useful to analyze sequences of decisions

Sonoralo Cellular Phones 3 strategies

Bayes’ Theorem Allows us to incorporate new information into the process.

Aggressive

Basic

Major commitment

Move production to existing facility

Major capital expenditure Large inventory Major global marketing campaign

Modify current line Maintain inventory for popular items Local/regional advertising

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Cautious

States of Nature

Use excess capacity

Strong Demand (S)

Minimize retooling Produce enough to satisfy demand

Weak Demand (W)

Advertise at discretion of local dealer

Sonoralo Cellular Phones Payoff table

A square node represents a point at which a decision must be made. Each line (branch) leading from a square represents a possible decision.

TREE PLAN A square node represents a point at which a decision must be made. A circular 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.

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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).

Terminology

Prior Probabilities Conditional Probabilities / Reliabilities Joint & Marginal Probabilities

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

Posterior Probabilities

A MARKET RESEARCH STUDY FOR CELLULAR PHONES

A MARKET RESEARCH STUDY FOR CELLULAR PHONES

Prior Probabilities:

Conditional Probabilities / Reliabilities:

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

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

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

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A MARKET RESEARCH STUDY FOR CELLULAR PHONES

A MARKET RESEARCH STUDY FOR CELLULAR PHONES

Conditional Probabilities / Reliabilities:

Conditional Probabilities / Reliabilities:

For example, P(E|S) is the conditional probability that marketing gives an encouraging report 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:

A MARKET RESEARCH STUDY FOR CELLULAR PHONES Posterior Probabilities:

P(E|S) = 0.6 P(D|S) = 1 - P(E|S) = 0.4

P(D|W) = 0.7 P(E|W) = 1 - P(D|W) = 0.3

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! S IV W A II

B

S V

C VI

P( E)

E

W S W

I ) D P(

D

A III

B

C

S VII W S VIII W S IX W

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

) P(S|D P(W|D) P(S|D) P(W|D)

P(S|D)

30 -8 20

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

7 5 15 30 -8 20 7 5

P(W |D )

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THE EXPECTED VALUE OF SAMPLE INFORMATION

Sonoralo Cellular Phones Payoff table (w/out sample info)

EVSI =

maximum possible expected return with sample information

maximum possible expected return without sample information

EVSI = 13.46 – 12.85 = $0.61 million.

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

THE EXPECTED VALUE OF PERFECT INFORMATION

EVPI =

expected return with perfect information

THE EXPECTED RETURN WITH PERFECT INFORMATION

maximum possible expected return without 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

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THE EXPECTED VALUE OF PERFECT INFORMATION

EVPI =

expected return with perfect information

maximum possible expected return without perfect information

EVSI =

maximum possible expected return with sample information

maximum possible expected return without sample information

EVPI = 21.75 – 12.85 = $8.90 million EVPI = EVPI is the maximum possible increase in the expected return that can be obtained from new information.

expected return with perfect information

maximum possible expected return without perfect information

Sequential Decisions: To 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 sequence of decisions and uncertain events that will follow the initial decision. This is called a sequential decision model. model

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