topic 1 decision analysis chapter 3

Topic 1

Decision Analysis (Chapter 3)

Source: Render et al., 2012. Quantitative Analysis Management, 11 editions, Pearson.

Learning Objectives

Students will be able to:

List the steps of the decision-making process.

Describe the types of decision-making environments.

Make decisions under uncertainty.

Use probability values to make decisions under risk.

Chapter Outline 3.1 Introduction

3.2 The Six Steps in Decision Theory

3.3 Types of Decision-Making Environments

3.4 Decision Making under Uncertainty

3.5 Decision Making under Risk

3.6 Decision Trees

3.7 How Probability Values Are Estimated by Bayesian Analysis

3.8 Utility Theory

Introduction

Decision theory is an analytical and systematic way to tackle problems.

A good decision is based on logic.

The Six Steps in Decision Theory

1. Clearly define the problem at hand.

2. List the possible alternatives.

3. Identify the possible outcomes.

4. List the payoff or profit of each combination of alternatives and outcomes.

5. Select one of the mathematical decision theory models.

6. Apply the model and make your decision.

Example: Thompson Lumber Company Case

John Thompson is the founder and president of Thompson Lumber Company, a profitable firm located in Portland, Oregon.

The problem that John Thompson identifies is whether to expand his product line by manufacturing and marketing a new product, backyard sheds.

Define problem To manufacture or market backyard storage

sheds

List alternatives 1. Construct a large new plant

2. A small plant

3. No plant at all

Identify outcomes The market could be favorable or unfavorable

for storage sheds

List payoffs List the payoff for each state of nature/decision

alternative combination

Select a model Decision tables and/or trees can be used to solve

the problem

Apply model and make

decision

Solutions can be obtained and a sensitivity

analysis used to make a decision


Alternative

State of Nature

Favorable

Market ($)

Unfavorable

Market ($)

Construct a large plant 200,000 -180,000

Construct a small plant 100,000 -20,000

Do nothing 0 0

Example: Thompson Lumber Company Case Decision Table

Types of Decision-Making Environments

Decision making under certainty.

Decision maker knows with certainty the consequences of every alternative or decision choice.

Decision making under uncertainty.

The decision maker does not know the probabilities of the various outcomes.

Decision making under risk.

The decision maker knows the probabilities of the various outcomes.

Decision Making under certainty:

Decision makers know within certainty the consequence of every alternative or decision choice.

E.g.: You have $1000 to invest for a 1-year period. Alternatives are to open a savings or a fixed deposit account paying 3% or 5% interest per year respectively. If both investments are secure and guaranteed, there is a certainty that a fixed deposit will pay a higher return.

Decision Making under Uncertainty: Criteria

Optimistic (Maximax)

Pessimistic (Maximin)

Criterion of Realism (Hurwicz)

Equally likely (Laplace Criterion)

Minimax Regret (Opportunity Loss or Regret)

Alternative

State of Nature

Favorable

Market ($)

Unfavorable

Market ($)

Construct a large plant 200,000 -180,000

Construct a small plant 100,000 -20,000

Do nothing 0 0

Decision Making under Uncertainty

Example: Thompson Lumber Company Case.

Alternative

State of Nature

Maximum in

Row Favorable

Market ($)

Unfavorable

Market ($)

Construct a large

plant 200,000 -180,000 200,000

Construct a small

plant 100,000 -20,000 100,000

Do nothing 0 0 0

Decision Making under Uncertainty: Optimistic (Maximax) Criterion


Optimistic (Maximax)

Alternative

State of Nature

Minimum in

Row Favorable

Market ($)

Unfavorable

Market ($)

Construct a large

plant 200,000 -180,000 -180,000

Construct a small

plant 100,000 -20,000 -20,000

Do nothing 0 0 0

Decision Making under Uncertainty: Pessimistic (Maximin) Criterion


Pessimistic (Maximin)

Weighted Average = (best in row) + (1- ) (worst in row)

where: is the coefficient of realism (or the degree of optimism of the decision maker), 0 1.

Alternative

State of Nature Weighted

Average in

Row

( = 0.8)

Favorable

Market ($)

Unfavorable

Market ($)

Construct a large

plant 200,000 -180,000 124,000

Construct a small

plant 100,000 -20,000 76,000

Do nothing 0 0 0

Decision Making under Uncertainty: Criterion of Realism (Hurwicz) Criterion

Example: Thompson Lumber Company Case (given = 0.8)

Criterion of Realism (Hurwicz)

Alternative

State of Nature

Average in

Row Favorable

Market ($)

Unfavorable

Market ($)

Construct a large

plant 200,000 -180,000 10,000

Construct a small

plant 100,000 -20,000 40,000

Do nothing 0 0 0

Decision Making under Uncertainty: Equally likely (Laplace) Criterion


Assume all states of nature to be equally likely, choose

maximum Average.

Equally likely

(Laplace)

Alternative

State of Nature

Favorable Market ($) Unfavorable Market ($)

Construct a

large plant 200,000 200,000 = 0 0 (180,000) = 180,000

Construct a

small plant 200,000 100,000 = 100,000 0 (20,000) = 20,000

Do nothing 200,000 0 = 200,000 0 0 = 0

Step 1: Create an opportunity loss table by subtracting each payoff in the column from the best payoff in the same column.

Opportunity Loss Optimal payoff Actual payoff

Decision Making under Uncertainty: Minimax Regret (Opportunity Loss) Criterion


Alternative

State of Nature Maximum in

Row Favorable Market

($)

Unfavorable

Market ($)

Construct a large

plant 0 180,000 180,000

Construct a

small plant 100,000 20,000 100,000

Do nothing 200,000 0 200,000

Step 2: Choose the minimum alternative out of all the maximum opportunity losses.

Decision Making under Uncertainty: Minimax Regret (Opportunity Loss) Criterion


Minimax Regret

(Opportunity Loss)

Decision Making under Risk Expected Monetary Value (EMV)

The probabilities of the states of nature P(S) are known.

EMV(Alternative) = Payoffs1*P(S1) + Payoffs2*P(S2) ++ Payoffsn*P(Sn)

Alternative

State of Nature Expected Monetary Value

(EMV) Favorable

Market ($)

Unfavorable

Market ($)

Construct a

large plant 200,000 180,000

(200,000)(0.5) +

(180,000)(0.5) = 10,000

Construct a

small plant 100,000 20,000

(100,000)(0.5) +

(20,000)(0.5) = 40,000

Do nothing 0 0 (0)(0.5) + (0)(0.5) = 0

Probabilities 0.50 0.50


Alternative

State of Nature

Expected Monetary Value

(EMV) Favorable

Market, S1 ($)

Unfavorable

Market, S2 ($)

Construct a large plant 200,000 -180,000 10,000

Construct a small plant 100,000 -20,000 40,000

Do nothing 0 0 0


With Perfect

Information

Best payoff

in S1

200,000

Best payoff in

S2

0

EVwPI = (200,000)(0.5)

+ (0)(0.5) = 100,000

Decision Making under Risk Expected Value with Perfect Information (EVwPI)


EVwPI = best Payoffs1*P(S1) + best Payoffs2*P(S2) ++ best Payoffsn*P(Sn)

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

EVPI = EVwPI Best EMV

where:

EVwPI is Expected Value with Perfect Information.

Best EMV is expected value without perfect information.

Decision Making under Risk Expected Value of Perfect Information (EVPI)

Decision Making under Risk Expected Value of Perfect Information (EVPI)

Hence, the EVPI = EVwPI best EMV

= 100,000 40,000 = $60,000

Expected Opportunity Loss (EOL)

EOL is the cost of not picking the best solution.

EOL = Expected Regret

Thompson Lumber: Payoff Table

Alternative

State of Nature

Favorable Market

($)

Unfavorable

Market ($)

Construct a large

plant 200,000 -180,000

Construct a small

plant 100,000 -20,000

Do nothing 0 0


Thompson Lumber: EOL The Opportunity Loss Table

Alternative

State of Nature

Favorable Market ($) Unfavorable Market

($)

Construct a large plant 200,000 200,000 0- (-180,000)

Construct a small

plant 200,000 - 100,000 0 (-20,000)

Do nothing 200,000 - 0 0-0


Thompson Lumber: Opportunity Loss Table

Alternative

State of Nature

Favorable Market

($)

Unfavorable

Market ($)

Construct a large

plant 0 180,000

Construct a small

plant 100,000 20,000

Do nothing 200,000 0


Thompson Lumber: EOL Solution

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

Note:

1. The minimum EOL and maximum EMV will suggest the same decision.

2. The value of EVPI will always equal to the minimum EOL value.

Thompson Lumber: Sensitivity Analysis

Let P = probability of favorable market EMV(Large Plant): = $200,000P + (-$180,000)(1-P) = 380,000P - 180,000 EMV(Small Plant): = $100,000P + (-$20,000)(1-P) = 120,000P - 20,000 EMV(Do Nothing): = $0P + 0(1-P) = 0

Thompson Lumber: Sensitivity Analysis (continued)

Decision Making with Uncertainty: Using the Decision Trees

Decision trees are most beneficial when a

sequence of decisions must be made.

All information included in a payoff table is

also included in a decision tree.

Five Steps to Decision Tree Analysis

1. Define the problem.

2. Structure or draw the decision tree.

3. Assign probabilities to the states of nature.

4. Estimate payoffs for each possible combination of alternatives and states of nature.

5. Solve the problem by computing expected monetary values (EMVs) at each state of nature node.

Structure of Decision Trees Trees start from left to right and represent decisions and

outcomes in sequential order.

A decision node (indicated by a square ) from which one of several alternatives may be chosen.

A state-of-nature node (indicated by a circle ) out of which one state of nature will occur.

Lines or branches connect the decisions nodes and the states of nature.

Thompsons Decision Tree

1

2

A Decision

Node

A State of

Nature Node Favorable Market

Unfavorable Market

Favorable Market

Unfavorable Market

Construct Small Plant

Step 1: Define the problem

Lets re-look at John Thompsons decision regarding storage sheds.

This simple problem can be depicted using a decision tree.

Step 2: Draw the tree


Step 3: Assign probabilities to the states of nature.

Step 4: Estimate payoffs.

1

2

A Decision

Node

A State of

Nature Node Favorable (0.5) Market

Unfavorable (0.5) Market

Favorable (0.5) Market



$200,000

-$180,000

$100,000

-$20,000

0

Alternatives

Outcomes Payoffs


1

2

A Decision

Node

A State of

Nature Node Favorable (0.5) Market


Favorable (0.5) Market



$200,000

-$180,000

$100,000

-$20,000

0

EMV

=$40,000

EMV

=$10,000

Step 5: Compute EMVs and make decision.

Thompsons Decision: A More Complex Problem

John Thompson has the opportunity of obtaining a market survey that will give additional information on the probable state of nature. Results of the market survey will likely indicate there is a percent change of a favorable market. Historical data show market surveys accurately predict favorable markets 78 % of the time.

P(Fav. Mkt | Results Favourable) = 0.78

Likewise, if the market survey predicts an unfavorable market, there is a 73 % chance of its occurring.

P(Unfav. Mkt | Results Negative) = 0.73

Assumed the cost of conduct market survey is $10,000, and is deducted from each payoff under Conduct Market Survey.


Now that we have redefined the problem (Step 1), lets use this additional data and redraw Thompsons decision tree (Step 2).

Thompsons Decision Tree Step 3: Assign the new probabilities to the states of nature.

Step 4: Estimate the payoffs.

Step 5: Compute the EMVs and make decision.

John Thompson Dilemma

John Thompson is not sure how much value to place on

market survey. He wants to determine the monetary

worth of the survey. John Thompson is also interested in

how sensitive his decision is to changes in the market

survey results. What should he do?

Expected Value of Sample Information

Sensitivity Analysis

Expected Value of Sample Information

o The survey cost $10,000.

o The expected value of $49,200 (when the survey is used) is based on payoffs after the $10,000 cost was subtracted.

o The expected value with sample information (EV with SI) is the expected value of using the survey assuming no cost to gather it. Thus, in this example:

EV with SI = $49,200 + $10,000 (cost) = $59,200.

o Without the sample information, the best expected value is $40,000. Thus, the expected value would increase by $19,200 if the survey was available free.

Expected Value of Sample Information (EVSI)

Expected value with

sample information

+ study cost

Expected value of best decision without sample information

EVSI ==

EVSI for Thompson Lumber = $59,200 - $40,000

= $19,200

Thompson could pay up to $19,200 for a market study.

Since it costs only $10,000, the survey is indeed worthwhile.

Sensitivity Analysis

How sensitive are the decisions to changes in the

probabilities?

e.g. If the probability of a favorable survey were

less than the current value (0.45), would the survey

still be selected? How low would this have to be to

cause a change in the decision?

Let p = probability of favorable market

Calculations for Thompson Lumber Sensitivity Analysis

2,400 $104,000

($2,400) ($106,400) 1) EMV(node

+ =

- + =

p

) p ( p 1

Equating the EMV with the survey to the EMV without the

survey, we have

0.36 $104,000

$37,600 or

$37,600 $104,000

$40,000 $2,400 $104,000

= =

=

= +

p

p

p

Hence, our decision will stay the same as long as the

probability of favorable survey results, p, is greater than 0.36.

Estimating Probability Values with Bayes Theorem

Management experience or intuition History Existing data Need to be able to revise probabilities based

upon new data

Posterior

probabilities Prior

probabilities Bayes Theorem

Information about accuracy

of sample information.

Bayesian Analysis

The probabilities of a favorable / unfavorable state of nature can be

obtained by analyzing the Market Survey Reliability in Predicting

Actual States of Nature.

STATE OF NATURE

RESULT OF SURVEY

FAVORABLE MARKET (FM)

UNFAVORABLE MARKET (UM)

Positive (predicts favorable market for product)

P (survey positive | FM)

= 0.70

P (survey positive | UM)

= 0.20

Negative (predicts unfavorable market for product)

P (survey negative | FM)

= 0.30

P (survey negative | UM)

= 0.80

Bayesian Analysis (continued): Favorable Survey

POSTERIOR PROBABILITY

STATE OF NATURE

CONDITIONAL PROBABILITY

P(SURVEY POSITIVE | STATE

OF NATURE) PRIOR

PROBABILITY JOINT

PROBABILITY

P(STATE OF NATURE | SURVEY

POSITIVE)

FM 0.70 X 0.50 = 0.35 0.35/0.45 = 0.78

UM 0.20 X 0.50 = 0.10 0.10/0.45 = 0.22

P(survey results positive) = 0.45 1.00

Bayesian Analysis (continued): Unfavorable Survey

POSTERIOR PROBABILITY

STATE OF NATURE

CONDITIONAL PROBABILITY

P(SURVEY NEGATIVE | STATE

OF NATURE) PRIOR

PROBABILITY JOINT

PROBABILITY

P(STATE OF NATURE | SURVEY

NEGATIVE)

FM 0.30 X 0.50 = 0.15 0.15/0.55 = 0.27

UM 0.80 X 0.50 = 0.40 0.40/0.55 = 0.73

P(survey results positive) = 0.55 1.00

Decision Making Using Utility Theory

There are many occasions in which people make decisions that would appear to be inconsistent with the EMV criterion.

E.g.: 1. When people buy insurance, the amount of the premium is greater than the expected payout. 2. A person involved in a law suit may choose to settle out of court rather than go trial even if the expected value of going to trial is greater hat the proposed settlement.

This is because monetary value is not always a true indicator of the overall value of the result of a decision.

The overall worth of a particular decision is called utility.

Rational people make decisions that maximize the expected utility.

Decision Making Using Utility Theory Utility assessment assigns the worst outcome a utility of 0,

and the best outcome, a utility of 1.

A standard gamble is used to determine utility values.

When you are indifferent, your utility values are equal.

Expected utility of alternative 2 = Expected utility of alternative 1

Utility of other outcome = (p)(utility of best outcome, which is

1) + (1p)(utility of the worst

outcome, which is 0)

Utility of other outcome = (p)(1) + (1p)(0) = p

Where p is the probability of obtaining the best outcome, and (1p) is the

probability of obtaining the worst

outcome.

Standard Gamble for Utility Assessment

Best outcome

Utility = 1

Worst outcome

Utility = 0

Other outcome

Utility = ??

p

1p

Real Estate Example: Utility Theory Jane Dickson wants to construct a utility curve revealing her

preference for money between $0 and $10,000.

A utility curve plots the utility value versus the monetary value.

An investment in a bank will result in $5,000.

An investment in real estate will result in $0 or $10,000.

Unless there is an 80% chance of getting $10,000 from the real estate deal, Jane would prefer to have her money in the bank.

So if p = 0.80, Jane is indifferent between the bank or the real estate investment.

Real Estate Example: Solution

$10,000

U($10,000) = 1.0

0

U(0) = 0

$5,000

U($5,000) = p = 0.80

p= 0.80

(1 p)= 0.20

Hence, Janes Utility for $5,000 = U($5,000) = p

= p*U($10,000) + (1 p)*U($0)

= (0.8)(1) + (0.2)(0) = 0.8

Real Estate Example: Solution

Utility for $7,000 = 0.90

Utility for $3,000 = 0.50

We can assess other utility values in the same way.

For Jane, let say these are:

Using the three utilities for different dollar amounts, she can construct a utility curve.

Note: In setting the value of probability p, one should be aware that utility assessment is completely subjective. It is a value set by the decision maker that cannot be measured on an objective scale.

Real Estate Example: Utility Curve

U ($7,000) = 0.90

U ($5,000) = 0.80

U ($3,000) = 0.50

U ($0) = 0

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

| | | | | | | | | | |

$0 $1,000 $3,000 $5,000 $7,000 $10,000

Monetary Value

Uti

lity

U ($10,000) = 1.0

topic 1 decision analysis chapter 3

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