Spreadsheet Modeling & Decision Analysis

Spreadsheet Modeling & Decision Analysis

A Practical Introduction to Management Science

4th edition

Cliff T. Ragsdale

15-2

Decision Analysis

Chapter 15

15-3

On Uncertainty and Decision-Making…

"Uncertainty is the most difficult thing about decision-making. In the face of uncertainty, some people react with paralysis, or they do exhaustive research to avoid making a decision. The best decision-making happens when the mental environment is focused. …That fined-tuned focus doesn’t leave room for fears and doubts to enter. Doubts knock at the door of our consciousness, but you don't have to have them in for tea and crumpets."

-- Timothy Gallwey, author of The Inner Game of Tennis and The Inner Game of Work.

15-4

Introduction to Decision Analysis• Models help managers gain insight and

understanding, but they can’t make decisions.

• Decision making often remains a difficult task due to:– Uncertainty regarding the future– Conflicting values or objectives

• Consider the following example...

15-5

Deciding Between Job Offers• Company A

– In a new industry that could boom or bust.– Low starting salary, but could increase rapidly.– Located near friends, family and favorite sports

team.• Company B

– Established firm with financial strength and commitment to employees.

– Higher starting salary but slower advancement opportunity.

– Distant location, offering few cultural or sporting activities.

• Which job would you take?

15-6

Good Decisions vs. Good Outcomes

• A structured approach to decision making can help us make good decisions, but can’t guarantee good outcomes.

• Good decisions sometimes result in bad outcomes.

15-7

Characteristics of Decision Problems• Alternatives - different courses of action intended

to solve a problem.– Work for company A – Work for company B– Reject both offers and keep looking

• Criteria - factors that are important to the decision maker and influenced by the alternatives.– Salary – Career potential– Location

• States of Nature - future events not under the decision makers control.– Company A grows– Company A goes bust– etc

15-8

An Example: Magnolia Inns• Hartsfield International Airport in Atlanta, Georgia,

is one of the busiest airports in the world.• It has expanded many times to handle increasing

air traffic. • Commercial development around the airport

prevents it from building more runways to handle future air traffic demands.

• Plans are being made to build another airport outside the city limits.

• Two possible locations for the new airport have been identified, but a final decision will not be made for a year.

Continued…

15-9

Magnolia Inns (continued)

• The Magnolia Inns hotel chain intends to build a new facility near the new airport once its site is determined.

• Land values around the two possible sites for the new airport are increasing as investors speculate that property values will increase greatly in the vicinity of the new airport.

• See data in file Fig15-1.xls

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The Decision Alternatives1) Buy the parcel of land at location

A.2) Buy the parcel of land at location

B.3) Buy both parcels.4) Buy nothing.

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The Possible States of Nature

1) The new airport is built at location A.

2) The new airport is built at location B.

15-12

Constructing a Payoff Matrix

See file Fig15-1.xls

15-13

Decision Rules• If the future state of nature (airport

location) were known, it would be easy to make a decision.

• Failing this, a variety of nonprobabilistic decision rules can be applied to this problem:– Maximax– Maximin– Minimax regret

• No decision rule is always best and each has its own weaknesses.

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The Maximax Decision Rule• Identify the maximum payoff for each alternative.• Choose the alternative with the largest maximum

payoff.• See file Fig15-1.xls

• Weakness – Consider the following payoff matrix

State of NatureDecision 1 2 MAX

A 30 -10000 30 <--maximumB 29 29 29

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The Maximin Decision Rule• Identify the minimum payoff for each alternative.• Choose the alternative with the largest minimum

payoff.• See file Fig15-1.xls


State of NatureDecision 1 2 MIN

A 1000 28 28B 29 29 29 <--maximum

15-16

The Minimax Regret Decision Rule

• Compute the possible regret for each alternative under each state of nature.

• Identify the maximum possible regret for each alternative.• Choose the alternative with the smallest maximum regret.• See file Fig15-1.xls

15-17

Anomalies with the Minimax Regret Rule

• Consider the following payoff matrix State of Nature

Decision 1 2 A 9 2 B 4 6


A 0 4 4 <--minimumB 5 0 5

• The regret matrix is:

• Note that we prefer A to B.• Now let’s add an alternative...

15-18

Adding an Alternative• Consider the following payoff matrix

• The regret matrix is:

State of NatureDecision 1 2

A 9 2 B 4 6C 3 9


A 0 7 7B 5 3 5 <--minimumC 6 0 6

• Now we prefer B to A???

15-19

Probabilistic Methods• At times, states of nature can be assigned probabilities that

represent their likelihood of occurrence. • For decision problems that occur more than once, we can

often estimate these probabilities from historical data. • Other decision problems (such as the Magnolia Inns problem)

represent one-time decisions where historical data for estimating probabilities don’t exist.

• In these cases, subjective probabilities are often assigned based on interviews with one or more domain experts.

• Interviewing techniques exist for soliciting probability estimates that are reasonably accurate and free of the unconscious biases that may impact an expert’s opinions.

• We will focus on techniques that can be used once appropriate probability estimates have been obtained.

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Expected Monetary Value• Selects alternative with the largest expected

monetary value (EMV)EMVi ij

jjr p

r i jij payoff for alternative under the th state of nature

p jj the probability of the th state of nature

• EMVi is the average payoff we’d receive if we faced the same decision problem numerous times and always selected alternative i.

• See file Fig15-1.xls

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EMV Caution• The EMV rule should be used with caution in

one-time decision problems.


State of NatureDecision 1 2 EMV

A 15,000 -5,000 5,000 <--maximumB 5,000 4,000 4,500

Probability 0.5 0.5

15-22

Expected Regret or Opportunity Loss

• Selects alternative with the smallest expected regret or opportunity loss (EOL)

EOLi ijj

jg pg i jij regret for alternative under the th state of nature

p jj the probability of the th state of nature

• The decision with the largest EMV will also have the smallest EOL.


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The Expected Value of Perfect Information

• Suppose we could hire a consultant who could predict the future with 100% accuracy.

• With such perfect information, Magnolia Inns’ average payoff would be:EV with PI = 0.4*$13 + 0.6*$11 = $11.8 (in millions)

• Without perfect information, the EMV was $3.4 million.

• The expected value of perfect information is therefore,

EV of PI = $11.8 - $3.4 = $8.4 (in millions)• In general,

EV of PI = EV with PI - maximum EMV• It will always the the case that,

EV of PI = minimum EOL

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A Decision Tree for Magnolia Inns

0

1

2

3

4

Buy A-18

Buy B-12

Buy A&B-30

Buy nothing0

Land Purchase Decision Airport Location

A

B

A

B

B

B

A

A

Payoff

13

-12

-8

11

5

-1

0

0

31

6

4

23

35

29

0

0

15-25

Rolling Back A Decision Tree

0

1

2

3

4

Buy A-18

Buy B-12

Buy A&B-30

Buy nothing0


A

B

A

B

B

B

A

A

Payoff

13

-12

-8

11

5

-1

0

0

31

6

4

23

35

29

0

0

0.4

0.60.4

0.6

0.4

0.60.4

0.6

EMV=-2

EMV=3.4

EMV=1.4

EMV= 0

EMV=3.4

15-26

Alternate Decision Tree

0

1

2

3

Buy A-18

Buy B-12

Buy A&B-30

Buy nothing0


A

B

A

B

B

A

Payoff

13

-12

-8

11

5

-1

31

6

4

23

35

29

0.4

0.60.4

0.6

0.4

0.6

EMV=-2

EMV=3.4

EMV=1.4EMV=3.4

0

15-27

Using TreePlan• TreePlan is an Excel add-in for decision trees.


15-28

About TreePlan• TreePlan is a shareware product

developed by Dr. Mike Middleton at the Univ. of San Francisco and distributed with this textbook at no charge to you.

• If you like this software package and plan to use for more than 30 days, you are expected to pay a nominal registration fee. Details on registration are available near the end of the TreePlan help file.

15-29

Multi-stage Decision Problems• Many problems involve a series of

decisions• Example

– Should you go out to dinner tonight?– If so,

»How much will you spend?»Where will you go?»How will you get there?

• Multistage decisions can be analyzed using decision trees

15-30

Multi-Stage Decision Example: COM-TECH• Steve Hinton, owner of COM-TECH, is considering whether

to apply for a $85,000 OSHA research grant for using wireless communications technology to enhance safety in the coal industry.

• Steve would spend approximately $5,000 preparing the grant proposal and estimates a 50-50 chance of receiving the grant.

• If awarded the grant, Steve would need to decide whether to use microwave, cellular, or infrared communications technology.

• Steve would need to acquire some new equipment depending on which technology is used…

Technology Equipment Cost

Microwave $4,000

Cellular $5,000Infrared $4,000

continued...

15-31

COM-TECH (continued)

• Steve needs to synthesize all the factors in this problem to decide whether or not to submit a grant proposal to OSHA.


Cost Prob. Cost Prob.Microwave $30,000 0.4 $60,000 0.6Cellular $40,000 0.8 $70,000 0.2Infrared $40,000 0.9 $80,000 0.1

Best Case Worst Case

• Steve knows he will also spend money in R&D, but he doesn’t know exactly what the R&D costs will be. Steve estimates the following best case and worst case R&D costs and probabilities, based on his expertise in each area.

15-32

Analyzing Risk in a Decision Tree• How sensitive is the decision in the COM-TECH

problem to changes in the probability estimates?• We can use Solver to determine the smallest

probability of receiving the grant for which Steve should still be willing to submit the proposal.

• Let’s go back to file Fig15-26.xls...

15-33

Risk Profiles

• A risk profile summarizes the make-up of an EMV

• This can also be summarized in a decision tree.


Event Probability Payoff

Receive grant, Low R&D costs 0.5*0.9=0.45 $36,000

Receive grant, High R&D costs 0.5*0.1=0.05 -$4,000

Don’t receive grant 0.5 -$5,000

EMV $13,500

• The $13,500 EMV for COM-TECH was created as follows:

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Using Sample Information in Decision Making

• We can often obtain information about the possible outcomes of decisions before the decisions are made.

• This sample information allows us to refine probability estimates associated with various outcomes.

15-35

Example: Colonial Motors• Colonial Motors (CM) needs to determine whether to build a

large or small plant for a new car it is developing. • The cost of constructing a large plant is $25 million and the

cost of constructing a small plant is $15 million. • CM believes a 70% chance exists that demand for the new

car will be high and a 30% chance that it will be low.

• The payoffs (in millions of dollars) are summarized below.

DemandFactory Size High Low

Large $175 $95

Small $125 $105• See decision tree in file Fig15-32.xls

15-36

Including Sample Information• Before making a decision, suppose CM conducts a

consumer attitude survey (with zero cost).• The survey can indicate favorable or unfavorable

attitudes toward the new car. Assume:P(favorable response) = 0.67P(unfavorable response) = 0.33

• If the survey response is favorable, this should increase CM’s belief that demand will be high. Assume:

P(high demand | favorable response)=0.9P(low demand | favorable response)=0.1

• If the survey response is unfavorable, this should increase CM’s belief that demand will be low. Assume:

P(low demand | unfavorable response)=0.7P(high demand | unfavorable response)=0.3

• See decision tree in file Fig15-33.xls

15-37

The Expected Value of Sample Information

• How much should CM be willing to pay to conduct the consumer attitude survey?

Expected Value of Sample Information

Expected Value with Sample Information

Expected Value without Sample Information= -

• In the CM example,E.V. of Sample Info. = $126.82 - $126 = $0.82 million

15-38

Computing Conditional Probabilities• Conditional probabilities (like those in the CM example)

are often computed from joint probability tables.High Low

Demand Demand TotalFavorable Response 0.600 0.067 0.667Unfavorable Response 0.100 0.233 0.333Total 0.700 0.300 1.000

• The joint probabilities indicate:P(F H) 0.6, P(F L) = 0.067

P(U H) = 0.1, P(U L) 0.233

• The marginal probabilities indicate:P(F) 0.667, P(U) = 0.333

P(H) = 0.700, P(L) 0.300

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Computing Conditional Probabilities (cont’d)

• In general, P(A|B) =

P(A B)

P(B)

• So we have,

P(H|F) =P(H F)

P(F)

0 60

0 6670 90

.

..

P(L|F) =P(L F)

P(F)

0 067

0 667010

.

..

P(H|U) =P(H U)

P(U)

010

0 3330 30

.

..

P(L|U) =P(L U)

P(U)

0 233

0 3330 70

.

..

High LowDemand Demand Total

Favorable Response 0.600 0.067 0.667Unfavorable Response 0.100 0.233 0.333Total 0.700 0.300 1.000

15-40

Bayes’s Theorem

• Bayes’s Theorem provides another definition of conditional probability that is sometimes helpful.

P(A|B) =P(B|A)P(A)

P(B|A)P(A) + P(B|A)P(A)

• For example,

P(H|F) =P(F|H)P(H)

P(F|H)P(H) + P(F|L)P(L)

( . )( . )

( . )( . ) ( . )( . ).

0 857 0 70

0 857 0 70 0 223 0 300 90

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Utility Theory• Sometimes the decision with the highest EMV is not

the most desired or most preferred alternative.• Consider the following payoff table,

State of NatureDecision 1 2 EMV

A 150,000 -30,000 60,000 <--maximumB 70,000 40,000 55,000

Probability 0.5 0.5

• Decision makers have different attitudes toward risk:Some might prefer decision alternative A,Others would prefer decision alternative B.

• Utility Theory incorporates risk preferences in the decision making process.

15-42

Common Utility Functions

0.00

0.25

0.50

0.75

1.00

Utility

Payoff

risk averse

risk neutral

risk seeking

15-43

Constructing Utility Functions• Assign utility values of 0 to the worst payoff and 1 to the best. • For the previous example,

U(-$30,000)=0 and U($150,000)=1• To find the utility associated with a $70,000 payoff identify the

value p at which the decision maker is indifferent between:Alternative 1: Receive $70,000 with certainty.Alternative 2: Receive $150,000 with probability p and lose

$30,000 with probability (1-p).

• If decision maker is indifferent when p=0.8:

U($70,000)=U($150,000)*0.8+U(-30,000)*0.2=1*0.8+0*0.2=0.8• When p=0.8, the expected value of Alternative 2 is:

$150,000*0.8 + $30,000*0.2 = $114,000• The decision maker is risk averse. (Willing to accept

$70,000 with certainty versus a risky situation with an expected value of $114,000.)

15-44

Constructing Utility Functions (cont’d)• If we repeat this process with different values in Alternative 1,

the decision maker’s utility function emerges (e.g., if U($40,000)=0.65):

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

-30 -20 -10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Payoff (in $1,000s)

Utility

15-45

Comments• Certainty Equivalent - the amount that is

equivalent in the decision maker’s mind to a situation involving risk.(e.g., $70,000 was equivalent to Alternative 2 with p

= 0.8)

• Risk Premium - the EMV the decision maker is willing to give up to avoid a risky decision. (e.g., Risk premium = $114,000-$70,000 = $44,000)

15-46

Using Utilities to Make Decisions

• Replace monetary values in payoff tables with utilities.

• Consider the utility table from the earlier example,

State of NatureExpectedDecision 1 2 Utility

A 1 0 0.500B 0.8 0.65 0.725 <--maximum

Probability 0.5 0.5

• Decision B provides the greatest utility even though it the payoff table indicated it had a smaller EMV.

15-47

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

-50 -25 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350

x

U(x)

R=100 R=200

R=300

The Exponential Utility Function• The exponential utility function is often used to model

classic risk averse behavior:U( ) = 1- - /Rx e x

15-48

Incorporating Utilities in TreePlan• TreePlan will automatically convert monetary values to

utilities using the exponential utility function.• We must first determine a value for the risk tolerance

parameter R.• R is equivalent to the maximum value of Y for which

the decision maker is willing to accept the following gamble:

Win $Y with probability 0.5,Lose $Y/2 with probability 0.5.

• Note that R must be expressed in the same units as the payoffs!

• In Excel, insert R in a cell named RT. (Note: RT must be outside the rectangular range containing the decision tree!)

• On TreePlan’s ‘Options’ dialog box select,‘Use Exponential Utility Function’


15-49

Multicriteria Decision Making• Decision problem often involve two or more

conflicting criterion or objectives:– Investing:

» risk vs. return– Choosing Among Job Offers:

» salary, location, career potential, etc.– Selecting a Camcorder:

» price, warranty, zoom, weight, lighting, etc.– Choosing Among Job Applicants:

» education, experience, personality, etc.

• We’ll consider two techniques for these types of problems:– The Multicriteria Scoring Model– The Analytic Hierarchy Process (AHP)

15-50

The Multicriteria Scoring Model• Score (or rate) each alternative on each criterion.• Assign weights the criterion reflecting their

relative importance.

w sii

ij

• For each alternative j, compute a weighted average score as:

wi = weight for criterion i

sij = score for alternative i on criterion j


15-51

The Analytic Hierarchy Process (AHP)

• Provides a structured approach for determining the scores and weights in a multicriteria scoring model.

• We’ll illustrate AHP using the following example:– A company wants to purchase a new payroll and

personnel records information system.– Three systems are being considered (X, Y and Z).– Three criteria are relevant:

»Price»User support»Ease of use

15-52

Pairwise Comparisons• The first step in AHP is to create a pairwise comparison matrix for

each alternative on each criterion using the following values:

Value Preference1 Equally Preferred2 Equally to Moderately Preferred3 Moderately Preferred4 Moderately to Strongly Preferred5 Strongly Preferred6 Strongly to Very Strongly Preferred7 Very Strongly Preferred8 Very Strongly to Extremely Preferred9 Extremely Preferred

• Pij = extent to which we prefer alternative i to j on a given criterion.

• We assume Pji = 1/Pij

• See price comparisons in file Fig15-45.xls

15-53

Normalization & Scoring• To normalize a pairwise comparison matrix,

1) Compute the sum of each column,

2) Divide each entry in the matrix by its column sum.

• The score (sj) for each alternative is given by

the average of each row in the normalized comparison matrix.


15-54

Consistency• We can check to make sure the decision maker was

consistent in making the comparisons.

• The consistency measure for alternative i is:

C

P s

si

ij jj

i

where

Pij = pairwise comparison of alternative i to j

sj = score for alternative j • If the decision maker was perfectly

consistent each Ci should equal to the number of alternatives in the problem.

15-55

Consistency (cont’d)

• Typically, some inconsistency exists.

• The inconsistency is not deemed a problem provided the Consistency Ratio (CR) is no more than 10%

CRCI

RI0.10

where,

CI = Ci

ni

n n

/ ( )1

n the number of alternatives

RI = 0.00 0.58 0.90 1.12 1.24 1.32 1.41

for n = 2 3 4 5 6 7 8

15-56

Obtaining Remaining Scores & Weights

• This process is repeated to obtain scores for the other criterion as well as the criterion weights.

• The scores and weights are then used as inputs to a multicriteria scoring model in the usual way.


15-57

End of Chapter 15

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Spreadsheet Modeling & Decision Analysis