Transcript
Page 1: Bba 3274 qm week 4 decision analysis

Decision AnalysisDecision AnalysisDecision AnalysisDecision Analysis

BBA3274 / DBS1084 QUANTITATIVE METHODS for BUSINESSBBA3274 / DBS1084 QUANTITATIVE METHODS for BUSINESS

byStephen Ong

Visiting Fellow, Birmingham City University Business School, UK

Visiting Professor, Shenzhen University

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Today’s Overview Today’s Overview

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Learning ObjectivesLearning Objectives

1.1. List the steps of the decision-making process.List the steps of the decision-making process.

2.2. Describe the types of decision-making Describe the types of decision-making environments.environments.

3.3. Make decisions under uncertainty.Make decisions under uncertainty.

4.4. Use probability values to make decisions under Use probability values to make decisions under risk.risk.

5.5. Develop accurate and useful decision trees.Develop accurate and useful decision trees.

6.6. Revise probabilities using Bayesian analysis.Revise probabilities using Bayesian analysis.

7.7. Use computers to solve basic decision-making Use computers to solve basic decision-making problems.problems.

8.8. Understand the importance and use of utility Understand the importance and use of utility theory in decision making.theory in decision making.

After this lecture, students will be able to:After this lecture, students will be able to:

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OutlineOutline3.13.1 IntroductionIntroduction

3.3.22 The Six Steps in Decision MakingThe Six Steps in Decision Making

3.3.33 Types of Decision-Making EnvironmentsTypes of Decision-Making Environments

3.3.44 Decision Making under UncertaintyDecision Making under Uncertainty

3.3.55 Decision Making under RiskDecision Making under Risk

3.3.66 Decision TreesDecision Trees

3.3.77 How Probability Values Are Estimated by How Probability Values Are Estimated by Bayesian AnalysisBayesian Analysis

3.3.88 Utility TheoryUtility Theory

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IntroductionIntroduction

What is involved in making a good What is involved in making a good decision?decision?

Decision theory is an analytic and Decision theory is an analytic and systematic approach to the study of systematic approach to the study of decision making.decision making.

A good decision is one that is based A good decision is one that is based on logic, considers all available data on logic, considers all available data and possible alternatives, and the and possible alternatives, and the quantitative approach described here.quantitative approach described here.

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The Six Steps in Decision MakingThe Six Steps in Decision Making

1.1. Clearly define the problem at hand.Clearly define the problem at hand.

2.2. List the possible alternatives.List the possible alternatives.

3.3. Identify the possible outcomes or states Identify the possible outcomes or states of nature.of nature.

4.4. List the List the payoffpayoff (typically profit) of each (typically profit) of each combination of alternatives and combination of alternatives and outcomes.outcomes.

5.5. Select one of the mathematical decision Select one of the mathematical decision theory models.theory models.

6.6. Apply the model and make your decision.Apply the model and make your decision.

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Thompson Lumber CompanyThompson Lumber Company

Step 1 –Step 1 – Define the problem.Define the problem. The company is considering expanding

by manufacturing and marketing a new product – backyard storage sheds.

Step 2 –Step 2 – List alternatives.List alternatives. Construct a large new plant. Construct a small new plant. Do not develop the new product line at

all.

Step 3 –Step 3 – Identify possible outcomes.Identify possible outcomes. The market could be favourable or

unfavourable.

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Thompson Lumber CompanyThompson Lumber CompanyStep 4 –Step 4 – List the payoffs.List the payoffs.

Identify conditional valuesconditional values for the profits for large plant, small plant, and no development for the two possible market conditions.

Step 5 –Step 5 – Select the decision model.Select the decision model. This depends on the environment and

amount of risk and uncertainty.

Step 6 –Step 6 – Apply the model to the data.Apply the model to the data. Solution and analysis are then used to

aid in decision-making.

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Thompson Lumber CompanyThompson Lumber Company

STATE OF NATURESTATE OF NATURE

ALTERNATIVEALTERNATIVEFAVOURABLE FAVOURABLE

MARKET ($)MARKET ($)UNFAVOURABLEUNFAVOURABLE

MARKET ($)MARKET ($)

Construct a large Construct a large plantplant 200,000200,000 ––180,000180,000

Construct a small Construct a small plantplant 100,000100,000 ––20,00020,000

Do nothingDo nothing 00 00

Table 3.1

Decision Table with Decision Table with Conditional Values for Conditional Values for

Thompson LumberThompson Lumber

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Types of Decision-Making EnvironmentsTypes of Decision-Making Environments

Type 1:Type 1: Decision making under certaintyDecision making under certainty The decision maker The decision maker knows with certaintyknows with certainty

the consequences of every alternative or the consequences of every alternative or decision choice.decision choice.

Type 2:Type 2: Decision making under uncertaintyDecision making under uncertainty The decision maker The decision maker does not knowdoes not know the the

probabilities of the various outcomes.probabilities of the various outcomes.

Type 3:Type 3: Decision making under riskDecision making under risk The decision maker The decision maker knows the knows the

probabilitiesprobabilities of the various outcomes. of the various outcomes.

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Decision Making Under UncertaintyDecision Making Under Uncertainty

1.1. Maximax (optimistic)Maximax (optimistic)

2.2. Maximin (pessimistic)Maximin (pessimistic)

3.3. Criterion of realism (Hurwicz)Criterion of realism (Hurwicz)

4.4. Equally likely (Laplace) Equally likely (Laplace)

5.5. Minimax regretMinimax regret

There are several criteria for There are several criteria for making decisions under making decisions under uncertainty:uncertainty:

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MaximaxMaximaxUsed to find the alternative Used to find the alternative that maximizes the maximum that maximizes the maximum payoff.payoff.

Locate the maximum payoff for each Locate the maximum payoff for each alternative.alternative.

Select the alternative with the maximum Select the alternative with the maximum number.number.

STATE OF NATURESTATE OF NATURE

ALTERNATIVEALTERNATIVEFAVORABLE FAVORABLE MARKET ($)MARKET ($)

UNFAVORABLE UNFAVORABLE MARKET ($)MARKET ($)

MAXIMUM IN MAXIMUM IN A ROW ($)A ROW ($)

Construct a large Construct a large plantplant 200,000200,000 ––180,000180,000 200,000200,000

Construct a small Construct a small plantplant 100,000100,000 ––20,00020,000 100,000100,000

Do nothingDo nothing 00 00 00

Table 3.2

MaximaxMaximax

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MaximinMaximinUsed to find the alternative that maximizes the minimum payoff.

Locate the minimum payoff for each alternative.

Select the alternative with the maximum number.

STATE OF NATURE

ALTERNATIVEALTERNATIVEFAVOURABLE FAVOURABLE

MARKET ($)MARKET ($)UNFAVOURABLUNFAVOURABLE MARKET ($)E MARKET ($)

MINIMUM IN MINIMUM IN A ROW ($)A ROW ($)

Construct a large Construct a large plantplant 200,000200,000 ––180,000180,000 ––180,000180,000

Construct a small Construct a small plantplant 100,000100,000 ––20,00020,000 ––20,00020,000

Do nothingDo nothing 00 00 00

Table 3.3 MaximinMaximin

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Criterion of Realism (Hurwicz)Criterion of Realism (Hurwicz)

This is a This is a wweightedeighted average average compromise compromise between optimism and pessimism.between optimism and pessimism.

Select a coefficient of realism Select a coefficient of realism , , withwith 0≤ 0≤ αα ≤1.≤1. A value of 1 is perfectly optimistic, while a A value of 1 is perfectly optimistic, while a

value of 0 is perfectly pessimistic.value of 0 is perfectly pessimistic. Compute the weighted averages for each Compute the weighted averages for each

alternative.alternative. Select the alternative with the highest value.Select the alternative with the highest value.

Weighted average =Weighted average = (maximum in row) (maximum in row) + (1 + (1 –– )(minimum in row))(minimum in row)

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Criterion of Realism (Hurwicz)Criterion of Realism (Hurwicz) For the large plant alternative using For the large plant alternative using

= 0.8:= 0.8:(0.8)(200,000) + (1 (0.8)(200,000) + (1 – 0.8)(–180,000) = – 0.8)(–180,000) = 124,000124,000

For the small plant alternative using For the small plant alternative using = 0.8: = 0.8: (0.8)(100,000) + (1 (0.8)(100,000) + (1 – 0.8)(–20,000) = – 0.8)(–20,000) = 76,00076,000

STATE OF NATURE

ALTERNATIVEALTERNATIVEFAVOURABLE FAVOURABLE

MARKET ($)MARKET ($)UNFAVOURABLE UNFAVOURABLE

MARKET ($)MARKET ($)

CRITERION CRITERION OF REALISM OF REALISM

(( = 0.8) $ = 0.8) $

Construct a large Construct a large plantplant 200,000200,000 ––180,000180,000 124,000124,000

Construct a small Construct a small plantplant 100,000100,000 ––20,00020,000 76,00076,000

Do nothingDo nothing 00 00 00

Table 3.4

RealismRealism

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Equally Likely (Laplace)Equally Likely (Laplace)Considers all the payoffs for each Considers all the payoffs for each alternative alternative

Find the average payoff for each Find the average payoff for each alternative.alternative.

Select the alternative with the Select the alternative with the highest average.highest average.

STATE OF NATURESTATE OF NATURE

ALTERNATIVEALTERNATIVEFAVOURABLFAVOURABLE MARKET ($)E MARKET ($)

UNFAVOURABLE UNFAVOURABLE MARKET ($)MARKET ($)

ROW ROW AVERAGE ($)AVERAGE ($)

Construct a large Construct a large plantplant 200,000200,000 ––180,000180,000 10,00010,000

Construct a small Construct a small plantplant 100,000100,000 ––20,00020,000 40,00040,000

Do nothingDo nothing 00 00 00

Table 3.5

Equally likelyEqually likely

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Minimax RegretMinimax RegretBased on Based on opportunity lossopportunity loss or or regretregret, this is the , this is the difference between the optimal profit and actual difference between the optimal profit and actual payoff for a decision.payoff for a decision.

Create an opportunity loss table by Create an opportunity loss table by determining the opportunity loss from not determining the opportunity loss from not choosing the best alternative.choosing the best alternative.

Opportunity loss is calculated by subtracting Opportunity loss is calculated by subtracting each payoff in the column from the best each payoff in the column from the best payoff in the column.payoff in the column.

Find the maximum opportunity loss for each Find the maximum opportunity loss for each alternative and pick the alternative with the alternative and pick the alternative with the minimum number.minimum number.

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Minimax RegretMinimax Regret

STATE OF NATURESTATE OF NATURE

FAVOURABLE FAVOURABLE MARKET ($)MARKET ($)

UNFAVOURABLE UNFAVOURABLE MARKET ($)MARKET ($)

200,000 200,000 – 200,000– 200,000 0 0 – (–180,000)– (–180,000)

200,000 200,000 – 100,000– 100,000 00 – (–20,000) – (–20,000)

200,000 200,000 – 0– 0 00 – 0 – 0

Table 3.6

Determining Opportunity Losses for Thompson Lumber

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Minimax RegretMinimax Regret

Table 3.7

STATE OF NATURESTATE OF NATURE

ALTERNATIVEALTERNATIVEFAVOURABLE FAVOURABLE MARKET ($)MARKET ($)

UNFAVOURABLE UNFAVOURABLE MARKET ($)MARKET ($)

Construct a large plantConstruct a large plant 00 180,000180,000

Construct a small plantConstruct a small plant 100,000100,000 20,00020,000

Do nothingDo nothing 200,000200,000 00

Opportunity Loss Table for Thompson Lumber

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Minimax RegretMinimax Regret

Table 3.8

STATE OF NATURESTATE OF NATURE

ALTERNATIVEALTERNATIVEFAVOURABLE FAVOURABLE

MARKET ($)MARKET ($)UNFAVOURABLE UNFAVOURABLE

MARKET ($)MARKET ($)MAXIMUM IN MAXIMUM IN

A ROW ($)A ROW ($)

Construct a large Construct a large plantplant 00 180,000180,000 180,000180,000

Construct a small Construct a small plantplant 100,000100,000 20,00020,000 100,000100,000

Do nothingDo nothing 200,000200,000 00 200,000200,000MinimaxMinimax

Thompson’s Minimax Decision Thompson’s Minimax Decision Using Opportunity LossUsing Opportunity Loss

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Decision Making Under RiskDecision Making Under Risk This is decision making when there are several This is decision making when there are several

possible states of nature, and the probabilities possible states of nature, and the probabilities associated with each possible state are known.associated with each possible state are known.

The most popular method is to choose the The most popular method is to choose the alternative with the highest alternative with the highest expected monetary expected monetary value (value (EMVEMV).).This is very similar to the This is very similar to the expected value expected value calculated in probability distribution.calculated in probability distribution.

EMVEMV (alternative (alternative ii)) = (payoff of first = (payoff of first state of nature)state of nature)x (probability of first state of x (probability of first state of nature)nature)+ (payoff of second state of + (payoff of second state of nature)nature)x (probability of second state of x (probability of second state of nature)nature)+ … + (payoff of last state of + … + (payoff of last state of nature)nature)x (probability of last state of x (probability of last state of nature)nature)

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EMVEMV for Thompson Lumber for Thompson Lumber Suppose each market outcome has a

probability of occurrence of 0.50. Which alternative would give the

highest EMV? The calculations are:EMVEMV (large plant)= ($200,000)(0.5) + ( (large plant)= ($200,000)(0.5) + (––

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

EMVEMV (small plant)= ($100,000)(0.5) + (– (small plant)= ($100,000)(0.5) + (–$20,000)(0.5)$20,000)(0.5) = $40,000= $40,000

EMVEMV (do nothing)= ($0)(0.5) + ($0)(0.5) (do nothing)= ($0)(0.5) + ($0)(0.5) = $0= $0

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EMVEMV for Thompson Lumber for Thompson Lumber

STATE OF NATURESTATE OF NATURE

ALTERNATIVEALTERNATIVEFAVOURABLE FAVOURABLE

MARKET ($)MARKET ($)UNFAVOURABLE UNFAVOURABLE

MARKET ($)MARKET ($) EMVEMV ($) ($)

Construct a large Construct a large plantplant 200,000200,000 ––180,000180,000 10,00010,000

Construct a small Construct a small plantplant 100,000100,000 ––20,00020,000 40,00040,000

Do nothingDo nothing 00 00 00

ProbabilitiesProbabilities 0.500.50 0.500.50

Table 3.9 Largest Largest EMVEMV

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Expected Value of Perfect Expected Value of Perfect Information (Information (EVPIEVPI))

EVPIEVPI places an upper bound on what you should places an upper bound on what you should pay for additional information.pay for additional information.

EVPIEVPI = = EVwPIEVwPI –– Maximum Maximum EMVEMV EVwPIEVwPI is the long run average return if we have is the long run average return if we have

perfect information before a decision is made.perfect information before a decision is made.

EVwPIEVwPI= (best payoff for first state of = (best payoff for first state of nature)nature)x (probability of first state of x (probability of first state of nature)nature)+ (best payoff for second state of + (best payoff for second state of nature)nature)x (probability of second state of x (probability of second state of nature)nature)+ … + (best payoff for last state of + … + (best payoff for last state of nature)nature)x (probability of last state of x (probability of last state of nature)nature)

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Expected Value of Perfect Expected Value of Perfect Information (Information (EVPIEVPI))

Suppose Scientific Marketing, Inc. Suppose Scientific Marketing, Inc. offers analysis that will provide offers analysis that will provide certainty about market conditions certainty about market conditions (favourable).(favourable).

Additional information will cost Additional information will cost $65,000.$65,000.

Should Thompson Lumber purchase Should Thompson Lumber purchase the information?the information?

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Expected Value of Perfect Expected Value of Perfect Information (Information (EVPIEVPI))

STATE OF NATURESTATE OF NATURE

ALTERNATIVEALTERNATIVEFAVOURABLFAVOURABLE MARKET ($)E MARKET ($)

UNFAVOURABLE UNFAVOURABLE MARKET ($)MARKET ($) EMV ($)EMV ($)

Construct a large Construct a large plantplant 200,000200,000 -180,000-180,000 10,00010,000

Construct a small Construct a small plantplant 100,000100,000 -20,000-20,000 40,00040,000

Do nothingDo nothing 00 00 00

With perfect With perfect informationinformation 200,000200,000 00 100,000100,000

ProbabilitiesProbabilities 0.50.5 0.50.5

Table 3.10

EVwPIEVwPI

Decision Table with Perfect Information

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Expected Value of Perfect Expected Value of Perfect Information (Information (EVPIEVPI))

The maximum EMV without additional information is $40,000.

EVPI = EVwPI – Maximum EMV= $100,000 - $40,000= $60,000

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Expected Value of Perfect Expected Value of Perfect Information (Information (EVPIEVPI))

The maximum The maximum EMVEMV without additional information without additional information is $40,000.is $40,000.

EVPIEVPI = = EVwPIEVwPI – Maximum – Maximum EMVEMV= $100,000 - $40,000= $100,000 - $40,000= $60,000= $60,000

So the maximum So the maximum Thompson should pay Thompson should pay for the additional for the additional information is information is $60,000.$60,000.Therefore, Thompson should not Therefore, Thompson should not

pay $65,000 for this information.pay $65,000 for this information.

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Expected Opportunity LossExpected Opportunity Loss Expected opportunity lossExpected opportunity loss ( (EOLEOL)) is the is the

cost of not picking the best solution.cost of not picking the best solution. First construct an opportunity loss table.First construct an opportunity loss table. For each alternative, multiply the For each alternative, multiply the

opportunity loss by the probability of that opportunity loss by the probability of that loss for each possible outcome and add loss for each possible outcome and add these together.these together.

Minimum Minimum EOLEOL will always result in the will always result in the same decision as same decision as maximum maximum EMV.EMV.

Minimum Minimum EOLEOL will always will always equal equal EVPI.EVPI.

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Expected Opportunity LossExpected Opportunity Loss

EOLEOL (large plant)= (0.50)($0) + (0.50)( (large plant)= (0.50)($0) + (0.50)($180,000) $180,000) = $90,000= $90,000

EOLEOL (small plant)= (0.50)($100,000) + (0.50) (small plant)= (0.50)($100,000) + (0.50)($20,000) = $60,000($20,000) = $60,000

EOLEOL (do nothing)= (0.50)($200,000) + (0.50)($0) (do nothing)= (0.50)($200,000) + (0.50)($0) = $100,000= $100,000

Table 3.11

STATE OF NATURESTATE OF NATURE

ALTERNATIVEALTERNATIVEFAVOURABLE FAVOURABLE

MARKET ($)MARKET ($)UNFAVOURABLE UNFAVOURABLE

MARKET ($)MARKET ($) EOLEOL

Construct a large plantConstruct a large plant 00 180,000180,000 90,00090,000

Construct a small Construct a small plantplant 100,000100,000 20,00020,000 60,00060,000

Do nothingDo nothing 200,000200,000 00 100,000100,000

ProbabilitiesProbabilities 0.500.50 0.500.50

Minimum Minimum EOLEOL

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Sensitivity AnalysisSensitivity Analysis Sensitivity analysis Sensitivity analysis

examines how the decision examines how the decision might change with different might change with different input data.input data.

For the Thompson Lumber For the Thompson Lumber example:example:

PP = probability of a = probability of a favourable marketfavourable market

(1 – (1 – PP) = probability of an ) = probability of an unfavourable marketunfavourable market

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Sensitivity AnalysisSensitivity Analysis

EMVEMV(Large Plant)(Large Plant) = $200,000= $200,000PP – – $180,000)($180,000)(1 1 – – PP))

= $200,000= $200,000PP –– $180,000 + $180,000 $180,000 + $180,000PP

= $380,000= $380,000PP –– $180,000 $180,000

EMVEMV(Small Plant)(Small Plant) = $100,000= $100,000PP – – $20,000)($20,000)(1 1 – – PP))

= $100,000= $100,000PP –– $20,000 + $20,000 $20,000 + $20,000PP

= $120,000= $120,000PP –– $20,000 $20,000

EMVEMV(Do Nothing)(Do Nothing) = $0= $0PP + 0( + 0(1 1 – – PP))

= $0= $0

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Sensitivity AnalysisSensitivity Analysis

$300,000

$200,000

$100,000

0

–$100,000

–$200,000

EMV Values

EMVEMV (large plant) (large plant)

EMVEMV (small plant) (small plant)

EMVEMV (do nothing) (do nothing)

Point 1

Point 2

.167 .615 1

Values of P

Figure 3.1

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Sensitivity AnalysisSensitivity AnalysisPoint 1:Point 1:

EMVEMV(do nothing) = (do nothing) = EMVEMV(small plant)(small plant)

000200001200 ,$,$ P 167000012000020

.,,

P

00018000038000020000120 ,$,$,$,$ PP

6150000260000160

.,,

P

Point 2:Point 2:EMVEMV(small plant) = (small plant) = EMVEMV(large plant)(large plant)

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Sensitivity AnalysisSensitivity Analysis

$300,000

$200,000

$100,000

0

–$100,000

–$200,000

EMV Values

EMVEMV (large plant) (large plant)

EMVEMV (small plant) (small plant)

EMVEMV (do nothing) (do nothing)

Point 1

Point 2

.167 .615 1

Values of P

BEST BEST ALTERNATIVEALTERNATIVE

RANGE OF RANGE OF PP VALUESVALUES

Do nothingDo nothing Less than 0.167Less than 0.167

Construct a small plantConstruct a small plant 0.167 – 0.6150.167 – 0.615

Construct a large plantConstruct a large plant Greater than 0.615Greater than 0.615

Figure 3.1

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Using ExcelUsing Excel

Program 3.1A

Input Data for the Thompson Lumber Input Data for the Thompson Lumber Problem Using Excel QMProblem Using Excel QM

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Using ExcelUsing ExcelOutput Results for the Thompson Output Results for the Thompson Lumber Problem Using Excel QMLumber Problem Using Excel QM

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Decision TreesDecision Trees

Any problem that can be presented in a Any problem that can be presented in a decision table can also be graphically decision table can also be graphically represented in a represented in a decision tree.decision tree.

Decision trees are most beneficial when a Decision trees are most beneficial when a sequence of decisions must be made.sequence of decisions must be made.

All decision trees contain All decision trees contain decision pointsdecision points or or nodes, nodes, from which one of several alternatives from which one of several alternatives may be chosen.may be chosen.

All decision trees contain All decision trees contain state-of-nature state-of-nature pointspoints or or nodes, nodes, out of which one state of out of which one state of nature will occur.nature will occur.

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Five Steps ofFive Steps ofDecision Tree AnalysisDecision Tree Analysis

1.1. Define the problem.Define the problem.

2.2. Structure or draw the decision tree.Structure or draw the decision tree.

3.3. Assign probabilities to the states of nature.Assign probabilities to the states of nature.

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

5.5. Solve the problem by computing expected Solve the problem by computing expected monetary values (monetary values (EMVEMVs) for each state of s) for each state of nature node.nature node.

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Structure of Decision TreesStructure of Decision Trees

Trees start from left to right.Trees start from left to right. Trees represent decisions and outcomes Trees represent decisions and outcomes

in sequential order.in sequential order. Squares represent decision nodes.Squares represent decision nodes. Circles represent states of nature nodes.Circles represent states of nature nodes. Lines or branches connect the decisions Lines or branches connect the decisions

nodes and the states of nature.nodes and the states of nature.

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ThompsonThompson’’s Decision Trees Decision Tree

Favourable Market

Unfavourable Market

Favourable Market

Unfavourable MarketDo Nothing

Constru

ct

Large

Plant

11

Construct

Small

Plant

22

Figure 3.2

A Decision Node

A State-of-Nature Node

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ThompsonThompson’’s Decision Trees Decision Tree

Favourable Market

Unfavourable Market

Favourable Market

Unfavourable MarketDo Nothing

Constru

ct

Large

Plant

1

Construct

Small

Plant

2

Alternative with Alternative with best best EMVEMV is is

selectedselected

Figure 3.3

EMVEMV for for Node 1 = Node 1 = $10,000$10,000

= (0.5)($200,000) + (0.5)(–$180,000)

EMV for Node 2 = $40,000

= (0.5)($100,000) + (0.5)(–$20,000)

PayoffsPayoffs

$200,000$200,000

––$180,000$180,000

$100,000$100,000

––$20,000$20,000

$0$0

(0.5)(0.5)

(0.5)(0.5)

(0.5)(0.5)

(0.5)(0.5)

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ThompsonThompson’’s Complex Decision Trees Complex Decision Tree

First First Decision Decision PointPoint

Second Second Decision PointDecision Point

Favourable Market (0.78)

Unfavourable Market (0.22)

Favourable Market (0.78)Unfavourable Market (0.22)

Favourable Market (0.27)

Unfavourable Market (0.73)

Favourable Market (0.27)Unfavourable Market (0.73)

Favourable Market (0.50)

Unfavourable Market (0.50)

Favourable Market (0.50)Unfavourable Market (0.50)

Large Plant

Small

PlantNo Plant

6

7

Con

duct

Mar

ket Su

rvey

Do Not Conduct Survey

Large Plant

Small

PlantNo Plant

2

3

Large Plant

Small

PlantNo Plant

4

5

1Res

ults

Favor

able

Results

Negative

Survey

(0.4

5)

Survey (0.55)

PayoffsPayoffs

–$190,000

$190,000190,000

$90,000–$30,000

–$10,000

–$180,000

$200,000

$100,000–$20,000

$0

–$190,000

$190,000

$90,000–$30,000

–$10,000

Figure 3.4

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ThompsonThompson’’s Complex Decision Trees Complex Decision Tree

1.1. Given favourable survey results,Given favourable survey results,EMVEMV(node 2)= (node 2)= EMVEMV(large plant | positive (large plant | positive survey)survey)= (0.78)($190,000) + (0.22)(–$190,000) = = (0.78)($190,000) + (0.22)(–$190,000) = $106,400$106,400EMVEMV(node 3)= (node 3)= EMVEMV(small plant | positive (small plant | positive survey)survey)

= (0.78)($90,000) + (0.22)(–$30,000) = = (0.78)($90,000) + (0.22)(–$30,000) = $63,600$63,600EMVEMV for no plant = –$10,000 for no plant = –$10,000

2.2. Given negative survey results,Given negative survey results,EMVEMV(node 4)= (node 4)= EMVEMV(large plant | negative (large plant | negative survey)survey)= (0.27)($190,000) + (0.73)(–$190,000) = –= (0.27)($190,000) + (0.73)(–$190,000) = –$87,400$87,400EMVEMV(node 5)= (node 5)= EMVEMV(small plant | negative (small plant | negative survey)survey)

= (0.27)($90,000) + (0.73)(–$30,000)= (0.27)($90,000) + (0.73)(–$30,000) = $2,400= $2,400

EMVEMV for no plant = –$10,000 for no plant = –$10,000

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ThompsonThompson’’s Complex Decision Trees Complex Decision Tree

3.3. Compute the expected value of the Compute the expected value of the market survey,market survey,EMVEMV(node 1)= (node 1)= EMVEMV(conduct survey)(conduct survey)

= (0.45)($106,400) + (0.55)($2,400)= (0.45)($106,400) + (0.55)($2,400)= $47,880 + $1,320 = $49,200= $47,880 + $1,320 = $49,200

4.4. If the market survey is not conducted,If the market survey is not conducted,EMVEMV(node 6)= (node 6)= EMVEMV(large plant)(large plant)

= (0.50)($200,000) + (0.50)(–$180,000) = (0.50)($200,000) + (0.50)(–$180,000) = $10,000= $10,000

EMVEMV(node 7)= (node 7)= EMVEMV(small plant)(small plant)= (0.50)($100,000) + (0.50)(–$20,000) = (0.50)($100,000) + (0.50)(–$20,000) = $40,000= $40,000

EMVEMV for no plant = $0 for no plant = $0

5.5. The best choice is to seek marketing The best choice is to seek marketing informationinformation

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ThompsonThompson’’s Complex Decision Trees Complex Decision Tree

Figure 3.5

First First Decision Decision PointPoint

Second Second Decision PointDecision Point

Favourable Market (0.78)

Unfavourable Market (0.22)

Favourable Market (0.78)Unfavourable Market (0.22)

Favourable Market (0.27)

Unfavourable Market (0.73)

Favourable Market (0.27)Unfavourable Market (0.73)

Favourable Market (0.50)

Unfavourable Market (0.50)

Favourable Market (0.50)Unfavourable Market (0.50)

Large Plant

Small

PlantNo Plant

Con

duct

Mar

ket Su

rvey

Do Not Conduct Survey

Large Plant

Small

PlantNo Plant

Large Plant

Small

PlantNo Plant

Results

Favor

able

Results

Negative

Survey

(0.4

5)

Survey (0.55)

PayoffsPayoffs

–$190,000

$190,000

$90,000–$30,000

–$10,000

–$180,000

$200,000

$100,000–$20,000

$0

–$190,000

$190,000

$90,000–$30,000

–$10,000

$4

0,0

00

$4

0,0

00

$2

,40

0$

2,4

00

$1

06

,40

0$

10

6,4

00

$4

9,2

00

$4

9,2

00

$106,400

$63,600

–$87,400

$2,400

$10,000

$40,000

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Expected Value of Sample InformationExpected Value of Sample Information

Suppose Thompson wants to know the Suppose Thompson wants to know the actual value of doing the survey.actual value of doing the survey.

EVSIEVSI = = ––

Expected valueExpected valuewithwith sample sample

information, assuminginformation, assumingno cost to gather itno cost to gather it

Expected valueExpected valueof best decisionof best decisionwithoutwithout sample sample

informationinformation

==((EVEV with sample information + cost) with sample information + cost)– – ((EVEV without sample information) without sample information)

EVSIEVSI = ($49,200 + $10,000) = ($49,200 + $10,000) – $40,000 = $19,200– $40,000 = $19,200

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Sensitivity AnalysisSensitivity Analysis

How sensitive are the How sensitive are the decisions to changes in the decisions to changes in the probabilities?probabilities? How sensitive is our decision to the How sensitive is our decision to the

probability of a favourable survey probability of a favourable survey result? result?

That is, if the probability of a That is, if the probability of a favourable result (favourable result (pp = 0.45) were to = 0.45) were to change, would we make the same change, would we make the same decision? decision?

How much could it change before How much could it change before we would make a different we would make a different decision?decision?

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Sensitivity AnalysisSensitivity Analysisp =probability of a favourable survey result(1 – p) = probability of a negative survey result

EMVEMV(node 1)(node 1) = ($106,400)= ($106,400)pp +($2,400) +($2,400)(1 – (1 – pp))

= $104,000= $104,000pp + $2,400 + $2,400We are indifferent when the We are indifferent when the EMVEMV of node 1 is of node 1 is the same as the the same as the EMVEMV of not conducting the of not conducting the survey, $40,000survey, $40,000

$104,000$104,000pp + $2,400 + $2,400= $40,000= $40,000

$104,000$104,000pp= $37,600= $37,600pp= $37,600/$104,000 = $37,600/$104,000 = 0.36= 0.36

If If pp<0.36, do not conduct the survey. If <0.36, do not conduct the survey. If pp>0.36, conduct the survey.>0.36, conduct the survey.

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Bayesian AnalysisBayesian Analysis There are many ways of There are many ways of

getting probability data. It getting probability data. It can be based on:can be based on: Management’s experience and intuition.Management’s experience and intuition. Historical data.Historical data. Computed from other data using Bayes’ Computed from other data using Bayes’

theorem.theorem.

Bayes’ theorem incorporates Bayes’ theorem incorporates initial estimates and initial estimates and information about the information about the accuracy of the sources.accuracy of the sources.

It also allows the revision of It also allows the revision of initial estimates based on new initial estimates based on new information.information.

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Calculating Revised Calculating Revised ProbabilitiesProbabilities

In the Thompson Lumber case In the Thompson Lumber case we used these four conditional we used these four conditional probabilities:probabilities:PP (favourable market( (favourable market(FMFM) | survey results positive) = 0.78) | survey results positive) = 0.78

PP (unfavourable market( (unfavourable market(UMUM) | survey results positive) = 0.22) | survey results positive) = 0.22PP (favourable market( (favourable market(FMFM) | survey results negative) = 0.27) | survey results negative) = 0.27

PP (unfavourable market( (unfavourable market(UMUM) | survey results negative) = 0.73) | survey results negative) = 0.73

But how were these calculated?But how were these calculated? The prior probabilities of these The prior probabilities of these

markets are:markets are:PP ( (FMFM) = 0.50) = 0.50PP ( (UMUM) = 0.50) = 0.50

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Calculating Revised Calculating Revised ProbabilitiesProbabilities

Through discussions with experts Thompson has learned the information in the table below.

He can use this information and Bayes’ theorem to calculate posterior probabilities.

STATE OF NATURESTATE OF NATURE

RESULT OF RESULT OF SURVEYSURVEY

FAVOURABLE MARKET FAVOURABLE MARKET ((FMFM))

UNFAVOURABLE MARKET UNFAVOURABLE MARKET ((UMUM))

Positive (predicts Positive (predicts favorable market favorable market for product)for product)

PP (survey positive | (survey positive | FMFM) ) = 0.70= 0.70

PP (survey positive | (survey positive | UMUM) ) = 0.20= 0.20

Negative (predicts Negative (predicts unfavorable unfavorable market for market for product)product)

PP (survey negative | (survey negative | FMFM) ) = 0.30= 0.30

PP (survey negative | (survey negative | UMUM) ) = 0.80= 0.80

Table 3.12

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3-53

Calculating Revised Calculating Revised ProbabilitiesProbabilities

Recall Bayes’ theorem:Recall Bayes’ theorem:

)()|()()|()()|(

)|(APABPAPABP

APABPBAP

wherewhereevents two anyBA,

AA of complement

For this example, For this example, AA will will represent a favourable market represent a favourable market and and BB will represent a positive will represent a positive survey.survey.

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Calculating Revised Calculating Revised ProbabilitiesProbabilities

PP ( (FMFM | survey positive) | survey positive)

P(UM)|UM)P(P(FM) |FM)P(FMPFMP

positive surveypositive survey

positive survey )()|(

780450350

500200500700500700

...

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

P(FM)|FM)P(P(UM) |UM)P(UMPUMP

positive surveypositive survey

positive survey )()|(

220450100

500700500200500200

...

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

PP ( (UMUM | survey positive) | survey positive)

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Calculating Revised Calculating Revised ProbabilitiesProbabilities

POSTERIOR PROBABILITYPOSTERIOR PROBABILITY

STATE OF STATE OF NATURENATURE

CONDITIONAL CONDITIONAL PROBABILITY PROBABILITY

PP(SURVEY (SURVEY POSITIVE | STATE POSITIVE | STATE

OF NATURE)OF NATURE)PRIOR PRIOR

PROBABILITYPROBABILITYJOINT JOINT

PROBABILITYPROBABILITY

PP(STATE OF (STATE OF NATURE | NATURE | SURVEY SURVEY

POSITIVE)POSITIVE)

FMFM 0.700.70 X 0.50X 0.50 == 0.350.35 0.35/0.45 = 0.35/0.45 = 0.780.78

UMUM 0.200.20 X 0.50X 0.50 == 0.100.10 0.10/0.45 = 0.10/0.45 = 0.220.22

PP(survey results positive) =(survey results positive) = 0.450.45 1.001.00

Table 3.13

Probability Revisions Given a Positive SurveyProbability Revisions Given a Positive Survey

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Calculating Revised Calculating Revised ProbabilitiesProbabilities

PP ( (FMFM | survey negative) | survey negative)

P(UM)|UM)P(P(FM) |FM)P(FMPFMP

negative surveynegative survey

negative survey )()|(

270550150

500800500300500300

...

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

P(FM)|FM)P(P(UM) |UM)P(UMPUMP

negative surveynegative survey

negative survey )()|(

730550400

500300500800500800

...

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

PP ( (UMUM | survey negative) | survey negative)

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Calculating Revised Calculating Revised ProbabilitiesProbabilities

POSTERIOR PROBABILITYPOSTERIOR PROBABILITY

STATE OF STATE OF NATURENATURE

CONDITIONAL CONDITIONAL PROBABILITY PROBABILITY

PP(SURVEY (SURVEY NEGATIVE | STATE NEGATIVE | STATE

OF NATURE)OF NATURE)PRIOR PRIOR

PROBABILITYPROBABILITYJOINT JOINT

PROBABILITYPROBABILITY

PP(STATE OF (STATE OF NATURE | NATURE | SURVEY SURVEY

NEGATIVE)NEGATIVE)

FMFM 0.300.30 X 0.50X 0.50 == 0.150.15 0.15/0.55 = 0.15/0.55 = 0.270.27

UMUM 0.800.80 X 0.50X 0.50 == 0.400.40 0.40/0.55 = 0.40/0.55 = 0.730.73

PP(survey results positive) =(survey results positive) = 0.550.55 1.001.00

Table 3.14

Probability Revisions Given a Negative Survey

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Using ExcelUsing ExcelFormulas Used for Bayes’ Calculations in ExcelFormulas Used for Bayes’ Calculations in Excel

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Using ExcelUsing ExcelResults of Bayes’ Calculations in Excel

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Potential Problems Using Potential Problems Using Survey ResultsSurvey Results

We can not always get the We can not always get the necessary data for necessary data for analysis.analysis.

Survey results may be Survey results may be based on cases where an based on cases where an action was taken.action was taken.

Conditional probability Conditional probability information may not be as information may not be as accurate as we would like.accurate as we would like.

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Utility Theory Utility Theory Monetary value is not always a true Monetary value is not always a true

indicator of the overall value of the indicator of the overall value of the result of a decision.result of a decision.

The overall value of a decision is The overall value of a decision is called called utility.utility.

Economists assume that rational Economists assume that rational people make decisions to maximize people make decisions to maximize their utility.their utility.

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HeadsHeads (0.5)

Tails (0.5)

$5,000,000

$0

Utility Theory Utility Theory

Accept Accept OfferOffer

Reject Reject OfferOffer

$2,000,000$2,000,000

EMV = $2,500,000Figure 3.6

Your Decision Tree for the Lottery TicketYour Decision Tree for the Lottery Ticket

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Utility Theory Utility Theory Utility assessmentUtility assessment assigns the worst outcome a utility assigns the worst outcome a utility

of 0, and the best outcome, a utility of 1.of 0, and the best outcome, a utility of 1. A A standard gamblestandard gamble is used to determine utility values. is used to determine utility values. When you are indifferent, your utility values are equal.When you are indifferent, your utility values are equal.

Expected utility of alternative Expected utility of alternative 2 =2 =Expected utility of Expected utility of alternative 1alternative 1Utility of other outcome =Utility of other outcome = ((pp))(utility of best outcome, which (utility of best outcome, which is 1)is 1)+ (1 – + (1 – pp)(utility of the worst )(utility of the worst outcome, which is 0)outcome, which is 0)Utility of other outcome =Utility of other outcome = ((pp))(1) + (1 – (1) + (1 – pp)(0) = )(0) = pp

Page 64: Bba 3274 qm week 4 decision analysis

Standard Gamble for Utility Standard Gamble for Utility Assessment Assessment

Best OutcomeUtility = 1

Worst OutcomeUtility = 0

Other OutcomeUtility = ?

(p)

(1 – p)

Altern

ative 1

Alternative 2

Figure 3.7

Page 65: Bba 3274 qm week 4 decision analysis

Investment ExampleInvestment Example Jane Dickson wants to construct a utility curve Jane Dickson wants to construct a utility curve

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

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

An investment in a bank will result in $5,000.An investment in a bank will result in $5,000. An investment in real estate will result in $0 or $10,000.An investment in real estate will result in $0 or $10,000. Unless there is an 80% chance of getting $10,000 from Unless there is an 80% chance of getting $10,000 from

the real estate deal, Jane would prefer to have her the real estate deal, Jane would prefer to have her money in the bank.money in the bank.

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

Page 66: Bba 3274 qm week 4 decision analysis

Investment ExampleInvestment Example

Figure 3.8

p = 0.80

(1 – p) = 0.20

Invest in

Real Esta

te

Invest in Bank

$10,000U($10,000) = 1.0

$0U($0.00) = 0.0

$5,000U($5,000) = p = 0.80

Utility for $5,000 = Utility for $5,000 = UU($5,000)= ($5,000)= pUpU($10,000) + (1 – ($10,000) + (1 – pp))UU($0)($0)= (0.8)(1) + (0.2)(0) = 0.8= (0.8)(1) + (0.2)(0) = 0.8

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Investment ExampleInvestment Example

Utility for $7,000 = 0.90Utility for $7,000 = 0.90Utility for $3,000 = 0.50Utility for $3,000 = 0.50

We can assess other utility values in the same way.

For Jane these are:

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

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Utility CurveUtility CurveU ($7,000) = 0.90

U ($5,000) = 0.80

U ($3,000) = 0.50

U ($0) = 0

Figure 3.9

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

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Utility CurveUtility Curve Jane’s utility curve is typical of a Jane’s utility curve is typical of a risk risk

avoideravoider.. She gets less utility from She gets less utility from

greater risk.greater risk. She avoids situations where She avoids situations where

high losses might occur.high losses might occur. As monetary value increases, As monetary value increases,

her utility curve increases at a her utility curve increases at a slower rate.slower rate.

A A risk seeker risk seeker gets more utility from gets more utility from greater riskgreater risk As monetary value increases, the As monetary value increases, the

utility curve increases at a faster utility curve increases at a faster rate.rate.

Someone with Someone with riskrisk indifferenceindifference will will have a linear utility curve.have a linear utility curve.

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Preferences for RiskPreferences for Risk

Figure 3.10

Monetary Outcome

Uti

lity

Risk Avoider

Risk

Indi

fferen

ce

Risk Seeker

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Utility as a Utility as a Decision-Making CriteriaDecision-Making Criteria

Once a utility curve has been developed it can be used in making decisions.

This replaces monetary outcomes with utility values.

The expected utility is computed instead of the EMV.

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Utility as a Utility as a Decision-Making CriteriaDecision-Making Criteria

Mark Simkin loves to gamble. He plays a game tossing

thumbtacks in the air. If the thumbtack lands point up,

Mark wins $10,000. If the thumbtack lands point down,

Mark loses $10,000. Mark believes that there is a 45%

chance the thumbtack will land point up.

Should Mark play the game (alternative 1)?

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Utility as a Utility as a Decision-Making CriteriaDecision-Making Criteria

Figure 3.11

Tack Lands Point Up (0.45)

Altern

ative 1

Mark Plays the Game

Alternative 2

$10,000

–$10,000

$0

Tack Lands Point Down (0.55)

Mark Does Not Play the Game

Decision Facing Mark SimkinDecision Facing Mark Simkin

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Utility as a Utility as a Decision-Making CriteriaDecision-Making Criteria

Step 1– Define Mark’s utilities.

UU ( (––$10,000) = 0.05$10,000) = 0.05UU ($0) = 0.15 ($0) = 0.15

UU ($10,000) = 0.30 ($10,000) = 0.30

Step 2 – Replace monetary values with

utility values.EE(alternative 1: play the game)(alternative 1: play the game) = (0.45)(0.30) + = (0.45)(0.30) + (0.55)(0.05)(0.55)(0.05)

= 0.135 + 0.027 = = 0.135 + 0.027 = 0.1620.162EE(alternative 2: don’t play the game)= 0.15(alternative 2: don’t play the game)= 0.15

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Utility Curve for Mark SimkinUtility Curve for Mark Simkin

Figure 3.12

1.00 –

0.75 –

0.50 –

0.30 –0.25 –

0.15 –

0.05 –0 –| | | | |

–$20,000 –$10,000 $0 $10,000 $20,000Monetary Outcome

Uti

lity

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Utility as a Utility as a Decision-Making CriteriaDecision-Making Criteria

Figure 3.13

Tack Lands Point Up (0.45)

Altern

ative 1

Mark Plays the Game

Alternative 2

0.30

0.05

0.15

Tack Lands Point Down (0.55)

Don’t Play

UtilityE = 0.162

Using Expected Utilities in Decision Making

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TutorialTutorial

Lab Practical : Spreadsheet Lab Practical : Spreadsheet

1 - 77

Page 78: Bba 3274 qm week 4 decision analysis

Further ReadingFurther Reading

Render, B., Stair Jr.,R.M. & Hanna, M.E. (2013) Quantitative Analysis for Management, Pearson, 11th Edition

Waters, Donald (2007) Quantitative Methods for Business, Prentice Hall, 4th Edition.

Anderson D, Sweeney D, & Williams T. (2006) Quantitative Methods For Business Thompson Higher Education, 10th Ed.

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QUESTIONS?QUESTIONS?


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