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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
Today’s Overview Today’s Overview
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:
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
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.
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.
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.
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.
3-9
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
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.
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:
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
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
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)
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
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.
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
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
3-20
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
3-21
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)
3-22
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= $0
3-23
EMVEMV for Thompson Lumber for Thompson Lumber
STATE OF NATURESTATE OF NATURE
ALTERNATIVEALTERNATIVEFAVOURABLE FAVOURABLE
MARKET ($)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)
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.$65,000.
Should Thompson Lumber purchase Should Thompson Lumber purchase the information?the information?
Expected Value of Perfect Expected Value of Perfect Information (Information (EVPIEVPI))
STATE OF NATURESTATE OF NATURE
ALTERNATIVEALTERNATIVEFAVOURABLFAVOURABLE MARKET ($)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
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
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.
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.
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
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
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
Sensitivity AnalysisSensitivity Analysis
$300,000
$200,000
$100,000
0
–$100,000
–$200,000
EMV Values
EMVEMV (large plant) (large plant)
EMVEMV (small plant) (small plant)
EMVEMV (do nothing) (do nothing)
Point 1
Point 2
.167 .615 1
Values of P
Figure 3.1
Sensitivity AnalysisSensitivity AnalysisPoint 1:Point 1:
EMVEMV(do nothing) = (do nothing) = EMVEMV(small plant)(small plant)
000200001200 ,$,$ P 167000012000020
.,,
P
00018000038000020000120 ,$,$,$,$ PP
6150000260000160
.,,
P
Point 2:Point 2:EMVEMV(small plant) = (small plant) = EMVEMV(large plant)(large plant)
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
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
Using ExcelUsing ExcelOutput Results for the Thompson Output Results for the Thompson Lumber Problem Using Excel QMLumber Problem Using Excel QM
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.
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.
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.
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
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)
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
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
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
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
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
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?
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.
3-50
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.
3-51
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
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
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.
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)
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
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)
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
Using ExcelUsing ExcelFormulas Used for Bayes’ Calculations in ExcelFormulas Used for Bayes’ Calculations in Excel
Using ExcelUsing ExcelResults of Bayes’ Calculations in Excel
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.
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.
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
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
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
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.
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
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.
3-68
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
3-69
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.
3-70
Preferences for RiskPreferences for Risk
Figure 3.10
Monetary Outcome
Uti
lity
Risk Avoider
Risk
Indi
fferen
ce
Risk Seeker
3-71
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.
3-72
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)?
3-73
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
3-74
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
3-75
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
3-76
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
TutorialTutorial
Lab Practical : Spreadsheet Lab Practical : Spreadsheet
1 - 77
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.
QUESTIONS?QUESTIONS?