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Chapter 8
Decision Analysis
Slides 8a: Introduction
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Decision Analysis
A set of alternative actions We may chose whichever we please
A set of possible states of nature Only one will be correct, but we dont know in
advance
A set of outcomes and a value for each Each is a combination of an alternative action and a
state of nature
Value can be monetary or otherwise
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Decision Analysis Certainty
Decision Maker knows with certainty what the state of
nature will be - only one possible state of nature Ignorance
Decision Maker knows all possible states of nature,but does not know probability of occurrence
Risk Decision Maker knows all possible states of nature,
and can assign probability of occurrence for eachstate
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Decision Making Under CertaintyDecision Variable
Units to build 150
Parameter Estimates
Cost to build (/unit) 6,000$Revenue (/unit) 14,000$
Demand (units) 250
Consequence Variables
Total Revenue 2,100,000$Total Cost 900,000$
Performance Measure
Net Revenue 1,200,000$
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Decision Making Under Ignorance
Payoff TableKelly Construction Payoff Table (Prob. 8-17)
Low (50 units) Medium (100 units) High (150 units)
Build 50 400,000 400,000 400,000
Build 100 100,000 800,000 800,000
Build 150 (200,000) 500,000 1,200,000
State of Nature
DemandAlternative
Actions
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Decision Making Under Ignorance Maximax
Select the strategy with the highest possible
return Maximin
Select the strategy with the smallest possibleloss
LaPlace-Bayes All states of nature are equally likely to occur.
Select alternative with best average payoff
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Maximax:The Optimistic Point of View
Select the best of the best strategy
Evaluates each decision by the maximum possiblereturn associated with that decision (Note: if cost datais used, the minimum return is best)
The decision that yields the maximum of thesemaximum returns (maximax) is then selected
For risk takers Doesnt consider the down side risk
Ignores the possible losses from the selectedalternative
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Maximax Example
Low (50 units) Medium (100 units) High (150 units) Max
Build 50 400,000 400,000 400,000 400,000
Build 100 100,000 800,000 800,000 800,000
Build 150 (200,000) 500,000 1,200,000 1,200,000
State of NatureMaximax
CriterionDemandAlternative
Actions
Kelly Construction
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Maximin:The Pessimistic Point of View
Select the best of the worst strategy Evaluates each decision by the minimum
possible return associated with the decision
The decision that yields the maximum valueof the minimum returns (maximin) is selected
For risk averse decision makers A protect strategy
Worst case scenario the focus
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Maximin
Low (50 units) Medium (100 units) High (150 units) Min
Build 50 400,000 400,000 400,000 400,000
Build 100 100,000 800,000 800,000 100,000
Build 150 (200,000) 500,000 1,200,000 (200,000)
State of NatureMaximin
CriterionDemandAlternative
Actions
Kelly Construction
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Decision Making Under Risk Expected Return (ER)*
Select the alternative with the highest (long term)
expected return A weighted average of the possible returns for
each alternative, with the probabilities used as
weights
*Also referred to as Expected Value (EV) or ExpectedMonetary Value (EMV)
**Note that this amount will not be obtained in the shortterm, or if the decision is a one-time event!
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Expected Return
Low (50 units) Medium (100 units) High (150 units) ER
Build 50 400,000 400,000 400,000 400,000
Build 100 100,000 800,000 800,000 660,000
Build 150 (200,000) 500,000 1,200,000 570,000
Probability 0.2 0.5 0.3 1.0
State of NatureExpected
ReturnDemandAlternative
Actions
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Expected Value of Perfect Information
EVPI measures how much better you could do onthis decision if you could always know when eachstate of nature would occur, where:
EVUPI = Expected Value Under Perfect Information
(also called EVwPI, the EV with perfect information, or
EVC, the EV under certainty)
EVUII = Expected Value of the best action with
imperfect information (also called EVBest )
EVPI = EVUPI EVUII
EVPI tells you how much you are willing to pay forperfect information (or is the upper limit for what youwould pay for additional imperfect information!)
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Expected Value of Perfect
Information
Low (50 units) Medium (100 units) High (150 units) ER
Build 50 400,000 400,000 400,000 400,000
Build 100 100,000 800,000 800,000 660,000
Build 150 (200,000) 500,000 1,200,000 570,000
Probability 0.2 0.5 0.3 1.0
Best Decision 400,000 800,000 1,200,000 840,000
EVPI 180,000
State of NatureExpected
ReturnDemandAlternative
Actions
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Using Excel to Calculate EVPI:
Formulas View
A B C D E
12
3 Payoffs States of Nature Expected Return
4 Alternatives Low (50 units) Medium (100 units) High (150 units) ER
5 Build 50 400000 400000 400000 =SUMPRODUCT(B5:D5,B$8:D$8)
6 Build 100 100000 800000 800000 =SUMPRODUCT(B6:D6,B$8:D$8)
7 Build 150 -200000 500000 1200000 =SUMPRODUCT(B7:D7,B$8:D$8)
8 Probability 0.2 0.5 0.39 Best Decision =MAX(B5:B7) =MAX(C5:C7) =MAX(D5:D7)
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11 EVwPI = =SUMPRODUCT(B9:D9,B8:D8)
12 EVBest = =MAX(E5:E7)
13 EVPI = =E11-E12
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Kelly Construction
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A newsvendor can buy the Wall Street Journalnewspapers for 40 cents each and sell them for 75cents.
However, he must buy the papers before he knowshow many he can actually sell. If he buys morepapers than he can sell, he disposes of the excess atno additional cost. If he does not buy enoughpapers, he loses potential sales now and possibly inthe future.
Suppose that the loss of future sales is captured by aloss of goodwill cost of 50 cents per unsatisfiedcustomer.
The Newsvendor ModelThe Newsvendor Model
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The demand distribution is as follows:
P0 = Prob{demand = 0} = 0.1
P1 = Prob{demand = 1} = 0.3
P2 = Prob{demand = 2} = 0.4
P3 = Prob{demand = 3} = 0.2
Each of these four values represent the states ofnature. The number of papers ordered is the decision.The returns or payoffs are as follows:
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State of Nature (Demand)
0 1 2 3Decision
0 0 -50 -100 -150
1 -40 35 -15 -65
2 -80 -5 70 20
3 -120 -45 30 105
Payoff = 75(# papers sold) 40(# papers ordered) 50(unmet demand)
Where 75 = selling price40 = cost of buying a paper
50 = cost of loss of goodwill
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Now, the ER is calculated for each decision:
State of Nature (Demand)
0 1 2 3Decision
0 0 -50 -100 -150 -85
1 -40 35 -15 -65 -12.5
2 -80 -5 70 20 22.5
3 -120 -45 30 105 7.5
ER
Prob. 0.1 0.3 0.4 0.2
ER1 = -40(0.1) + 35(0.3) 15(0.4) 65(0.2) = -12.5
ER2 = -80(0.1) 5(0.3) + 70(0.4) + 20(0.2) = 22.5
ER3 = -120(0.1) 45(0.3) + 30(0.4) 105(0.2) = 7.5
ER0
= 0(0.1) 50(0.3) 100(0.4) 150(0.2) = -85
Of these four ERs,
choose the maximum,
and order 2 papers
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ER(new) = 0(0.1) + 35(0.3) + 70(0.4) + 105(0.2)
State of Nature
0 1 2 3Decision
0 0 -50 -100 -1501 -40 35 -15 -65
2 -80 -5 70 20
3 -120 -45 30 105Prob. 0.1 0.3 0.4 0.2
= 59.5
ER(current) = 22.5
EVPI = 59.5 22.5 = 37.0 cents
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The decision that yields the maxmaximum of these maxmaximum
returns (maximax) is then selected.
This method evaluates each decision by the maximummaximum
possible return associated with that decision.
Maximax Criterion:Maximax Criterion: The Maximax criterion is anoptimistic decision making criterion.
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Then, the decision that yields the maxmaximum value of theminminimum returns (maximin) is selected.
Maximin Criterion:Maximin Criterion: The Maximin criterion is anextremely conservative, or pessimistic, approach to
making decisions.
Maximin evaluates each decision by the minimumminimum possible
return associated with the decision.
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So, using the 3 criteria, we made the followingdecisions regarding the newsvendor data:
CriteriaCriteria DecisionDecision
Maximin Cash Flow Order 1 paper
Expected Return Order 2 papers
Maximax Cash Flow Order 3 papers
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Most people are riskrisk--averseaverse, which means theywould feel that the loss of a certain amount of
money would be more painful than the gain ofthe same amount of money.
Utility functionsUtility functions in decision analysis measure theattractiveness of money.
Utility can be thought of as a measure ofsatisfaction.
THE RATIONALE FOR UTILITYTHE RATIONALE FOR UTILITY
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Utility
1.00.9100.8500.775
0.680
0.524
100 200 300 400 500 600 Dollars
Typical risk-averse utility function:
Go from $400 to
$500 results in
A gainin
utilityof 0.06
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To illustrate, first suppose you have $100 and someonegives you an additional $100. Note that your utility
increases by
U(200) U(100) = 0.680 0.524 = 0.156
Now suppose you start with $400 and someone gives youan additional $100. Now your utility increases by
U(500) U(400) = 0.910 0.850 = 0.060
This illustrates that an additional $100 is less attractive ifyou have $400 on hand than it is if you start with $100.
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Utilities and Decisions under RiskUtilities and Decisions under Risk
Summary:Summary:
UtilityUtility is a way to incorporate risk aversion into theexpected return calculation.
Calculating a utility function is out of the scope ofthis course, but it can be calculated by a series oflottery questions (e.g., Would you prefer one milliondollars or a 50% chance of earning five million?).