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Transcript of Decision Analysis1 DSC 3120 Generalized Modeling Techniques with Applications Part III. Decision...
Decision Analysis 1
DSC 3120 Generalized Modeling
Techniques with Applications
Part III. Decision Analysis
Decision Analysis 2
Decision Analysis A Rational and Systematic Approach to
Decision Making
Decision Making: choose the “best” from several available alternative courses of action
Key Element is Uncertainty of the outcome• We, as decision maker, control the decision• Outcome of the decision is uncertain to and
uncontrolled by decision maker (controlled by nature)
Decision Analysis 3
Example of Decision Analysis
You have $10,000 for investing in one of the three options: Stock, Mutual Fund, and CD. What is the best choice?
Question: Do you know the choices?
Do you know the best choice?
What is the uncertainty?
How do you make your choice?
Decision Analysis 4
Components of Decision Problem Alternative Actions -- Decisions
• There are several alternatives from which we want to choose the best
States of Nature -- Outcomes• There are several possible outcomes but which
one will occur is uncertain to us
Payoffs• Numerical (monetary) value representing the
consequence of a particular alternative action we choose and a state of nature that occurs later on
Decision Analysis 5
Payoff TableState of Nature
Alternative S1 S2 Sm
A1 r11 r12 r1m
A2 r21 r22 r2m
An rn1 rn2 rnm
Decision Analysis 6
An Example
S1
RainS2
No Rain
A1
UmbrellaOutcome (A1S1)Don’t get wet but alittle inconvenient.(80)
Outcome (A1S2)Cumbersome andinconvenient(-20)
A2
No UmbrellaOutcome (A2S1)Get wet (and possiblyget sick) (-40)
Outcome (A2S2)The best!(100)
State of Nature
Alt
ern
ativ
e
Decision Analysis 7
Three Classes of Decision Models Decision Making Under Certainty
• Only one state of nature (or we know with 100% sure what will happen)
Decision Making Under Uncertainty (ignorance)
• Several possible states of nature, but we have no idea about the likelihood of each possible state
Decision Making Under Risk• Several possible states of nature, and we have an
estimate of the probability for each state
Decision Analysis 8
Decision Making Under Uncertainty LaPlace (Assume Equal Likely States of Nature)
• Select alternative with best average payoff
Maximax (Assume The Best State of Nature)• Select alternative that will maximize the maximum payoff
(expect the best outcome--optimistic) Maximin (Assume The Worst State of Nature)
• Select alternative that will maximize the minimum payoff (expect the worst situation--pessimistic)
Minimax Regret (Don’t Want to Regret Too Much)• Select alternative that will minimize the maximum regret
Decision Analysis 9
Payoff Table
Example: Newsboy Problem
State of Nature (Demand)
Alternative(Order) 0 1 2 3
0 0 -50 -100 -150
1 -40 35 -15 -65
2 -80 -5 70 20
3 -120 -45 30 105
Decision Analysis 10
LaPlace Criterion State of Nature (Demand)
Alternative(Order) 0 1 2 3 Mean
0 0 -50 -100 -150 -75
1 -40 35 -15 -65 -21.25
2* -80 -5 70 20 1.25*
3 -120 -45 30 105 -7.5
Example: Newsboy Problem
Decision Analysis 11
Maximax Criterion
Example: Newsboy Problem
State of Nature (Demand)
Alternative(Order) 0 1 2 3 Max
0 0 -50 -100 -150 0
1 -40 35 -15 -65 35
2 -80 -5 70 20 70
3* -120 -45 30 105 105*
Decision Analysis 12
Maximin Criterion
Example: Newsboy Problem
State of Nature (Demand)
Alternative(Order) 0 1 2 3 Min
0 0 -50 -100 -150 -150
1* -40 35 -15 -65 -65*
2 -80 -5 70 20 -80
3 -120 -45 30 105 -120
Decision Analysis 13
Minimax Regret Criterion: Step 1
Example: Newsboy Problem
State of Nature (Demand)
Alternative(Order) 0 1 2 3
0 0 -50 -100 -150
1 -40 35 -15 -65
2 -80 -5 70 20
3 -120 -45 30 105
Best 0 35 70 105
Decision Analysis 14
Minimax Regret: Step 2 (Regret or Opportunity Loss Table)
Example: Newsboy Problem
State of Nature (Demand)
Alternative(Order) 0 1 2 3 Max
0 0 85 170 255 255
1 40 0 85 170 170
2* 80 40 0 85 85*
3 120 80 40 0 120
Decision Analysis 15
Decision Making Under Risk
State of NatureAlternative S1 S2 Sm
A1 r11 r12 r1m
A2 r21 r22 r2m
An rn1 rn2 rnm
Probability p1 p2 pm
• In this situation, we have more information about the uncertainty--probability
Decision Analysis 16
Decision Making Under Risk Maximize Expected Return (ER)
ERi = (pj rij) = p1ri1 + p2ri2 +…+ pmrim
Where ERi = Expected return if choosing the ith
alternative (Ai), (i = 1, 2, …, n)
pj = The probability of state j (Sj)
rij = The payoff if we choose alternative Ai
and Sj state of nature occurs
Decision Analysis 17
Expected Return & VarianceExample: Newsboy Problem
State of Nature (Demand)
Alternative(Order) 0 1 2 3 ER Variance
0 0 -50 -100 -150 -85 2025
1 -40 35 -15 -65 -12.5 1306.25
2* -80 -5 70 20 22.5* 2181.25
3 -120 -45 30 105 7.5 4556.25
Probability 0.1 0.3 0.4 0.2
Decision Analysis 18
Decision Making Under Risk High return is good, but on the other hand,
low risk is also important Variance -- a measure of the risk
Variancei = pj (rij - ERi)2
Where pj = The probability of state j (Sj)
rij = The payoff if choose Ai and Sj occurs
ERi= Expected return for alternative Ai
Decision Analysis 19
Expected Value of Perfect Information
EVPI measures the maximum worth (value) of the “Perfect Information” that we should pay for in order to improve our decisions
EVPI = ER w/ perfect info. - ER w/o perfect info.
• ER w/ perfect info. = pj max(rij)
• ER w/o perfect info. = max(ERi)
= max( pj rij)
Decision Analysis 20
Calculate EVPIExample: Newsboy Problem
State of Nature (Demand)
Alternative(Order)
0 1 2 3 ER
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
Best 0 35 70 105 59.5
Probability 0.1 0.3 0.4 0.2 37.0
ER w/ PI
ER w/o PI
EVPI
Decision Analysis 21
Expected Opportunity Loss (EOL) We can also use EOL to choose the best
alternative
Minimizing EOL = Maximizing ER
• both criteria yield the same best alternative
EOLi = pj OLij
where pj = The probability of state j (Sj)
OLij = The opportunity loss if choose Ai and Sj occurs
min(EOLi) = EVPI
Decision Analysis 22
Expected Opportunity LossExample: Newsboy Problem
State of Nature (Demand)
Alternative(Order) 0 1 2 3 EOL
0 0 85 170 255 144.5
1 40 0 85 170 72
2* 80 40 0 85 37*
3 120 80 40 0 52
Probability 0.1 0.3 0.4 0.2
EVPI
Decision Analysis 23
Decision Making with Utilities Problem with Monetary Payoffs
• People do not always just look at the highest expected monetary return to make decisions; they often evaluate the risk
• Example: A company wants to decide to develop a new product or not
Success Fail ER
Develop $1,000,000 -$400,000 $20,000
Don't develop 0 0 0
Probability 0.3 0.7
Decision Analysis 24
Decision Making with Utilities Utility -- combines monetary return with people’s
attitude toward risk Utility Function -- a mathematical function that
transforms monetary values into utility values• Three general types of utility functions
0 MV
Utility
0 MV
Utility
0 MV
Utility
(1) Risk-Averse (2) Risk-Neutral (3) Risk-Seeking
Decision Analysis 25
Risk-Averse Utility FunctionUtility
Properties of Risk-averse Utility Function • non-decreasing: more money is always better
• concave: utility increase for unit ($100, e.g.) increase of money is decreasing (extra money is less attractive)
100 200 300 400 500 Dollars0
0.524
0.680
0.7750.8500.910
Decision Analysis 26
How to Create Utility Function Method I. Equivalent Lottery
Start with two endpoints A (the worst possible payoff) and B (the best possible payoff) and assign U(A) = 0 and U(B) = 1
Then to find the utility for a possible payoff z between A and B, select the probability p (=U(z)) such that you are indifferent between the following two alternatives
– receive a payoff of z for sure
– receive a payoff of B with probability p or a payoff of A with probability 1 - p
Decision Analysis 27
How to Create Utility Function Method II. Exponential Utility Function
rxexU /1)( where x is the monetary value, r>0 is an adjustable
parameter called risk tolerance First, the value of r can be estimated such that we are
indifferent between the following choices a payoff of zero a payoff of r dollars or a loss of r/2 dollars with 50-50 chance
Then the utility for a particular monetary value x can be found using the above assumed exponential utility function
Decision Analysis 28
Expected UtilityExample: Newsboy Problem
State of Nature (Demand)
Alternative(Order) 0 1 2 3 EU
0 0 -0.65 -1.72 -3.48 -1.58
1 -0.49 0.30 -0.16 -0.92 -0.21
2* -1.23 -0.05 0.50 0.18 0.10*
3 -2.32 -0.57 0.26 0.65 -0.17
Probability 0.1 0.3 0.4 0.2