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



Maximax Criterion



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*


Maximin Criterion



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


Minimax Regret Criterion: Step 1



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


Minimax Regret: Step 2 (Regret or Opportunity Loss Table)



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 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 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


Expected Return & VarianceExample: Newsboy Problem


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 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


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)


Calculate EVPIExample: Newsboy Problem


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


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


Expected Opportunity LossExample: Newsboy Problem


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


EVPI


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 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


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


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


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


Expected UtilityExample: Newsboy Problem


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


Decision Analysis1 DSC 3120 Generalized Modeling Techniques with Applications Part III. Decision...

Documents

Transcript of Decision Analysis1 DSC 3120 Generalized Modeling Techniques with Applications Part III. Decision...