Decision Analysis.lecfinal

8/3/2019 Decision Analysis.lecfinal

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*Problem Formulation

*Decision Making without Probabilities

*Decision Making with Probabilities*Risk Analysis and Sensitivity Analysis

*Decision Analysis with Sample Information

*Computing Branch Probabilities

*Utility and Decision Making


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*A decision problem is characterized by decisionalternatives, states of nature, and resulting payoffs.

*The decision alternatives are the different possiblestrategies the decision maker can employ.

*The states of nature refer to future events, not underthe control of the decision maker, which willultimately affect decision results. States of natureshould be defined so that they are mutually exclusiveand contain all possible future events that couldaffect the results of all potential decisions.


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*Decision theory problems are

generally represented as one ofthe following:

*Influence Diagram

* Payoff Table

*Decision Tree


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*An influence diagram is a graphical deviceshowing the relationships among the

decisions, the chance events, and theconsequences.

*Squares or rectangles depict decision nodes.

*Circles or ovals depict chance nodes.*Diamonds depict consequence nodes.

*Lines or arcs connecting the nodes show thedirection of influence.


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*The consequence resulting from a specific

combination of a decision alternative and astate of nature is a payoff.

*A table showing payoffs for all combinationsof decision alternatives and states of nature isa payoff table.

*Payoffs can be expressed in terms of profit,cost, time, distance or any other appropriate

measure.


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**

A decision tree is a chronological representation ofthe decision problem.

*Each decision tree has two types of nodes; roundnodes correspond to the states of nature while

square nodes correspond to the decisionalternatives.

*The branches leaving each round node represent thedifferent states of nature while the branches leavingeach square node represent the different decision

alternatives.

*At the end of each limb of a tree are the payoffsattained from the series of branches making up thatlimb.


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*

A developer must decide how large a

luxury condominium complex to build small, medium, or large. Theprofitability of this complex depends

upon the future level of demand forthe complexs condominiums.


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* States of nature: The states of nature could

be defined as low demand and high demand.* Alternatives: CAL could decide to build asmall, medium, or large condominium complex.

* Payoffs: The profit for each alternativeunder each potential state of nature is going tobe determined.


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Alternatives Low HighSmall 8 8Medium 5 15

Large -11 22

States of Nature

(payoffs in millions of dollars)

THIS IS A PROFIT PAYOFF TABLE


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Medium Complex

8

8

5

15

22

-11


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*Three commonly used criteria fordecision making when probability

information regarding the likelihood ofthe states of nature is unavailable are:

*the optimistic approach

*the conservative approach*the minimax regret approach.


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**The optimistic approach would be used by anoptimistic decision maker.

*The decision with the best possible payoff is

chosen.*If the payoff table was in terms of costs, thedecision with the lowest cost would bechosen.

*If the payoff table was in terms of profits,the decision with the highest profit would bechosen.


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**For each decision the worst payoff is listed and thenthe decision corresponding to the best of these worstpayoffs is selected. (Hence, the worst possiblepayoff is maximized.)

*If the payoff was in terms of costs, the maximumcosts would be determined for each decision andthen the decision corresponding to the minimum ofthese maximum costs is selected. (Hence, the

maximum possible cost is minimized.)*If the payoff was in terms of profits, the minimumprofits would be determined for each decision andthen the decision corresponding to the maximum ofthese minimum profits is selected. (Hence, the

minimum possible profit is maximized.)


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1. The minimax regret approach requires theconstruction of a regret table or anopportunity loss table. This is done by

calculating for each state of nature thedifference between each payoff and the bestpayoff for that state of nature.

2. Then, using this regret table, the maximum

regret for each possible decision is listed.3. The decision chosen is the onecorresponding to the minimum of themaximum regrets.


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*If the optimistic approach is selected:

STATES OF NATUREBEST

Alternatives Low HighPROFIT

Small 8 8 8

Medium 5 15 15

Large -11 22 22

MaximaxMaximaxpayoffpayoff

MaximaxMaximaxdecisiondecision


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*If the conservative approach is selected:

STATES OF NATUREWORST

Alternatives Low HighPROFIT

Small 8 8 8

Medium 5 15 5

Large -11 22 -11

MaximinMaximinpayoffpayoff

MaximinMaximindecisiondecision

The decision with the best profit from the column of worst profits is selected.


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*If the minimax regret approach is selected:

Step 1: Determine the best payoff for each state ofnature and create a regret table.

STATES OF NATURE

Alternatives Low High

Small

8 8Medium 5 15

Large -11 22

Best Profit

for Low8

Best Profit

for High22


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**If the minimax regret approach is selected:

Step 1: Create a regret table (continued).STATES OF NATURE

Alternatives Low High

Small 0 14

Medium 3 7Large 19 0

For a profit payoff

table,entries in

the regret tablerepresent profits

that could have

been earned.

If they knew in advanced that the demand would be low, they would have built a

small complex. Without this psychic insight, if they decided to build a medium

facility and the demand turned out to be low, they would regret building a medium

complex because they only made 5 million dollars instead of 8 million had they built

a small facility instead. They regret their decision by 3 million dollars.


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Step 2: Create a regret table (continued).Step 3: Determine the maximum regret for eachdecision.

STATES OF NATUREMax

Alternatives Low HighRegret

Small 0 14 14

Medium 3 7 7

Large 19 0 19Regret not getting a profit

of 19 more than not making

a profit of 0.


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Step 4: Select the decision with the minimum valuefrom the column of max regrets.

STATES OF NATUREMax

Alternatives Low HighRegret

Small 0 14 14

Medium 3 7 7

Large 19 0 19

MinimaxMinimaxRegretRegretpayoffpayoff

MinimaxMinimaxRegretRegret

decisiondecision


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*Expected Value Approach

*If probabilistic information regarding the states of

nature is available, one may use the expected value

(EV) approach.*Here the expected return for each decision is

calculated by summing the products of the payoff

under each state of nature and the probability of the

respective state of nature occurring.*The decision yielding the best expected return is

chosen.


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*The expected value of a decision alternative is the sum ofweighted payoffs for the decision alternative.

*The expected value (EV) of decision alternative di isdefined as:

where: N= the number of states of nature

P(sj ) = the probability of state of nature sjVij = the payoff corresponding to decision

alternative di and state of nature sj

EV( ) ( )d P s V i j ijj

N!

!

1

EV( ) ( )d P s V i j ijj

N!

!

1


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Burger Prince Restaurant is contemplating opening anew restaurant on Main Street. It has three differentmodels, each with a different seating capacity. BurgerPrince estimates that the average number of customers

per hour will be 80, 100, or 120. The payoff table(profits) for the three models is on the next slide.


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*Payoff Table

Average Number of Customers Per Hour

s1 = 80 s2 =100 s3 =

120

Model A $10,000 $15,000 $14,000

Model B $ 8,000 $18,000 $12,000

Model C $ 6,000 $16,000 $21,000


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*Expected Value Approach

Calculate the expected value foreach decision.

Here d1, d2, d3 represent the decisionalternatives of models A, B, C, and s1,

s2, s3 represent the states of nature of80, 100, and 120.

Suppose that optimism leads to an initialsubjective probability assessment of .4,

.2 & .4 respectively.


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*Example: BurgerP

rince

*Decision Tree

1

.2

.4

.4

.4

.2

.4

.4

.2

.4

d1

d2

d3

s1

s1

s1

s2

s3

s2

s2

s3

s3

Payoffs10,000

15,000

14,000

8,000

18,000

12,000

6,00016,000

21,000

2

3

4


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Example: Burger Prince

Expected Value For Each Decision

Choose the model with largest EV, Model C.

3

d1

d2

d3

EMV = .4(10,000) + .2(15,000) + .4(14,000)= $12,600

EMV = .4(8,000) + .2(18,000) + .4(12,000)= $11,600

EMV = .4(6,000) + .2(16,000) + .4(21,000)

= $14,000

Model A

Model B

Model C

2

1

4


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*Suppose market research was

conducted in the community wherethe complex will be built. Thisresearch allowed the company toestimate that the probability of low

demand will be 0.35, and theprobability of high demand will be0.65. Which decision alternativeshould they select.


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STATES OF NATUREAlternatives Low (0.35) High (0.65)

Small 8 8

Medium 5 15Large -11 22


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STATES OF NATUREAlternatives Low High

(0.35) (0.65) Expectedvalue (EV)

Small

8 8 8(0.35) +8(0.65) = 8

Medium 5 15 5(0.35) +15(0.65) = 11.5

Large -11 22 -11(0.35) +22(0.65) = 10.45

Recall that this is a profit payoff table. Thus since the decision to build a medium

complex has the highest expected profit, this is our best decision.


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*Frequently information is available which canimprove the probability estimates for the statesof nature.

*The expected value of perfect information(EVPI) is the increase in the expected profit thatwould result if one knew with certainty which

state of nature would occur.*The EVPI provides an upper bound on theexpected value of any sample or surveyinformation.


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*EVPI Calculation

*Step 1:

Determine the optimal return corresponding to each

state of nature.*Step 2:

Compute the expected value of these optimal

returns.

*Step 3:

Subtract the EV of the optimal decision from the

amount determined in step (2).


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*Expected Value ofPerfect Information

Calculate the expected value for theoptimum payoff for each state of nature andsubtract the EV of the optimal decision.

EVPI= .4(10,000) + .2(18,000) + .4(21,000) -14,000 = $2,000


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*If a small change in the value of one of the

inputs causes a change in therecommended decision alternative, extraeffort and care should be taken inestimating the input value.

*One approach to sensitivity analysis is toarbitrarily assign different values to theprobabilities of the states of nature and/orthe payoffs and resolve the problem. If

the recommended decision changes, thenyou know that the solution is sensitive tothe changes

35


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*For the special case of two states of nature, agraphical technique can be used to determine

how sensitive the solution is to theprobabilities associated with the states ofnature.

*This problem has two states of nature.

Previously, we stated that CAL Condominiumsestimated that the probability of future lowdemand is 0.35 and 0.65 is the probability ofhigh demand. These probabilities yielded the

recommended decision to build the mediumcomplex.


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*In order to see how sensitive thisrecommendation is to changingprobability values, we will let pequal the probability of low demand.

Thus (1-p) is the probability of highdemand. Therefore

EV( small) = 8*p + 8*(1-p)= 8

EV( medium) = 5*p + 15*(1-p) = 15 10p

EV( large) = -11*p + 22*(1-p) = 22 33p


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*Next we will plot the expected value

lines for each decision by plotting pon the x axis and EV on the y axis.

*EV( small) = 8

EV( medium) = 15 10p

EV( large) = 22 33p


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0 0 .2 0 .4 0 .6 0 .8 10

5

1 0

1 5

2 0

2 5

EV( small)


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*Since CAL condominiums list

payoffs are in terms of profits,we know that the highest profitsis desirable.

*Look over the entire range of p(p=0 to p=1) and determine therange over which each decisionyields the highest profits.


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0 0 .2 0 .4 0 .6 0 .8 10

5

1 0

1 5

2 0

2 5

EV( small)

B1 B2


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*Do not estimate the values of B1 or B2 (the points where theintersection of lines occur). Determine the exact intersectionpoints.

*B1 is the point where the EV( large) line intersects with the EV( medium) line:

To find this point set these two lines equal to each other andsolve for p.

22-33p= 15-10p

7= 23p

p=7/23= 0.3403

*B2 is the point where the EV( medium) line intersects with theEV( small) line:

15-10p = 8

7 = 10p

p = 0.7

So B1 equals 0.3403

So B2 equals 0.7


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0 0 .2 0 .4 0 .6 0 .8 10

5

1 0

1 5

2 0

2 5

EV( small)

0.3403 0.7


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*From the graph we see that if the probabilityof low demand (p) is between 0 and 0.3403,

we recommend building a large complex.*From the graph we see that if the probabilityof low demand (p) is between 0.3403 and 0.7,we recommend building a medium complex.

*From the graph we see that if the probabilityof low demand (p) is between 0.7 and 1, werecommend building a large complex.

*


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See your textbook for moreexamples and detailed explanations

of all topics discussed in these notes.

Date post:	06-Apr-2018
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Decision Analysis.lecfinal

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