Decision analysis decision trees
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Decision Analysis (Decision Trees ). Y. İlker TOPCU , Ph .D. www.ilkertopcu. net www. ilkertopcu .org www. ilkertopcu . info www. facebook .com/ yitopcu twitter .com/ yitopcu. Decision Trees. A decision tree is a diagram consisting of decision nodes (squares)

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Decision Analysis (Decision Trees )

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Decision analysis decision trees

Decision Analysis(Decision Trees)

Y. İlker TOPCU, Ph.D.

www.ilkertopcu.net www.ilkertopcu.org www.ilkertopcu.info

www.facebook.com/yitopcu

twitter.com/yitopcu


Decision trees

Decision Trees

  • A decision tree is a diagram consisting of

    • decision nodes (squares)

    • chance nodes (circles)

    • decision branches (alternatives)

    • chance branches (state of natures)

    • terminal nodes (payoffsorutilities)


Representing decision table as decision tree

Representing decision table as decision tree

q1

x11

a1

qn

x1n

a2

am

q1

xm1

qn

xmn


Decision tree method

Decision Tree Method

  • Define the problem

  • Structure / draw the decision tree

  • Assign probabilities to the states of nature

  • Calculate expected payoff (or utility) for the corresponding chance node – backward, computation

  • Assign expected payoff (or utility) for the corresponding decision node – backward, comparison

  • Represent the recommendation


Example 1

Example 1

A chancenode

Favorable market(0.6)

$200,000

1

Unfav. market(0.4)

Constructlargeplant

-$180,000

A decisonnode

Favorable market(0.6)

$100,000

Constructsmallplant

2

Unfav. market

(0.4)

-$20,000

Do nothing

$0


Decision analysis decision trees

A chancenode

Favorable market(0.6)

$200,000

1

Unfav. market(0.4)

EV =

$48,000

Constructlargeplant

-$180,000

A decisonnode

Favorable market(0.6)

$100,000

Constructsmallplant

2

Unfav. market

(0.4)

EV =

$52,000

-$20,000

Do nothing

$0


Example 2

Example 2

184

220

130

%60

%60

%60

186

210

%40

%40

%40

150

170

162

150


Sequential decision tree

Sequential Decision Tree

  • A sequential decision tree is used to illustrate a situation requiring a series of decisions (multi-stage decision making) and it is used where a payoff matrix (limited to a single-stage decision) cannot be used


Example 3

Example 3

  • Let’s say that DM has two decisions to make, with the second decision dependent on the outcome of the first.

  • Before deciding about building a new plant, DM has the option of conducting his own marketing research survey, at a cost of $10,000.

  • The information from his survey could help him decide whether to construct a large plant, a small plant, or not to build at all.


Decision analysis decision trees

  • Before survey, DM believes that the probability of a favorable market is exactly the same as the probability of an unfavorable market: each state of nature has a 50% probability

  • There is a 45% chance that the survey results will indicate a favorable market

  • Such a market survey will not provide DM with perfect information, but it may help quite a bit nevertheless by conditional (posterior) probabilities:

    • 78% is the probability of a favorable market given a favorable result from the market survey

    • 27% is the probability of a favorable market given a negative result from the market survey


Example

Example


Example1

Example


Example 4

Example 4

  • A manager has to decide whether to market a new product nationally and whether to test market the product prior to the national campaign.

  • The costs of test marketing and national campaign are respectively $20,000 and $100,000.

  • Their payoffs are respectively $40,000 and $400,000.

  • A priori, the probability of the new product's success is 50%.

  • If the test market succeeds, the probability of the national campaign's success is improved to 80%.

  • If the test marketing fails, the success probability of the national campaign decreases to 10%.


Decision analysis decision trees

[240]

S(.8)

320

C

[240]

F(.2)

~C

S(.5)

-80

20

[110]

S(.1)

[-80]

280

F(.5)

T

[-20]

F(.9)

C

[110]

-120

~C

~T

-20

[100]

[100]

C

S(.5)

300

~C

F(.5)

-100

0


Expected value of sample information

Expected Value of Sample Information

EVSI

= EV of best decision withsample information, assuming no cost to gather it

– EV of best decision without sample information

= EV with sample info. + cost – EV without sample info.

DM could pay up to EVSI for a survey.

If the cost of the survey is less than EVSI, it is indeed worthwhile.

In the example:

EVSI = $49,200 + $10,000 – $40,000 = $19,200


Estimating probability values by bayesian analysis

Estimating Probability Values by Bayesian Analysis

Bayes Theorem

Posterior

probabilities

Prior

probabilities

New data

  • Management experience or intuition

  • History

  • Existing data

  • Need to be able to reviseprobabilities based upon new data


Bayesian analysis

Bayesian Analysis

Example:

  • Market research specialists have told DM that, statistically, of all new products with a favorable market, market surveys were positive and predicted success correctly 70% of the time.

  • 30% of the time the surveys falsely predicted negative result

  • On the other hand, when there was actually an unfavorable market for a new product, 80% of the surveys correctly predicted the negative results.

  • The surveys incorrectly predicted positive results the remaining 20% of the time.


Market survey reliability

Market Survey Reliability

Actual States of Nature

Result of Survey

Favorable

Unfavorable

Market (FM)

Market (UM)

(survey positive|FM)

(survey positive|UM)

Positive (predicts

P

P

= 0.70

=

0.20

favorable market

for product)

(survey

(survey negative|UM)

Negative (predicts

P

P

negative|FM) = 0.30

= 0.80

unfavorable

market for

product)


Calculating posterior probabilities

Calculating Posterior Probabilities

P(BA) P(A)

P(AB) =

P(BA) P(A) + P(BA’) P(A’)

where A and B are any two events, A’ is the complement of A

P(FMsurvey positive) =

[P(survey positiveFM)P(FM)] /

[P(survey positiveFM)P(FM) + P(survey positiveUM)P(UM)]

P(UMsurvey positive) =

[P(survey positiveUM)P(UM)] /

[P(survey positiveFM)P(FM) + P(survey positiveUM)P(UM)]


Decision analysis decision trees

Probability Revisions Given a Positive Survey

Conditional

Probability

State

P(Survey positive|State of Nature

Prior Probability

Joint Probability

Posterior Probability

of

Nature

0.35

= 0.78

FM

0.70

* 0.50

0.35

0.45

0.10

= 0.22

0.20

0.10

* 0.50

UM

0.45

1.00

0.45


Decision analysis decision trees

Probability Revisions Given a Negative Survey

Conditional

Probability

State

P(Survey

Prior Probability

Joint Probability

Posterior Probability

of

negative|State

Nature

of Nature)

0.15

= 0.27

0.15

0.30

* 0.50

FM

0.55

0.40

= 0.73

0.40

UM

0.80

* 0.50

0.55

0.55

1.00


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