decision analysis-decision trees

-1- HMP654

Decision Analysis-Decision Trees

• A decision tree is a graphical representation of every possible sequence of decision and random outcomes (states of nature) that can occur within a given decision making problem.

• A decision tree is composed of a collection of nodes (represented by circles and squares) interconnected by branches (represented by lines).

-2- HMP654

Decision Analysis-Decision TreesGeneral Form of a Decision Tree

-3- HMP654


• A square node is called a decision node because it represents a decision. Branches emanating from a decision node represent the different alternatives for a particular decision.

Alternative A

Alternative B

Alternative C

Decision Node

-4- HMP654


• A circular node in a decision tree is called an event node because it represents an uncertain event. The branches emanating from an event node correspond to the possible states of nature or the possible outcomes of an uncertain event.

State of Nature 1

State of Nature 2

State of Nature 3

Event Node

-5- HMP654


Case Problem - (A) p. 38 (continued)

-6- HMP654


-7- HMP654


• In a maximization problem, the value assigned to a decision node is the maximum of the values of the adjacent nodes.

Evaluation of Nodes

V1

V2

V3

V4

V4 = MAX(V1, V2, V3, .....)

-8- HMP654


• The value assigned to an event node is the expectation of the values that correspond to adjacent nodes.

Evaluation of Nodes

V1

V2

V3

V4

p1

p2

p3

V4 = V1 x p1 + V2 x p2 + V3 x p3

-9- HMP654


-10- HMP654


The agency has been in operation for more than a year and is now reassessing itsperformance and staffing. In reviewing demand data compiled in its information system,the agency learns that monthly demand has actually been slightly different from what hadoriginally been anticipated. In fact, the agency now feels that monthly demand is morerealistically modeled by the following probability distribution:

Monthly Demand Probability30 0.1090 0.27140 0.33150 0.30

The home health agency now has several things to consider as it plans how it willprovide physical therapy services for its clients in the coming year. First of all, a newindependent contractor has approached the agency offering to provide PT services for aflat rate of $55 per visit. No fringe benefits or other costs would be incurred.

In addition, this contractor has also developed a new marketing program that it hassuccessfully applied in a number of other cities. This program consists of an intensivemonth-long campaign to recruit additional clients followed by a brief market researchstudy to determine the success of the effort. The agency has the option of purchasing thismarketing program whether or not it hires the contractor to provide PT services.

The agency has surveyed a number of organizations that have utilized thismarketing program. The results of this survey indicate that the contractor has a 72percent success rate in increasing demand for PT services. However, in the remaining 28percent of the cases there was actually a decrease in demand for PT services because ofthe negative reaction by potential clients to the contractor's hard-sell marketingapproached.

The home health agency carefully analyzes the results of this quick but methodicalsurvey and derives two additional probability distributions for demand for PT services-one that is expected to hold if the marketing campaign to recruit additional clients issuccessful, and a second distribution applicable if the marketing campaign is a failure. Thedistribution of demand created by a successful marketing campaign is given below:

Monthly Demand Probability140 0.5150 0.5

On the other hand, when the marketing campaign is not successful, the demand isexpected to be described by the following distribution:

Monthly Demand Probability30 0.590 0.5

The home health agency now has several decisions to make. First of all, it mustdecide whether to negotiate with the new independent contractor to perform the

Case Problem (A) p. 64

-11- HMP654


marketing campaign and follow-up market research study. the cost of this program is$300 per month (for the 12-month planning period currently under study).

If the home health agency does decide to contract for the marketing program, thenit will receive a marketing research report indicating whether the marketing program wasa success or a failure. The agency must decide for each reported outcome whether it willcontinue utilizing its salaried PT or utilize the contractor to provide the PT services. Thecosts associated with these two options are the same as those outlined above. In all cases,the average payment for a PT home visit is $75 per visit, and the agency is trying tomaximize expected net profit.

The home health agency realizes that the optimum approach is dependent upon thecost of the marketing program (currently set at $300 per month), so another objective is toinvestigate the sensitivity of the solution to this cost. Upon realizing that they mustperform a multistage decision analysis, the agency staff turns their attention to the detailsof constructing an appropriate decision tree model.

-12- HMP654

Decision Analysis-Decision Trees0.1

Demand = 30

0.27Demand = 90

Salaried PT

0.33Demand = 140

0.3Demand = 150

No Campaign1

0.1Demand = 30

0.27Demand = 90

Purchase PT Services

0.33Demand = 140

0.3Demand = 150

10.5

Demand = 30

Salaried PT

0.5Demand = 90

0.28Campaign is a Failure

10.5

Demand = 30


0.5Demand = 90

Campaign

0.5Demand = 140

Salaried PT

0.5Demand = 150

0.72Campaign is a Success

10.5

Demand = 140


0.5Demand = 150

-13- HMP654

Decision Analysis-Decision Trees0.1

Demand = 30-2360

2040 -2360

0.27Demand = 90

1720Salaried PT 6120 1720

-4400 3658 0.33Demand = 140

51209520 5120

0.3Demand = 150

5800No Campaign 10200 5800

10 3658 0.1

Demand = 30600

600 600

0.27Demand = 90

1800Purchase PT Services 1800 1800

0 2370 0.33Demand = 140

28002800 2800

0.3Demand = 150

30003000 3000

23967.2 0.5

Demand = 30-2660

Salaried PT 2040 -2660

-4400 -620 0.5Demand = 90

0.28 1420Campaign is a Failure 6120 1420

20 900 0.5

Demand = 30300

Purchase PT Services 600 300

0 900 0.5Demand = 90

1500Campaign 1800 1500

-300 3967.2 0.5Demand = 140

4820Salaried PT 9520 4820

-4400 5160 0.5Demand = 150

0.72 5500Campaign is a Success 10200 5500

10 5160 0.5

Demand = 1402500

Purchase PT Services 2800 2500

0 2600 0.5Demand = 150

27003000 2700

-14- HMP654

Decision Analysis - TreeplanCtrl-t activates Treeplan

Decision 10

0 0

1 0.50 Event 3

0Decision 2 0 0

0 0 0.5Event 4

00 0

-15- HMP654

Decision Analysis - Treeplan

-16- HMP654

Decision Analysis - Probability

F re q u e n c y T a b le

S S S TC B 1 2 0 1 5 1 3 5C R 1 0 8 5 9 5

1 3 0 1 0 0 2 3 0

J o in t P ro b a b il i ty D is t r ib u t io n

S S S TC B 0 .5 2 0 .0 7 0 .5 9C R 0 .0 4 0 .3 7 0 .4 1

0 .5 7 0 .4 3 1

230

15STCBp

p S S 1 3 02 3 0

p C R 9 52 3 0

-17- HMP654

Decision Analysis Conditional Probability

C o n d i t i o n a l P r o b a b i l i t i e s C o l o r g i v e n S h a p e

S S S TC B 0 . 9 2 0 . 1 5C R 0 . 0 8 0 . 8 5

1 1

S h a p e g i v e n c o l o r

S S S TC B 0 . 8 9 0 . 1 1 1C R 0 . 1 1 0 . 8 9 1

43.0

37.0

STp

STCRpSTCRp

41.0

37.0

CRp

STCRpCRSTp

C o m p a r e t o p ( C R ) = 0 . 4 1

C o m p a r e t o p ( S T ) = 0 . 4 3

-18- HMP654

Decision AnalysisPerfect Information

Perfect Information

Frequency Table Joint Probability Distribution

SS ST SS STCB 135 0 135 CB 0.59 0.00 0.59CR 0 95 95 CR 0.00 0.41 0.41

135 95 230 0.59 0.41 1

Color given Shape Shape given Color

SS ST SS STCB 1 0 CB 1 0CR 0 1 CR 0 1

-19- HMP654

Decision AnalysisNo Information

No Information

Frequency Table Joint Probability Distribution

SS ST SS STCB 413 177 590 CB 0.41 0.18 0.59CR 287 123 410 CR 0.29 0.12 0.41

700 300 1000 0.70 0.30 1

Color given Shape Shape given Color

SS ST SS STCB 0.59 0.59 CB 0.7 0.3CR 0.41 0.41 CR 0.7 0.3

-20- HMP654

Decision AnalysisPerfect Information

0.59Draw blue

10Predict blue 0 10

0 3.85 0.41Draw red

-5Don't use shape info 0 -5

10 3.85 0.59

Draw blue-5

Predict red 0 -5

0 1.15 0.41Draw red

100 10

1Draw blue

10Predict blue 0 10

210 0 10 0

Draw red0.59 -5

Draw square 0 -51

0 10 1Draw blue

-5Predict red 0 -5

0 -5 0Draw red

10Use shape info 0 10

0 10 0Draw blue

10Predict blue 0 10

0 -5 1Draw red

0.41 -5Draw triangle 0 -5

20 10 0

Draw blue-5

Predict red 0 -5

0 10 1Draw red

100 10

-21- HMP654

Decision AnalysisNo Information

0.59Draw blue

10Predict blue 0 10

0 3.85 0.41Draw red


10 3.85 0.59

Draw blue-5

Predict red 0 -5

0 1.15 0.41Draw red

100 10

0.59Draw blue

10Predict blue 0 10

13.85 0 3.85 0.41

Draw red0.7 -5

Draw square 0 -51

0 3.85 0.59Draw blue

-5Predict red 0 -5

0 1.15 0.41Draw red


0 3.85 0.59Draw blue

10Predict blue 0 10

0 3.85 0.41Draw red


10 3.85 0.59

Draw blue-5

Predict red 0 -5

0 1.15 0.41Draw red

100 10

-22- HMP654

Decision AnalysisImperfect Information

0.59Draw blue

10Predict blue 0 10

0 3.85 0.41Draw red


10 3.85 0.59

Draw blue-5

Predict red 0 -5

0 1.15 0.41Draw red

100 10

0.92Draw blue

10Predict blue 0 10

28.3485 0 8.8 0.08

Draw red0.57 -5

Draw square 0 -51

0 8.8 0.92Draw blue

-5Predict red 0 -5

0 -3.8 0.08Draw red


0 8.3485 0.15Draw blue

10Predict blue 0 10

0 -2.75 0.85Draw red


20 7.75 0.15

Draw blue-5

Predict red 0 -5

0 7.75 0.85Draw red

100 10

-23- HMP654

Decision Analysis Bayes Theorem

etc.

then

shown that becan it y,similar wa aIn

CRpCRSSpCBpCBSSp

CBpCBSSp

SSp

CBSSpSSCBp

CRpCRSTpCBpCBSTpSTp

CRpCRSSpCBpCBSSpSSp

CRpCRSSpCRSSpCRp

CRSSpCRSSp

CBpCBSSpCBSSpCBp

CBSSpCBSSp

CRSSpCBSSpSSp

-24- HMP654

Decision Analysis-Decision TreesModified Case Problem - Imperfect Information

• Assume that it is possible for the market research report to be wrong. Thus, the content of the report does not provide the decision maker with certain knowledge about the true outcome of the campaign.

Outcome ofMarketingResearch Report

Result is really asuccess (S)

Result is really afailure (F)

Report says“success” (RS)

0.85 0.25

Report says“failure” (RF)

0.15 0.75

Conditional probabilities of ‘report outcomes’ given ‘actual outcomes’

-25- HMP654


Demand = 30

Demand = 90

Salaried PT

Demand = 140

Demand = 150

No Campaign1

Demand = 30

Demand = 90


Demand = 140

Demand = 150

Demand = 30

Campaign is a failure

Demand = 90

Salaried PT1

Demand = 140

Campaign is a success

Demand = 150

Report says "Failure"1

Demand = 30


Demand = 90


Demand = 140


Demand = 150

Campaign

Demand = 30


Demand = 90

Salaried PT

Demand = 140


Demand = 150

Report says "Success"1

Demand = 30


Demand = 90


Demand = 140


Demand = 150

-26- HMP654


RFp

FpFRFpRFFp

RSp

FpFRSpRSFp

RFp

SpSRFpRFSp

RSp

SpSRSpRSSp

FpFRFpSpSRFpRFp

FpFRSpSpSRSpRSp

FRFp

FRSp

FpSRFp

SpSRSp

75.0

25.0

28.0 15.0

72.0 85.0

-27- HMP654


0.85 0.25

0.15 0.75

S F

RS

RF

0.72 0.28

0.318

0.682

Probabilities of “report outcome” given “actual outcome”

p(S) p(F)

p(RS)

p(RF)

0.8974 0.1026

0.3396 0.6604

S F

RS

RF

Probabilities of “actual outcome” given “report outcome”

-28- HMP654


0.1Demand = 30

-2,3602,040 -2,360

0.27Demand = 90

1,720Salaried PT 6,120 1,720

-4,400 3,658 0.33Demand = 140

5,1209,520 5,120

0.3Demand = 150

5,800No Campaign 10,200 5,800

10 3,658 0.1

Demand = 30600

600 600

0.27Demand = 90

1,800Purchase PT Services 1,800 1,800

0 2,370 0.33Demand = 140

2,8002,800 2,800

0.3Demand = 150

3,0003,000 3,000

Next Page

Modified Case Problem - Imperfect Information

-29- HMP654

Decision Analysis-Decision TreesModified Case Problem- Imperfect Information

0.5Demand = 30

0.6604 -2,660Campaign is a failure 2,040 -2,660

0 -620 0.5Demand = 90

1,420Salaried PT 6,120 1,420

13,658 -4,400 1,343 0.5

Demand = 1400.3396 4,820

Campaign is a success 9,520 4,820

0 5,160 0.5Demand = 150

0.318 5,500Report says "Failure" 10,200 5,500

20 1,477 0.5

Demand = 300.6604 300

Campaign is a failure 600 300

0 900 0.5Demand = 90


0 1,477 0.5Demand = 140

0.3396 2,500Campaign is a success 2,800 2,500

0 2,600 0.5Demand = 150

2,700Campaign 3,000 2,700

-300 3,584 0.5Demand = 30


0 -620 0.5Demand = 90

1,420Salaried PT 6,120 1,420

-4,400 4,567 0.5Demand = 140


0 5,160 0.5Demand = 150

0.682 5,500Report says "Success" 10,200 5,500

10 4,567 0.5

Demand = 300.1026 300


0 900 0.5Demand = 90


0 2,426 0.5Demand = 140


0 2,600 0.5Demand = 150

2,7003,000 2,700

Previous Page

-30- HMP654

Decision Analysis-Decision TreesImperfect Information-Sensitivity Analysis

0.90 0.15

0.10 0.85

S F

RS

RF

0.72 0.28

0.31

0.69

Probabilities of “report outcome” given “actual outcome”

p(S) p(F)

p(RS)

p(RF)

0.9391 0.0609

0.2323 0.7677

S F

RS

RF

Probabilities of “actual outcome” given “report outcome”

-31- HMP654


0.1Demand = 30

-2,3602,040 -2,360

0.27Demand = 90

1,720Salaried PT 6,120 1,720

-4,400 3,658 0.33Demand = 140

5,1209,520 5,120

0.3Demand = 150

5,800No Campaign 10,200 5,800

10 3,658 0.1

Demand = 30600

600 600

0.27Demand = 90


0 2,370 0.33Demand = 140

2,8002,800 2,800

0.3Demand = 150

3,0003,000 3,000

Next Page

-32- HMP654


0.5Demand = 30


0 -620 0.5Demand = 90

1,420Salaried PT 6,120 1,420

23,719 -4,400 723 0.5

Demand = 1400.2323 4,820

Campaign is a success 9,520 4,820

0 5,160 0.5Demand = 150

0.31 5,500Report says "Failure" 10,200 5,500

20 1,295 0.5

Demand = 300.7677 300


0 900 0.5Demand = 90


0 1,295 0.5Demand = 140


0 2,600 0.5Demand = 150

2,700Campaign 3,000 2,700

-300 3,719 0.5Demand = 30


0 -620 0.5Demand = 90

1,420Salaried PT 6,120 1,420

-4,400 4,808 0.5Demand = 140


0 5,160 0.5Demand = 150

0.69 5,500Report says "Success" 10,200 5,500

10 4,808 0.5

Demand = 300.0609 300


0 900 0.5Demand = 90


0 2,496 0.5Demand = 140


0 2,600 0.5Demand = 150

2,7003,000 2,700

Previous Page

decision analysis-decision trees

Documents