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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).
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Decision Analysis-Decision TreesGeneral Form of a Decision Tree
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Decision Analysis-Decision Trees
• 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
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Decision Analysis-Decision Trees
• 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
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Decision Analysis-Decision TreesCase Problem - (A) p. 38 (continued)
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Decision Analysis-Decision Trees
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Decision Analysis-Decision Trees
• 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, .....)
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Decision Analysis-Decision Trees
• 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
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Decision Analysis-Decision Trees
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Decision Analysis-Decision Trees
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
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Decision Analysis-Decision Trees
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.
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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
Purchase PT Services
0.5Demand = 90
Campaign
0.5Demand = 140
Salaried PT
0.5Demand = 150
0.72Campaign is a Success
10.5
Demand = 140
Purchase PT Services
0.5Demand = 150
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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
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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
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Decision Analysis - Treeplan
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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 t y 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
23015
STCBp
p S S 1 3 02 3 0
p C R 9 52 3 0
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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
STpSTCRp
STCRp
41.0
37.0
CRpSTCRp
CRSTp
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
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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
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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
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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
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Decision AnalysisNo 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
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
10Use shape info 0 10
0 3.85 0.59Draw blue
10Predict blue 0 10
0 3.85 0.41Draw red
0.3 -5Draw triangle 0 -5
10 3.85 0.59
Draw blue-5
Predict red 0 -5
0 1.15 0.41Draw red
100 10
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Decision AnalysisImperfect 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
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
10Use shape info 0 10
0 8.3485 0.15Draw blue
10Predict blue 0 10
0 -2.75 0.85Draw red
0.43 -5Draw triangle 0 -5
20 7.75 0.15
Draw blue-5
Predict red 0 -5
0 7.75 0.85Draw red
100 10
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Decision Analysis Bayes Theorem
etc.
then
shown that becan it y,similar wa aIn
CRpCRSSpCBpCBSSpCBpCBSSp
SSpCBSSpSSCBp
CRpCRSTpCBpCBSTpSTp
CRpCRSSpCBpCBSSpSSp
CRpCRSSpCRSSpCRpCRSSpCRSSp
CBpCBSSpCBSSpCBpCBSSpCBSSp
CRSSpCBSSpSSp
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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’
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Decision Analysis-Decision TreesModified Case Problem - Imperfect Information
Demand = 30
Demand = 90
Salaried PT
Demand = 140
Demand = 150
No Campaign1
Demand = 30
Demand = 90
Purchase PT Services
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
Campaign is a failure
Demand = 90
Purchase PT Services
Demand = 140
Campaign is a success
Demand = 150
Campaign
Demand = 30
Campaign is a failure
Demand = 90
Salaried PT
Demand = 140
Campaign is a success
Demand = 150
Report says "Success"1
Demand = 30
Campaign is a failure
Demand = 90
Purchase PT Services
Demand = 140
Campaign is a success
Demand = 150
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Decision Analysis-Decision TreesModified Case Problem - Imperfect Information
RFpFpFRFp
RFFpRSp
FpFRSpRSFp
RFpSpSRFp
RFSpRSp
SpSRSpRSSp
FpFRFpSpSRFpRFp
FpFRSpSpSRSpRSp
FRFp
FRSp
FpSRFp
SpSRSp
75.0
25.0
28.0 15.0
72.0 85.0
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Decision Analysis-Decision TreesModified Case Problem - Imperfect Information
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”
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Decision Analysis-Decision Trees
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
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Modified Case Problem - Imperfect Information
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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
1,500Purchase PT Services 1,800 1,500
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
0.1026 -2,660Campaign is a failure 2,040 -2,660
0 -620 0.5Demand = 90
1,420Salaried PT 6,120 1,420
-4,400 4,567 0.5Demand = 140
0.8974 4,820Campaign is a success 9,520 4,820
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
Campaign is a failure 600 300
0 900 0.5Demand = 90
1,500Purchase PT Services 1,800 1,500
0 2,426 0.5Demand = 140
0.8974 2,500Campaign is a success 2,800 2,500
0 2,600 0.5Demand = 150
2,7003,000 2,700
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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”
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Decision Analysis-Decision TreesImperfect Information-Sensitivity Analysis
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
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Decision Analysis-Decision TreesImperfect Information-Sensitivity Analysis
0.5Demand = 30
0.7677 -2,660Campaign is a failure 2,040 -2,660
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
Campaign is a failure 600 300
0 900 0.5Demand = 90
1,500Purchase PT Services 1,800 1,500
0 1,295 0.5Demand = 140
0.2323 2,500Campaign is a success 2,800 2,500
0 2,600 0.5Demand = 150
2,700Campaign 3,000 2,700
-300 3,719 0.5Demand = 30
0.0609 -2,660Campaign is a failure 2,040 -2,660
0 -620 0.5Demand = 90
1,420Salaried PT 6,120 1,420
-4,400 4,808 0.5Demand = 140
0.9391 4,820Campaign is a success 9,520 4,820
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
Campaign is a failure 600 300
0 900 0.5Demand = 90
1,500Purchase PT Services 1,800 1,500
0 2,496 0.5Demand = 140
0.9391 2,500Campaign is a success 2,800 2,500
0 2,600 0.5Demand = 150
2,7003,000 2,700
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