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).
-73-
HMP654
Decision Analysis-Decision Trees
General Form of a Decision Tree
-74-
HMP654
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
-75-
HMP654
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
-76-
HMP654
Decision Analysis-Decision Trees
Case Problem - (A) p. 38 (continued)
-77-
HMP654
Decision Analysis-Decision Trees
-78-
HMP654
Decision Analysis-Decision Trees
Evaluation of Nodes
V1
V2
V4
V3
V4 = MAX(V1, V2, V3, .....)
• In a maximization problem, the
value assigned to a decision
node is the maximum of the
values of the adjacent nodes.
-79-
HMP654
Decision Analysis-Decision Trees
Evaluation of Nodes
p1
V4
p2
p3
V1
V2
V3
V4 = V1 x p1 + V2 x p2 + V3 x p3
• The value assigned to an event
node is the expectation of the
values that correspond to
adjacent nodes.
-80-
HMP654
Decision Analysis-Decision Trees
-81-
HMP654
Decision Analysis-Decision Trees
Case Problem (A) p. 64
The agency has been in operation for more than a year and is now reassessing its
performance and staffing. In reviewing demand data compiled in its information system,
the agency learns that monthly demand has actually been slightly different from what had
originally been anticipated. In fact, the agency now feels that monthly demand is more
realistically modeled by the following probability distribution:
Monthly Demand
30
90
140
150
Probability
0.10
0.27
0.33
0.30
The home health agency now has several things to consider as it plans how it will
provide physical therapy services for its clients in the coming year. First of all, a new
independent contractor has approached the agency offering to provide PT services for a
flat 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 has
successfully applied in a number of other cities. This program consists of an intensive
month-long campaign to recruit additional clients followed by a brief market research
study to determine the success of the effort. The agency has the option of purchasing this
marketing program whether or not it hires the contractor to provide PT services.
The agency has surveyed a number of organizations that have utilized this
marketing program. The results of this survey indicate that the contractor has a 72
percent success rate in increasing demand for PT services. However, in the remaining 28
percent of the cases there was actually a decrease in demand for PT services because of
the negative reaction by potential clients to the contractor's hard-sell marketing
approached.
The home health agency carefully analyzes the results of this quick but methodical
survey and derives two additional probability distributions for demand for PT servicesone that is expected to hold if the marketing campaign to recruit additional clients is
successful, and a second distribution applicable if the marketing campaign is a failure. The
distribution of demand created by a successful marketing campaign is given below:
Monthly Demand
140
150
Probability
0.5
0.5
On the other hand, when the marketing campaign is not successful, the demand is
expected to be described by the following distribution:
Monthly Demand
30
90
Probability
0.5
0.5
The home health agency now has several decisions to make. First of all, it must
decide whether to negotiate with the new independent contractor to perform the
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HMP654
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, then
it will receive a marketing research report indicating whether the marketing program was
a success or a failure. The agency must decide for each reported outcome whether it will
continue utilizing its salaried PT or utilize the contractor to provide the PT services. The
costs 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 to
maximize expected net profit.
The home health agency realizes that the optimum approach is dependent upon the
cost of the marketing program (currently set at $300 per month), so another objective is to
investigate the sensitivity of the solution to this cost. Upon realizing that they must
perform a multistage decision analysis, the agency staff turns their attention to the details
of constructing an appropriate decision tree model.
-83-
HMP654
Decision Analysis-Decision Trees
0.1
Dem and = 30
0.27
Dem and = 90
Salaried PT
0.33
Dem and = 140
0.3
Dem and = 150
No Cam paign
1
0.1
Dem and = 30
0.27
Dem and = 90
Purchase PT Serv ices
0.33
Dem and = 140
0.3
Dem and = 150
1
0.5
D em and = 30
S alaried P T
0.5
D em and = 90
0.28
C am paign is a F ailure
1
0.5
D em and = 30
P urchase P T S erv ices
0.5
D em and = 90
C am paign
0.5
D em and = 140
S alaried P T
0.5
D em and = 150
0.72
C am paign is a S uccess
1
0.5
D em and = 140
P urchase P T S erv ices
0.5
D em and = 150
-84-
HMP654
Decision Analysis-Decision Trees
0.1
Demand = 30
-2360
2040
-2360
0.27
Demand = 90
1720
6120
Salaried PT
-4400
1720
0.33
Demand = 140
3658
5120
9520
5120
0.3
Demand = 150
5800
10200
No Campaign
5800
1
0
0.1
Demand = 30
3658
600
600
600
0.27
Demand = 90
1800
Purchase PT Services
0
1800
1800
0.33
Demand = 140
2370
2800
2800
2800
0.3
Demand = 150
3000
3000
3000
2
3967.2
0.5
Dem and = 30
-2660
Salaried PT
-4400
2040
-620
0.28
Cam paign is a Failure
-2660
0.5
Dem and = 90
1420
6120
1420
2
0
900
0.5
Dem and = 30
300
Purchase PT Services
0
900
600
300
0.5
Dem and = 90
1500
Cam paign
-300
1800
3967.2
1500
0.5
Dem and = 140
4820
Salaried PT
-4400
9520
5160
0.72
Cam paign is a Success
4820
0.5
Dem and = 150
5500
10200
5500
1
0
5160
0.5
Dem and = 140
2500
Purchase PT Services
0
2600
2800
2500
0.5
Dem and = 150
2700
3000
-85-
2700
HMP654
Decision Analysis - Treeplan
Ctrl-t activates Treeplan
Decision 1
0
0
0
1
0.5
Event 3
0
0
Decision 2
0
0
0
0
0.5
Event 4
0
0
-86-
0
HMP654
Decision Analysis - Treeplan
-87-
HMP654
Decision Analysis - Probability
Frequency Table
CB
CR
SS
120
10
130
ST
15
85
100
135
95
230
Joint Probability Distribution
CB
CR
SS
0.52
0.04
0.57
ST
0.07
0.37
0.43
p CB  ST  
15
230
0.59
0.41
1
p  CR  
130
p  SS  
230
-88-
95
230
HMP654
Decision Analysis
Conditional Probability
Conditional Probabilities
Color given Shape
CB
CR
SS
0.92
0.08
1
pCR  ST  0.37
pCR ST  

pST 
0.43
ST
0.15
0.85
1
Compare to p(CR) = 0.41
Shape given color
CB
CR
SS
0.89
0.11
ST
0.11
0.89
1
1
pST CR  
Compare to p(ST) = 0.43
-89-
pCR  ST  0.37

pCR 
0.41
HMP654
Decision Analysis
Perfect Information
Perfect Information
Frequency Table
CB
CR
SS
135
0
135
ST
0
95
95
Joint Probability Distribution
135
95
230
CB
CR
Color given Shape
CB
CR
SS
1
0
SS
0.59
0.00
0.59
ST
0.00
0.41
0.41
0.59
0.41
1
Shape given Color
ST
0
1
CB
CR
-90-
SS
1
0
ST
0
1
HMP654
Decision Analysis
No Information
No Information
Frequency Table
CB
CR
SS
413
287
700
ST
177
123
300
Joint Probability Distribution
590
410
1000
CB
CR
Color given Shape
CB
CR
SS
0.59
0.41
SS
0.41
0.29
0.70
ST
0.18
0.12
0.30
0.59
0.41
1
Shape given Color
ST
0.59
0.41
CB
CR
-91-
SS
0.7
0.7
ST
0.3
0.3
HMP654
Decision Analysis
Perfect Information
0.59
Draw blue
10
Predict blue
0
0
3.85
10
0.41
Draw red
-5
Don't use shape info
0
-5
1
0
3.85
0.59
Draw blue
-5
Predict red
0
0
1.15
-5
0.41
Draw red
10
0
10
1
Draw blue
10
Predict blue
0
10
2
10
0
10
0
Draw red
0.59
Draw square
-5
0
-5
1
0
10
1
Draw blue
-5
Predict red
0
0
-5
-5
0
Draw red
10
Use shape info
0
0
10
10
0
Draw blue
10
Predict blue
0
0
-5
10
1
Draw red
0.41
Draw triangle
-5
0
-5
2
0
10
0
Draw blue
-5
Predict red
0
0
10
-5
1
Draw red
10
0
-92-
10
HMP654
Decision Analysis
No Information
0.59
Draw blue
10
Predict blue
0
0
3.85
10
0.41
Draw red
-5
Don't use shape info
0
-5
1
0
3.85
0.59
Draw blue
-5
Predict red
0
0
1.15
-5
0.41
Draw red
10
0
10
0.59
Draw blue
10
Predict blue
0
10
1
3.85
0
3.85
0.7
Draw square
0.41
Draw red
-5
0
-5
1
0
3.85
0.59
Draw blue
-5
Predict red
0
0
1.15
-5
0.41
Draw red
10
Use shape info
0
0
3.85
10
0.59
Draw blue
10
Predict blue
0
0
3.85
0.3
Draw triangle
10
0.41
Draw red
-5
0
-5
1
0
3.85
0.59
Draw blue
-5
Predict red
0
0
1.15
-5
0.41
Draw red
10
0
-93-
10
HMP654
Decision Analysis
Imperfect Information
0.59
Draw blue
10
Predict blue
0
0
3.85
10
0.41
Draw red
-5
Don't use shape info
0
-5
1
0
3.85
0.59
Draw blue
-5
Predict red
0
0
1.15
-5
0.41
Draw red
10
0
10
0.92
Draw blue
10
Predict blue
0
10
2
8.3485
0
8.8
0.57
Draw square
0.08
Draw red
-5
0
-5
1
0
8.8
0.92
Draw blue
-5
Predict red
0
0
-3.8
-5
0.08
Draw red
10
Use shape info
0
0
8.3485
10
0.15
Draw blue
10
Predict blue
0
0
-2.75
0.43
Draw triangle
10
0.85
Draw red
-5
0
-5
2
0
7.75
0.15
Draw blue
-5
Predict red
0
0
7.75
-5
0.85
Draw red
10
0
-94-
10
HMP654
Decision Analysis
Bayes Theorem
pSS   pSS  CB   pSS  CR 
p SS CB  
pSS  CB 
 pSS  CB   p SS CB  pCB 
pCB 
p SS CR  
pSS  CR 
 pSS  CR   p SS CR  pCR 
pCR 
 pSS   p SS CB  pCB   p SS CR  pCR 
In a similar wa y, it can be shown that
pST   p ST CB  pCB   pST CR  pCR 
then
p CB SS  
p SS CB  pCB 
pSS  CB 

pSS 
pSS CB  pCB   p SS CR  pCR 
etc.
-95-
HMP654
Decision Analysis-Decision Trees
Modified 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 of
Marketing
Research Report
Result is really a
success
(S)
Result is really a
failure
(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’
-96-
HMP654
Decision Analysis-Decision Trees
Modified Case Problem - Imperfect Information
D em and = 30
D em and = 90
S alaried P T
D em and = 140
D em and = 150
No C am paign
1
D em and = 30
D em and = 90
P urc hase P T S ervic es
D em and = 140
D em and = 150
D em and = 30
C am paign is a failure
D em and = 90
S alaried P T
1
D em and = 140
C am paign is a suc c ess
D em and = 150
R eport say s "F ailure"
1
D em and = 30
C am paign is a failure
D em and = 90
P urc hase P T S ervic es
D em and = 140
C am paign is a suc c ess
D em and = 150
C am paign
D em and = 30
C am paign is a failure
D em and = 90
S alaried P T
D em and = 140
C am paign is a suc c ess
D em and = 150
R eport say s "S uc c ess"
1
D em and = 30
C am paign is a failure
D em and = 90
P urc hase P T S ervic es
D em and = 140
C am paign is a suc c ess
D em and = 150
-97-
HMP654
Decision Analysis-Decision Trees
Modified Case Problem - Imperfect Information
p RS S   0.85
pS   0.72
p RF S   0.15
pF   0.28
p RS F   0.25
p RF F   0.75
pRS   p RS S  pS   p RS F  pF 
pRF   p RF S  pS   p RF F  pF 
p S RS  
p F RS  
p RS S  pS 
pRS 
p RS F  pF 
pRS 
-98-
p S RF  
p F RF  
p RF S  pS 
pRF 
p RF F  pF 
pRF 
HMP654
Decision Analysis-Decision Trees
Modified Case Problem - Imperfect Information
Probabilities of “report outcome” given “actual outcome”
S
RS
RF
F
0.85
0.25
0.15
0.75
0.72
p(S)
0.28
p(F)
0.682 p(RS)
0.318 p(RF)
Probabilities of “actual outcome” given “report outcome”
S
F
RS 0.8974 0.1026
RF 0.3396 0.6604
-99-
HMP654
Decision Analysis-Decision Trees
Modified Case Problem - Imperfect Information
0.1
Demand = 30
-2,360
2,040
-2,360
0.27
Demand = 90
1,720
Salaried PT
-4,400
6,120
3,658
1,720
0.33
Demand = 140
5,120
9,520
5,120
0.3
Demand = 150
5,800
No Campaign
10,200
5,800
1
0
3,658
0.1
Demand = 30
600
600
600
0.27
Demand = 90
1,800
Purchase PT Services
0
2,370
1,800
1,800
0.33
Demand = 140
2,800
2,800
2,800
0.3
Demand = 150
3,000
3,000
3,000
Next Page
-100-
HMP654
Decision Analysis-Decision Trees
Modified Case Problem- Imperfect Information
0.5
Demand = 30
0.6604
Campaign is a failure
Previous Page
0
-620
-2,660
2,040
-2,660
0.5
Demand = 90
1,420
Salaried PT
6,120
1,420
1
3,658
-4,400
1,343
0.5
Demand = 140
0.3396
Campaign is a success
0
5,160
0.318
Report says "Failure"
4,820
9,520
4,820
0.5
Demand = 150
5,500
10,200
5,500
2
0
1,477
0.5
Demand = 30
0.6604
Campaign is a failure
0
900
300
600
300
0.5
Demand = 90
1,500
Purchase PT Services
0
1,800
1,477
1,500
0.5
Demand = 140
0.3396
Campaign is a success
0
2,600
2,500
2,800
2,500
0.5
Demand = 150
2,700
Campaign
-300
3,000
3,584
2,700
0.5
Demand = 30
0.1026
Campaign is a failure
0
-620
-2,660
2,040
-2,660
0.5
Demand = 90
1,420
Salaried PT
-4,400
6,120
4,567
1,420
0.5
Demand = 140
0.8974
Campaign is a success
0
5,160
0.682
Report says "Success"
4,820
9,520
4,820
0.5
Demand = 150
5,500
10,200
5,500
1
0
4,567
0.5
Demand = 30
0.1026
Campaign is a failure
0
900
300
600
300
0.5
Demand = 90
1,500
Purchase PT Services
0
1,800
2,426
1,500
0.5
Demand = 140
0.8974
Campaign is a success
0
2,600
2,500
2,800
2,500
0.5
Demand = 150
2,700
3,000
-101-
2,700
HMP654
Decision Analysis-Decision Trees
Imperfect Information-Sensitivity Analysis
Probabilities of “report outcome” given “actual outcome”
S
RS
RF
F
0.90
0.15
0.10
0.85
0.72
p(S)
0.28
p(F)
0.69
p(RS)
0.31
p(RF)
Probabilities of “actual outcome” given “report outcome”
S
F
RS 0.9391 0.0609
RF 0.2323 0.7677
-102-
HMP654
Decision Analysis-Decision Trees
Imperfect Information-Sensitivity Analysis
0.1
Demand = 30
-2,360
2,040
-2,360
0.27
Demand = 90
1,720
Salaried PT
-4,400
6,120
3,658
1,720
0.33
Demand = 140
5,120
9,520
5,120
0.3
Demand = 150
5,800
No Campaign
10,200
5,800
1
0
3,658
0.1
Demand = 30
600
600
600
0.27
Demand = 90
1,800
Purchase PT Services
0
2,370
1,800
1,800
0.33
Demand = 140
2,800
2,800
2,800
0.3
Demand = 150
3,000
3,000
3,000
Next Page
-103-
HMP654
Decision Analysis-Decision Trees
Imperfect Information-Sensitivity Analysis
0.5
Demand = 30
Previous Page
0.7677
Campaign is a failure
0
-620
-2,660
2,040
-2,660
0.5
Demand = 90
1,420
6,120
Salaried PT
1,420
2
-4,400
3,719
0.5
Demand = 140
723
0.2323
Campaign is a success
0
5,160
4,820
9,520
4,820
0.5
Demand = 150
5,500
0.31
Report says "Failure"
10,200
5,500
2
0
0.5
Demand = 30
1,295
0.7677
Campaign is a failure
0
900
300
600
300
0.5
Demand = 90
1,500
1,800
Purchase PT Services
0
1,500
0.5
Demand = 140
1,295
0.2323
Campaign is a success
0
2,600
2,500
2,800
2,500
0.5
Demand = 150
2,700
3,000
Campaign
-300
3,719
2,700
0.5
Demand = 30
0.0609
Campaign is a failure
0
-620
-2,660
2,040
-2,660
0.5
Demand = 90
1,420
Salaried PT
-4,400
6,120
4,808
1,420
0.5
Demand = 140
0.9391
Campaign is a success
0
5,160
0.69
Report says "Success"
4,820
9,520
4,820
0.5
Demand = 150
5,500
10,200
5,500
1
0
4,808
0.5
Demand = 30
0.0609
Campaign is a failure
0
900
300
600
300
0.5
Demand = 90
1,500
Purchase PT Services
0
1,800
2,496
1,500
0.5
Demand = 140
0.9391
Campaign is a success
0
2,600
2,500
2,800
2,500
0.5
Demand = 150
2,700
3,000
-104-
2,700
HMP654