Introduction to Value Tree
Analysis
Evatech seminar
eLearning resources / MCDA team
Director prof. Raimo P. Hämäläinen
Helsinki University of Technology
Systems Analysis Laboratory
http://www.eLearning.sal.hut.fi
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Contents
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About the introduction
Basic concepts
A job selection problem
Problem structuring
Preference elicitation
Results and sensitivity analysis
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
About the introduction
 This is a brief introduction to multiple criteria
decision analysis and specifically to value tree
analysis
 After reading the material you should know
 basic concepts of value tree analysis
 how to construct a value tree
 how to use the Web-HIPRE software in simple
decision making problems to support your decision
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Basic concepts
Objective
 is a statement of something that one desires to
achieve
 for example; “more wealth”
Attribute
 indicates the level to which an objective is achieved
in a given decision alternative
 for example by selecting a certain job offer you may
get 3000 €/month
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Basic concepts
Value function
 Value function v(x) assigns a number i.e. value to each attribute
level x.
 Value describes subjective desirability of the corresponding
attribute level.
 For example:
value
value
1
1
Size of the ice cream cone
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Working hours / day
Basic concepts
Value tree
In a value tree objectives are organised hierarchically
overall objective
sub-objectives
attributes
alternatives
Top speed
Driving
Citroen
Acceleration
Ideal car
VW Passat
Price
Audi A4
Economy
Expenses
• Each objective is defined by sub-objectives or attributes
• There can be several layers of objectives
• Attributes are added under the lowest level of objectives
• Decision alternatives are connected to the attributes
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Basic concepts
Phases of value tree analysis
The aim of the Problem structuring is to create
a better understanding of the problem
Decision context is a setting in which the
decision occurs
In Preference elicitation DM’s preferences
over a set of objectives is estimated and
measured
The aim of the Sensitivity analysis is to explore
how changes in the model influence the
recommended decision
Note: Only the highlighted parts are
covered in this mini intro
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Problem structuring
Decision context is the setting
in which the decision occurs
Use the figure to define the decision context for the Job selection problem.
· Start with the easiest.
· Proceed to more
complicated areas.
· At the end, select and
highlight the most
important ones.
How does the nature
of possible job
opportunities affect
the decision context?
See the “Problem structuring / Defining the decision context” section in the theory part.
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Problem structuring
Identifying decision alternatives
 Identify possible decision alternatives
 To stimulate the process
a) use fundamental objectives
 If there were only one objective, two objectives...
b) use means objectives
c) remove constraints
 If time were no concern...
c) use different perspectives
 How would you see the situation after ten years?
See the “Problem structuring / Generating and identifying decision alternatives” section
in the theory part.
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
A job selection problem
Assume that you have four job offers to choose between;
1) a place as a researcher in a governmental research institute
2) a place as a consultant in a multinational consulting firm
3) a place as a decision analyst in a large domestic firm
4) a place in a small IT firm
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Problem structuring
Generating objectives
 List all the objectives that you find relevant
 Specify their meaning carefully
 object
 direction
 You may use
 Wish list
 Alternatives:
 What makes the difference between the alternatives?
 Consequences
 Different perspectives
See the “Problem structuring / Identifying and generating objectives” section in the theory part.
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Possible objectives
with their descriptions
objective
description
Maximise new contacts with persons and bodies who can potentially
influence your personal career opportunitites.
continuing education Maximise possibilities for continuing education.
fit with interests
Maximise the match between tasks and personal interests.
tasks diversity
Maximise possibilities for carrying out different tasks.
networking
Maximise the correspondence between task requirements and
professional skills and opportunities for further professional growth.
working environment Maximise the positive effect of working environment.
atmosphere
Maximise the positive effect of corporate culture and atmosphere.
Maximise the positive effect of facilities and physical working
facilities
environment.
starting salary
Maximise the starting salary.
expected salary in 3
Maximise the expected salary in three years.
years
fringe benefits
Maximise fringe benefits.
challenge
effects on leisure time Mimimise the extent to which the work constrains the leisure time.
working hours
daily commuting
business travel
Minimise working hours.
Minimise daily commuting.
Minimise the amount of extended trips.
What other objectives might there be?
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Problem structuring
Hierarchical organisation of objectives
1) Identify the overall objective.
2) Clarify its meaning with more specific sub-objectives. Add the subobjectives to the next level of the hierarchy.
3) Continue recursively until an attribute can be associated with each
lowest level objective.
4) Add the decision alternatives to the hierarchy and link them to the
attributes.
5) Iterate the steps 1- 4, until you are satisfied with the structure.
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Problem structuring - Hierarchical organisation of objectives
A preliminary objectives hierarchy with
alternatives illustrated with Web-HIPRE
Note:
• Alternatives are shown
in yellow in Web-HIPRE.
• Only the fundamental objectives
are included.
• All objectives are assumed to
be preferentially independent.
Is there anything you would like to
change?
Does the value tree satisfy the
conditions listed in the “Checking
the structure” section?
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Problem structuring - Hierarchical organisation of objectives
Checking the structure
The hierarchy requires further modification;
 Networking may be difficult to measure and there is
no real information available on it either.
 According to the DM
 Task diversity is not relevant; tasks are likely to change
over time, and all job offers have some variability.
 Facilities have only a minor importance.
 Daily commuting may be neglected because it is almost the
same for all jobs.
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Problem structuring
The objectives hierarchy
for the job selection problem
Overall objective
Decision alternatives
Sub-objectives
Attributes
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Video Clip: Structuring a value tree in
Web-HIPRE
with sound (.avi 3.3MB)
no sound (.avi 970KB )
animation (.gif 475KB)
Problem structuring
Specifying attributes
 Attributes measure the degree to which objectives are
achieved.
 Attributes should be
 comprehensive and understandable
 Attribute levels define unambiguously the extent to which an objective is achieved.
 measurable
 It is possible to measure DM’s preferences for different attribute levels.
1) Specify attributes for each lowest level objective.
2) Assess the alternatives’ consequences with respect to
those attributes.
For more see the “Specification of attributes” section in the theory part.
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Problem structuring
Consequences
Attribute
Research Institute Consulting Firm Large Corporation
continuing
3
3
1
education
starting salary/€
1900
2700
2200
expected salary
2500
3500
2800
in 3 years/€
hours / week
37.5
55
40
atmosphere
3.2
2.5
3.7
travelling days /
20
160
100
year
Small IT Firm
2
2300
3000
42.5
4.5
30
Video Clip: Entering the consequences
of the alternatives in Web-HIPRE
with sound (.avi 1.33 MB)
no sound (.avi 230 KB)
animation (.gif 165 KB)
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Preference elicitation: an overview
The aim is to measure DM’s preferences on each objective.
Value elicitation
First, single attribute value functions
vi are determined for all attributes Xi.
Weight elicitation
1/4
1/8
3/8
1/4
Value
1
vi(x)  [0,1]
Attribute level
Second, the relative weights of
the attributes wi are determined.
Finally, the total value of an alternative a with consequences Xi(a)=xi
is calculated as
n
V ( x1 , x2 ,..., xn )   wi vi ( xi )
i 1
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
(i=1..n)
Preference elicitation
Single attribute value
function elicitation in brief
1) Set attribute ranges
 All alternatives should be within
the range.
 Large range makes it difficult to
discriminate between alternatives.
 New alternatives may lay
outside the range if it is too small.
Possible ranges for the “working hours/d“ attribute
2) Estimate value functions for attributes
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
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


Assessing the form of value function
Direct rating
Bisection
Difference standard sequence
Category estimation
Note:
Ratio estimation
Methods used in this case are shown in bold
AHP
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Preference elicitation
Setting attributes’ ranges
 No new job offers expected
 Analysis is used to compare only the existing
alternatives
small ranges are most appropriate
Attribute Research Institute Consulting Firm Large Corporation Small IT Firm
Range
continuing
3
3
1
2
1-3
education
starting
1900
2700
2200
2300
1900 - 2300
salary/€
expected
salary in 3
2500
3500
2800
3000
2500 - 3500
years/€
hours / week
37.5
55
40
42.5
37.5 - 55
atmosphere
3.2
2.5
3.7
4.5
2.5 - 4.5
travelling days
20
160
100
30
20 - 160
/ year
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Preference elicitation
Assessing the form of value function
Value scale
Is the value function
• increasing or decreasing?
• linear?
Is an increase at the end of the
attribute scale more important than
a same sized increase at the
beginning of the scale?
You can use Bisection method to
ease the assessment.
More about the Bisection method (optional)
Attribute level scale
In the following video clip the Bisection method is used to estimate a point from the value curve.
Web-HIPRE uses exponential approximation to estimate the rest of the value function.
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Video Clip: Assessing the form of the value
function with bisection method in Web-HIPRE
with sound (.avi 1.69 MB)
no sound (.avi 303 KB)
animation (.gif 180 KB)
Preference elicitation
Direct rating
1) Rank the alternatives
2) Give 100 points to the best alternative
3) Give 0 points to the worst alternative
4) Rate the remaining alternatives between 0 and 100
Note that direct rating:
• is most appropriate when the performance levels of an attribute
can be judged only with subjective measures
• can be used also for weight elicitation
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Video Clip: Using direct rating in WebHIPRE
with sound (.avi 1.17 MB)
no sound (.avi 217 KB)
animation (.gif 142 KB)
Preference elicitation
About weight elicitation
In the Job selection case hierarchical weighting is used.
1) Weights are defined for
each hierarchical level...
0.6
2) ...and multiplied down to get
the final lower level weights.
0.6
0.4
0.4
Multiply
0.7
0.3
0.2
0.6
0.2
0.7
0.3
0.2
0.6
0.2
0.42
0.18
0.08
0.24
0.08
To improve the quality of weight estimates
• use several weight elicitation methods
• iterate until satisfactory weights are reached
In the following the use of different weight elicitation
methods is presented...
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Preference elicitation
SMART
1) Assign 10 points to the least important attribute
(objective)
wleast = 10
2) Compare other attributes with xleast and weigh them
accordingly
wi > 10, i  least
3) Normalise the weights
w’k = wk/(iwi ), i =1...n, n=number of attributes (subobjectives)
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Video Clip: Using SMART in WebHIPRE
with sound (.avi 1.12 MB)
no sound (.avi 209 KB)
animation (.gif 133 KB)
Preference elicitation
AHP
1) Compare each pair of
 sub-objectives or attributes under an objective
2) Store preference ratios in a comparison matrix
 for every i and j, give rij, the ratio of importance between
the ith and jth objective (or attribute, or alternative)
 Assign A(i,j) = rij
A=
r11
...
r1n
...
...
...
rn1
... rnn
3) Check the consistency measure (CM)
 If CM > 0.20 identify and eliminate inconsistencies
in preference statements
Video Clip: Using AHP in Web-HIPRE
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
with sound (.avi 1.97 MB)
no sound (.avi 377 KB)
animation (.gif 204 KB)
Weight Elicitation Methods: SMART
Web-HIPRE example
 The weights for the attributes under the “Compensation” objective
in the job selection problem are determined with the SMART
method.
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Weight Elicitation Methods: SMART
Weighting attributes under the
“Compensation” objective
• ”Fringe benefits” is the least
important attribute 10 points
• ”Starting salary” is the second
most important with 40 points
• ”Expexted salary in 3 years” is
the most important attribute
with 65 points.
points
normalised weights
SMART
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
• with sound (1.2Mb)
• no sound (200Kb)
• animation (130Kb)
Results & sensitivity analysis
Used preference elicitation methods
 The job selection value tree with used preference elicitation
methods shown in Web-HIPRE:
Direct rating
Assessing the form of the value
function (Bisection method)
SMART
AHP
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Note:
Only the highlighted
methods are
covered in this
introduction.
Results & sensitivity analysis
Recommended decision

Small IT firm is the
recommended alternative with
the highest total value (0.442)

Large corporation and consulting
firm options are almost equally
preferred (total values 0.407 and
0.405 respectively)

Research Institute is clearly the
least preferred alternative (total
value of 0.290)
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Solution of the job selection problem in WebHIPRE. Only first-level objectives are shown.
Video Clip: Viewing the results in WebHIPRE
with sound (.avi 1.58 MB)
no sound (.avi 286 KB)
animation (.gif 213 KB)
Results & sensitivity analysis
One-way sensitivity analysis
 What happens to the solution of the job selection problem if one of
the parameters affecting the solution changes? What if, for
example the working hours in the IT firm alternative increase to 50
h/week or the salary in the Research Institute rises to 2900
euros/month?
 In other words, how sensitive our solution is to changes in the
objective weights, single attribute value functions or attribute
ratings
 In one-way sensitivity analysis one parameter is varied at time
 Total values of decision alternatives are drawn as a function of the
variable under consideration
 Next, we apply one-way sensitivity analysis to the job selection
case
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Results & sensitivity analysis
Changes in “working hours” attribute

If working hours in the IT firm rise
to 53 h/week or over and nothing
else in the model changes, Large
Corporation becomes the most
preferred alternative

If working hours in the Consulting
firm were 47 h/week or less
instead of the current 55 h/week,
it would be considered the best
alternative
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Results & sensitivity analysis
Changes in “working hours” attribute


Changes in the weekly working
hours in Large corporation‘s job
offer would not affect the
recommended solution even if
they decreased to zero. The
ranking order of the other
alternatives would change
though.
Changes in the weekly working
hours in the Research Institute‘s
job offer don‘t have any effect on
the solution or on the preference
order of rest of the alternatives.
Video Clip: Sensitivity analysis in Web-HIPRE
with sound (.avi 1.60 MB)
no sound (.avi 326 KB)
animation (.gif 239 KB)
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA
Results & Sensitivity Analysis
Conclusion
 Small IT Firm is the recommended solution, i.e.
the most preferred alternative
 The solution is not sensitive to changes in
 the weights of the first level objectives or
 weekly working hours of any single alternative
 Sensitivity to other aspects of the model
requires further studying, however
Systems Analysis Laboratory
Helsinki University of Technology eLearning / MCDA