Transcript Decision support by interval SMART/SWING Methods to incorporate uncertainty into multiattribute analysis

Decision support by interval
SMART/SWING
Methods to incorporate uncertainty into
multiattribute analysis
Jyri Mustajoki
Raimo P. Hämäläinen
Ahti Salo
Systems Analysis Laboratory
Helsinki University of Technology
www.sal.hut.fi
S ystems
Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 1
Multiattribute value tree analysis
• Value tree:
• Value of an alternative x:
n
v( x)   wi vi ( xi )
i 1
wi is the weight of attribute i
vi(xi) is the component value of an alternative x with
respect to attribute i
S ystems
Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 2
Ratio methods in weight elicitation
SWING
• 100 points to the most important attribute range
change from lowest level to the highest level
• Fewer points to other attributes reflecting their
relative importance
• Weights by normalizing the sum to one
SMART
• 10 points to the least important attribute
• otherwise similar
S ystems
Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 3
Questions of interest
• Role of the reference attribute
• What if other than worst/best =
SMART/SWING?
• How to incorporate preferential uncertainty?
• Uncertain replies modelled as intervals of
ratios instead of pointwise estimates
• Are there behavioral or procedural benefits?
S ystems
Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 4
Generalized SMART and SWING
Allow:
1. the reference attribute to be any attribute
2. the DM to reply with intervals instead of exact
point estimates
3. also the reference attribute to have an interval
 A family of Interval SMART/SWING methods
• Mustajoki, Hämäläinen and Salo, 2001
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Mustajoki, Hämäläinen and Salo
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Generalized SMART and SWING
Reference attribute
Reference
Elicitation
Name
Least important
10 (or any number)
Point estimates
SMART
Most important
100 (or any number)
Point estimates
SWING
Any
Any number of points
Point estimates
SMART/SWING with a free
reference attribute
Least important
10 (or any number)
Intervals of points
Interval SMART
Most important
100 (or any number)
Intervals of points
Interval SWING
Any
Any number of points
Intervals of points
Interval SMART/SWING
Any
Any interval
Intervals of points
Interval SMART/SWING
with inteval reference
attribute
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Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 6
Some interval methods
• Preference Programming (Interval AHP)
• Arbel, 1989; Salo and Hämäläinen 1995
• PAIRS (Preference Assessment by
Imprecise Ratio Statements)
• Salo and Hämäläinen, 1992
• PRIME (Preference Ratios In Multiattribute
Evaluation)
• Salo and Hämäläinen, 1999
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Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 7
Classification of ratio methods
Minimum number
of judgments
Exact point
estimates
SMART,
SWING
Interval
estimates
Interval
SMART/SWING
More than
minimum number
of judgments
AHP,
Regression
analysis
PAIRS,
Preference
programming
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Analysis Laboratory
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Decision support by interval SMART/SWING / 8
Interval SMART/SWING =
Simple PAIRS
• PAIRS
wA
• Constraints on any
weight ratios
 Feasible region S
• Interval
SMART/SWING
• Constraints from the
ratios of the points
w =2w
A
C
wA= wB
S
w =3w
B
A
w
B
wB= 3 wC
wC= 4 wA
wC= 3 wB
w
C
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Decision support by interval SMART/SWING / 9
1. Relaxing the reference attribute
• Reference attribute allowed to be any attribute
• Compare to direct rating
• Weight ratios calculated as ratios of the given
points
 Technically no difference to SMART and
SWING
• Possibility of behavioral biases
• How to guide the DM?
• Experimental research needed
S ystems
Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 10
2. Interval judgments about ratio
estimates
• Interval SMART/SWING
• The reference attribute given any (exact)
number of points
• Points to non-reference attributes given as
intervals
S ystems
Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
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Interval judgments about ratio
estimates
• Max/min ratios of points constraint the
feasible region of weights
• Can be calculated with PAIRS
• Pairwise dominance
• A dominates B pairwisely, if the value of A is
greater than the value of B for every feasible
weight combination
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Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 12
Choice of the reference attribute
• Only the weight ratio constraints including
the reference attribute are given
 Feasible region depends on the choice of
the reference attribute
• Example
• Three attributes: A, B, C
1) A as reference attribute
2) B as reference attribute
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Analysis Laboratory
Helsinki University of Technology
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Example: A as reference
• A given 100 points
• Point intervals given to the other attributes:
• 50-200 points to attribute B
• 100-300 points to attribute C
• Weight ratio between B and C not yet given by
the DM
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Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
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Feasible region S
wA

2
wB
wA
1
1
3 
wC
1
2
wB
 
2
wC
1
6
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Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 15
Example: B as reference
• A given 50-200 points
• Ratio between A and B as before
• The DM gives a pointwise ratio between B
and C = 200 points for C
• Less uncertainty in results  smaller feasible
region
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Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 16
Feasible region S'
wB

2
wA
wC
2
wB
1
2
wC
1 
4
wA
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Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 17
Which attribute to choose as a
reference attribute?
• Attribute agaist which one can give the most
precise comparisons
• Easily measurable attribute, e.g. money
• The aim is to eliminate the remaining
uncertainty as much as possible
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Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
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3. Using an interval on the
reference attribute
• Meaning of the intervals
• Uncertainty related to the measurement scale
of the attribute
• not to the ratio between the attributes (as when
using an pointwise reference attribute)
• Ambiguity of the attribute itself
• Feasible region from the max/min ratios
• Every constraint is bounding the feasible
region
S ystems
Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 19
Interval reference
A: 50-100 points
B: 50-100 points
C: 100-150 points
S ystems
Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 20
Implies additional constraints
• Feasible region S:
wA

2
wB
wA
1
1
3 
wC
wB
1
1
3 
wC
1
2
S ystems
Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 21
Using an interval on the
reference attribute
• Are the DMs able to compare against intevals?
• Two helpful procedures:
1. First give points with
pointwise reference
attribute and then
extend these to
intervals
2. Use of external anchoring attribute, e.g. money
S ystems
Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
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WINPRE software
• Weighting methods
• Preference programming
• PAIRS
• Interval SMART/SWING
• Interactive graphical user interface
• Instantaneous identification of dominance
 Interval sensitivity analysis
• Available free for academic use:
www.decisionarium.hut.fi
S ystems
Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
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Vincent Sahid's job selection example
(Hammond, Keeney and Raiffa, 1999)
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Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 24
Consequences table
Job A
Job B
Job C
Job D
Job E
Monthly salary
$2,000
$2,400
$1,800
$1,900
$2,200
Flexibility of
work schedule
Moderate
Low
High
Moderate
None
Business skills
development
Computer
Manage
people,
computer
Operations,
computer
Organization
Time
management,
multiple
tasking
Vacation
(annual days)
14
12
10
15
12
Benefits
Health, dental,
retirement
Health, dental
Health
Health,
retirement
Health, dental
Enjoyment
Great
Good
Good
Great
Boring
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Decision support by interval SMART/SWING / 25
Imprecise rating of the alternatives
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Analysis Laboratory
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Decision support by interval SMART/SWING / 26
Interval SMART/SWING weighting
S ystems
Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 27
Value intervals
• Jobs C and E
dominated
 Can be
eliminated
• Process continues by narrowing the ratio
intervals of attribute weights
• Easier as Jobs C and E are eliminated
S ystems
Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 28
Conclusions
• Interval SMART/SWING
• An easy method to model uncertainty by
intervals
• Linear programming algorithms involved
• Computational support needed
• WINPRE software available for free
• How do the DMs use the intervals?
• Procedural and behavioral aspects should be
addressed
S ystems
Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 29
References
Arbel, A., 1989. Approximate articulation of preference and priority derivation,
European Journal of Operational Research 43, 317-326.
Hammond, J.S., Keeney, R.L., Raiffa, H., 1999. Smart Choices. A Practical Guide
to Making Better Decisions, Harvard Business School Press, Boston, MA.
Mustajoki, J., Hämäläinen, R.P., Salo, A., 2005. Decision support by interval
SMART/SWING – Incorporating imprecision in the SMART and SWING
methods, Decision Sciences, 36(2), 317-339.
Salo, A., Hämäläinen, R.P., 1992. Preference assessment by imprecise ratio
statements, Operations Research 40 (6), 1053-1061.
Salo, A., Hämäläinen, R.P., 1995. Preference programming through approximate
ratio comparisons, European Journal of Operational Research 82, 458-475.
Salo, A., Hämäläinen, R.P., 2001. Preference ratios in multiattribute evaluation
(PRIME) - elicitation and decision procedures under incomplete information.
IEEE Trans. on SMC 31 (6), 533-545.
Downloadable publications at www.sal.hut.fi/Publications
S ystems
Analysis Laboratory
Helsinki University of Technology
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 30