mustajoki, hämäläinen and salo decision support by interval smart/swing / 1 s ystems analysis...
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Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 1
S ystemsAnalysis LaboratoryHelsinki University of Technology
Decision support by interval Decision support by interval SMART/SWINGSMART/SWING
Methods to incorporate uncertainty into Methods to incorporate uncertainty into multiattribute analysismultiattribute analysis
Jyri MustajokiRaimo P. Hämäläinen
Ahti SaloSystems Analysis Laboratory
Helsinki University of Technologywww.sal.hut.fi
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
Multiattribute value tree analysisMultiattribute value tree analysis
• Value tree:
• Value of an alternative x:
wi is the weight of attribute i
vi(xi) is the component value of an alternative x with respect to attribute i
n
iiii xvwxv
1
)()(
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
Ratio methods in weight elicitationRatio methods in weight elicitationSWING
• 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
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
Questions of interest 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?
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
Generalized SMART and SWINGGeneralized 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
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
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 freereference 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/SWINGwith inteval referenceattribute
Generalized SMART and SWINGGeneralized SMART and SWING
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
Some interval methodsSome 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
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
Classification of ratio methodsClassification of ratio methods
Exact pointestimates
Intervalestimates
Minimum numberof judgments
SMART,SWING
IntervalSMART/SWING
More thanminimum numberof judgments
AHP,Regressionanalysis
PAIRS,Preferenceprogramming
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
wA
wB
wC
S
wA= 2 w
C
wC
= 4 wA
wA= w
B
wB
= 3 wA
wB
= 3 wC
wC
= 3 wB
Interval SMART/SWING = Interval SMART/SWING = Simple PAIRSSimple PAIRS
• PAIRS• Constraints on any
weight ratios
Feasible region S
• Interval SMART/SWING• Constraints from the
ratios of the points
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 10
S ystemsAnalysis LaboratoryHelsinki University of Technology
1. Relaxing the reference attribute 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
Mustajoki, Hämäläinen and Salo
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2. Interval judgments about ratio 2. Interval judgments about ratio estimatesestimates
• Interval SMART/SWING
• The reference attribute given any (exact) number of points
• Points to non-reference attributes given as intervals
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
Interval judgments about ratio Interval judgments about ratio estimatesestimates
• 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
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
Choice of the reference attributeChoice 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
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
Example: Example: A as referenceA 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
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
2
1
2
61
31
21
C
B
C
A
B
A
w
w
w
ww
w
Feasible region SFeasible region S
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
Example: Example: B as referenceB 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
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
41
2
221
A
C
B
C
A
B
w
w
w
ww
w
Feasible region S'Feasible region S'
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
Which attribute to choose as a Which attribute to choose as a reference attribute?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
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
3. Using an interval on the 3. Using an interval on the reference attributereference 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
Mustajoki, Hämäläinen and Salo
Decision support by interval SMART/SWING / 20
S ystemsAnalysis LaboratoryHelsinki University of Technology
Interval referenceInterval reference
A: 50-100 points
B: 50-100 points
C: 100-150 points
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
Implies additional constraintsImplies additional constraints
• Feasible region S:
1
1
2
31
31
21
C
B
C
A
B
A
w
ww
ww
w
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
Using an interval on the Using an interval on the reference attributereference 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
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
WINPRE softwareWINPRE 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
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
Vincent Sahid's job selection exampleVincent Sahid's job selection example(Hammond, Keeney and Raiffa, 1999)
Mustajoki, Hämäläinen and Salo
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Consequences tableConsequences table
Job A Job B Job C Job D Job E
Monthly salary $2,000 $2,400 $1,800 $1,900 $2,200
Flexibility ofwork schedule
Moderate Low High Moderate None
Business skillsdevelopment
Computer Managepeople,computer
Operations,computer
Organization Timemanagement,multipletasking
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
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
Imprecise rating of the alternativesImprecise rating of the alternatives
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
Interval SMART/SWING weightingInterval SMART/SWING weighting
Mustajoki, Hämäläinen and Salo
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Value intervalsValue 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
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
ConclusionsConclusions
• 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
Mustajoki, Hämäläinen and Salo
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S ystemsAnalysis LaboratoryHelsinki University of Technology
ReferencesReferences
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.
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