decision analysis by interval smart/swing
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Decision analysis by interval SMART/SWING. Jyri Mustajoki Raimo P. Hämäläinen Systems Analysis Laboratory Helsinki University of Technology www.sal.hut.fi. Multiattribute Value Tree Analysis. Value tree: Value of an alternative x (additive): w i is the weight of attribute i - PowerPoint PPT PresentationTRANSCRIPT
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 1
Decision analysis by interval Decision analysis by interval SMART/SWINGSMART/SWING
Jyri MustajokiRaimo P. Hämäläinen
Systems Analysis LaboratoryHelsinki University of Technology
www.sal.hut.fi
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 2
Multiattribute Value Tree AnalysisMultiattribute Value Tree Analysis
• Value tree:
• Value of an alternative x (additive):
wi is the weight of attribute ivi(xi) is the component value of an alternative x in respect of an attribute i
n
iiii xvwxv
1
)()(
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 3
Ratio methods in weight elicitationRatio methods in weight elicitation
Questions of interest - new alternative ways:• Reference attribute (Are there other than
worst/best = SMART/SWING?)• Relationship to direct weighting?• Uncertain replies modelled as intervals• Uncertain reference considered as an interval• Behavioral and procedural benefits and
problems
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 4
Attribute weightingAttribute weightingSWING• 100 points to the most important attribute
change from its lowest level to the highest level• Fewer points to other attributes denoting their
relative importance• Weights elicited by normalizing the sum of the
points to oneSMART• 10 points to the least important attribute
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 5
Interval decision analysis methodsInterval decision analysis methods• Intervals used to describe impreciseness • 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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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 6
Generalizing SMART and SWINGGeneralizing SMART and SWING
• Relaxing the reference attribute to be any attribute
• Allowing the DM to reply with intervals instead of exact point estimates
• Allowing also the reference attribute to be an interval
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Generalizing SMART and SWINGGeneralizing SMART and SWINGReference attribute Reference points Elicitation Name
Least important 10 (or 1) Point estimates SMART
Most important 100 (or 1) Point estimates SWING
Any Any number of points Point estimates (Generalized) RATIO method
Least important 10 (or 1) Intervals of points Interval SMART
Most important 100 (or 1) Intervals of points Interval SWING
Any Any number of points Intervals of points Interval RATIO method(Interval SMART/SWING)
Any Any interval Point estimates RATIO method with intevalreference attribute
Any Any interval Intervals of points Interval RATIO method withinterval reference attribute
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 8
wA
wB wC
S
wA= 2 wC
wC= 4 wA
wA= wB
wB= 3 wA
wB= 3 wC wC= 3 wB
Simplified PAIRSSimplified PAIRS
• PAIRS• Constraints on any
weight ratios Feasible region S
• Generalized ratio methods simplified cases of PAIRS
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 9
Relaxing the reference attribute to Relaxing the reference attribute to be any attributebe any attribute
• Generalization of SMART/SWING or direct weighting
• Weight ratios calculated as ratios of the given points Technically no difference to SMART and SWING
• Possibility of behavioral biases• Proper guidance to the DMs• More research needed
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 10
Interval SMART/SWINGInterval SMART/SWING
• The reference attribute given any (exact) number of points
• Points to non-reference attributes given as intervals
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 11
Interval SMART/SWINGInterval SMART/SWING
• Max/min ratios of points constraint the feasible region of weights• Values 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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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 12
An exampleAn example
• Three attributes: A, B, C• Preferences of the DM:
• Two cases considered:1. A chosen as reference attribute (100 points) Other attributes (B, C) given 50-200 points2. B chosen as reference attribute (100 points) A given 50-200 points, C given 100 points
1,2,2 21
21
C
B
C
A
B
A
ww
ww
ww
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 13
Reference attributeReference attribute• A as a reference attribute
2,2 21
21
C
A
B
A
ww
ww
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 14
Feasible regionFeasible region
wA
wB wC
S
wA= 2 wC
wC= 2 wA
wA= 2 wB
wB= 2 wA
wB= 4 wC wC= 4 wB
• A as a reference attribute
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 15
Reference attributeReference attribute• B as a reference attribute
1,221
C
B
A
B
ww
ww
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 16
Feasible regionFeasible region
wA
wB wC
S'
wA= 2 wC
wC= 2 wA
wA= 2 wB
wB= 2 wA
wB= wC
• B as a reference attribute
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 17
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
• Choice of the reference attribute?• Attribute with least uncertainty• Easily measurable attribute, e.g. money
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 18
Using an interval on the reference Using an interval on the reference attributeattribute
• Meaning of the intervals• Ambiguity
• Constraints for the weight ratios:
• Every constraint is bounding the feasible region
y
x
y
x
y
x
ww
minmax
maxmin
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 19
Using an interval on the reference Using an interval on the reference attributeattribute
• An example
A B C0
50
100
150
200
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 20
wA
wB wC
S
wA= 2 wC
wC= 4 wA
wA= wB
wB= 4 wA
wB= 4 wC wC= 2 wB
Using an interval on the reference Using an interval on the reference attributeattribute
• Feasible region S
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 21
Using an interval on the reference Using an interval on the reference attributeattribute
• Are the DMs able to compare the intevals?
• The final step of generalizations is to relax the weight ratio constraints to be any constraints PAIRS method
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 22
WINPRE softwareWINPRE software
• Weighting methods• Preference programming• PAIRS• Interval SMART/SWING
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An exampleAn example
• Vincent Sahid's job selection (Hammond, Keeney and Raiffa, 1999)
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
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 24
Value TreeValue Tree
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 25
Imprecise rating of the alternativesImprecise rating of the alternatives
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 26
Interval SMART/SWING weightingInterval SMART/SWING weighting
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 27
PAIRSPAIRS
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Mustajoki and HämäläinenDecision analysis by interval SMART/SWING / 28
The resultsThe results
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The resultsThe results
• Jobs C and E dominated Eliminated from subsequent analyses
• Process could be continued by defining the attributes more accurately• Easier as fewer alternative
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ConclusionsConclusions
• Interval SMART/SWING• An easy method to model uncertainty by
intervals• Linear programming algorithms involved
• Software needed• WINPRE introduced
• Does the DMs understand the intervals?• More research needed
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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.
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., 1999. PRIME - Preference ratios in multiattribute evaluation. Manuscript. Downloadable at http://www.sal.hut.fi/ Publications/pdf-files/Prime.pdf