Decision analysis by interval SMART/SWING

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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



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

)()(



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



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



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



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



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



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



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



Interval SMART/SWINGInterval SMART/SWING

• The reference attribute given any (exact) number of points

• Points to non-reference attributes given as intervals



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



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



Reference attributeReference attribute• A as a reference attribute

2,2 21

21

C

A

B

A

ww

ww



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



Reference attributeReference attribute• B as a reference attribute

1,221

C

B

A

B

ww

ww



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



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



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




• An example

A B C0

50

100

150

200



wA

wB wC

S

wA= 2 wC

wC= 4 wA

wA= wB

wB= 4 wA

wB= 4 wC wC= 2 wB


• Feasible region S




• 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



WINPRE softwareWINPRE software

• Weighting methods• Preference programming• PAIRS• Interval SMART/SWING



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



Value TreeValue Tree



Imprecise rating of the alternativesImprecise rating of the alternatives



Interval SMART/SWING weightingInterval SMART/SWING weighting



PAIRSPAIRS



The resultsThe results



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



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



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

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Decision analysis by interval SMART/SWING

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