Jyri Mustajoki Raimo P. Hämäläinen Ahti Salo Systems Analysis Laboratory

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




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

)()(




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




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?




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




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




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




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




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




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




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




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




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




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




2

1

2

61

31

21

C

B

C

A

B

A

w

w

w

ww

w

Feasible region SFeasible region S




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




41

2

221

A

C

B

C

A

B

w

w

w

ww

w

Feasible region S'Feasible region S'




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




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




Interval referenceInterval reference

A: 50-100 points

B: 50-100 points

C: 100-150 points




Implies additional constraintsImplies additional constraints

• Feasible region S:

1

1

2

31

31

21

C

B

C

A

B

A

w

ww

ww

w




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




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




Vincent Sahid's job selection exampleVincent Sahid's job selection example(Hammond, Keeney and Raiffa, 1999)




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




Imprecise rating of the alternativesImprecise rating of the alternatives




Interval SMART/SWING weightingInterval SMART/SWING weighting




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




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




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.

Downloadable publications at www.sal.hut.fi/Publications

Jyri Mustajoki Raimo P. Hämäläinen Ahti Salo Systems Analysis Laboratory

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