Helsinki University of Technology Systems Analysis Laboratory

Incomplete Ordinal Information in Incomplete Ordinal Information in Value Tree AnalysisValue Tree Analysis

Antti Punkka and Ahti SaloSystems Analysis Laboratory

Helsinki University of TechnologyP.O. Box 1100, 02015 TKK, Finland

http://www.sal.tkk.fi/forename.surname@tkk.fi

Helsinki University of Technology Systems Analysis Laboratory

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m alternatives X={x1,…,xm} n twig-level attributes A={a1,…,an} Additive value

Set of possible non-normalized scores

– vi(xi0)=0 and vi(xi

*) = wi

Value tree analysisValue tree analysis

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jii

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non-normalized formnon-normalized form normalized formnormalized form]1,0[iw

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Preference informationPreference information

Complete information– Point estimates for weights and scores– Examples

» SWING (von Winterfeldt and Edwards 1986)» SMART (Edwards 1977)

Incomplete information– Modeled through linear constraints on weights and scores– Provides dominance relations and value intervals for alternatives– Supports ex ante sensitivity analysis in view of feasible

parameters– Examples

» PAIRS (Salo and Hämäläinen 1992)» PRIME (Salo and Hämäläinen 2001)

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Incomplete weight informationIncomplete weight information

Forms of incomplete information (Park and Kim 1997):1. weak ranking wi wj

2. strict ranking wi – wj 3. ranking with multiples wi wj

4. interval form ≤ wi ≤ + 5. ranking of differences

wi – wj wk – wl

A feasible region forattribute weights Sw

3w

2w

1w

wS23 ww

)0,1,0(

)1,0,0(

)0,0,1(

23 3ww

13 2ww

13 4ww

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116,

112,

113

1912,

194,

193

21 5.1 ww 21 25.0 ww

2/1/4/1,1/3/1

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32

wwww

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Preference Programming with ordinal informationPreference Programming with ordinal information

Incomplete ordinal information about:1. Relative importance of attributes2. Values of alternatives w.r.t.

- A single twig-level attribute- Several attributes (e.g. higher-level attributes)- All attributes (holistic comparisons)

Dominance relations,Decision rules

andOverall value intervals

MILPmodel

Decision

Other forms of incomplete information:1. Weak ranking2. Strict ranking3. Ranking with multiples4. Interval form5. Ranking of differences

Results sufficientlyconclusive

for the DM?

Additionalinformation

yesyes

nono

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Ordinal preference informationOrdinal preference information

Comparative statements between objects– No information about how much ’better’ or more important an object is

than another– Can be useful in evaluation w.r.t. qualitative attributes– Complete information = a rank-ordering of all attributes or alternatives

Uses in preference elicitation– Rank attributes in terms of relative importance

» Obtain point estimates through, e.g., rank sum weights (Stillwell et al. 1981), rank order centroid (Barron 1992)

– Rank alternatives with regard to one or several attributes– Holistic comparisons: ”alternative x1 preferred to alternative x2 overall”

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Incomplete ordinal preference informationIncomplete ordinal preference information

A complete rank-ordering, too, may be difficult to obtain– Identification of best performing alternative with regard to some attribute

» which office facility has the best public transport connections?– Comparison of attributes

» which attribute is the most important one?

Rank Inclusion in Criteria Hierarchies (RICH) – Salo and Punkka (2005), European Journal of Operational Research

163/2, pp. 338-356– Admits incomplete ordinal information about the importance of attributes

» ”the most important attribute is either cost or durability”» ”environmental factors is among the three most important attributes”

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Non-convex feasible region RICHNon-convex feasible region RICH ”Either attribute a1 or a2 is the most

important of the three attributes”

Four rank-orderings compatible with this statement

Supported by RICH Decisions ©, http://www.decisionarium.tkk.fi http://www.rich.tkk.fi

Selection of risk management methods (Ojanen et al. 2005)

Participatory priority-setting for a Scandinavian research program (Salo and Liesiö 2006)

3w

2w

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)0,1,0(

)1,0,0(

23 ww

13 ww

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)2,1,3(r

)2,3,1(r

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RICHER - RICH with Extended RankingsRICHER - RICH with Extended Rankings

Admits incomplete ordinal information about alternatives– ”Alternatives x1, x2 and x3 are the three most preferred ones with regard to

environmental factors”– ”Alternative x1 is the least preferred among x1, x2 and x3 w.r.t. cost”– ”Alternative x1 is not among the three most preferred ones overall”

Ordinal statements w.r.t. different attribute sets – Twig-level attributes– Higher-level attributes A’ A– Holistic statements w.r.t. all attributes

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Modeling of incomplete ordinal information (1/3)Modeling of incomplete ordinal information (1/3)

The smaller the ranking, the more preferred the alternative– r(x4)=1 the ranking of x4 is 1 it is the most preferred

Rank-orderings r=(r1, ..., rm’) on alternatives X’ X– Bijections from alternatives X’ X to corresponding rankings 1,...,|X’|=m’– Notation: ri = r(xj), s.t. j is the i-th smallest index in X’– Convex feasible region

» A’={ai}

)()()()( kjkj xvxvxrxr

)}()( if )()(|{)( 0kjk

iijii xrxrxvxvSsrS

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Modeling of incomplete ordinal information (2/3)Modeling of incomplete ordinal information (2/3)

Specified as a set of alternatives I X ’ X and corresponding rankings J {1,...,m’}– X’ = subset of alternatives under comparison and m’ = |X’| its cardinality

If |I|<|J|, alternatives in I have their rankings in J– x4 and x5 belong to the three most preferred alternatives– I = {x4, x5}, J = {1,2,3}

If |I||J|, rankings in J are attained by alternatives in I– The least preferred alternative in X={x1,...,x10} is among x1, x2, x3, x4

– I = {x1, x2, x3, x4}, J = {10}

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Modeling of incomplete ordinal information (3/3)Modeling of incomplete ordinal information (3/3)

Sets I and J lead to compatible rank-orderings R(I,J) for each combination of X’, A’

Feasible region associated with compatible rank-orderings

Sets S(I,J) have several useful properties, for example– S(I,J) = S(IC,JC), where IC is the complement of I in X’– Set inclusions: I2 I1, |Ii||J| => S(I2,J) S(I1,J)

)(),(),(rSJIS

JIRr

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),(1

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Linear inequality formulation for Linear inequality formulation for SS((II,,JJ) (1/3)) (1/3)

Values of alternatives with rankings k and k+1 are separated by milestone variable zk

– If the ranking of xj is ”worse” than k, its value is at most zk

– Binary variable yk(xj)=1 iff the value of xj is at least zk

– Milestone, binary and value variables subjected to A’ and X’

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j

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There are exactly k alternatives whose ranking is k or better

If the ranking of xj is better than k-1, it is also better than k

Linear inequality formulation for Linear inequality formulation for SS((II,,JJ) (2/3)) (2/3)

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decreasing valuedecreasing value

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Linear inequality formulation for Linear inequality formulation for SS((II,,JJ) (3/3)) (3/3)

Feasible region S(I,J) characterized by linear constraints on binary variables

By using milestone and binary variables for each set pair (A’, X’) used in elicitation, all constraints are in the same linear model

Characteristics of incomplete ordinal information used to enhance computational properties– E.g., only the relevant milestone and binary variables are

introduced» given a statement that alternatives x1 and x2 are the two most

preferred, only z2 is needed

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Pairwise dominancePairwise dominance Value intervals may overlap, but

Example with two attributes– Interval statement on weights

– Point estimates for scores– x1 dominates x2– x3 is also non-dominated

Non-dominated alternatives– Calculation through LP

7.04.0 1 w

V

w1 0.4 0.7w2 0.6 0.3

x1 dominates x2

)( 1xV

)( 2xV

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0)]()(min[ jk xVxVand strictly positive with some and strictly positive with some feasible scores and weightsfeasible scores and weights

Alternative xk dominates xj

Valueintervals

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Decision rulesDecision rules

Maximize max overall value (’maximax’) => x1

Maximize min overall value (’maximin’) => x3

Maximize avg of max and min values (’central values’) => x1

Minimize greatest possible loss relative to another alternative (’minimax regret’) => x1

V

w10.4 0.7w20.6 0.3

)( 1xV )( 3xV maximaxmaximax

maximinmaximin

central valuescentral valuesminimax regretminimax regret

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RICHER RICHER

Key features – Extends preference elicitation techniques by admitting incomplete ordinal

information about attributes and alternatives – Converts preference statements into a linear inequality formulation

» can thus be combined with any other Preference Programming methods – Offers recommendations through pairwise dominance and decision rules

Decision support tools– Experiments suggest that MILP model is reasonably efficient – Software implementation of RICHER Decisions© ongoing

Future research directions – Sorting / classification procedures in score elicitation– Analyses of voting behavior (e.g., acceptance voting)

Submitted manuscript downloadable at http://www.sal.hut.fi/Publications/pdf-files/mpun04.pdf

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ReferencesReferencesBarron, F. H., “Selecting a Best Multiattribute Alternative with Partial Information about Attribute Weights”, Acta

Psychologica 80 (1992) 91-103

Edwards, W., “How to Use Multiattribute Utility Measurement for Social Decision Making”, IEEE Transactions on Systems, Man, and Cybernetics 7 (1977) 326-340.

Ojanen, O., Makkonen, S. and Salo, A., “A Multi-Criteria Framework for the Selection of Risk Analysis Methods at Energy Utilities”, International Journal of Risk Assessment and Management 5 (2005) 16-35.

Park, K. S. and Kim, S. H., “Tools dor Interactive Decision Making with Incompletely Identified Information”, European Journal of Operational Research 98 (1997) 111-123.

Salo, A. and Hämäläinen, R. P., "Preference Assessment by Imprecise Ratio Statements”, Operations Research 40 (1992) 1053-1061.

Salo, A. and Hämäläinen, R. P., “Preference Ratios in Multiattribute Evaluation (PRIME) - Elicitation and Decision Procedures under Incomplete Information”, IEEE Transactions on Systems, Man, and Cybernetics 31 (2001) 533-545.

Salo, A. and Liesiö, J., “A Case Study in Participatory Priority-Setting for a Scandinavian Research Program”, International Journal of Information Technology and Decision Making (to appear).

Salo, A. and Punkka, A., “Rank Inclusion in Criteria Hierarchies”, European Journal of Operations Research 163 (2005) 338-356.

Stillwell, W. G., Seaver, D. A. and Edwards, W., “A Comparison of Weight Approximation Techniques in Multiattribute Utility Decision Making”, Organizational Behavior and Human Performance 28 (1981) 62-77.

von Winterfeldt, D., Edwards, W., ”Decision Analysis and Behavioral Research”, Cambridge University Press (1986).