helsinki university of technology systems analysis laboratory richer – a method for exploiting...

Helsinki University of Technology Systems Analysis Laboratory

RICHER – A Method for Exploiting Incomplete RICHER – A Method for Exploiting Incomplete Ordinal Information in Value TreesOrdinal Information in Value Trees

Antti Punkka and Ahti Salo

Systems Analysis Laboratory

Helsinki University of Technology

P.O. Box 1100, 02015 HUT, Finland

[email protected]

http://www.sal.hut.fi/


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m alternatives: X={x1,…,xm} , n attributes: A={a1,…,an} Additive value function

or

Value tree analysisValue tree analysis

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jii

j xvxV1

)()(

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ji

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j xvwxVi

1

)()(

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iiw

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non-normalized formnon-normalized form

normalized formnormalized form


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Preference elicitation Preference elicitation

Complete information– Point estimates, e.g. w1=0.5

– E.g., SMART (Edwards 1977)

Incomplete information: preference programming methods– Weight ratio and weight intervals

»

– Intervals for normalized scores»

– PAIRS (Salo and Hämäläinen 1992), PRIME (Salo and Hämäläinen 2001), Arbel’s approach (1989)

Ordinal information– Rank attributes in terms of relative importance

» point estimates through, e.g., rank sum weights (Stillwell et al. 1981)» incomplete ordinal information (RICH; Salo and Punkka 2004)

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Incomplete preference informationIncomplete preference information

Complete information hard to acquire – Relative importance of attributes– Alternatives’ properties– Incomplete information

»

Overall value intervals for alternatives– Smallest and largest possible value from

LP

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i

ji

Ni

Swxvw

iw 1

)(maxmin/ where Sw is the feasible region for the attribute weights

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Pairwise dominance relationPairwise dominance relation

Alternative xk dominates xj in the sense of pairwise dominance– Two attributes,

Several alternatives may remain non-dominated– Additional preference statements to make

the feasible region smaller– Decision rules assist the DM in selection

of the most preferred one

0)]()(min[ jk xVxV

V

w1 0.4 0.7w20.6 0.3

x1 dominates x2

)( 1xV

)( 2xV

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and positive with some and positive with some feasible scores and weightsfeasible scores and weights


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Incomplete ordinal preference informationIncomplete ordinal preference information

Complete ordinal information is a complete rank-ordering of attributes or alternatives– Rankings are exactly known for each alternative– Leads to a convex set of feasible scores and weights, when interpreted as

incomplete preference information

The RICH (Rank Inclusion in Criteria Hierarchies) method– Incomplete ordinal statements about relative importance of attributes– ”Cost is the most important attribute”– ”Environmental factors is among the three most important attributes”– Several rank-orderings can be compatible with the preference statements

» e.g.: either attribute a1 or a2 is the most important of the three attributes aa33 is either the second or the least important one is either the second or the least important one

» may lead to non-convex feasible region of the attribute weights


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

important of the three attributes”

Calculation by dividing into compatible rank-orderings

– Extreme points readily computed– Lower bounds for weights wi b 0

Full support provided by RICH Decisions ©, http://www.decisionarium.hut.fi/

– Applications» evalution of risk management tools

(Ojanen et al. 2004)» support for setting priorities for a

research programme in wood material science (Salo and Liesiö 2004)

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The RICHER (RICH with Extended Rankings) methodThe RICHER (RICH with Extended Rankings) method

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

environmental factors”– ”Alternative x1 is not among the three most preferred ones”– ”Considering alternatives x1, x2 and x3, the least preferred with regard to

cost is x1”

Statements about attribute weights incorporated as well

Comparison to the RICH method– Suitable also for statements about alternatives– Computationally much more efficient– Includes all features of RICH– Allows evaluation within subsets– Applicable in conjunction with other preference programming methods


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Modeling of incomplete ordinal information (1/4)Modeling of incomplete ordinal information (1/4)

Rank-ordering –function r– Bijection from (sub)set of alternatives X’X (or (sub)set of attributes) to set of

rankings– E.g., r=(r(x1), r(x2), r(x3))=(1,3,2)

– The smaller the ranking, the better the alternative

» e.g., ”r(x4)=1 the ranking of x4 is 1, i.e. it is the most preferred”

– Several rank-orderings may be compatible with the preference information

– Incomplete ordinal statements about alternatives can be expressed with regard to different sets of attributes A’

» single attribute, (sub)set of attributes or holistic statements considering all attributes

» e.g., one can subject statements to cost and environmental factors together

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Non-normalized form of value function

Set of feasible values V includes score vectors v=(v(x1),..., v(xm)) – v(xk) denotes the value of xk with regard to some set of attributes

» e.g., if A’={a2}, then v(xk)=v2(x2k)

» e.g., if A’=A, then v(xk)=V(xk)

– Restricted by preference statements

Feasible region associated with a rank-ordering is convex

}',),()(if)()(|{)( ' XxxxrxrxvxvvrS jkjkjkX V

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Elicitation of the preference statements is carried out through an alternative set IX’X and a ranking set J{1,...,m’}, where m’=|X’|

– If |I||J|, the rankings in J are attained by alternatives in I– If |I|<|J|, the alternatives in I have their rankings in J– Sets subjected to X’ and A’ denoted by I(A’,X’) and J(A’,X’)

Examples

– x1 and x2 are among the three most preferred ones with regard to cost attribute a1. Now A’={a1} and X’=X, I({a1}; X)={x1,x2}, J({a1}; X)={1,2,3}

– Holistically (A’=A) the two least preferred are among x4,x5, x8,x9: I(A; X)= {x4,x5, x8,x9}, J(A; X)={m-1,m}

– Holistically the most preferred of the set X’={x1, x2, x7} is x1: I(A; X’)={x1}, J(A; X’)={1}


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Sets I and J lead to compatible rank-orderings R(I,J)

The feasible region associated with many compatible rank-orderings is usually non-convex

Statements can be given with regard to different attribute sets– Several rank-orderings may be compatible with each of these sets

)(),(),(rSJIS

JIRr


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Mixed integer linear programming model (1/5)Mixed integer linear programming model (1/5)

Overcoming the non-convexity

– Continuous ”milestone variable” zk distinguishes between the values of alternatives with rankings k and k+1

– If xj’s ranking is at most k, its value is at least zk and we let yk(xj)=1, else 0

– There are exactly k alternatives whose ranking is at most k» e.g., the three rankings 1, 2 and 3 are at most 3

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1z 2z 3z 4z 5z

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Mixed integer linear programming modelMixed integer linear programming model (2/5) (2/5)

Formally

For the sake of interpretational and computational matters, we set

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Adding preference statements into the model Assumption |J|≤|I|

– For all rankings jJ, the respective alternative belongs to I

Because of the uniqueness of the rankings, there is exactly one alternative in I, for which yj-1(xi)=0, and yj(xi)=1. For other alternatives, yj-1(xi) and yj(xi) get same values

E.g., I=(x1, x2, x5), J={2,4}, exactly one of the alternatives has the ranking 2 it is the only one with different values for y2(xi) and y1(xi)

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ij

Ixi

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Some milestone and binary variables and the respective constraints are redundant– Given a statement that alternatives x1 and x2 are the two most preferred,

for example variables z1, z3 and y1(xj), y3(xj) are not needed

» actually only z2 and y2(xj) are needed

If set J is ”sequential”, i.e., it constitutes of consecutive positive integers, the number of variables and constraints can be substantially decreased– For example sets {3,4,5} and {1} are sequential, set {1,3} not

For the compatible rank-orderings associated with sets I and J, |I||J|, it holds

k

i

k

iii JJJIRJIR

1 1

),,(),(


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Partitioning of J into a minimal number of sequential sets Ji– For example, J={1,2,6,7} is partitioned into J1={1,2} and J2={6,7}– At most 2 milestone and 2m’ binary variables needed to represent the

statement associated to a sequential set Ji

Representation of the feasible region S(I,J) as the intersection of the feasible regions S(I,Ji)– Constraints for all pairs I, Ji are set in the same model

If contradictionary to the assumption it holds |J|>|I|, the feasible region is constructed with the help of complement sets IC=X’\I and JC={1,...,m’}\J– Now |JC|≤|IC|– S(I,J)=S(IC,JC)

All linear inequalities can be included in the model


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An illustrative example (1/6)An illustrative example (1/6)

A company is about to choose the facility for a new office– Eight attributes relevant: a1: size of the office, a2: rental costs, a3: renovation

need, a4: car park opportunities, a5: means of communication, a6: distance to city center, a7: other facilities in the neighborhood, a8: habitability

– 12 alternatives

Public Distancealt. Area Rent / € Renovation need Car park Garage transport to center Other facilities Habitabilityx1 180 m2 2000 considerable 13 no quite bad 12 km intermediate greatx2 240 m2 3000 none 13 no good 15 km good badx3 210 m2 2800 intermediate 2 no great 0 km great goodx4 214 m2 2000 very small 13 yes bad 25 km intermediate goodx5 300 m2 3200 considerable 13 yes good 4 km good or great very goodx6 170 m2 1800 considerable 5 no good 0 km great goodx7 250 m2 2600 small / considerable 13 yes intermediate 7 km good intermediatex8 260 m2 2650 intermediate 10 no good 10 km intermediate intermediatex9 262 m2 2400 big 13 yes good 10 km intermediate very goodx10 241 m2 2500 small 11 yes intermediate 7 km good goodx11 198 m2 2200 considerable 13 no bad 17 km good goodx12 201 m2 2000 intermediate 7 no bad 22 km quite bad intermediate


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Attributes size, rental costs, car park opportunities and distance to city center are assessed through [0,1]-normalized scores or score intervals

alt. v1 v2 v4 v6x1 0.13 0.86 [0.90,0.95] [0.27,0.40]x2 0.66 0.14 [0.90,0.95] [0.17,0.30]x3 0.43 0.29 [0.20,0.30] [1.00,1.00]x4 0.47 0.86 [1.00,1.00] [0.00,0.00]x5 1.00 0.00 [1.00,1.00] [0.70,0.80]x6 0.00 1.00 [0.40,0.50] [1.00,1.00]x7 0.73 0.43 [1.00,1.00] [0.50,0.65]x8 0.79 0.39 [0.80,0.88] [0.30,0.45]x9 0.80 0.57 [1.00,1.00] [0.30,0.45]x10 0.67 0.50 [0.95,0.98] [0.50,0.65]x11 0.32 0.71 [0.90,0.95] [0.10,0.20]x12 0.35 0.86 [0.55,0.65] [0.01,0.05]


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Other information is turned into incomplete ordinal statements– E.g., alternative x2 is the only one with no renovation need (a3), hence the ranking 1

– E.g., alternative x2 is the least preferred w.r.t. habitability (a8)

attr. I J attr. I Ja3 {x2} {1} a7 {x3,x6} {1,2,3}

{x4} {2} {x5} {1,2,3,4,5,6,7}{x10} {3,4} {x2,x7,x10,x11} {3,4,5,6,7}{x7} {3,4,8,9,10,11,12} {x1,x4,x8,x9} {8,9,10,11}{x3,x8,x12} {4,5,6,7} {x12} {12}{x9} {7,8}{x1,x5,x6,x11} {8,9,10,11,12}

a5 {x3} {1} a8 {x1} {1}{x2,x5,x6,x8,x9} {2,3,4,5,6} {x5,x9} {2,3}{x7,x10} {7,8} {x3,x4,x6,x10,x11} {4,5,6,7,8}{x1} {9} {x7,x8,x12} {9,10,11}{x4,x11,x12} {10,11,12} {x2} {12}


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Information on attributes’ relative importance– Complete rank-ordering of the attributes is r(a1,a2,...,a8)=(1,2,...,8)

– A weight of 0.50 is assigned to the most important attribute, size of the office– Weights are lower bounded by wi 1/3n

A holistic preference for x4 over x1 over x3

)()()( 314 xVxVxV

24

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Pairwise bounds (minima of overall value differences) indicate that there are 5 non-dominated alternatives x5, x7, x8, x9 and x10

alternative xx1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12

PB(x1,x) - -0.263 0.000 -0.282 -0.428 -0.058 -0.350 -0.354 -0.323 -0.368 -0.089 -0.153PB(x2,x) 0.033 - 0.033 -0.123 -0.312 -0.025 -0.171 -0.178 -0.274 -0.199 -0.056 -0.048PB(x3,x) -0.040 -0.263 - -0.282 -0.428 -0.058 -0.350 -0.354 -0.323 -0.368 -0.089 -0.153PB(x4,x) 0.000 -0.244 0.000 - -0.376 0.007 -0.229 -0.236 -0.314 -0.264 -0.071 0.004PB(x5,x) 0.143 0.049 0.143 0.029 - 0.137 -0.051 -0.061 -0.155 -0.038 0.105 0.102PB(x6,x) -0.238 -0.390 -0.238 -0.397 -0.655 - -0.464 -0.481 -0.561 -0.482 -0.308 -0.263PB(x7,x) 0.076 -0.043 0.076 -0.079 -0.309 0.058 - -0.137 -0.217 -0.150 0.016 0.035PB(x8,x) 0.119 -0.089 0.119 -0.115 -0.215 0.103 -0.177 - -0.188 -0.192 0.067 0.039PB(x9,x) 0.172 -0.042 0.172 -0.028 -0.173 0.203 -0.129 -0.135 - -0.106 0.167 0.087PB(x10,x) 0.143 -0.070 0.143 -0.099 -0.239 0.085 -0.158 -0.067 -0.164 - 0.054 0.102PB(x11,x) -0.152 -0.298 -0.152 -0.310 -0.569 -0.150 -0.382 -0.388 -0.465 -0.399 - -0.183PB(x12,x) -0.115 -0.349 -0.115 -0.361 -0.480 -0.079 -0.433 -0.439 -0.418 -0.452 -0.177 -


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To discriminate between non-dominated alternatives, decision rules are applied

Each rule recommends alternative x5

XPress-MP was used in solving the example

Calculation of the example (pairwise bounds, overall value intervals for each 12 alternatives; 156 MILPs) took 14 seconds on a Pentium III at 800 MHz with 256 MB RAM

x5 x7 x8 x9 x10maximax 0.892 0.758 0.762 0.834 0.768maximin 0.571 0.505 0.511 0.556 0.485central values 0.732 0.632 0.637 0.695 0.627minimax regret 0.155 0.309 0.215 0.173 0.239


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ConclusionConclusion

The DM can give incomplete ordinal information about the alternatives with regard to a single attribute, a set of attributes or holistically

Statements about relative importance of attributes are allowed, as well

Based on a linear model and hence it can be used in conjunction with other preference programming methods

Computationally far more efficient than RICH, and more flexible as it contains all features of RICH

Software implementation of RICHER Decisions © ongoing

Future research directions – Modeling of classification procedures with RICHER methodology (cf. the example in this

presentation)– Application of RICHER methodology to voting or other group decision processes– Application of incompelete ordinal information in Robust Portfolio Modeling (RPM)


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Related referencesRelated referencesArbel, A., “Approximate Articulation of Preference and Priority DerivationApproximate Articulation of Preference and

Priority Derivation”, European Journal of Operations Research 43 (1989) 317-326.

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 (to appear).

Salo, A. ja R. P. Hämäläinen, "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 Programme”,

submitted manuscript.

Salo, A. and Punkka, A., “Rank Inclusion in Criteria Hierarchies”, European Journal of Operations Research (to appear).

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.

helsinki university of technology systems analysis laboratory richer – a method for exploiting...

Documents

n attributes

n incomplete information

rich n

attributes n calculation

n n additive value function

important attributes

attribute weights

nonconvex feasible region