Robust ordinal regression for outranking methodsidss.cs.put.poznan.pl/site/fileadmin/seminaria/2010/kadzinski-seminar2011.pdfOutline Robust ordinal regression for outranking methods

Outline

Robust ordinal regressionfor outranking methods

Salvatore Greco1 Miłosz Kadzinski2

Vincent Mousseau3 Roman Słowinski2

1Faculty of Economics, University of Catania, Italy

2Institute of Computing Science, Poznan University of Technology, Poland

3Laboratoire de Genie Industriel, Ecole Centrale Paris, France

S. Greco, M. Kadzinski, V. Mousseau, R. Słowinski Robust ordinal regression for outranking methods

Outline

Outline

1 Introduction

2 Robust ordinal regression for outranking methods

3 Robust ordinal regression for group decision

4 Extreme ranking analysis

5 Representative set of parameters

6 Conclusions


IntroductionRobust ordinal regression for outranking methods

Robust ordinal regression for group decisionExtreme ranking analysis

Representative set of parametersConclusions

Multiple criteria problems

CharacteristicsActions described by evaluation vectorsFamily of criteria is supposed to satisfythe consistency conditions

RankingRank the actions from the best to theworst according to DM’s preferencesRanking can be complete or partial

ChoiceChoose the subset of best actions





Outranking relationDefinition

Outranking relation S groups three basic preferencerelations: S = {∼, .,�}aSb means “action a is at least as good as action b”Non-compensatory preference model usedin the ELECTRE family of MCDA methodsAccept incomparability, no completeness nor transitivityOutranking relation on set of actions A is constructed viaconcordance and discordance tests





Outranking relation

Concordance and discordance

Concordance test: checks if the coalitionof criteria concordant with the hypothesis aSbis strong enough:

C(a,b) =∑m

j=1 kj · Cj (a,b)/∑m

j=1 kj =[k1C1(a,b) + . . .+ kmCm(a,b)]/(k1 + . . .+ km)

Coalition is composed of two subsets of criteria:these being clearly in favor of aSb, i.e.Cj (a,b) = 1, if gj (a) ≥ gj (b)− qj ,these that do not oppose to aSb, i.e. suchthat b . a





Outranking relation


Since (k1 + . . .+ km) = 1, we can consider:

C(a,b) = ψ1(a,b) + . . .+ ψm(a,b),

where ψj (a,b) = kj · Cj (a,b), j = 1, . . . ,m,is a monotone, non-decreasing functionw.r.t. gj (a)− gj (b)

Concordance test is positive if: C(a,b) ≥ λ,where λ is a cutting level (concordance threshold)Cutting level λ is required to be not less than 0.5





Outranking relation


Discordance test: checks if among criteriadiscordant with the hypothesis aSb there isa strong opposition against aSb:

gj (b)− gj (a) ≥ vj (for gain-type criterion)gj (a)− gj (b) ≥ vj (for cost-type criterion)

Conclusion: aSb is true if and only if C(a,b) ≥ λand there is no criterion strongly opposed(making veto) to the hypothesisFor each couple (a,b) ∈ A× A, one obtainsrelation S either true (1) or false (0)





Outranking methods

Two major problems raised in the literatureElicitation of preference information:

Rather technical parameters, precise numerical valuesIntra-criterion parameters vs. inter criteria parametersDisaggregation-aggregation proceduresMainly in terms of sorting problems(e.g., ELECTRE TRI)

Robustness analysis:Examination of the impact of each parameteron the final outcomeIndication of the solutions which are good (bad) for differentinstances of a preference model





Robust ordinal regression

Main assumptionsTake into account all instances of a preference modelcompatible with the preference information given by the DMSupply the DM with two kinds of results:

necessary results specify recommendations worked out onthe basis of all compatible instances of a preference modelconsidered jointlypossible results identify all possible recommendationsmade by at least one compatible instance of a preferencemodel considered individually

Methods that use value function as a preference model:UTAGMS, GRIP, UTADISGMS (additive value function),robust ordinal regression applied to Choquet integral





Robust ordinal regression for outranking methods

QuestionsDoes a outrank b for all compatible outranking models?Does a outrank b for at least one compatible outrankingmodel?





ELECTREGKMS - Preference InformationPairwise comparisons

Set of pairwise comparisons of reference actions(a,b) ∈ BR ⊂ AR × AR

aSb or aSCb

Intra-criterion preference information

[qj,∗,q∗j ] - the range of indifference thresholdvalues allowed by the DM[pj,∗,p∗j ] - the range of preference thresholdvalues allowed by the DMa ∼j b ⇔ “the difference between gj (a) and gj (b)is not significant for the DM”a �j b ⇔ “the difference between gj (a) and gj (b)is significant for the DM”





ELECTREGKMS - Compatibility and Results

CompatibilityAn outranking model is called compatible, if it is able to restoreall pairwise comparisons from BR for provided impreciseintra-criterion preference information

The necessary and the possibleIn result, one obtains two outranking relations on set A, suchthat for any pair of actions (a,b) ∈ A× A:

1 a necessarily outranks b (aSNb)if a outranks b for all compatible outranking models

2 a possibly outranks b (aSPb)if a outranks b for at least one compatible outranking model





ELECTREGKMS - Compatible outranking model (1)

Set of concordance indices C(a,b), cutting levels λ,indifference qj , preference pj , and veto thresholds vj ,j = 1, . . . ,m, satisfying the foll. set of constraints EAR

:

If aSb for (a,b) ∈ BR

C(a,b) =∑m

j=1 ψj (a,b) ≥ λ

gj (b)− gj (a) + ε ≤ vj , j = 1, . . . ,m

If aSCb for (a,b) ∈ BR

C(a,b) =∑m

j=1 ψj (a,b) + ε ≤ λ+ M0(a,b)

gj (b)− gj (a) ≥ vj − δMj (a,b)

Mj (a,b) ∈ {0,1}, j = 0, . . . ,m,∑m

j=0 Mj (a,b) ≤ m

where δ is a big given value






1 ≥ λ ≥ 0.5, vj ≥ p∗j + ε

vj ≥ gj (b)− gj (a) + ε, vj ≥ gj (a)− gj (b) + ε if a ∼j b

normalization:∑mj=1 ψj (a∗j ,aj,∗) = 1

monotonicity: for all a,b, c,d ∈ A and j = 1, . . . ,m :

ψj (a,b) ≥ ψj (c,d) if gj (a)− gj (b) > gj (c)− gj (d)

ψj (a,b) = ψj (c,d) if gj (a)− gj (b) = gj (c)− gj (d)

EAR






partial concordance: for all (a,b) ∈ A× A and j = 1, . . . ,m :

[1] ψj (a,b) = 0 if gj (a)− gj (b) ≤ −p∗j[2] ψj (a,b) ≥ ε if gj (a)− gj (b) > −pj,∗

[3] ψj (a,b) + ε ≤ ψj (a∗j ,aj,∗) if gj (a)− gj (b) < −q∗j[4] ψj (a,b) = ψj (a∗j ,a,∗) if gj (a)− gj (b) ≥ −qj,∗

[1] ψj (a,b) = 0 if b �j a

[4] ψj (a,b) = ψj (a∗j ,aj,∗), ψj (b,a) = ψj (a∗j ,aj,∗) if a ∼j b

EAR





ELECTREGKMS - Preference information

Extensions of the preference model

Consideration of thresholds dependent on gj(a)(e.g., affine functions)Pairwise comparisons of reference actions in terms ofrelations of preference, indifference, or incomparability:

a � b, aIb, or a?bInter-criteria preference information:

Interval weights of the criteria kj (e.g., k1 > 0.2, k2 < 0.5)Pairwise comparisons of the weights of the criteriaInterval cutting level λ ∈ [λ∗, λ

∗] (e.g., λ > 0.75)Veto thresholds vj (e.g., v1 ∈ [8.2,9.8], v2 = 5.6, a ��3 b)





ELECTREGKMS - Checking the truth of SP

IdeaProve that aSb is possible in the set of all compatible outranking models

aSPb ⇔ ε∗ > 0

where: ε∗ = max ε

EAR

C(a,b) =∑m

j=1 ψi (a,b) ≥ λ

gj (b)− gj (a) + ε ≤ vj , j = 1, . . . ,m

ResultIf ε∗ > 0 and the set of constraints is feasible, then a outranks bfor at least one compatible outranking model (aSPb)





ELECTREGKMS - Checking the truth of SN

Idea

Prove that aSCb is not possible in the set of all compatible outrankingmodel

aSNb ⇔ ε∗ ≤ 0

where: ε∗ = max ε

EAR

C(a,b) =∑m

j=1 ψj (a,b) + ε ≤ λ+ M0(a,b)

gj (b)− gj (a) ≥ vj − δMj (a,b)

Mj (a,b) ∈ {0,1}, j = 0, . . . ,m,∑m

j=0 Mj (a,b) ≤ m

ResultIf ε∗ ≤ 0 or the set of constraints is infeasible, then a outranks bfor all compatible outranking model (aSNb)





ELECTREGKMS - Properties of relations SP and SN

Properties

SP and SN are reflexive, intransitive, and incompleteSP ⊇ SN

aSNb ⇔ not(aSCPb) and aSPb ⇔ not(aSCNb)From SN and SP , one can obtain indifference I, preference�= {P ∪Q}, and incomparability R, in a usual way, e.g.:

if aSNb and bSNb, then aINbif aSPb and not(bSPb), then a �P b

Possible relations between actions a and b for a singleinstance of compatible outranking model conditioned by thetruth or falsity of SN and SP , e.g.:

if aSNb and bSNa, then aINbif not(aSPb) and not(bSNa), then b �P a or bRPa





ELECTREGKMS - Partial conclusionsMain distinguishing features

Taking into account all instances of the outranking modelcompatible with the provided preference informationConsidering the marginal concordance functionsas general non-decreasing ones, defined in the “spirit”of ELECTRE methodsHandling of preference information composed of pairwisecomparisons and of imprecise intra-criterion preferenceinformationConstructing two relations on the set of actions:necessary and possible





ELECTREGKMS - Illustrative exampleProblem statement and given data

Actions: 10 buses originally considered by the teamof Professor Jacek Zak (FMT, PUT)Criteria: 5 criteria: Price (th. euro), Exploitation costs(th. zl/100k km), Comfort (pts), Safety (pts), Modernity (pts)Evaluation table:

Price Exploit. Comfort Safety Modern.Bus name [euro] costs [zl] [pts] [pts] [pts]Autosan 209 87.5 7.64 9.04 7.8Bova Futura 231 88 7.74 8.39 8.8Ikarus EAG 207 92 5.67 4.44 5.6Jelcz T 102 79.7 2.75 5.23 3.9Setra S315 266 89.8 5.08 7.62 6.1MAN Lion’s 239 83.4 5.18 7.07 4.8... ... ... ... ... ...





ELECTREGKMS - Illustrative exampleExample

Step1: Ask the DM for preference informationThe DM provides imprecise intra-criterion preference information:

qj,∗ q∗j pj,∗ p∗jPrice 5 10 20 40Exploitation 3 5 10 16Comfort 0.3 0.6 1.4 2.8Safety 0.2 0.4 1.0 2.0Modernity 0.3 0.7 1.8 2.8

... and pairwise comparisons:

MAN S Volvo, Neoplan S Bova

Volvo SC Autosan, Setra SC MAN





ELECTREGKMS - Illustrative exampleExampleStep 2: Determine the necessary and possible outranking relationsPossible outranking matrix:

SP1 A B I J T M E N R V

A 1 1 1 0 1 1 1 1 1 1B 1 1 1 0 1 1 1 1 1 1I 1 1 1 0 1 1 1 1 1 1J 1 1 1 1 1 1 1 1 1 1T 0 1 1 0 1 0 1 1 1 1M 1 1 1 1 1 1 1 1 1 1E 1 1 1 1 1 1 1 1 1 1N 1 1 1 1 1 1 1 1 1 1R 1 1 1 0 1 1 1 1 1 1V 0 1 1 0 1 1 1 1 1 1





ELECTREGKMS - Illustrative exampleExampleStep 2: Determine the necessary and possible outranking relationsNecessary outranking matrix - converg. index = |SP = SN |% = 0.42:

SN1 A B I J T M E N R V

A 1 1 1 0 1 0 1 1 1 1B 0 1 0 0 1 0 1 0 1 1I 0 0 1 0 0 0 0 0 0 0J 0 0 0 1 0 0 0 0 0 0T 0 0 0 0 1 0 0 0 0 0M 0 0 0 0 1 1 0 0 0 1E 0 1 0 0 1 1 1 0 0 1N 0 1 0 0 1 1 0 1 1 1R 0 0 0 0 0 0 0 0 1 0V 0 0 0 0 1 0 0 0 0 1





ELECTREGKMS - Illustrative exampleExample

Step 3: Incremental specification of pairwise comparisons:

Analyze the necessary (SN and SCN ) and possible (SP and SCP) resultsIn the following iterations state aSb or aSCb for pairs (a,b) ∈ A× A,for which the possible relation SP (or SCP) was true, but not thenecessary SN (or SCN ) one

The DM provides additional pairwise comparisons:

Mercedes S Ikarus, Ikarus S MAN

Bova SC Neoplan





ELECTREGKMS - Illustrative exampleExampleStep 4: Determine the necessary and possible outranking relationsPossible outranking matrix SP

1 ⊇ SP2 :

SP2 A B I J T M E N R V

A 1 1 1 0 1 1 1 1 1 1B 1 1 1 0 1 1 1 0 1 1I 0 0 1 0 1 1 1 0 1 1J 1 1 1 1 1 1 1 1 1 1T 0 1 1 0 1 0 0 0 1 0M 1 1 1 1 1 1 1 1 1 1E 1 1 1 1 1 1 1 1 1 1N 1 1 1 1 1 1 1 1 1 1R 1 1 1 0 1 1 1 1 1 1V 0 1 1 0 1 1 1 1 1 1





ELECTREGKMS - Illustrative exampleExampleStep 4: Determine the necessary and possible outranking relationsNecessary outranking matrix SN

1 ⊆ SN2 - converg. index = 0.56:

SN2 A B I J T M E N R V

A 1 1 1 0 1 1 1 1 1 1B 0 1 0 0 1 1 1 0 1 1I 0 0 1 0 0 1 0 0 0 0J 0 0 0 1 0 0 0 0 0 0T 0 0 0 0 1 0 0 0 0 0M 0 0 0 0 1 1 0 0 0 1E 0 1 1 0 1 1 1 0 0 1N 0 1 0 0 1 1 0 1 1 1R 0 1 0 0 1 0 0 0 1 1V 0 0 0 0 1 0 0 0 0 1





ELECTREGKMS - Basic exploitation proceduresRecommendation in case of choice problems

Identify kernel K N of the necessary outranking graph SN

Identify such a ∈ A : ∀b ∈ A, b 6= a it holds not(bSPa)

Recommendation in case of ranking problems

Net Flow Score: NFS(a) = strength(a)− weakness(a)





ELECTREGKMS - Illustrative exampleStep 4: Final recommendation





ELECTREGKMS - Extensions

Analysis of incompatibility

Associate a binary variable va,b with each couple of referenceactions (a,b) ∈ BR :

aSb ⇔ C(a,b) =∑m

j=1 ψj (a,b) + Mva,b ≥ λand gj (b)− gj (a) + ε ≤ vj (a) + Mva,b, j = 1, . . . ,m,

where M is a big positive value (transform EARinto EAR

v )

If va,b = 1, then the corresponding constraint is always satisfied

Identify a minimal subset of troublesome exemplary decisions:

min f =∑

(a,b)∈BR va,b, subject to EAR

v





ELECTREGKMS - Gradual confidence levelsValued possible and necessary outranking relations

BR1 ⊆ BR

2 ⊆ . . . ⊆ BRs - embedded sets of pairwise comparisons

SAR

1 ⊇ SAR

2 ⊇ . . . ⊇ SAR

s - sets of compatible outranking modelsLet θt be the confidence level assigned to pairwise comparisonsconcerning pairs ((a,b) ∈ BR

t and (a,b) /∈ BRt−1), t = 1, . . . , s :

1 = θ1 > θ2 > . . . > θs > 0

SNval : A× A→ {θ1, θ2, . . . , θs,0}:

if ∃t : aSNt b, then SN

val (a,b) = max{θt : aSNt b, t = 1, . . . , s}

if @t : aSNt b, then SN

val (a,b) = 0SP

val : A× A→ {1− θ1,1− θ2, . . . ,1− θs,1}:if ∃t : aSP

t b, then SPval (a,b) = min{1− θt : not(aSP

t b), t = 1, . . . , s}if ∀t : aSP

t b, then SPval (a,b) = 1





ELECTREGKMS - Illustrative exampleStep 5: Graph of the valued necessary relation after third iteration





The ELECTREGKMS −GROUP method

CharacteristicsSeveral DMs D = {d1, . . . ,ds}cooperate in a decision problemDMs share the same “description”of the decision problemsThe collective results (ranking or subsetof the best actions) should account forpreferences expressed by each DMAvoid discussions of DMson technical parametersReason in terms of necessaryand possible relations and coalitions





The ELECTREGKMS −GROUP method

CharacteristicsFor each dh ∈ D′ ⊆ D who expresses herindividual preferences as in ELECTREGKMS,calculate the necessary and possibleoutranking relationsWith respect to all DMs four situations areconsidered:

aSN,ND′ b : aSN

dhb for all dh ∈ D′

aSP,ND′ b : aSP

dhb for all dh ∈ D′

aSN,PD′ b : aSN

dhb for at least one dh ∈ D′

aSP,PD′ b : aSP

dhb for at least one dh ∈ D′





ELECTREGKMS −GROUP - Extensions

Consider preferences of DMs individuallyA is small, DMs have outlook of the whole set A,interrelated preferencesAnalyze statements of DMs individuallyExamine the spaces of agreement and disagreement

Consider preferences of DMs simultaneouslyA is numerous,DMs are experts only w.r.t. to its small disjoint subsetsCombine knowledge of DMs into preference informationof a single fictitious DM






Consider preferences of all DMs simultaneously

Suppose that SAR

D of compatible outrankingmodels is not emptyOne obtains two outranking relations:SND and SP

DDifference between SN

D and SN,ND

aSNDb ⇔ aSb for all outranking models

compatible with all preferences of all DMsfrom DaSN,ND b ⇔ aSb for all compatible

outranking models of each DM from DIf SAR

D 6= ∅, then for all a,b ∈ A,aSN,ND b ⇒ aSN

Db and and aSNDb ⇒ aSP,N

D b






Consider preferences of all DMs simultaneously

Suppose that SAR

D of compatible outrankingmodels is not emptyOne obtains two outranking relations:SND and SP

DDifference between SP

D and SP,PD

aSPDb ⇔ aSb for at least one outranking

model compatible with all preferences of allDMs from DaSP,PD b ⇔ aSb for at least one compatible

outranking model of at least one DM from DIf SAR

D 6= ∅, then for all a,b ∈ A,aSPDb ⇒ aSP,P

D b and aSPDb ⇒ aSP,N

D b





PROMETHEE - Main principlesPreference function and degree

Compute unicriterion preference degree for every pair ofactions: πj(a,b), j = 1, . . . ,mCompute global preference degree for every pair of actions(a,b) ∈ A× A:

π(a,b) =∑m

j=1 kj · πj(a,b),where kj is a weight expressing relative importance of gj





PROMETHEE - Main principlesOutranking flows

The positive outranking flow Φ+(a):

Φ+(a) = 1/(n − 1)∑

b∈A π(a,b)

The negative outranking flow Φ−(a):

Φ−(a) = 1/(n − 1)∑

b∈A π(b,a)

Net outranking flow:

Φ(a) = Φ+(a)− Φ−(a)

PROMETHEE-II:aPb if Φ(a) > Φ(b)

PROMETHEE-I:aPb if Φ+(a) ≥ Φ+(b) and Φ−(a) ≤ Φ−(b)

and Φ+(a)− Φ−(a) > Φ+(b)− Φ−(b)





PROMETHEEGKS - Compatible outranking model (1)

Pairwise comparisons (constr.): if aSSb for (a,b) ∈ BR : π(a,b) ≥ π(b,a)

Pairwise comparisons (exploit.): if aSEb for (a,b) ∈ BR : Φ(a) ≥ Φ(b)

Normalization:∑m

j=1 πj (a∗j ,aj,∗) = 1

Monotonicity: for all a,b, c,d ∈ A and j = 1, . . . ,m :

πj (a,b) ≥ πj (c,d) if gj (a)− gj (b) > gj (c)− gj (d)

πj (a,b) = πj (c,d) if gj (a)− gj (b) = gj (c)− gj (d)

EAR

S





PROMETHEEGKS - Compatible outranking model (2)

Partial preference: for all (a,b) ∈ A× A and j = 1, . . . ,m :

[1] πj (a,b) = 0 if gj (a)− gj (b) ≤ qj,∗

[2] πj (a,b) ≥ ε if gj (a)− gj (b) > q∗j[3] πj (a,b) + ε ≤ πj (a∗j ,aj,∗) if gj (a)− gj (b) < pj,∗

[4] πj (a,b) = πj (a∗j ,aj,∗) if gj (a)− gj (b) ≥ p∗j[1] πj (a,b) = 0, πj (b,a) = 0 if a ∼j b

[4] πj (a,b) = πj (a∗j ,aj,∗) if a �j b

EAR

S





Extreme ranking analysisMotivation

Binary relations vs. rankingComplete rankings are intuitive, easy to understand, and popularThe DM is interested in ranks and scores of the actionsExamine how different are all rankings compatiblewith preferences of the DMCompute the highest and the lowest rankattained by each actionCompute the best and the worst score of each action





Extreme ranking analysisThe highest rank in robust multiple criteria ranking

Assume that a ∈ A is in the top of the rankingIdentify the minimal subset of actions that aresimultaneously not worse than a:

min f posmax =

∑b∈A\{a}

vb

EAR

S

Φ(a) > Φ(b)−Mvb, ∀b ∈ A \ {a}

EAR

S,max

where M is a big positive valueP∗(a) of action a is indicated by (f pos

max + 1)





Extreme ranking analysisThe lowest rank in robust multiple criteria ranking

Assume that a ∈ A is in the bottom of the rankingIdentify the minimal subset of actions that aresimultaneously not better than a:

min f posmin =

∑b∈A\{a}

vb

EAR

S

Φ(b) > Φ(a)−Mvb, ∀b ∈ A \ {a}

EAR

S,min

where M is a big positive value and ε is a smallpositive valueP∗(a) of action a is indicated by (|A| − f pos

min )





Extreme ranking analysis

Extreme net outranking flows

The highest outranking net flow: Φ∗(a) = max Φ(a), s.t. EAR

S

The lowest outranking net flow: Φ∗(a) = min Φ(a), s.t. EAR

S





Extreme ranking analysis - ExtensionsIncremental specification of preference information

SAR

t ⊆ SAR

t−1, for all t = 2, . . . , s,

P∗t (a) ≥ P∗t−1(a) and P∗,t (a) ≤ P∗,t−1(a)

Φ∗t (a) ≤ Φ∗t−1(a) and Φ∗,t (a) ≥ Φ∗,t−1(a)

Interval orders

Preference �rank and indifference ∼rank relations w.r.t. the intervalsof ranking positions [P∗(a),P∗(a)], e.g.:

a �rank b ⇔ P∗(a) < P∗(b) and P∗(a) < P∗(b)

a ∼rank b ⇔ [P∗(a),P∗(a)] ⊂ [P∗(b),P∗(b)] or

[P∗(a),P∗(a)] ⊃ [P∗(b),P∗(b)]





Extreme ranking analysis - ExtensionsApplication to multiple criteria choice problems

Define additional conditions for being in BThe most appealing approach: B = {a ∈ A : P∗(a) = 1}Possible approach: B = {a ∈ A : P∗(a) ≤ 3 and P∗(a) ≤ |A|/2}Group actions: indifference classes which stemfrom the ranking based on the best (or the worst) positions

Application to multiple criteria problems of different type

Group decision in the spirit of “necessary and possible”:

PND(a) =

⋂dr∈D

Pdr (a) and PPD(a) =

⋃dr∈D

Pdr (a)

Set of parameters corresponding to the extreme ranks

Assume that a is ranked either at its best or its worst position





PROMETHEEGKS - Illustrative exampleProblem statement and given data - Liveability of cities

Actions: 15 cities originally considered by Financial TimesCriteria: 5 criteria: stability (g1), healthcare (g2), culture (g3),education (g4), infrastructure (g5)Evaluation table

City g1 g2 g3 g4 g5Vancouver 95.0 100.0 100.0 100.0 96.4Vienna 95.9 100.0 96.5 100.0 100.0Melbourne 95.0 100.0 95.1 100.0 100.0Toronto 100.0 100.0 97.2 100.0 89.3Calgary 100.0 100.0 89.1 100.0 96.4Osaka 90.0 100.0 93.5 100.0 96.4Tokyo 90.0 100.0 94.4 100.0 92.9Hong Kong 95.0 87.5 85.9 100.0 96.4. . . . . . . . . . . . . . . . . .





PROMETHEEGKS - Illustrative examplePreference infromationThe DM provides imprecise intra-criterion preference information:

qj,∗ q∗j pj,∗ p∗jStability 2 2 5 5Healthcare 3 5 6 8Culture 0 1 2 3Education 3 5 6 8Infrastracture 0 1 2 3

... and pairwise comparisons:

Vancouver � Vienna

Tokyo � Hong Kong

London � Seoul

Lagos � Port Moresby





PROMETHEEGKS - Illustrative exampleExtreme ranking results for PROMETHEEGKS

City P∗(a) P∗(a) Φ∗(a) Φ∗(a)Vancouver 1 2 13.999 6.727Vienna 2 4 10.727 6.000Melbourne 3 5 9.307 4.500Toronto 1 6 11.799 2.000Calgary 5 9 7.999 −7.999Osaka 4 8 7.999 0.500Tokyo 4 8 7.999 1.500Hong Kong 8 11 2.118 −4.999Singapore 8 12 1.571 −7.999London 3 11 10.999 −3.999New York 8 11 2.499 −3.999Seoul 11 12 −0.500 −6.999Lagos 13 14 −10.538 −12.999Port Moresby 14 15 −11.000 −13.999Harare 14 15 −10.000 −13.999





Representative set of parameters

MotivationAssign precise values to variables of the modelIdentify the representative set of parameters withoutloosing the advantage of knowing all compatible outrankingmodelsExtend robust ordinal regression in its capacity ofexplaining the final outputGet synthetic representation of the robust resultsExploit outranking relation for these parameters in order toobtain representative results






QuestionWhich set of parameters is representative for the set of all outrankingmodels compatible with the preference infromation?

Representativeness

In the sense of robustness preoccupation

“One for all, all for one”






ProcedurePre-defined targets built on results of robust ordinalregression and extreme ranking analysisEnhancement of differences between actions

Emphasize the evident advantageof some actions over the otherse.g., C(a,b) ≥N λ, C(a,b) <N λ,

Φ(a) >N Φ(b), P∗(a) < P∗(b)Reduce the ambiguity in the statement of such advantagee.g., C(a,b) ≥P λ and C(a,b) <P λ,

Φ(a) >P Φ(b) and Φ(b) >P Φ(a),P∗(a) < P∗(b) and P∗(a) > P∗(b)





Representative set of parametersProcedure

Targets consist in maximization or minimization of thedifference between scores of actions, e.g.:

max ε, such that Φ(a) ≥ Φ(b) + ε, if Φ(a) >N Φ(b)min δ, such that |Φ(a)− Φ(b)| ≤ δ,

if Φ(a) >P Φ(b) and Φ(b) >P Φ(a)

DM is left the freedom of assigning priorities to targetsTargets may be attained:

One after another, according to a given priority orderSo as to find a compromise solution, e.g.:max ε, such that Φ(a)− Φ(b) ≥ |Φ(c)− Φ(d)|+ εif Φ(a) >N Φ(b) and Φ(c) >P Φ(d) and Φ(d) >P Φ(c)






Innovation of this proposalDealing with consequences of preference informationprovided by the DMInvolving the DM in the process of specifying the targetsGeneral framework for considering a few targets indifferent configurationsGeneral monotonic functions taking all criteria values ascoordinates of characteristic pointsAutonomous method vs. confrontation with results ofrobust ordinal regression





Summary

New family of outranking-based methods

The preference information is used within a robust ordinalregression approach to build a complete set of compatibleoutranking models

Identification of possible and necessary consequences ofprovided information

Identification of extreme ranking results

Representative set of parameters built on relations definedw.r.t. the whole set of compatible preference models





Future research

The necessary and the possible for robust multiple criteriasorting for outranking methods

Eliciting and selecting compatible instances of thepreference model on the basis of rank related information


Documents

Robust ordinal regression for outranking methodsidss.cs.put.poznan.pl/site/fileadmin/seminaria/2010/kadzinski-seminar2011.pdfOutline Robust ordinal regression for outranking methods