[PPT] - Ranking with Multiple reference Points Efficient Elicitation and PowerPoint Presentation

SLIDE 1

Ranking with Multiple reference Points

Efficient Elicitation and Learning Procedure Khaled Belahcène1 - Vincent Mousseau1 - Wassila Ouerdane1 - Marc Pirlot2 - Olivier Sobrie2

1CentraleSupélec - Université Paris-Saclay 2University of Mons - Faculty of engineering

June 21, 2019

University of Mons

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1 Introduction 2 Ranking with Multiple reference Points (RMP) 3 Inferring the parameters of an RMP model 4 MAX-SAT formulation for inferring an RMP Model 5 Experimental results 6 Conclusion and further research

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Introduction

1 Introduction 2 Ranking with Multiple reference Points (RMP) 3 Inferring the parameters of an RMP model 4 MAX-SAT formulation for inferring an RMP Model 5 Experimental results 6 Conclusion and further research

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SLIDE 4

Introduction

Ranking alternatives/objects

Problem

◮ Ranking alternative/object by preference ◮ e.g. ranking of cars

≻ ≻

MCDA ranking methods/models

◮ UTilités Additives (UTA) ◮ ELimination and Choice Expressing REality (ELECTRE II) ◮ Ranking with Multiple reference Points (RMP)

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SLIDE 5

Ranking with Multiple reference Points (RMP)

1 Introduction 2 Ranking with Multiple reference Points (RMP) 3 Inferring the parameters of an RMP model 4 MAX-SAT formulation for inferring an RMP Model 5 Experimental results 6 Conclusion and further research

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K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019

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Ranking with Multiple reference Points (RMP)

Ranking with Multiple reference Points (RMP) I

C B A Brakes Comfort Price Acceleration

2 4 12 kA C 31 s. 8 8 18 kA C 28 s.

r 1 r 2

◮ Equi-important

criteria

◮ Reference points

R = {r 1, r 2} A A A B C A A A

≻

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SLIDE 7

Ranking with Multiple reference Points (RMP)

Ranking with Multiple reference Points (RMP) II

C B A Brakes Comfort Price Acceleration

2 4 12 kA C 31 s. 8 8 18 kA C 28 s.

r 1 r 2

◮ Equi-important

criteria

◮ Reference points

R = {r 1, r 2} A A A B C A A A B A B A

≻ ≻ ≻

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SLIDE 8

Ranking with Multiple reference Points (RMP)

Ranking with Multiple reference Points (RMP) III

Some characteristics of RMP

◮ Model introduced by Antoine Rolland (Rolland, 2013) ◮ Transitivity ensured

≻ ≻ ≻

◮ Safe regarding rank-reversal

≻ ≻ ≻

◮ No need for commensurate scales

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SLIDE 9

Inferring the parameters of an RMP model

1 Introduction 2 Ranking with Multiple reference Points (RMP) 3 Inferring the parameters of an RMP model 4 MAX-SAT formulation for inferring an RMP Model 5 Experimental results 6 Conclusion and further research

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SLIDE 10

Inferring the parameters of an RMP model

Inferring the parameters of an RMP model I

RMP model

C B A

≻ ≻ ≻ ∼ ∼ ≻ ... Learning set Algorithm

Brakes Comfort Price Acceleration

2 4 12 kA C 31 s. 8 8 18 kA C 28 s.

r 1 r 2

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Inferring the parameters of an RMP model

Inferring the parameters of an RMP model II

Existing algorithms

◮ MIP-based algorithms (Zheng et al., 2012; Liu, 2016)

◮ S-RMP model (RMP with additive weights) ◮ Mixed Integer Program ◮ Minimization of Kemeny distance (Kemeny, 1959)

◮ Metaheuristic algorithm (Liu et al., 2014; Liu, 2016)

◮ S-RMP model (RMP with additive weights) ◮ Evolutionnary algorithm ◮ Reasonable computing time

Limitations of the existing algorithms

◮ Additive representation of criteria importance relation ◮ MIP only able to handle very limited datasets ◮ Metaheuristic cannot always restore a S-RMP model

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MAX-SAT formulation for inferring an RMP Model

1 Introduction 2 Ranking with Multiple reference Points (RMP) 3 Inferring the parameters of an RMP model 4 MAX-SAT formulation for inferring an RMP Model 5 Experimental results 6 Conclusion and further research

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SLIDE 13

MAX-SAT formulation for inferring an RMP Model

SAT formulation for inferring an RMP Model I

Boolean Satisfiability problem

◮ Boolean variables V ; ◮ Logical proposition about these variables f : {0, 1}V → {0, 1} ; ◮ SATisfiable if v∗ exists such that f (v∗) = 1 ◮ f can be expressed as conjunction of clauses C :

f =

c∈C c ; ◮ Each clause c ∈ C is a disjunction of their variables or their negation :

∀c ∈ C, ∃c+, c− ∈ P(V ) : c =

v∈c+ v ∨ v∈c− ¬v ; ◮ NP-complete problem BUT efficient SAT algorithms exist

SAT for learning an RMP model

◮ Expression of constraints as a SAT problem ◮ Limited to strict preferences (a ≻ b)

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SLIDE 14

MAX-SAT formulation for inferring an RMP Model

SAT formulation for inferring an RMP Model II

ϕ := ϕscales ∧ ϕprofiles ∧ ϕorder ∧ ϕoutranking ∧ ϕlexicography ∧ ϕpreference

1. ϕscales : Monotonicity of criteria scales

ϕscales :=

i∈N
k′<k∈Xi

(xi,h,k ∨ ¬xi,h,k′) ◮ xi,h,k : equal to 1 if value k above reference point rh on criterion i ◮ N : set of criteria indices ◮ Xi : set of values on criterion i

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SLIDE 15

MAX-SAT formulation for inferring an RMP Model

SAT formulation for inferring an RMP Model II

ϕ := ϕscales ∧ ϕprofiles ∧ ϕorder ∧ ϕoutranking ∧ ϕlexicography ∧ ϕpreference

2. ϕprofiles : Dominance of the profiles

ϕprofiles := ϕprofiles1 ∧ ϕprofiles2 ϕprofiles1 :=

h=h′∈H
i∈N
k∈Xi

(xi,h′,k ∨ ¬xi,h,k ∨ ¬dh,h′) ϕprofiles2 :=

h<h′∈H

(dh,h′ ∨ dh′,h) ◮ N : set of criteria indices ◮ Xi : set of values on criterion i ◮ H : set of reference points indices ◮ dh,h′ : equal to 1 if value if rh dominates rh′ ◮ xi,h,k : equal to 1 if value k above reference point rh on criterion i

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SLIDE 16

MAX-SAT formulation for inferring an RMP Model

SAT formulation for inferring an RMP Model II

ϕ := ϕscales ∧ ϕprofiles ∧ ϕorder ∧ ϕoutranking ∧ ϕlexicography ∧ ϕpreference

3. ϕorder : Order among criteria sets

ϕorder := ϕPareto ∧ ϕcompleteness ∧ ϕtransitivity ϕPareto :=

A⊆B∈P(N)

(yB,A) ϕcompleteness :=

A,B∈P(N)

(yA,B ∨ yB,A) ϕtransitivity :=

A,B,C∈P(N)

(¬yA,B ∨ ¬yB,C ∨ yA,C ) ◮ P(N) : set of possible criteria coalitions ◮ yA,B : equal to 1 if criteria coalition A is more important than criteria coalition B

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SLIDE 17

MAX-SAT formulation for inferring an RMP Model

SAT formulation for inferring an RMP Model II

ϕ := ϕscales ∧ ϕprofiles ∧ ϕorder ∧ ϕoutranking ∧ ϕlexicography ∧ ϕpreference

4. ϕoutranking : Outranking relation between pairs

ϕoutranking := ϕoutranking1 ∧ ϕoutranking2 ∧ ϕoutranking3 ϕoutranking1 :=

A,B∈P(N)
j∈J
h∈H

(

i /

∈A

xi,h,pj

i ∨

i∈B

¬xi,h,nj

i ∨ yA,B ∨ ¬zj,h)

◮ pj ≻ nj : pairwise comparison j ◮ J : set of pairwise comparisons indices ◮ P(N) : set of possible criteria coalitions ◮ H : set of reference points indices ◮ xi,h,k : equal to 1 if value k above reference point rh on criterion i ◮ yA,B : equal to 1 if criteria coalition A is more important than criteria coalition B ◮ zj : equals to 1 if criteria set on which pj above rh is more important than the criteria set

n which nj is above rh

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SLIDE 18

MAX-SAT formulation for inferring an RMP Model

SAT formulation for inferring an RMP Model II

ϕ := ϕscales ∧ ϕprofiles ∧ ϕorder ∧ ϕoutranking ∧ ϕlexicography ∧ ϕpreference

4. ϕoutranking : Outranking relation between pairs

ϕoutranking := ϕoutranking1 ∧ ϕoutranking2 ∧ ϕoutranking3 ϕoutranking2 :=

A,B∈P(N)
j∈J
h∈H

(

i /

∈A

xi,h,nj

i ∨

i∈B

¬xi,h,pj

i ∨ yA,B ∨ ¬z′

j,h)

◮ pj ≻ nj : pairwise comparison j ◮ J : set of pairwise comparisons indices ◮ P(N) : set of possible criteria coalitions ◮ H : set of reference points indices ◮ xi,h,k : equal to 1 if value k above reference point rh on criterion i ◮ yA,B : equal to 1 if criteria coalition A is more important than criteria coalition B ◮ z′

j : equals to 1 if criteria set on which nj above rh is more important than the criteria set

n which pj is above rh

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SLIDE 19

MAX-SAT formulation for inferring an RMP Model

SAT formulation for inferring an RMP Model II

ϕ := ϕscales ∧ ϕprofiles ∧ ϕorder ∧ ϕoutranking ∧ ϕlexicography ∧ ϕpreference

4. ϕoutranking : Outranking relation between pairs

ϕoutranking := ϕoutranking1 ∧ ϕoutranking2 ∧ ϕoutranking3 ϕoutranking3 :=

A,B∈P(N)
j∈J
h∈H

(

i∈A

¬xi,h,pj

i ∨

i /

∈A

xi,h,pj

i ∨

i∈B

¬xi,h,nj

i ∨

i /

∈B

xi,h,nj

i

∨ ¬yB,A ∨ z′

j,h)

◮ pj ≻ nj : pairwise comparison j ◮ J : set of pairwise comparisons indices ◮ P(N) : set of possible criteria coalitions ◮ H : set of reference points indices ◮ xi,h,k : equal to 1 if value k above reference point rh on criterion i ◮ yA,B : equal to 1 if criteria coalition A is more important than criteria coalition B ◮ z′

j : equals to 1 if criteria set on which nj above rh is more important than the criteria set

n which pj is above rh

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SLIDE 20

MAX-SAT formulation for inferring an RMP Model

SAT formulation for inferring an RMP Model II

ϕ := ϕscales ∧ ϕprofiles ∧ ϕorder ∧ ϕoutranking ∧ ϕlexicography ∧ ϕpreference

5. ϕlexicography : Lexicography of reference points

ϕlexicography := ϕlexicography1 ∧ ϕlexicography2 ∧ ϕlexicography3 ϕlexicography1 :=

j∈J
h≤h′∈H

(zj,h ∨ ¬sj,h′) ϕlexicography2 :=

j∈J
h<h′∈H

(z′

j,h ∨ ¬sj,h′)

ϕlexicography3 :=

h∈H

(¬z′

j,h ∨ ¬sj,h)

◮ J : set of pairwise comparisons indices ◮ H : set of reference points indices ◮ zj : equals to 1 if criteria set on which pj above rh is more important than the criteria set

n which nj is above rh

◮ z′

j : equals to 1 if criteria set on which nj above rh is more important than the criteria set

n which pj is above rh

◮ sj,h : equals to 1 if pj indifferent to nj for all reference points rh′ such that h′ < h and strictly outranks nj at reference point rh

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SLIDE 21

MAX-SAT formulation for inferring an RMP Model

SAT formulation for inferring an RMP Model II

ϕ := ϕscales ∧ ϕprofiles ∧ ϕorder ∧ ϕoutranking ∧ ϕlexicography ∧ ϕpreference

6. ϕpreference : Strict preference

ϕpreference :=

j∈J

(

h∈H

sj,h) ◮ J : set of pairwise comparisons indices ◮ H : set of reference points indices ◮ sj,h : equals to 1 if pj indifferent to nj for all reference points rh′ such that h′ < h and strictly outranks nj at reference point rh

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SLIDE 22

MAX-SAT formulation for inferring an RMP Model

MAX-SAT formulation

Goal

◮ Handle incompatibilities in the learning set

(e.g. a ≻ b and b ≻ a) How ?

◮ By allowing the violation of some clauses ◮ Weighting the clauses :

◮ Some cannot be violated (e.g. monotonicity of the scales)

⇒ Huge weights for these clauses

◮ Others can be violated (e.g. pairwise comparison)

⇒ Small weights for these clauses

MAX-SAT solvers

◮ Lot of different ones (MAX-SAT competition every year) ◮ Chosen solvers : MaxHS (Davies, 2013), maxino (Alviano et al., 2015).

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Experimental results

1 Introduction 2 Ranking with Multiple reference Points (RMP) 3 Inferring the parameters of an RMP model 4 MAX-SAT formulation for inferring an RMP Model 5 Experimental results 6 Conclusion and further research

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Experimental results

Experimental results I

M0

(3, 5, 2, 7) ? (2, 7, 3, 9) (1, 2, 3, 4) ? (4, 1, 2, 1) (4, 3, 6, 5) ? (2, 7, 5, 9) (5, 9, 7, 6) ? (2, 8, 3, 1) (2, 6, 4, 7) ? (2, 7, 1, 4) ...

Pairs of alternatives

(3, 5, 2, 7) ≻ (2, 7, 3, 9) (1, 2, 3, 4) ≺ (4, 1, 2, 1) (4, 3, 6, 5) ≻ (2, 7, 5, 9) (5, 9, 7, 6) ≺ (2, 8, 3, 1) (2, 6, 4, 7) ≻ (2, 7, 1, 4) ...

Learning set

(3, 5, 2, 7) ≻ (2, 7, 3, 9) (1, 2, 3, 4) ≻ (4, 1, 2, 1) (4, 3, 6, 5) ≻ (2, 7, 5, 9) (5, 9, 7, 6) ≺ (2, 8, 3, 1) (2, 6, 4, 7) ≻ (2, 7, 1, 4) ...

Learning set Mlearned SAT-RMP

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Experimental results

Experimental results II

Computing time

200 400 600 800 1,000 101 102 103 104 Number of pairwise comparisons Time (in seconds) 3 criteria 4 criteria 5 criteria 6 criteria

◮ 2 reference points ◮ Computing time grows fast when the number of criteria increases ◮ More efficient than MIP based algorithms

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Experimental results

Experimental results III

Computing time

200 400 600 800 1,000 101 102 103 Number of pairwise comparisons Time (in seconds) 1 ref. pt. 2 ref. pt. 3 ref. pt.

◮ 5 criteria ◮ Computing time also significantly impacted by the number of reference

points

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Experimental results

Experimental results IV

M0 Mlearned

(3, 5, 2, 7) ? (2, 7, 3, 9) (1, 2, 3, 4) ? (4, 1, 2, 1) (4, 3, 6, 5) ? (2, 7, 5, 9) (5, 9, 7, 6) ? (2, 8, 3, 1) (2, 6, 4, 7) ? (2, 7, 1, 4) ...

Test set

(3, 5, 2, 7) ≻ (2, 7, 3, 9) (1, 2, 3, 4) ≺ (4, 1, 2, 1) (4, 3, 6, 5) ≻ (2, 7, 5, 9) (5, 9, 7, 6) ≺ (2, 8, 3, 1) (2, 6, 4, 7) ≻ (2, 7, 1, 4) ...

Comparisons

(3, 5, 2, 7) ≻ (2, 7, 3, 9) (1, 2, 3, 4) ≻ (4, 1, 2, 1) (4, 3, 6, 5) ≻ (2, 7, 5, 9) (5, 9, 7, 6) ≺ (2, 8, 3, 1) (2, 6, 4, 7) ≺ (2, 7, 1, 4) ...

Comparisons

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Experimental results

Experimental results V

Model retrieval

200 400 600 800 1,000 0.7 0.8 0.9 1 Number of pairwise comparisons Prediction accuracy 3 criteria 4 criteria 5 criteria 6 criteria

◮ 2 reference points ◮ Accuracy above 90% with barely 300 pairwise comparisons

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Experimental results

Experimental results VI

Model retrieval

200 400 600 800 1,000 0.7 0.8 0.9 1 Number of pairwise comparisons Prediction accuracy 1 ref. pt. 2 ref. pt. 3 ref. pt.

◮ 5 criteria ◮ Number of reference points hasn’t lot of impact on the accuracy when

the number of pairwise comparisons is greater than 300

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SLIDE 30

Experimental results

Experimental results VII

M0

(3, 5, 2, 7) ? (2, 7, 3, 9) (1, 2, 3, 4) ? (4, 1, 2, 1) (4, 3, 6, 5) ? (2, 7, 5, 9) (5, 9, 7, 6) ? (2, 8, 3, 1) (2, 6, 4, 7) ? (2, 7, 1, 4) ...

Pairs of alternatives

(3, 5, 2, 7) ≻ (2, 7, 3, 9) (1, 2, 3, 4) ≺ (4, 1, 2, 1) (4, 3, 6, 5) ≻ (2, 7, 5, 9) (5, 9, 7, 6) ≺ (2, 8, 3, 1) (2, 6, 4, 7) ≻ (2, 7, 1, 4) ...

Learning set

(3, 5, 2, 7) ≻ (2, 7, 3, 9) (1, 2, 3, 4) ≻ (4, 1, 2, 1) (4, 3, 6, 5) ≻ (2, 7, 5, 9) (5, 9, 7, 6) ≺ (2, 8, 3, 1) (2, 6, 4, 7) ≺ (2, 7, 1, 4) ...

Learning set* Mlearned MAXSAT-RMP

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Experimental results

Experimental results VIII

Tolerance for errors

10 20 30 100 102 104 Percentage of errors Time (in seconds) 3 criteria 4 criteria 5 criteria

◮ 1 profile ◮ 500 pairwise comparisons ◮ Computing time increases quickly when the number of inversion

increases

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SLIDE 32

Experimental results

Experimental results IX

Tolerance for errors

10 20 30 0.96 0.98 1 Percentage of errors Prediction accuracy 3 criteria 4 criteria 5 criteria

◮ 1 profiles ◮ 500 pairwise comparisons ◮ MAX-SAT formulation identifies errors

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SLIDE 33

Conclusion and further research

1 Introduction 2 Ranking with Multiple reference Points (RMP) 3 Inferring the parameters of an RMP model 4 MAX-SAT formulation for inferring an RMP Model 5 Experimental results 6 Conclusion and further research

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SLIDE 34

Conclusion and further research

Conclusion

◮ Efficient formulation for problems involving less than 7 criteria (1000

pairwise comparisons)

◮ Computing time increases when there are errors in the learning set

Further research

◮ Support for indifference ◮ Formalization of SAT clauses (demonstration)

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SLIDE 35

İlginiz için teşekkürler !

(Thank you for your attention !)

SLIDE 36

References

References I

Alviano, M., Dodaro, C., and Ricca, F. (2015). A MaxSAT algorithm using cardinality constraints of bounded size. In Proceedings of the 24th International Joint Conference on Artificial Intelligence (IJCAI 2015), pages 92677–2683. Davies, J. (2013). Solving MAXSAT by Decoupling Optimization and Satisfaction. PhD thesis, University of Toronto, Department of Computer Science. Kemeny, J. G. (1959). Mathematics without numbers. Daedalus, 88(4) :577–591. Liu, J. (2016). Preference elicitation for multi-criteria ranking with multiple reference points. PhD thesis, Université Paris-Saclay, CentraleSupélec.

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SLIDE 37

References

References II

Liu, J., Ouerdane, W., and Mousseau, V. (2014). A metaheuristic approach for preference learning in multicriteria ranking based on reference points. In Proceeding of the 2nd wokshop from multiple criteria Decision aid to Preference Learning (DA2PL), pages 76–86, Chatenay-Malabry, France. Rolland, A. (2013). Reference-based preferences aggregation procedures in multi-criteria decision making. European Journal of Operational Research, 225(3) :479–486. Zheng, J., Rolland, A., and Mousseau, V. (2012). Preference elicitation for a ranking method based on multiple reference profiles. Technical report, Laboratoire Génie Industriel, Ecole Centrale Paris. Research report 2012-05.

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