Capacitive MR-Sort model Preference modeling and learning Olivier - PowerPoint PPT Presentation

Capacitive MR-Sort model Preference modeling and learning Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 1 École Centrale de Paris - Laboratoire de Génie Industriel 2 University of Mons - Faculty of engineering November 20, 2014 Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 - November 20, 2014 University of Mons - Ecole Centrale Paris 1 / 29

1 Introductory example 2 MR-Sort 3 Capacitive MR-Sort 4 Learning a Capacitive MR-Sort model 5 Experimentations 6 Comments and Conclusion Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 - November 20, 2014 University of Mons - Ecole Centrale Paris 2 / 29

Introductory example 1 Introductory example 2 MR-Sort 3 Capacitive MR-Sort 4 Learning a Capacitive MR-Sort model 5 Experimentations 6 Comments and Conclusion Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 - November 20, 2014 University of Mons - Ecole Centrale Paris 3 / 29

Introductory example Introductory example ◮ Admission/Refusal of student ◮ Students are evaluated in 4 courses ◮ Admission condition : score above 10/20 in all the courses of one the minimal winning coalitions. Minimal winning coalitions Maximal loosing coalitions ◮ {math, physics} ◮ {math, history} ◮ {math, chemistry} ◮ {physics, chemistry} ◮ {chemistry, history} ◮ {physics, history} Math Physics Chemistry History A/R James 11 11 9 9 A Marc 11 9 11 9 A Robert 9 9 11 11 A John 11 9 9 11 R Paul 9 11 9 11 R Pierre 9 11 11 9 R Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 - November 20, 2014 University of Mons - Ecole Centrale Paris 4 / 29

MR-Sort 1 Introductory example 2 MR-Sort 3 Capacitive MR-Sort 4 Learning a Capacitive MR-Sort model 5 Experimentations 6 Comments and Conclusion Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 - November 20, 2014 University of Mons - Ecole Centrale Paris 5 / 29

MR-Sort MR-Sort I Characteristics ◮ Allows to sort alternatives in ordered classes on basis of their performances on monotone criteria ◮ MCDA method based on outranking relations ◮ Simplified version of ELECTRE TRI Parameters ◮ Profiles performances ( b h , j for b 3 h = 1 , ..., p − 1 ; j = 1 , ..., n ) C 3 ◮ Criteria weights ( w j ≥ 0 for b 2 C 2 n = 1 , ..., n ) b 1 ◮ Majority threshold ( λ ) C 1 b 0 crit 1 crit 2 crit 3 crit 4 crit 5 Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 - November 20, 2014 University of Mons - Ecole Centrale Paris 6 / 29

MR-Sort MR-Sort II Parameters ◮ Profiles performances ( b h , j for b 3 C 3 h = 1 , ..., p − 1 ; j = 1 , ..., n ) b 2 ◮ Criteria weights ( w j ≥ 0 for C 2 b 1 n = 1 , ..., n ) C 1 ◮ Majority threshold ( λ ) b 0 crit 1 crit 2 crit 3 crit 4 crit 5 Assignment rule � � a ∈ C h ⇐ ⇒ w j ≥ λ and w j < λ j : a j ≥ b h − 1 , j j : a j ≥ b h , j Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 - November 20, 2014 University of Mons - Ecole Centrale Paris 7 / 29

MR-Sort MR-Sort applied to the examples ◮ Profile fixed at 10/20 on each criterion ◮ Admission condition : score above 10/20 in all the courses of one the minimal winning coalitions : ◮ {math, physics}  w math + w physics ≥ λ   ◮ {math, chemistry} ⇒ w math + w chemistry ≥ λ  ◮ {chemistry, history} w chemistry + w history ≥ λ  ◮ Maximal loosing coalitions : ◮ {math, history}  w math + w history < λ   ◮ {physics, chemistry} ⇒ w physics + w chemistry < λ  ◮ {physics, history} w physics + w history < λ  ◮ w math + w physics + w chemistry + w history = 1 Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 - November 20, 2014 University of Mons - Ecole Centrale Paris 8 / 29

MR-Sort MR-Sort applied to the examples ◮ Profile fixed at 10/20 on each criterion ◮ Admission condition : score above 10/20 in all the courses of one the minimal winning coalitions : ◮ {math, physics}  w math + w physics ≥ λ   ◮ {math, chemistry} ⇒ w math + w chemistry ≥ λ  ◮ {chemistry, history} w chemistry + w history ≥ λ  ◮ Maximal loosing coalitions : ◮ {math, history}  w math + w history < λ   ◮ {physics, chemistry} ⇒ w physics + w chemistry < λ  ◮ {physics, history} w physics + w history < λ  ◮ w math + w physics + w chemistry + w history = 1 ◮ w math + w physics ≥ λ and w chemistry + w history ≥ λ ⇒ λ ≤ 1 2 Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 - November 20, 2014 University of Mons - Ecole Centrale Paris 8 / 29

MR-Sort MR-Sort applied to the examples ◮ Profile fixed at 10/20 on each criterion ◮ Admission condition : score above 10/20 in all the courses of one the minimal winning coalitions : ◮ {math, physics}  w math + w physics ≥ λ   ◮ {math, chemistry} ⇒ w math + w chemistry ≥ λ  ◮ {chemistry, history} w chemistry + w history ≥ λ  ◮ Maximal loosing coalitions : ◮ {math, history}  w math + w history < λ   ◮ {physics, chemistry} ⇒ w physics + w chemistry < λ  ◮ {physics, history} w physics + w history < λ  ◮ w math + w physics + w chemistry + w history = 1 ◮ w math + w physics ≥ λ and w chemistry + w history ≥ λ ⇒ λ ≤ 1 2 ◮ w math + w history < λ and w physics + w chemistry < λ ⇒ λ > 1 2 Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 - November 20, 2014 University of Mons - Ecole Centrale Paris 8 / 29

MR-Sort MR-Sort applied to the examples ◮ Profile fixed at 10/20 on each criterion ◮ Admission condition : score above 10/20 in all the courses of one the minimal winning coalitions : ◮ {math, physics}  w math + w physics ≥ λ   ◮ {math, chemistry} ⇒ w math + w chemistry ≥ λ  ◮ {chemistry, history} w chemistry + w history ≥ λ  ◮ Maximal loosing coalitions : ◮ {math, history}  w math + w history < λ   ◮ {physics, chemistry} ⇒ w physics + w chemistry < λ  ◮ {physics, history} w physics + w history < λ  ◮ w math + w physics + w chemistry + w history = 1 ◮ w math + w physics ≥ λ and w chemistry + w history ≥ λ ⇒ λ ≤ 1 2 ◮ w math + w history < λ and w physics + w chemistry < λ ⇒ λ > 1 2 ◮ Impossible to represent all the coalitions with a MR-Sort model Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 - November 20, 2014 University of Mons - Ecole Centrale Paris 8 / 29

Capacitive MR-Sort 1 Introductory example 2 MR-Sort 3 Capacitive MR-Sort 4 Learning a Capacitive MR-Sort model 5 Experimentations 6 Comments and Conclusion Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 - November 20, 2014 University of Mons - Ecole Centrale Paris 9 / 29

Capacitive MR-Sort Capacitive MR-Sort Characteristic ◮ Take criteria interactions into account ◮ Improvement of the expressivity of the model ◮ Non Compensatory Sorting Model [Bouyssou and Marchant, 2007] Capacity ◮ F = { 1 , ..., n } : set of criteria ◮ A capacity is a function µ : 2 F → [ 0 , 1 ] such that : ◮ µ ( B ) ≥ µ ( A ) , for all A ⊆ B ⊆ F (monotonicity) ; ◮ µ ( ∅ ) = 0 and µ ( F ) = 1 (normalization). New assignment rule a ∈ C h ⇐ ⇒ µ ( { j ∈ F : a j ≥ b h − 1 , j } ) ≥ λ and µ ( { j ∈ F : a j ≥ b h , j } ) < λ Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 - November 20, 2014 University of Mons - Ecole Centrale Paris 10 / 29

Learning a Capacitive MR-Sort model 1 Introductory example 2 MR-Sort 3 Capacitive MR-Sort 4 Learning a Capacitive MR-Sort model 5 Experimentations 6 Comments and Conclusion Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 - November 20, 2014 University of Mons - Ecole Centrale Paris 11 / 29

Learning a Capacitive MR-Sort model Learning a Capacitive MR-Sort model - MIP I Mixed Integer Programming ◮ Objective : Finding a model compatible with as much example as possible ◮ MIP to learn an MR-Sort model in [Leroy et al., 2011] ◮ Limitation to 2-additive capacities ◮ For Capacitive MR-Sort, more constraints and binary variable are required Table: Max number of constraints MIP MR-Sort MIP Capacitive MR-Sort # binary variables n ( 2 m + 1 ) n ( 2 m + 1 + 2 m ( m + 1 )) 2 n ( 5 m + 1 ) + n ( p − 3 ) + 1 + 2 m ( n 2 + 1 ) + n 2 # constraints 2 n ( 5 m + 1 ) + n ( p − 3 ) + 1 ◮ Too much variables and constraints to be used with large datasets Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 - November 20, 2014 University of Mons - Ecole Centrale Paris 12 / 29

Learning a Capacitive MR-Sort model Learning a Capacitive MR-Sort model - MIP II Application to the introductory example ◮ Admission condition : score above 10/20 in all the courses of one these coalitions : ◮ {math, physics} ◮ {math, chemistry} ◮ {chemistry, history} ◮ MIP is able to find a model matching all the rules m ( J ) m ( J ) J J {math} 0 {math, physics} 0.3 {physics} 0 {math, chemistry} 0.3 {chemistry} 0 {math, history} 0 {history} 0 {physic, chemistry} 0 {physic, history} 0 {chemistry, history} 0.4 λ = 0 . 3 Olivier Sobrie 1 , 2 - Vincent Mousseau 1 - Marc Pirlot 2 - November 20, 2014 University of Mons - Ecole Centrale Paris 13 / 29