A new efficient SAT formulation for learning NCS models : numerical - PowerPoint PPT Presentation

A new efficient SAT formulation for learning NCS models : numerical results Kh. Belahcène, O. Khaled, V. Mousseau, W. Ouerdane, A. Tlili DA2PL’2018

Introductory Example NCS Learning NCS model Computational study Discussion and conclusions Contents Introductory Example NonCompensatory Sorting (NCS) Learning NCS model SAT formulation based on coalitions SAT formulation based on pairwise separation Computational study Discussion and conclusions 2/25

Introductory Example NCS Learning NCS model Computational study Discussion and conclusions Introductory Example Project Category a b c d 5 6 6 5 ? p 1 p 2 3.5 1 3 9 ? p 3 7.5 2 1 3 ? p 4 2 8 2.5 7 ? p 5 3 8.5 3 8.5 ? p 6 8 4 1.5 1.5 ? ⋆ < 4 < 3 < 2 < 2 boundary between ⋆ and ⋆⋆ ⋆⋆ [4,7[ [3,8[ [2,5[ [2,8[ ⋆ ⋆ ⋆ ≥ 7 ≥ 8 ≥ 5 ≥ 8 boundary between ⋆⋆ and ⋆ ⋆ ⋆ 3/25

Introductory Example NCS Learning NCS model Computational study Discussion and conclusions Introductory Example 1 st phase : criterion-wise sorting project Category a b c d p 1 ? ⋆⋆ ⋆⋆ ⋆ ⋆ ⋆ ⋆⋆ p 2 ? ⋆ ⋆ ⋆⋆ ⋆ ⋆ ⋆ p 3 ? ⋆ ⋆ ⋆ ⋆ ⋆ ⋆⋆ p 4 ? ⋆ ⋆ ⋆ ⋆ ⋆⋆ ⋆⋆ p 5 ? ⋆ ⋆ ⋆ ⋆ ⋆⋆ ⋆ ⋆ ⋆ p 6 ? ⋆ ⋆ ⋆ ⋆⋆ ⋆ ⋆ < 4 < 3 < 2 < 2 boundary between ⋆ and ⋆⋆ ⋆ [4,7[ [3,8[ [2,5[ [2,8[ ⋆⋆ ≥ 7 ≥ 8 ≥ 5 ≥ 8 boundary between ⋆⋆ and ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ 4/25

Introductory Example NCS Learning NCS model Computational study Discussion and conclusions Introductory Example 2 nd phase : noncompensatory multi criteria aggregation Sufficient coalitions project Category a b c d p 1 ? ⋆⋆ ⋆⋆ ⋆ ⋆ ⋆ ⋆⋆ p 2 ? ⋆ ⋆ ⋆⋆ ⋆ ⋆ ⋆ ? p 3 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆⋆ p 4 ? ⋆ ⋆ ⋆ ⋆ ⋆⋆ ⋆⋆ p 5 ? ⋆ ⋆ ⋆ ⋆ ⋆⋆ ⋆ ⋆ ⋆ p 6 ? ⋆ ⋆ ⋆ ⋆⋆ ⋆ ⋆ Insufficient coalitions ◮ Getting an overall ⋆⋆ or ⋆ ⋆ ⋆ requires getting ⋆⋆ or ⋆ ⋆ ⋆ on a sufficient coalition of criteria ◮ Getting an overall ⋆ ⋆ ⋆ requires getting ⋆ ⋆ ⋆ on a sufficient coalition of criteria 5/25

Introductory Example NCS Learning NCS model Computational study Discussion and conclusions Introductory Example 2 nd phase : noncompensatory multi criteria aggregation Sufficient coalitions project Category a b c d p 1 ⋆⋆ ⋆⋆ ⋆ ⋆ ⋆ ⋆⋆ ⋆⋆ p 2 ⋆ ⋆ ⋆⋆ ⋆ ⋆ ⋆ ⋆ p 3 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆⋆ ⋆⋆ p 4 ⋆ ⋆ ⋆ ⋆ ⋆⋆ ⋆⋆ ⋆⋆ p 5 ⋆ ⋆ ⋆ ⋆ ⋆⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ p 6 ⋆ ⋆ ⋆ ⋆⋆ ⋆ ⋆ ⋆ Insufficient coalitions ◮ Getting an overall ⋆⋆ or ⋆ ⋆ ⋆ requires getting ⋆⋆ or ⋆ ⋆ ⋆ on a sufficient coalition of criteria ◮ Getting an overall ⋆ ⋆ ⋆ requires getting ⋆ ⋆ ⋆ on a sufficient coalition of criteria 6/25

Introductory Example NCS Learning NCS model Computational study Discussion and conclusions Contents Introductory Example NonCompensatory Sorting (NCS) Learning NCS model SAT formulation based on coalitions SAT formulation based on pairwise separation Computational study Discussion and conclusions 7/25

Introductory Example NCS Learning NCS model Computational study Discussion and conclusions The Noncompensatory Sorting Model (NCS) ◮ MCDA method based on outranking relations ◮ Characterized by [Bouyssou and Marchant, 2007] An object is assigned to a category if : ◮ It is better than the lower limit of the category on a sufficiently strong subset of criteria ◮ While this is not the case when comparing the object to the upper limit of the category 8/25

Introductory Example NCS Learning NCS model Computational study Discussion and conclusions The Noncompensatory Sorting Model (NCS) Simplest case : 2 categories ◮ 2 categories : Good ( G ), Bad ( B ) ◮ objects to be sorted : X = � i ∈N X i with N = { 1 , . . . , n } ◮ � i total preorder on X i ◮ limit profile b = ( b 1 , . . . , b n ) ◮ F = family of sufficient coalitions, which is a subset of 2 N up-closed by inclusion Assignment rule : For all x = ( x 1 , . . . , x n ) ∈ X x ∈ G { i ∈ N : x i � i b i } ∈ F iff 9/25

Introductory Example NCS Learning NCS model Computational study Discussion and conclusions The Noncompensatory Sorting Model (NCS) More than 2 categories : ◮ an ordered set C 1 ≺ · · · ≺ C p of p categories. ◮ objects to be sorted : X = � i ∈N X i with N = { 1 , . . . , n } ◮ � i total preorder on X i , i ∈ N , ◮ limit profiles b h = ( b h i � i b h − 1 1 , . . . , b h n ) such that b h , with i i ∈ N , h = 1 .. p − 1 ◮ b h is the upper limit of C h , and the lower limit of C h + 1 ◮ p − 1 embedded families of sufficient coalitions F 1 ⊆ F 2 ⊆ ... ⊆ F p − 1 (subsets of 2 N up-closed by inclusion). Assignment rule : For all x = ( x 1 , . . . , x n ) ∈ X i } ∈ F h and { i ∈ N : x i � i b h + 1 x ∈ C h { i ∈ N : x i � i b h ∈ F h + 1 } / iff i 10/25

Introductory Example NCS Learning NCS model Computational study Discussion and conclusions Variants of the NCS Model Preference parameters ◮ Nested intervals of values sufficient at level C h , (i.e, b 1 , b 2 , ..., b p − 1 ) ◮ Nested upsets of coalitions of criteria sufficient at level C h , (i.e, F 1 ⊆ F 2 ⊆ ... ⊆ F p − 1 ) Particular cases ◮ U c -NCS : using a Unique set of sufficient coalitions of criteria (i.e, F 1 = F 2 = ... = F p − 1 ) ◮ U v -NCS : using a Unique set of sufficient value (i.e, b 1 = b 2 = ... = b p − 1 ) ◮ k - NCS : representing sufficient coalitions with a k -additive capacity ◮ 1-U c -NCS = MR-Sort, [Leroy et al., 2011] 11/25

Introductory Example NCS Learning NCS model Computational study Discussion and conclusions Contents Introductory Example NonCompensatory Sorting (NCS) Learning NCS model SAT formulation based on coalitions SAT formulation based on pairwise separation Computational study Discussion and conclusions 12/25

Introductory Example NCS Learning NCS model Computational study Discussion and conclusions Learning NCS model Learning NCS using a MIP formulation [Leroy et al. 2011] ◮ Best restoration of the learning set ◮ Solvable for small instances only Learning NCS using an heuristic [Sobrie et al. 2015] ◮ No guarantee about the inferred model ◮ handle large learning sets Learning NCS using SAT formulations 13/25

Introductory Example NCS Learning NCS model Computational study Discussion and conclusions A SAT formulation based on coalitions Binary Variables ◮ x i , h , k : on criterion i is the value k sufficient at level h or not? ◮ y B : is coalition B ⊆ N sufficient or not? Clauses 1. Ascending scales : for all criteria, frontiers and ordered pairs of values, k < k ′ , x i , h , k ′ ∨ ¬ x i , h , k 2. Hierarchy of profiles : for all criteria, values and ordered pairs of frontiers, h < h ′ , x i , h , k ∨ ¬ x i , h ′ , k 3. Coalitions strength : for all ordered pairs of coalitions, B ⊂ B ′ , y B ′ ∨ ¬ y B 4. Alternatives are outranked by boundary above them : for all coalitions, frontiers and alternatives a assigned immediately below the frontier ( � i ∈ B ¬ x i , h , u i ) ∨ ¬ y B 5. Alternatives outrank the boundary below them : for all coalitions, frontiers and alternatives b assigned immediately below the frontier ( � i ∈ B x i , h , a i ) ∨ y N \ B 14/25

Introductory Example NCS Learning NCS model Computational study Discussion and conclusions Contents Introductory Example NonCompensatory Sorting (NCS) Learning NCS model SAT formulation based on coalitions SAT formulation based on pairwise separation Computational study Discussion and conclusions 15/25

Introductory Example NCS Learning NCS model Computational study Discussion and conclusions A SAT formulation based on pairwise separation The simplest case : 2 categories Good ( G ), Bad ( B ) α : X → { Good , Bad } : An assignment of alternatives to categories. Binary Variables ◮ x i , k : is the value k ∈ X i sufficiently good, i ∈ N , ◮ z i , g , b : for i ∈ N , g a good alternative ( g ∈ α − 1 ( Good ) ) and b a bad alternative ( b ∈ α − 1 ( Bad ) ), criterion i distinguishes g i and b i . Clauses 1. Ascending scales : k ′ � i k ( x i , k ′ ∨ ¬ x i , k ) � � i ∈N 2. Pairwise Separation : � i ∈N , g ∈ α − 1 ( Good ) , b ∈ α − 1 ( Bad ) ( ¬ z i , g , b ∨ ¬ x i , b i ) � i ∈N , g ∈ α − 1 ( Good ) , b ∈ α − 1 ( Bad ) ( ¬ z i , g , b ∨ x i , g i ) � g ∈ α − 1 ( Good ) , b ∈ α − 1 ( Bad ) ( � i ∈N z i , g , b ) 16/25

Introductory Example NCS Learning NCS model Computational study Discussion and conclusions A SAT formulation based on pairwise separation More than two categories 1. Ascending scales : k ′ � i k ∈ X ⋆ ( x i , h , k ′ ∨ ¬ x i , h , k ) � � i ∈N , h ∈ [ 2 .. p ] 2. Hierarchy of profiles : i ∈N , h < h ′ ∈ [ 2 .. p ] , k ∈ X ⋆ ( x i , h , k ∨ ¬ x i , h ′ , k ) � 3. Pairwise Separation : ∈ α − 1 ( G � h ) ( ¬ z i , h , g , b ∨ ¬ x i , h , b i ) � � g ∈ α − 1 ( G � h ) , b / i ∈N , h ∈ [ 2 .. p ] � � ∈ α − 1 ( G � h ) ( ¬ z i , h , g , b ∨ x i , h , g i ) i ∈N , h ∈ [ 2 .. p ] g ∈ α − 1 ( G � h ) , b / � � ∈ α − 1 ( G � h ) ( � i ∈N z i , h , g , b ) h ∈ [ 2 .. p ] g ∈ α − 1 ( G � h ) , b / 17/25