Lattice polynomial functions and their use in qualitative decision - PowerPoint PPT Presentation

. . Lattice polynomial functions and their use in qualitative decision making AAA83 . . . . . Miguel Couceiro Jointly with D. Dubois, J.-L. Marichal, T. Waldhauser, . . . University of Luxembourg March 2012

Decision making DM Main Problem: Model preference!

Decision making DM Main Problem: Model preference! Model: R on X 1 × · · · × X n is represented by f : X 1 × · · · × X n → X : ⇐ ⇒ f ( x ) ≤ f ( y ) x R y

Decision making DM Main Problem: Model preference! Model: R on X 1 × · · · × X n is represented by f : X 1 × · · · × X n → X : ⇐ ⇒ f ( x ) ≤ f ( y ) x R y Limitation: The role of local preferences is not explicit!

. . . . . . . . . . . . . . . − → y = A ( x 1 , . . . , x n ) Aggregation: x 1 , . . . , x n

− → y = A ( x 1 , . . . , x n ) Aggregation: x 1 , . . . , x n . Let X be a scale (bounded chain). . . . An aggregation function on X is a mapping A : X n → X such that: . . 1 A is order-preserving: for every x , y ∈ X n . x ≤ y = ⇒ A ( x ) ≤ A ( y ) . . . 2 A preserves the boundaries: x ∈ X n A ( x ) = inf X inf and x ∈ X n A ( x ) = sup X . sup . . . . . Traditionally: X is a real interval I ⊆ R , e.g., I = [ 0, 1 ] .

Aggregation in decision making DM Numerical representation of relations: f : X 1 × · · · × X n → I ⊆ R : x R y ⇐ ⇒ f ( x ) ≤ f ( y ) DM: Preference on criteria i is represented by a local utility function ϕ i : X i → I . Preference on X 1 × · · · × X n is represented by an overall utility function: F ( x 1 , . . . , x n ) : = A ( ϕ 1 ( x 1 ) , . . . , ϕ n ( x n )) where A : I n → I is an aggregation function.

Examples of aggregation functions: . . . 1 Arithmetic means: For x ∈ I n , AM ( x ) : = 1 n ∑ x i 1 ≤ i ≤ n . 2 Weighted arithmetic means: For x ∈ I n and ∑ w i = 1, . . WAM ( x ) : = ∑ w i x i 1 ≤ i ≤ n . . . 3 Choquet integrals: For x ∈ I n , ∧ ∑ C ( x ) : = a I · x i I ⊆{ 1, ... , n } i ∈ I

Qualitative decision making QDM In the qualitative approach: The underlying sets X 1 , . . . , X n and X are finite chains (ordinal scales), e.g., X = { very bad, bad, satisfactory, good, very good } QDM: Preference relation on X i is represented by ϕ i : X i → X . Preference relation on X 1 × · · · × X n is represented by F ( x 1 , . . . , x n ) : = A ( ϕ 1 ( x 1 ) , . . . , ϕ n ( x n )) where A : X n → X is an aggregation function.

Capacities Let X be a chain with least and greatest elements 0 and 1, respectively. A capacity is a mapping v : 2 [ n ] → X , [ n ] = { 1, . . . , n } , such that . . . 1 v ( I ) ≤ v ( J ) whenever I ⊆ J , . . . 2 v ( ∅ ) = 0 and v ([ n ]) = 1.

Order simplexes of X n Let σ be a permutation on [ n ] = { 1, . . . , n } ( σ ∈ S n ) { } x = ( x 1 , . . . , x n ) ∈ X n : x σ ( 1 ) � · · · � x σ ( n ) X n σ = Example: X = [ 0, 1 ] and n = 2 ✻ � � � x 1 � x 2 � � � � x 1 � x 2 � ✲ � � 2 ! = 2 permutations ( 2 simplexes )

Sugeno integral The (discrete) Sugeno integral on X w.r.t. v is defined by ∨ S v ( x ) : = v ( { σ ( i ) , . . . , σ ( n ) } ) ∧ x σ ( i ) i ∈ [ n ] { ( x 1 , . . . , x n ) ∈ X n : x σ ( 1 ) � · · · � x σ ( n ) } for every x ∈ X n σ = . . Example . . . If x 3 � x 1 � x 2 , then x σ ( 1 ) = x 3 , x σ ( 2 ) = x 1 , x σ ( 3 ) = x 2 , and S v ( x 1 , x 2 , x 3 ) = ( v ( { 1, 2, 3 } ) ∧ x 3 ) ∨ ( v ( { 1, 2 } ) ∧ x 1 ) ∨ ( v ( { 2 } ) ∧ x 2 ) � �� � . = 1 . . . .

Qualitative decision making QDM Setting: . . . 1 n criteria on finite chains X 1 , . . . , X n . . 2 scores in a common finite chain X by local utility functions . ϕ i : X i → X We will assume that each ϕ i is order-preserving . . . . 3 Preference relation on X 1 × · · · × X n is represented by F ( x 1 , . . . , x n ) : = A ( ϕ 1 ( x 1 ) , . . . , ϕ n ( x n )) where A : X n → X is a Sugeno integral. We shall refer to these overall utility functions as Sugeno utility functions.

. . . . . . . . . . . . Outline . . 1 Preliminaries: Sugeno integrals as lattice polynomial functions. . . . 2 Characterizations of lattice polynomial functions. .

. . . . . . Outline . . 1 Preliminaries: Sugeno integrals as lattice polynomial functions. . . . . 2 Characterizations of lattice polynomial functions. . . 3 Generalization of polynomial functions: Sugeno utility functions. . . . . 4 Sugeno utility functions: characterizations and factorizations.

Outline . . . 1 Preliminaries: Sugeno integrals as lattice polynomial functions. . . . 2 Characterizations of lattice polynomial functions. . . 3 Generalization of polynomial functions: Sugeno utility functions. . . . . 4 Sugeno utility functions: characterizations and factorizations. . . . 5 Axiomatic approach to qualitative decision-making QDM . . . . 6 Further research directions and open problems.

Preliminaries Let X be a distributive (finite) lattice with . . . 1 operations ∧ and ∨ , . . . 2 least and greatest elements 0 and 1, respectively. V G N D B

Lattice polynomial functions A (lattice) polynomial function (on X ) is any map p : X n → X , n ≥ 1, obtainable by finitely many applications of the rules: . . . 1 The projections x �→ x i , i ∈ [ n ] , and the constant functions x �→ c , c ∈ X , are polynomial functions. . 2 If f : X n → X and g : X n → X are polynomial functions, then f ∧ g . . and f ∨ g are polynomial functions. . Example . . . median ( x 1 , x 2 , x 3 ) = ( x 1 ∧ x 2 ) ∨ ( x 2 ∧ x 3 ) ∨ ( x 3 ∧ x 1 ) . . . . .

. . . . . . . . . Representations: Disjunctive Normal Form A function f : X n → X has a disjunctive normal form ( DNF ) if ∨ ( ∧ ) f ( x ) = a I ∧ x i . i ∈ I I ⊆ [ n ]

Representations: Disjunctive Normal Form A function f : X n → X has a disjunctive normal form ( DNF ) if ∨ ( ∧ ) f ( x ) = a I ∧ x i . i ∈ I I ⊆ [ n ] . Proposition (Goodstein’67) . . . A function p : X n → X is a polynomial function iff it has the DNF : ∨ ( ∧ ) p ( x ) = p ( 1 I ) ∧ x i i ∈ I I ⊆ [ n ] where 1 I denotes the “characteristic tuple” of I ⊆ [ n ] . . . . . .

. . . . . . . . . Sugeno integrals as lattice polynomial functions The Sugeno integral on a chain X w.r.t. v : 2 [ n ] → X is defined by ∨ S v ( x ) : = v ( { σ ( i ) , . . . , σ ( n ) } ) ∧ x σ ( i ) i ∈ [ n ] { ( x 1 , . . . , x n ) ∈ X n : x σ ( 1 ) � · · · � x σ ( n ) } for every x ∈ X n σ = .

Sugeno integrals as lattice polynomial functions The Sugeno integral on a chain X w.r.t. v : 2 [ n ] → X is defined by ∨ S v ( x ) : = v ( { σ ( i ) , . . . , σ ( n ) } ) ∧ x σ ( i ) i ∈ [ n ] { ( x 1 , . . . , x n ) ∈ X n : x σ ( 1 ) � · · · � x σ ( n ) } for every x ∈ X n σ = . . Theorem (Marichal) . . . A function q : X n → X is the Sugeno integral S v iff ∨ ( ∧ ) q ( x ) = v ( I ) ∧ x i . i ∈ I I ⊆ [ n ] Since, q ( 1 I ) = v ( I ) , and v ( ∅ ) = 0 and v ([ n ]) = 1, Sugeno integrals coincide with idempotent polynomial functions: q ( x , . . . , x ) = x . . . . . .

General properties of polynomial functions . Fact . . . Every polynomial function (in part., Sugeno integral) is order-preserving . . . . . . . However... . . . The function f ( 0 ) = f ( a ) = 0 and f ( 1 ) = 1 is order-preserving on { 0, a , 1 } , but it is not a polynomial function, hence not a Sugeno integral! . . . . .

Median decomposability (Marichal) x c For c ∈ X and i ∈ [ n ] , set i = ( x 1 , . . . , x i − 1 , c , x i + 1 , . . . , x n ) . A function f : X n → X is median decomposable if for each i ∈ [ n ] ( ) f ( x 0 i ) , x i , f ( x 1 for every x ∈ X n . f ( x ) = median i ) ,

Median decomposability (Marichal) x c For c ∈ X and i ∈ [ n ] , set i = ( x 1 , . . . , x i − 1 , c , x i + 1 , . . . , x n ) . A function f : X n → X is median decomposable if for each i ∈ [ n ] ( ) f ( x 0 i ) , x i , f ( x 1 for every x ∈ X n . f ( x ) = median i ) , t = f ( x 1 i ) s = f ( x 0 i )

. . . . . . . . . . . . . . . Characterization of polynomial functions . Fact . . . Every median decomposable function is order-preserving. . . . . .

Characterization of polynomial functions . Fact . . . Every median decomposable function is order-preserving. . . . . . . Theorem (Marichal) . . . A function p : X n → X is . . . 1 a polynomial function iff it is median decomposable. . . . 2 a Sugeno integral iff it is idempotent and median decomposable. . . . . .

General characterization of lattice polynomial classes . General criterion (C. & Marichal) . . . Let C be a class of functions such that (i) the unary members of C are polynomial functions; (ii) any g : X → X obtained from f : X n → X ∈ C by fixing n − 1 arguments is in C . Then C is a class of polynomial functions. . . . . .

. . . . . . . . . Extensions: pseudo-polynomial functions Let X : = X 1 × · · · × X n , where each X i is a finite distributive lattice.