UTA-poly and UTADIS-poly: using polynomial marginal utility - PowerPoint PPT Presentation

UTA-poly and UTADIS-poly: using polynomial marginal utility functions in UTA and UTADIS Olivier Sobrie 1 , 2 - Nicolas Gillis 2 - Vincent Mousseau 1 - Marc Pirlot 2 1 École Centrale de Paris - Laboratoire de Génie Industriel 2 University of Mons - Faculty of engineering July 15, 2015 University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 15, 2015 1 / 26

1 UTA methods 2 Motivations for UTA-poly and UTADIS-poly 3 UTA-poly and UTADIS-poly 4 Experiments 5 Conclusion University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 15, 2015 2 / 26

UTA methods Additive utility function model ◮ A marginal utility function u j is associated to each criterion j ◮ Marginal utility functions are monotonic ◮ Marginal utility functions are normalized between 0 and 1, s.t. u j ( g j ) = 0 and u j ( g j ) = 1 ◮ A weight w j is associated to each criterion j , s.t. � j w j = 1 u j 1 ◮ Utility of an alternative a : u j ( a j ) n � U ( a ) = w j · u j ( a j ) j = 1 0 g j j + a j g j University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 15, 2015 3 / 26

UTA methods Additive utility function model ◮ A marginal utility function u j is associated to each criterion j ◮ Marginal utility functions are monotonic ◮ Marginal utility functions are normalized between 0 and 1, s.t. u j ( g j ) = 0 and u j ( g j ) = 1 ◮ A weight w j is associated to each criterion j , s.t. � j w j = 1 u ∗ j w j ◮ We also have : u ∗ j ( a ) = w j · u j ( a j ) and u ∗ j ( g j ) = w j ◮ Utility of an alternative a : u ∗ j ( a j ) n n � � u ∗ U ( a ) = w j · u j ( a j ) = j ( a j ) j = 1 j = 1 0 g j j + a j g j University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 15, 2015 3 / 26

UTA methods UTA : Presentation ◮ Not easy to elicit directly the marginal utility functions ◮ Disaggregation procedure proposed by [Jacquet-Lagrèze and Siskos, 1982] ◮ Utility functions are computed on basis of a ranking given in input ◮ Linear programming University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 15, 2015 4 / 26

UTA methods UTA - Constraints I ◮ Two type of information (pairwise comparison) : 1. a is preferred to b , i.e. U ( a ) > U ( b ) ⇒ ( a , b ) ∈ P 2. a is indifferent to b , i.e. U ( a ) = U ( b ) ⇒ ( a , b ) ∈ I ◮ A potential error is introduced for each alternative utility U ( a ) , s.t. U ′ ( a ) = U ( a ) + σ + ( a ) − σ − ( a ) ◮ Constraints of the linear program :  U ( a ) − U ( b ) + σ + ( a ) − σ − ( a )   − σ + ( b ) + σ − ( b ) 0  > ∀ ( a , b ) ∈ P ,    U ( a ) − U ( b ) + σ + ( a ) − σ − ( a )     − σ + ( b ) + σ − ( b )  = 0 ∀ ( a , b ) ∈ I ,  � n j = 1 u ∗ j ( g j ) = 1 ,  � n  j = 1 u ∗ j ( g j ) = 0 ,     σ + ( a ) , σ − ( a )  0 ∀ a ∈ A ∗ , ≥     u ∗ monotonic ∀ j ∈ N .  j University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 15, 2015 5 / 26

UTA methods UTA - Constraints II ◮ Monotonicity is ensured by using piecewise linear functions for the marginal utility functions u ∗ j ◮ Domain of the criterion split in k equal w j + u ∗ 3 parts + j u ∗ 2 + j ◮ Position of the g l j , for l = 0 , . . . , k fixed a u ∗ j ( a j ) u ∗ 1 priori (equidistant) + j 0 g j j + + + + + a j g 1 g 2 g 3 g j j j j ◮ Marginal utility value of an alternative a : � � � a j − g L − 1 � j u ∗ j ( a ) = u ∗ L − 1 u ∗ L − u ∗ L − 1 + j j j j − g L − 1 g L j with g L j the first breakpoint s.t. a j ≤ g L j University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 15, 2015 6 / 26

UTA methods UTADIS ◮ Sorting problems ◮ Comparison of alternatives to thresholds delimiting the categories ◮ Disaggregation though linear programming ◮ Similar approach as for UTA : utility functions are modelled through piecewise linear functions University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 15, 2015 7 / 26

Motivations for UTA-poly and UTADIS-poly Motivations for UTA-poly and UTADIS-poly I Drawbacks of UTA methods ◮ Marginal utility functions are not natural close to the breakpoints of the piecewise linear functions ◮ Breakpoints at pre-defined position : limit the flexibility of the model Existing works ◮ Bugera, V., Konno, H., and Uryasev, S. (2002). Credit cards scoring with quadratic utility functions. Journal of Multi-Criteria Decision Analysis , 11(4-5):197–211 ◮ Słowínski, R., Greco, S., and Mousseau, V. (2005). Multi-criteria ranking of a finite set of alternatives using ordinal regression and additive utility functions - a new UTA-GMS method. In Practical Approaches to Multi-Objective Optimization University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 15, 2015 8 / 26

Motivations for UTA-poly and UTADIS-poly Motivations for UTA-poly and UTADIS-poly II UTA-poly and UTADIS-poly ◮ We propose to replace piecewise linear functions by polynomial ones by using semi-definite programming (SDP) ◮ Degree of the polynomial chosen a priori u ∗ u ∗ j j w j w j + u ∗ 3 + j u ∗ 2 + j u ∗ j ( a j ) u ∗ j ( a j ) ⇒ u ∗ 1 + j 0 g j j + + + + + 0 g j j + a j g 1 g 2 g 3 g j a j g j j j j � a j − g 1 � � j ( a ) = p 0 + p 1 · a j + p 2 · a 2 j + . . . + p D · a D u ∗ � j j ( g 1 u ∗ 2 − u ∗ 1 u ∗ j ( a ) = u ∗ j )+ j j j g 2 j − g 1 j University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 15, 2015 9 / 26

UTA-poly and UTADIS-poly Semi-Definite Programming (SDP) I Theorem A polynomial F ( z ) , with z ∈ R n is nonnegative if it is possible to decompose it as a sum of squares (SOS) : f 2 � with f s ( z ) ∈ R n . F ( z ) = s ( z ) s Not every non-negative polynomial is a Sum Of Squares (SOS), but : Theorem (Hilbert) A non-negative polynomial in one variable is always a SOS. University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 15, 2015 10 / 26

UTA-poly and UTADIS-poly Semi-Definite Programming (SDP) II ◮ Consider the following polynomial of degree D : D p ( x ) = p 0 + p 1 x + p 2 x 2 + ... + p D x D = � p i x i . i = 0 p ( x ) non-negative ⇐ ⇒ it can be decomposed as a SOS. � D s ] and x T = [ 1 , x , ..., x d ] , the , b T s = [ b 0 s , b 1 ◮ Let d = � s , ..., b d 2 polynomial reads : � d � � 2 � q 2 b T � � � b i s x i � p ( x ) = s ( x ) = = s x j s s i = 0 s �� � � x T b s b T s x = x T b s b T x = x T Qx = s s s University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 15, 2015 11 / 26

UTA-poly and UTADIS-poly Semi-Definite Programming (SDP) III p ( x ) = p 0 + p 1 x + p 2 x 2 + . . . + p D x D T       1 q 0 , 0 q 0 , 1 q 0 , 2 · · · q 0 , d 1 x q 1 , 0 q 1 , 1 q 1 , 2 · · · q 1 , d x       x 2 x 2       = x T Qx = q 2 , 0 q 2 , 1 q 2 , 2 q 2 , d · · · .        .   . . . .   .  ... . . . . . .       . . . . . .       x d x d q d , 0 q d , 1 q d , 2 · · · q d , d ◮ p 0 , p 1 , . . . , p D are obtained by summing the off-diagonal entries of Q :  p 0 = q 0 , 0 ,    p 1 = q 1 , 0 + q 0 , 1 ,      p 2 = q 2 , 0 + q 1 , 1 + q 0 , 2 ,   . . .     p 2 d − 1 = q d , d − 1 + q d − 1 , d ,      p 2 d = q d , d , if D is odd, we have p 2 d = 0 .  University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 15, 2015 12 / 26

UTA-poly and UTADIS-poly UTA-poly and UTADIS-poly ◮ Marginal utility functions defined as polynomials of degree D : D � p j , i · a i u ∗ j ( a j ) = j . i = 0 ◮ To ensure monotonicity of the function, u ∗ ′ j ( a j ) has to be nonnegative. ⇒ u ∗ ′ should be a SOS : j ∂ u ∗ j = p j , 1 + 2 p j , 2 · a j + 3 p j , 3 · a 2 j + ... + Dp j , n · a D − 1 j ∂ a j = ∂ � � a j T Qa j ∂ a j ◮ Using a SDP solver, we impose Q to be semidefinite positive. University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 15, 2015 13 / 26

UTA-poly and UTADIS-poly UTA-poly - Example I x y a 1 10 7 a 1 ≻ a 2 ≻ a 3 a 2 6 8 a 3 7 5 ◮ We define u ∗ 1 ( x ) and u ∗ 2 ( y ) as second degree polynomials : 1 ( x ) = p x , 0 + p x , 1 · x + p x , 2 · x 2 , u ∗ 2 ( y ) = p y , 0 + p y , 1 · y + p y , 2 · y 2 . u ∗ ◮ Utilities of a 1 , a 2 and a 3 are given by : U ( a 1 ) = p x , 0 + 10 p x , 1 + 100 p x , 2 + p y , 0 + 7 p y , 1 + 49 p y , 2 , U ( a 2 ) = p x , 0 + 6 p x , 1 + 36 p x , 2 + p y , 0 + 8 p y , 1 + 64 p y , 2 , U ( a 3 ) = p x , 0 + 7 p x , 1 + 49 p x , 2 + p y , 0 + 5 p y , 1 + 25 p y , 2 . University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 15, 2015 14 / 26