Multiplecriteria ranking using an additive utility function constructed

Multiple-criteria ranking using an additive utility function constructed via ordinal regresion : UTA method Roman Słowiński Poznań University of Technology, Poland Roman Słowiński

Problem statement g 2(x) g 2 max A n Consider a finite set A of alternatives (actions, solutions) g 2 min g 1(x) evaluated by n criteria from a consistent family G={g 1, . . . , gn} g 1 min g 1 max 2

Problem statement g 2(x) n Consider a finite set A of alternatives (actions, solutions) A evaluated by n criteria from a consistent family G={g 1, . . . , gn} g 1(x) 3

Problem statement n Taking into account preferences of a Decision Maker (DM), rank all the alternatives of set A from the best to the worst A x * * x x * * x x x 4

Basic concepts and notation n n Xi – domain of criterion gi (Xi is finite or countably infinte) – evaluation space n x, y X – profiles of alternatives in evaluation space n – weak preference (outranking) relation on X: for each x, y X x y „x is at least as good as y” x y [x y and not y x] „x is preferred to y” x~y [x y and y x] „x is indifferent to y” 5

Basic concepts and notation n For simplicity: Xi , for all i=1, …, n n For each gi, Xi=[ i, i] is the criterion evaluation scale, i i , where i and i, are the worst and the best (finite) evaluations, resp. n Thus, A is a finite subset of X and where g(a) is the vector of evaluations of alternative a A on n criteria n Additive value (or utility) function on X: for each a X where ui are non-decreasing marginal value functions, ui : Xi , i=1, . . . , n 6

Criteria aggregation model = preference model n To solve a multicriteria decision problem one needs a criteria aggregation model, i. e. a preference model n Traditional aggregation paradigm: The criteria aggregation model is first constructed and then applied on set A to get information about the comprehensive preference n Disaggregation-aggregation (or regression) paradigm: The comprehensive preference on a subset AR A is known a priori and a consistent criteria aggregation model is inferred from this information 7

Criteria aggregation model = preference model n The disaggregation-aggregation paradigam has been introduced to MCDA by Jacquet-Lagreze & Siskos (1982) in the UTA method – the inferred criteria aggregation model is the additive value function with piecewise-linear marginal value functions n The disaggregation-aggregation paradigam is consistent with the „posterior rationality” principle by March (1988) and the inductive learning used in artificial intelligence and knowledge discovery 8

Principle of the UTA method n (Jacquet-Lagreze & Siskos, 1982) The comprehensive preference information is given in form of a complete preorder on a subset of reference alternatives AR A, AR={a 1, a 2, . . . , am} – the reference alternatives are rearranged such that ak ak+1 , k=1, . . . , m-1 a 1 A a 2 AR a 3 a 4 a 5 a 6 a 7 9

Principle of the UTA method n Example: Let AR={a 1, a 2, a 3}, G={Gain_1, Gain_2} Evaluation of reference alternatives on criteria Gain_1, Gain_2: Gain_1 Gain_2 a 1 4 6 a 2 5 5 a 3 6 4 Reference ranking: a 1 a 2 a 3 10

Principle of the UTA method 11

Principle of the UTA method 12

Principle of the UTA method n Let’s change the reference ranking: a 1 n a 2 a 3 a 2 One linear piece per each marginal value function u 1, u 2 is not enough Gain_1 Gain_2 a 1 4 6 a 2 5 5 a 3 6 4 u 1=w 1 Gain_1, u 2=w 2 Gain_2, U=u 1+u 2 For a 1 a 3, w 2>w 1, but for a 3 a 2, w 1>w 2, thus, marginal value functions cannot be linear 13

Principle of the UTA method 14

Principle of the UTA method n The inferred value of each reference alternative a AR: where is a calculated value function, is a value function compatible with the reference ranking, + and - are potential errors of over- and under-estimation of the compatible value function, respectively. n The intervals [ i, i] are divided into ( i– 1) equal sub-intervals with the end points (i=1, . . . , n) 15

Principle of the UTA method n The marginal value of alternative a A is approximated by a linear interpolation: for 16

Principle of the UTA method n Ordinal regression principle if then one of the following holds N. B. In practice, „ 0” is replaced here by a small positive number that may influence the result n Monotonicity of preferences n Normalization 17

Principle of the UTA method n The marginal value functions (breakpoint variables) are estimated by solving the LP problem (C) 18

Principle of the UTA method n If F*=0, then the polyhedron of feasible solutions for ui(gi) is not empty and there exists at least one value function U[g(a)] compatible with the complete preorder on AR n If F*>0, then there is no value function U[g(a)] compatible with the complete preorder on AR – three possible moves: n increasing the number of linear pieces i for ui(gi) n revision of the complete preorder on AR n post optimal search for the best function with respect to Kendall’s in the area F F*+ polyhedron of constraints (C) Jacquet-Lagreze & Siskos (1982) F F*+ F= F* 19

Współczynnik Kendalla n Do wyznaczania odległości między preporządkami stosuje się miarę Kendalla n Przyjmijmy, że mamy dwie macierze kwadratowe R i R* o rozmiarze m m, gdzie m = |AR|, czyli m jest liczbą wariantów referencyjnych n macierz R jest związana z porządkiem referencyjnym podanym przez decydenta, n macierz R* jest związana z porządkiem dokonanym przez funkcję użyteczności wyznaczoną z zadania PL (zadania regresji porządkowej) n Każdy element macierzy R, czyli rij (i, j=1, . . , m), może przyjmować wartości: n To samo dotyczy elementów macierzy R* n Tak więc w każdej z tych macierzy kodujemy pozycję (w porządku) wariantu a względem wariantu b 20

Współczynnik Kendalla n Następnie oblicza się współczynnik Kendalla : gdzie dk(R, R*) jest odległością Kendalla między macierzami R i R*: n Stąd -1, 1 n Jeżeli = -1, to oznacza to, że porządki zakodowane w macierzach R i R* są zupełnie odwrotne, np. macierz R koduje porządek a b c d, a macierz R* porządek d c b a n Jeżeli = 1, to zachodzi całkowita zgodność porządków z obydwu macierzy. W tej sytuacji błąd estymacji funkcji użyteczności F*=0 n W praktyce funkcję użyteczności akceptuje się, gdy 0. 75 21

Example of UTA+ n Ranking of 6 means of transportation 1 2 22

Preference attitude: „economical” 23

Preference attitude: „hurry” 24

Preference attitude: „hurry” 25