Choquet Integral and Applications

Transcription

1 Choquet Integral and Applications Brice Mayag École Centrale Paris 14 Janvier 2011 Brice Mayag (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

2 Plan 1 Preliminaries 2 MultiAttribute Utility Theory 3 Additive model: Weighted Arithmetic Mean 4 A non-additive model: the Choquet integral 5 The 2-additive Choquet Integral Binary Actions Elicitation of a 2-additive capacity 6 Applications: complex system design Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

3 Plan Preliminaries 1 Preliminaries 2 MultiAttribute Utility Theory 3 Additive model: Weighted Arithmetic Mean 4 A non-additive model: the Choquet integral 5 The 2-additive Choquet Integral Binary Actions Elicitation of a 2-additive capacity 6 Applications: complex system design Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

4 Preliminaries Notations DM: Decision-Maker X = the set of all alternatives N = {1,...,n} the finite set of n criteria X 1,...,X n represent the set of points of view or attributes An alternative or option x = (x 1,...,x n ) is identified to an element of X = X 1 X n Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

5 Preliminaries Example (Evaluation of students in the tv program Nouvelle Star ) Candidates 1 : Choreography 2 : Song 3 : Music a : Yvanessa b : Michaël c : Jessica 8 70 d : Frank 8 60 e : Suzanne f : Désiré Problem: Give a ranking of all the six students. We have: N = {1,2,3} X = the set of all students X 1 = [ ; ] X 2 = [0;20] X 3 = [0;100] X = {a,b,c,d} = a part of X Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

6 Plan MultiAttribute Utility Theory 1 Preliminaries 2 MultiAttribute Utility Theory 3 Additive model: Weighted Arithmetic Mean 4 A non-additive model: the Choquet integral 5 The 2-additive Choquet Integral Binary Actions Elicitation of a 2-additive capacity 6 Applications: complex system design Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

7 MultiAttribute Utility Theory MultiAttribute Utility Theory (MAUT) Goal MAUT aims at representing numerically the DM s preferences in the form of a complete preorder X with a function u : X R called overall utility function and such that: x,y X, x X y u(x) u(y) The function u is constructed so that the larger the overall utility associated to an alternative is, the greater this alternative is preferred by the DM. Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

8 MultiAttribute Utility Theory MultiAttribute Utility Theory (MAUT) MAUT s hypothesis: X is representable by an overall utility function u: x X y u(x) u(y) In general, we suppose u = F U where U(x) = (u 1 (x 1 ),...,u n (x n )), u i : X i R is an utility function on i, F : R n R is an aggregation function. Hence we have: (x 1,...,x n ) X, u(x 1,...,x n ) := F(U(x 1,...,x n )) Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

9 MultiAttribute Utility Theory MultiAttribute Utility Theory (MAUT) Remark Generally, The utility functions u i and the aggregation function F are not unique; The construction of utility functions u i and the determination of the aggregation function F are done separately; How to construct u i? It is not an easy task. For instance somme models need commensurability between criteria. How to choose de best aggregation function? Usually, one use as aggregation function the well-known arithmetic mean (weighted sum). Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

10 Plan Additive model: Weighted Arithmetic Mean 1 Preliminaries 2 MultiAttribute Utility Theory 3 Additive model: Weighted Arithmetic Mean 4 A non-additive model: the Choquet integral 5 The 2-additive Choquet Integral Binary Actions Elicitation of a 2-additive capacity 6 Applications: complex system design Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

11 Additive model: Weighted Arithmetic Mean Additive model: definition Definition The additive model is defined by the existence of utility functions u i : X i R such that: (x 1,...,x n ) X, u(x 1,...,x n ) := i N u i (x i ). (1) The functions u i can be determined by some methods like UTA. Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

12 Additive model: Weighted Arithmetic Mean Weighted Arithmetic Mean (WAM) Weighted Sum Definition The Weighted sum is a particular case of an additive model. It is defined by the existence of utility functions u i : X i R and real numbers w i (weight of criterion i) such that: (x 1,...,x n ) X, u(x 1,...,x n ) := i Nw i u i (x i ). (2) Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

13 Additive model: Weighted Arithmetic Mean Example (Evaluation of students in the tv program Nouvelle Star ) X = {a,b,c,d} Candidates 1 : Choreography 2 : Song 3 : Music a : Yvanessa b : Michaël c : Jessica 8 70 d : Frank 8 60 e : Suzanne f : Désiré If two students are good in Song and Music, then the jury prefers strictly the student who have a best evaluation in Choreography. b X a; If two students are bad in Song, then the jury prefers strictly the student who have a best evaluation in Music. c X d; Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

14 Additive model: Weighted Arithmetic Mean Example (Evaluation of students in the tv program Nouvelle Star ) Candidates 1 : Choreography 2 : Song 3 : Music a : Yvanessa b : Michaël c : Jessica 8 70 d : Frank 8 60 e : Suzanne f : Désiré X = {a,b,c,d}. If F weighted sum, b P a u 1 ( ) w 1 +u 3 (60) w 3 > u 1 ( ) w 1 +u 3 (70) w 3 (3) c P d u 1 ( ) w 1 +u 3 (70) w 3 > u 1 ( ) w 1 +u 3 (60) w 3 (4) Conclusion: Weighted Sum criteria are independent (no interaction). Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

15 Plan A non-additive model: the Choquet integral 1 Preliminaries 2 MultiAttribute Utility Theory 3 Additive model: Weighted Arithmetic Mean 4 A non-additive model: the Choquet integral 5 The 2-additive Choquet Integral Binary Actions Elicitation of a 2-additive capacity 6 Applications: complex system design Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

16 A non-additive model: the Choquet integral Capacity Definition A capacity (or fuzzy measure) on N is a set function µ : 2 N [0,1] satisfying the three properties: 1 µ( ) = 0 2 µ(n) = 1 3 A,B 2 N, [A B µ(a) µ(b)] (monotonicity). Notations i,j N,i j, µ = µ( ), µ i = µ({i}) and µ ij = µ({i,j}) Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

17 A non-additive model: the Choquet integral Definition The Choquet integral of x := (x 1,...,x n ) R n w.r.t. a capacity µ is defined by: C µ (x) := n (x τ(i) x τ(i 1) )µ({τ(i),...,τ(n)}) (5) i=1 where τ is a permutation on N such that x τ(1) x τ(2) x τ(n 1) x τ(n), and x τ(0) := 0 Remark Choquet integral to ensure commensurability between criteria Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

18 A non-additive model: the Choquet integral Example (Evaluation of students in the tv program Nouvelle Star ) The same example with commensurate scales Candidates 1 : Choreography 2 : Song 3 : Music a : Yvanessa b : Michaël c : Jessica d : Frank e : Suzanne f : Désiré If we consider the capacity µ : 2 N [0,1] defined by: µ(n) = 1, µ( ) = 0, µ({2}) = µ({3}) = µ({2,3}) = µ({1,3}) = 1 2, µ({1}) = 0, µ({1,2}) = 1, then we obtain: C µ (a) = 7+7 µ({2,3})+3 µ({2}) = 12 C µ (b) = 9+3 µ({2,3})+5 µ({2}) = 13 C µ (c) = 7+1 µ({2,3})+6 µ({3}) = 10.5 C µ (d) = 8+1 µ({1,3})+3 µ({3}) = 10 Hence we have now: b X a and c X d. Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

19 A non-additive model: the Choquet integral Interaction index Definition Given a capacity µ, the interaction index for any subset A N is defined by A N, I(A) := K N\A (n k A )!k! (n A +1)! ( 1) A L µ(k L). (6) L A Definition Let µ be a capacity. The interaction index for any pair of criteria i and j is given by the following expression I ij := K N\{i,j} (n k 2)!k! [µ(k {i, j}) µ(k {i}) µ(k {j})+µ(k)] (7) (n 1)! Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

20 A non-additive model: the Choquet integral Importance index Definition Let µ be a capacity. The importance index for a criterion i is given by the following expression: v i = K N\i (n k 1)!k! n! (µ(k i) µ(k)). (8) Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

21 Plan The 2-additive Choquet Integral 1 Preliminaries 2 MultiAttribute Utility Theory 3 Additive model: Weighted Arithmetic Mean 4 A non-additive model: the Choquet integral 5 The 2-additive Choquet Integral Binary Actions Elicitation of a 2-additive capacity 6 Applications: complex system design Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

22 The 2-additive Choquet Integral k-additive capacity Definition Let µ be a capacity on N. The Möbius transform of µ is a function m : 2 N R defined by m(t) := K T( 1) T\K µ(k) T 2 N. Definition µ is said to be k-additive, k > 0, if its Möbius transform m satisfied 1 T 2 N, m(t) = 0 if T > k 2 B 2 N such that B = k and m(b) 0. Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

23 The 2-additive Choquet Integral 2-additive capacity Definition µ is said to be 2-additive if its Möbius transform m satisfied 1 T 2 N, m(t) = 0 if T > 2 2 B 2 N such that B = 2 and m(b) 0. Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

24 2-additive capacity The 2-additive Choquet Integral Lemma If the coefficients µ i and µ ij are given for all i,j N, then the necessary and sufficient conditions that µ is a 2-additive capacity are: µ i = 1 (9) {i,j} Nµ ij (n 2) i N µ i 0, i N (10) A N, A 2, k A (µ ik µ i ) ( A 2)µ k (monotonicity). (11) i A\{k} Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

25 The 2-additive Choquet Integral The Choquet integral w.r.t. a 2-additive capacity Definition For any z = (z 1,z 2,...,z n ) R n +, the expression of the Choquet integral w.r.t. 2-additive capacity (2-additive Choquet integral) is: C µ (z) = n v i z i 1 2 i=1 I ij z i z j {i,j} N where v i = µ i I ik is the importance of the criterion i k N\i I ij = µ ij µ i µ j is the interaction index between criteria i and j. Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

26 The 2-additive Choquet Integral Example (Evaluation of students in the tv program Nouvelle Star ) Candidates 1 : Choreography 2 : Song 3 : Music a : Yvanessa b : Michaël c : Jessica d : Frank e : Suzanne f : Désiré X = {a,b,c,d}. If we consider the 2-additive capacity µ : 2 N [0,1] defined such that: µ(n) = 1, µ( ) = 0, µ({2}) = µ({3}) = µ({2,3}) = µ({1,3}) = 1 2, µ({1}) = 0, µ({1,2}) = 1, then we have I 12 = 1 2, I 13 = 0, I 23 = 1 2, v 1 = 1 4, v 2 = 1 2, v 3 = 1 4 : C µ(a) = 7 v v v 3 1 (I I I ) = 12 2 C µ(b) = 9 v v v 3 1 (I I I ) = 13 2 C µ(c) = 7 v 1 +8 v v 3 1 (I I I ) = C µ(d) = 9 v 1 +8 v v 3 1 (I I I ) = 10 2 Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

27 The 2-additive Choquet Integral Interaction index Interpretation of I ij I ij = 0 independence between i and j; I ij > 0 complementary among i and j; This means that for the DM, both criteria have to be satisfactory in order to get a satisfactory alternative, the satisfaction of only one criterion being useless. I ij < 0 substitutability among i and j; This means that for the DM, the satisfaction of one of the two criteria is sufficient to have a satisfactory alternative, satisfying both being useless.. Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

28 The 2-additive Choquet Integral Interest of the 2-additive model The 2-additive Choquet integral 1 is very used in many applications such that the evaluation of discomfort in sitting position (see Grabisch et al. (2002)); the construction of performance measurement systems model in a supply chain context (see Berrah and Clivillé (2007), Clivillé et al. (2007)); complex system design (Labreuche and Pignon (2007)); 2 offers a good compromise between flexibility of the model and complexity; 3 requires to be able to compare any element of one point of view with any element of any other point of view (commensurateness between criteria); Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

29 The 2-additive Choquet Integral Remark C µ (z 1,z 2,...,z n ) = n v i z i 1 2 i=1 I ij z i z j {i,j} N We have C µ (0,0,...,0,0) = µ = 0; C µ (0,0,...,1,0,...,0,0) = µ i = v i 1 2 I ik, when 1 is in position i; k N, k i C µ (0,0,...,1,0,...,1,0,...,0,0) = µ ij = v i +v j 1 2 where the two numbers 1 are in positions i and j. (I ik +I jk ) k N, k {i,j} Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

30 The 2-additive Choquet Integral Binary Actions Binary actions { an element 0i X On each criterion i we identify: i that is neutral an element 1 i X i that is satisfactory Definition A binary action is an element of the set where B = {0 N, (1 i,0 N i ), (1 ij,0 N ij ), i,j N,i j} 0 N = (1,0 N ) =: a 0 is the action considered neutral on all criteria. (1 i,0 N i ) =: a i is an action considered satisfactory on criterion i and neutral on the other criteria. (1 ij,0 N ij ) =: a ij is an action considered satisfactory on criteria i and j and neutral on the other criteria. Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

31 Binary actions The 2-additive Choquet Integral Elicitation of a 2-additive capacity 1 N 1 i N i 1 i j N ij 0 a 0 0 a i 0 a ij Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

32 The 2-additive Choquet Integral Elicitation of a 2-additive capacity Why binary actions? They allow us to determine: the interaction between two criteria the importance of a criterion We have for all i,j N C µ (U(a 0 )) = µ = 0 C µ (U(a i )) = µ i = v i 1 2 k N, k i C µ (U(a ij )) = µ ij = v i +v j 1 2 I ik k N, k {i,j} (I ik +I jk ) Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

33 The 2-additive Choquet Integral Elicitation of a 2-additive capacity Ordinal information Using pairwise comparisons, the DM gives a preferential information on B allowing the construction of these relations: P = {(x,y) B B : DM strictly prefers x to y} I = {(x,y) B B : DM is indifferent between x and y} Remark We can have P I P 1 B B. Definition The ordinal information on B is the structure {P, I}. Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

34 Example The 2-additive Choquet Integral Elicitation of a 2-additive capacity Example N = {1,2,3}, B = {a 0,a 1,a 2,a 3,a 12,a 13,a 23 } P = {(a 23,a 2 );(a 2,a 0 );(a 23,a 12 )} I = {(a 13,a 1 );(a 3,a 12 )} Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

35 The 2-additive Choquet Integral Elicitation of a 2-additive capacity Definition {P,I} is representable by a 2-additive Choquet integral C µ if x,y B, x P y C µ (U(x)) > C µ (U(y)), (12) x,y B, x I y C µ (U(x)) = C µ (U(y)). (13) Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

36 Plan Applications: complex system design 1 Preliminaries 2 MultiAttribute Utility Theory 3 Additive model: Weighted Arithmetic Mean 4 A non-additive model: the Choquet integral 5 The 2-additive Choquet Integral Binary Actions Elicitation of a 2-additive capacity 6 Applications: complex system design Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

37 Applications: complex system design We present a Thales s application. The aim: choose a best complex system for a site protection The criteria and sub-scriteria are: 1 Mission success 2 System complexity: 1 Skills 2 Work load 3 System deployment 3 System safety 4 Economic data The alternatives are unknown here Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

38 Applications: complex system design Figure: Criteria tree Brice Mayag (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

39 Applications: complex system design Example Aggregation of system cpmlexity node: an example of ordinal information B = {a 0,a 1,a 2,a 3,a 12,a 13,a 23 }, where 1 = Skills; 2 = Work load; 3 = System deployment. The elements of B are fictitious sytems a 1 : corresponds to a system requesting a high time of deployment with a huge workload and requiring no skills to use; a 13 I a 1 : a system requiring a high workload and better on other criteria is equivalent to a better system solely on the criterion skills a 12 P a 3 : the expert prefers a system with a very good management of human resources to a system that has none, even the time deployment of the first system is very important. Brice Mayag brice.mayag@ecp.fr (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

40 Applications: complex system design Figure: Ordinal information for system complexity node Brice Mayag (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

41 Applications: complex system design Figure: Parameters of system complexity node Brice Mayag (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

42 Applications: complex system design Figure: Aggregation node of site protection assessment Brice Mayag (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43

43 Applications: complex system design Figure: Parameters of aggregation node of site protection assessment Brice Mayag (École Centrale Paris) Choquet Integral and Applications 14 Janvier / 43