Robust ordinal regression for outranking methods

Transcription

1 Outline Robust ordinal regression for outranking methods Salvatore Greco 1 Miłosz Kadziński 2 Vincent Mousseau 3 Roman Słowiński 2 1 Faculty of Economics, University of Catania, Italy 2 Institute of Computing Science, Poznań University of Technology, Poland 3 Laboratoire de Génie Industriel, Ecole Centrale Paris, France

2 Outline Outline 1 Introduction

3 Multiple criteria problems Characteristics Actions described by evaluation vectors Family of criteria is supposed to satisfy the consistency conditions Ranking Rank the actions from the best to the worst according to DM s preferences Ranking can be complete or partial Choice Choose the subset of best actions

4 Outranking relation Definition Outranking relation S groups three basic preference relations: S = {,, } asb means action a is at least as good as action b Non-compensatory preference model used in the ELECTRE family of MCDA methods Accept incomparability, no completeness nor transitivity Outranking relation on set of actions A is constructed via concordance and discordance tests

5 Outranking relation Concordance and discordance Concordance test: checks if the coalition of criteria concordant with the hypothesis asb is strong enough: C(a, b) = m j=1 k j C j (a, b)/ m j=1 k j = [k 1 C 1 (a, b) k m C m (a, b)]/(k k m ) Coalition is composed of two subsets of criteria: these being clearly in favor of asb, i.e. C j (a, b) = 1, if g j (a) g j (b) q j, these that do not oppose to asb, i.e. such that b a

6 Outranking relation Concordance and discordance Since (k k m ) = 1, we can consider: C(a, b) = ψ 1 (a, b) ψ m (a, b), where ψ j (a, b) = k j C j (a, b), j = 1,..., m, is a monotone, non-decreasing function w.r.t. g j (a) g j (b) Concordance test is positive if: C(a, b) λ, where λ is a cutting level (concordance threshold) Cutting level λ is required to be not less than 0.5

7 Outranking relation Concordance and discordance Discordance test: checks if among criteria discordant with the hypothesis asb there is a strong opposition against asb: g j (b) g j (a) v j (for gain-type criterion) g j (a) g j (b) v j (for cost-type criterion) Conclusion: asb is true if and only if C(a, b) λ and there is no criterion strongly opposed (making veto) to the hypothesis For each couple (a, b) A A, one obtains relation S either true (1) or false (0)

8 Outranking methods Two major problems raised in the literature Elicitation of preference information: Rather technical parameters, precise numerical values Intra-criterion parameters vs. inter criteria parameters Disaggregation-aggregation procedures Mainly in terms of sorting problems (e.g., ELECTRE TRI) Robustness analysis: Examination of the impact of each parameter on the final outcome Indication of the solutions which are good (bad) for different instances of a preference model

9 Robust ordinal regression Main assumptions Take into account all instances of a preference model compatible with the preference information given by the DM Supply the DM with two kinds of results: necessary results specify recommendations worked out on the basis of all compatible instances of a preference model considered jointly possible results identify all possible recommendations made by at least one compatible instance of a preference model considered individually Methods that use value function as a preference model: UTA GMS, GRIP, UTADIS GMS (additive value function), robust ordinal regression applied to Choquet integral

10 Questions Does a outrank b for all compatible outranking models? Does a outrank b for at least one compatible outranking model?

11 ELECTRE GKMS - Preference Information Pairwise comparisons Set of pairwise comparisons of reference actions (a, b) B R A R A R asb or as C b Intra-criterion preference information [q j,, q j ] - the range of indifference threshold values allowed by the DM [p j,, p j ] - the range of preference threshold values allowed by the DM a j b the difference between g j (a) and g j (b) is not significant for the DM a j b the difference between g j (a) and g j (b) is significant for the DM

12 ELECTRE GKMS - Compatibility and Results Compatibility An outranking model is called compatible, if it is able to restore all pairwise comparisons from B R for provided imprecise intra-criterion preference information The necessary and the possible In result, one obtains two outranking relations on set A, such that for any pair of actions (a, b) A A: 1 a necessarily outranks b (as N b) if a outranks b for all compatible outranking models 2 a possibly outranks b (as P b) if a outranks b for at least one compatible outranking model

13 ELECTRE GKMS - Compatible outranking model (1) Set of concordance indices C(a, b), cutting levels λ, indifference q j, preference p j, and veto thresholds v j, j = 1,..., m, satisfying the foll. set of constraints E AR : If asb for (a, b) B R C(a, b) = m j=1 ψ j(a, b) λ g j (b) g j (a) + ε v j, j = 1,..., m If as C b for (a, b) B R C(a, b) = m j=1 ψ j(a, b) + ε λ + M 0 (a, b) g j (b) g j (a) v j δm j (a, b) M j (a, b) {0, 1}, j = 0,..., m, m j=0 M j(a, b) m where δ is a big given value

14 ELECTRE GKMS - Compatible outranking model (2) 1 λ 0.5, v j p j + ε v j g j (b) g j (a) + ε, v j g j (a) g j (b) + ε if a j b normalization: m j=1 ψ j(a j, a j, ) = 1 monotonicity: for all a, b, c, d A and j = 1,..., m : ψ j (a, b) ψ j (c, d) if g j (a) g j (b) > g j (c) g j (d) ψ j (a, b) = ψ j (c, d) if g j (a) g j (b) = g j (c) g j (d) E AR

15 ELECTRE GKMS - Compatible outranking model (3) partial concordance: for all (a, b) A A and j = 1,..., m : [1] ψ j (a, b) = 0 if g j (a) g j (b) p j [2] ψ j (a, b) ε if g j (a) g j (b) > p j, [3] ψ j (a, b) + ε ψ j (a j, a j, ) if g j (a) g j (b) < q j E AR [4] ψ j (a, b) = ψ j (a j, a, ) if g j (a) g j (b) q j, [1] ψ j (a, b) = 0 if b j a [4] ψ j (a, b) = ψ j (a j, a j, ), ψ j (b, a) = ψ j (a j, a j, ) if a j b

16 ELECTRE GKMS - Preference information Extensions of the preference model Consideration of thresholds dependent on g j (a) (e.g., affine functions) Pairwise comparisons of reference actions in terms of relations of preference, indifference, or incomparability: a b, aib, or a?b Inter-criteria preference information: Interval weights of the criteria k j (e.g., k 1 > 0.2, k 2 < 0.5) Pairwise comparisons of the weights of the criteria Interval cutting level λ [λ, λ ] (e.g., λ > 0.75) Veto thresholds v j (e.g., v 1 [8.2, 9.8], v 2 = 5.6, a 3 b)

17 ELECTRE GKMS - Checking the truth of S P Idea Prove that asb is possible in the set of all compatible outranking models where: E AR as P b ε > 0 ε = max ε C(a, b) = m j=1 ψ i(a, b) λ g j (b) g j (a) + ε v j, j = 1,..., m Result If ε > 0 and the set of constraints is feasible, then a outranks b for at least one compatible outranking model (as P b)

18 ELECTRE GKMS - Checking the truth of S N Idea Prove that as C b is not possible in the set of all compatible outranking model where: E AR as N b ε 0 ε = max ε C(a, b) = m j=1 ψ j(a, b) + ε λ + M 0 (a, b) g j (b) g j (a) v j δm j (a, b) M j (a, b) {0, 1}, j = 0,..., m, m j=0 M j(a, b) m Result If ε 0 or the set of constraints is infeasible, then a outranks b for all compatible outranking model (as N b)

19 ELECTRE GKMS - Properties of relations S P and S N Properties S P and S N are reflexive, intransitive, and incomplete S P S N as N b not(as CP b) and as P b not(as CN b) From S N and S P, one can obtain indifference I, preference = {P Q}, and incomparability R, in a usual way, e.g.: if as N b and bs N b, then ai N b if as P b and not(bs P b), then a P b Possible relations between actions a and b for a single instance of compatible outranking model conditioned by the truth or falsity of S N and S P, e.g.: if as N b and bs N a, then ai N b if not(as P b) and not(bs N a), then b P a or br P a

20 ELECTRE GKMS - Partial conclusions Main distinguishing features Taking into account all instances of the outranking model compatible with the provided preference information Considering the marginal concordance functions as general non-decreasing ones, defined in the spirit of ELECTRE methods Handling of preference information composed of pairwise comparisons and of imprecise intra-criterion preference information Constructing two relations on the set of actions: necessary and possible

21 ELECTRE GKMS - Illustrative example Problem statement and given data Actions: 10 buses originally considered by the team of Professor Jacek Zak (FMT, PUT) Criteria: 5 criteria: Price (th. euro), Exploitation costs (th. zl/100k km), Comfort (pts), Safety (pts), Modernity (pts) Evaluation table: Price Exploit. Comfort Safety Modern. Bus name [euro] costs [zl] [pts] [pts] [pts] Autosan Bova Futura Ikarus EAG Jelcz T Setra S MAN Lion s

22 ELECTRE GKMS - Illustrative example Example Step1: Ask the DM for preference information The DM provides imprecise intra-criterion preference information:... and pairwise comparisons: q j, qj p j, pj Price Exploitation Comfort Safety Modernity MAN S Volvo, Neoplan S Bova Volvo S C Autosan, Setra S C MAN

23 ELECTRE GKMS - Illustrative example Example Step 2: Determine the necessary and possible outranking relations Possible outranking matrix: S1 P A B I J T M E N R V A B I J T M E N R V

24 ELECTRE GKMS - Illustrative example Example Step 2: Determine the necessary and possible outranking relations Necessary outranking matrix - converg. index = S P = S N % = 0.42: S1 N A B I J T M E N R V A B I J T M E N R V

25 ELECTRE GKMS - Illustrative example Example Step 3: Incremental specification of pairwise comparisons: Analyze the necessary (S N and S CN ) and possible (S P and S CP ) results In the following iterations state asb or as C b for pairs (a, b) A A, for which the possible relation S P (or S CP ) was true, but not the necessary S N (or S CN ) one The DM provides additional pairwise comparisons: Mercedes S Ikarus, Ikarus S MAN Bova S C Neoplan

26 ELECTRE GKMS - Illustrative example Example Step 4: Determine the necessary and possible outranking relations Possible outranking matrix S P 1 SP 2 : S2 P A B I J T M E N R V A B I J T M E N R V

27 ELECTRE GKMS - Illustrative example Example Step 4: Determine the necessary and possible outranking relations Necessary outranking matrix S1 N SN 2 - converg. index = 0.56: S2 N A B I J T M E N R V A B I J T M E N R V

28 ELECTRE GKMS - Basic exploitation procedures Recommendation in case of choice problems Identify kernel K N of the necessary outranking graph S N Identify such a A : b A, b a it holds not(bs P a) Recommendation in case of ranking problems Net Flow Score: NFS(a) = strength(a) weakness(a)

29 ELECTRE GKMS - Illustrative example Step 4: Final recommendation

30 ELECTRE GKMS - Extensions Analysis of incompatibility Associate a binary variable v a,b with each couple of reference actions (a, b) B R : asb C(a, b) = m j=1 ψ j(a, b) + Mv a,b λ and g j (b) g j (a) + ε v j (a) + Mv a,b, j = 1,..., m, where M is a big positive value (transform E AR into E AR v ) If v a,b = 1, then the corresponding constraint is always satisfied Identify a minimal subset of troublesome exemplary decisions: min f = (a,b) B R v a,b, subject to E AR v

31 ELECTRE GKMS - Gradual confidence levels Valued possible and necessary outranking relations B R 1 BR 2... BR s - embedded sets of pairwise comparisons S1 AR S2 AR... Ss AR - sets of compatible outranking models Let θ t be the confidence level assigned to pairwise comparisons concerning pairs ((a, b) Bt R and (a, b) / Bt 1 R ), t = 1,..., s : 1 = θ 1 > θ 2 >... > θ s > 0 S N val : A A {θ 1, θ 2,..., θ s, 0}: if t : as N t b, then S N val (a, b) = max{θ t : as N t b, t = 1,..., s} if t : ast N b, then Sval N (a, b) = 0 Sval P : A A {1 θ 1, 1 θ 2,..., 1 θ s, 1}: if t : as P t b, then S P val (a, b) = min{1 θ t : not(as P t b), t = 1,..., s} if t : ast P b, then Sval P (a, b) = 1

32 ELECTRE GKMS - Illustrative example Step 5: Graph of the valued necessary relation after third iteration

33 The ELECTRE GKMS GROUP method Characteristics Several DMs D = {d 1,..., d s } cooperate in a decision problem DMs share the same description of the decision problems The collective results (ranking or subset of the best actions) should account for preferences expressed by each DM Avoid discussions of DMs on technical parameters Reason in terms of necessary and possible relations and coalitions

34 The ELECTRE GKMS GROUP method Characteristics For each d h D D who expresses her individual preferences as in ELECTRE GKMS, calculate the necessary and possible outranking relations With respect to all DMs four situations are considered: as N,N D b : asn d h b for all d h D as P,N D b : asp d h b for all d h D as N,P D b : asn d h b for at least one d h D as P,P D b : asp d h b for at least one d h D

35 ELECTRE GKMS GROUP - Extensions Consider preferences of DMs individually A is small, DMs have outlook of the whole set A, interrelated preferences Analyze statements of DMs individually Examine the spaces of agreement and disagreement Consider preferences of DMs simultaneously A is numerous, DMs are experts only w.r.t. to its small disjoint subsets Combine knowledge of DMs into preference information of a single fictitious DM

36 ELECTRE GKMS GROUP - Extensions Consider preferences of all DMs simultaneously Suppose that SD AR of compatible outranking models is not empty One obtains two outranking relations: SD N and SP D Difference between SD N and SN,N D asd N b asb for all outranking models compatible with all preferences of all DMs from D as N,N D b asb for all compatible outranking models of each DM from D If S AR D as N,N D, then for all a, b A, b asn D b and and asn D b asp,n D b

37 ELECTRE GKMS GROUP - Extensions Consider preferences of all DMs simultaneously Suppose that SD AR of compatible outranking models is not empty One obtains two outranking relations: SD N and SP D Difference between SD P and SP,P D asd P b asb for at least one outranking model compatible with all preferences of all DMs from D as P,P D b asb for at least one compatible outranking model of at least one DM from D If S AR D, then for all a, b A, as P D b asp,p D b and asp D b asp,n D b

38 PROMETHEE - Main principles Preference function and degree Compute unicriterion preference degree for every pair of actions: π j (a, b), j = 1,..., m Compute global preference degree for every pair of actions (a, b) A A: π(a, b) = m j=1 k j π j (a, b), where k j is a weight expressing relative importance of g j

39 PROMETHEE - Main principles Outranking flows The positive outranking flow Φ + (a): Φ + (a) = 1/(n 1) b A π(a, b) The negative outranking flow Φ (a): Φ (a) = 1/(n 1) b A π(b, a) Net outranking flow: PROMETHEE-II: apb if Φ(a) > Φ(b) Φ(a) = Φ + (a) Φ (a) PROMETHEE-I: apb if Φ + (a) Φ + (b) and Φ (a) Φ (b) and Φ + (a) Φ (a) > Φ + (b) Φ (b)

40 PROMETHEE GKS - Compatible outranking model (1) Pairwise comparisons (constr.): if as S b for (a, b) B R : π(a, b) π(b, a) Pairwise comparisons (exploit.): if as E b for (a, b) B R : Φ(a) Φ(b) Normalization: m j=1 π j(a j, a j, ) = 1 Monotonicity: for all a, b, c, d A and j = 1,..., m : π j (a, b) π j (c, d) if g j (a) g j (b) > g j (c) g j (d) π j (a, b) = π j (c, d) if g j (a) g j (b) = g j (c) g j (d) E AR S

41 PROMETHEE GKS - Compatible outranking model (2) Partial preference: for all (a, b) A A and j = 1,..., m : [1] π j (a, b) = 0 if g j (a) g j (b) q j, [2] π j (a, b) ε if g j (a) g j (b) > q j [3] π j (a, b) + ε π j (a j, a j, ) if g j (a) g j (b) < p j, E AR S [4] π j (a, b) = π j (a j, a j, ) if g j (a) g j (b) p j [1] π j (a, b) = 0, π j (b, a) = 0 if a j b [4] π j (a, b) = π j (a j, a j, ) if a j b

42 Motivation Binary relations vs. ranking Complete rankings are intuitive, easy to understand, and popular The DM is interested in ranks and scores of the actions Examine how different are all rankings compatible with preferences of the DM Compute the highest and the lowest rank attained by each action Compute the best and the worst score of each action

43 The highest rank in robust multiple criteria ranking Assume that a A is in the top of the ranking Identify the minimal subset of actions that are simultaneously not worse than a: E AR S min f pos max = b A\{a} v b Φ(a) > Φ(b) Mv b, b A \ {a} E AR S,max where M is a big positive value P (a) of action a is indicated by (f pos max + 1)

44 The lowest rank in robust multiple criteria ranking Assume that a A is in the bottom of the ranking Identify the minimal subset of actions that are simultaneously not better than a: E AR S min f pos min = b A\{a} v b Φ(b) > Φ(a) Mv b, b A \ {a} E AR S,min where M is a big positive value and ε is a small positive value P (a) of action a is indicated by ( A f pos min )

45 Extreme net outranking flows The highest outranking net flow: Φ (a) = max Φ(a), s.t. E AR S The lowest outranking net flow: Φ (a) = min Φ(a), s.t. E AR S

46 - Extensions Incremental specification of preference information S AR t St 1 AR, for all t = 2,..., s, P t (a) P t 1(a) and P,t (a) P,t 1 (a) Φ t (a) Φ t 1(a) and Φ,t (a) Φ,t 1 (a) Interval orders Preference rank and indifference rank relations w.r.t. the intervals of ranking positions [P (a), P (a)], e.g.: a rank b P (a) < P (b) and P (a) < P (b) a rank b [P (a), P (a)] [P (b), P (b)] or [P (a), P (a)] [P (b), P (b)]

47 - Extensions Application to multiple criteria choice problems Define additional conditions for being in B The most appealing approach: B = {a A : P (a) = 1} Possible approach: B = {a A : P (a) 3 and P (a) A /2} Group actions: indifference classes which stem from the ranking based on the best (or the worst) positions Application to multiple criteria problems of different type Group decision in the spirit of necessary and possible : PD(a) N = P dr (a) and PD(a) P = P dr (a) d r D d r D Set of parameters corresponding to the extreme ranks Assume that a is ranked either at its best or its worst position

48 PROMETHEE GKS - Illustrative example Problem statement and given data - Liveability of cities Actions: 15 cities originally considered by Financial Times Criteria: 5 criteria: stability (g 1 ), healthcare (g 2 ), culture (g 3 ), education (g 4 ), infrastructure (g 5 ) Evaluation table City g 1 g 2 g 3 g 4 g 5 Vancouver Vienna Melbourne Toronto Calgary Osaka Tokyo Hong Kong

49 PROMETHEE GKS - Illustrative example Preference infromation The DM provides imprecise intra-criterion preference information: q j, qj p j, pj Stability Healthcare Culture Education Infrastracture and pairwise comparisons: Vancouver Vienna Tokyo Hong Kong London Seoul Lagos Port Moresby

50 PROMETHEE GKS - Illustrative example Extreme ranking results for PROMETHEE GKS City P (a) P (a) Φ (a) Φ (a) Vancouver Vienna Melbourne Toronto Calgary Osaka Tokyo Hong Kong Singapore London New York Seoul Lagos Port Moresby Harare

51 Motivation Assign precise values to variables of the model Identify the representative set of parameters without loosing the advantage of knowing all compatible outranking models Extend robust ordinal regression in its capacity of explaining the final output Get synthetic representation of the robust results Exploit outranking relation for these parameters in order to obtain representative results

52 Question Which set of parameters is representative for the set of all outranking models compatible with the preference infromation? Representativeness In the sense of robustness preoccupation One for all, all for one

53 Procedure Pre-defined targets built on results of robust ordinal regression and extreme ranking analysis Enhancement of differences between actions Emphasize the evident advantage of some actions over the others e.g., C(a, b) N λ, C(a, b) < N λ, Φ(a) > N Φ(b), P (a) < P (b) Reduce the ambiguity in the statement of such advantage e.g., C(a, b) P λ and C(a, b) < P λ, Φ(a) > P Φ(b) and Φ(b) > P Φ(a), P (a) < P (b) and P (a) > P (b)

54 Procedure Targets consist in maximization or minimization of the difference between scores of actions, e.g.: max ε, such that Φ(a) Φ(b) + ε, if Φ(a) > N Φ(b) min δ, such that Φ(a) Φ(b) δ, if Φ(a) > P Φ(b) and Φ(b) > P Φ(a) DM is left the freedom of assigning priorities to targets Targets may be attained: One after another, according to a given priority order So as to find a compromise solution, e.g.: max ε, such that Φ(a) Φ(b) Φ(c) Φ(d) + ε if Φ(a) > N Φ(b) and Φ(c) > P Φ(d) and Φ(d) > P Φ(c)

55 Innovation of this proposal Dealing with consequences of preference information provided by the DM Involving the DM in the process of specifying the targets General framework for considering a few targets in different configurations General monotonic functions taking all criteria values as coordinates of characteristic points Autonomous method vs. confrontation with results of robust ordinal regression

56 Summary New family of outranking-based methods The preference information is used within a robust ordinal regression approach to build a complete set of compatible outranking models Identification of possible and necessary consequences of provided information Identification of extreme ranking results built on relations defined w.r.t. the whole set of compatible preference models

57 Future research The necessary and the possible for robust multiple criteria sorting for outranking methods Eliciting and selecting compatible instances of the preference model on the basis of rank related information