ELECTRE TRI-C: A Multiple Criteria Sorting Method Based on Central Reference Actions

Transcription

1 ELECTRE TRI-C: A Multiple Criteria Sorting Method Based on Central Reference Actions Juscelino ALMEIDA DIAS 12, José Rui FIGUEIRA 12, Bernard ROY 1 1 LAMSADE, Université Paris-Dauphine, France 2 CEG-IST, Technical University of Lisbon, Portugal Supported by FCT-Portugal (Grant: SFRH/BD/22985/2005) Doctoral School: Algorithmic Decision Theory MCDA, Data Mining, and Rough Sets Troina, Italy April, 2008 JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

2 Contents 1 Introduction 2 Concepts, assumptions, and requirements 3 ELECTRE TRI-C: Assignment rules and properties 4 Comparison with ELECTRE TRI-B 5 Conclusions 6 References JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

3 Introduction Sorting problematic Sorting problematic: the set of categories emerges naturally from the decision aiding context through an interaction process with the decision-makers. Nature of the categories: each category is defined in order to assign actions which will be subject to the same treatment or analysis. Absolute evaluation: the assignment of an action only takes into account the intrinsic evaluation of this action on all the criteria and does not depend on nor influence the category to which another action should be assigned. JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

4 Introduction Motivation examples Example (Credit analysis) Actions: Credit demand files Categories: Accepted without additional information Accepted with additional information Sent to a particular department further analysis Rejected under certain conditions Rejected with no conditions at all Example (Medical diagnosis) Actions: Patients waiting for treatment Categories: Set of pathologies studied JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

5 Introduction Reference actions Profile limits Each category is bounded by a lower and an upper profile. The well-known method called up to now ELECTRE TRI based on profile limits, or boundary actions, will be designated here by ELECTRE TRI-B. Central reference actions Each category is defined by a central reference action. ELECTRE TRI-C is, therefore, the designation of the procedures based on central reference actions. JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

6 Key concepts Actions, criteria, and credibility index a 1, a 2,... are the potential actions. The set of such actions, A, can be partially known a priori. F = {g 1,..., g j,..., g n } is a coherent family of criteria, with n 2. Ω j (a, a ) is the advantage of a over a on criterion g j F, Ω j (a, a ) = { gj (a) g j (a ) if g j is to be maximized g j (a ) g j (a) if g j is to be minimized Each criterion g j F will be considered as a pseudo-criterion. Definition σ(a, a ) is the credibility of the comprehensive outranking of a over a when taking all the criteria from F into account. Definition JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

7 Basic assumptions ELECTRE TRI-C C = {C 1,..., C h,..., C q } is the set of pre-defined and ordered categories, where C 1 is the worst category, and C q the best one, with q 2. Each category C h is defined by a central reference action b h, h = 1,...,q. B = {b 0, b 1,...,b h,..., b q, b q+1 } is the set of (q + 2) reference actions. b 0 is a particular reference action with the worst possible evaluation g j (b 0 ) on criterion g j, for all g j F. b q+1 is a particular reference action with the best possible evaluation g j (b q+1 ) on criterion g j, for all g j F. JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

8 Basic assumptions Strict separability condition Definition (Dominance) The set of reference actions, B, fulfills the (strict) dominance relation if and only if j, Ω j (b h+1, b h ) 0 and j, Ω j (b h+1, b h ) > 0; h = 0,...,q. Definition (Strict separability condition) The set of reference actions, B, fulfills the strict separability condition if and only if Ω j (b h+1, b h ) > p j ; j = 1,...,n; h = 0,...,q. Weak JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

9 Structural requirements 1 Conformity: Each central reference action, b h, must be assigned to the category, C h, h = 1,...,q. 2 Monotonicity: If an action a strictly dominates a, then a is assigned to a category at least as good as the category a is assigned to. 3 Homogeneity: Two actions must be assigned to the same category when they compare themselves in an identical manner with the reference actions. 4 Stability: After a modification of the set B by applying either a merging or a splitting procedure, the non-adjacent categories to the modified ones will remain with the same actions as before the modification. JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

10 Stability requirement Definition (Basic modification procedures) 1 Merging procedure: The distinction between two consecutive categories, C h 1 and C h, will be ignored by introducing a new central reference action, b h, such that: Ω j (b h, b h 1) 0, for all g j F Ω j (b h, b h ) 0, for all g j F 2 Splitting procedure: The category C h can be split into two new consecutive categories by introducing two new central reference actions, b h and b h, such that: b h+1 F b h, b h F b h, and b h F b h 1 Ω j (b h, b h) 0, for all g j F Ω j (b h, b h ) 0, for all g j F JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

11 Slackness functions Definition (Slackness functions) Let λ [0.5, 1] denote the chosen majority level: 1 Direct slackness function: ξ + h (a,λ) = σ(a, b h) λ, h = (q + 1),..., 0. 2 Reverse slackness function: ξ h (a,λ) = σ(b h, a) λ, h = 0,...,(q + 1). Proposition ( ) does not decrease when moving from a given category to a worst one. 1 ξ + h ( ) does not decrease when moving from a given category to a best one. 2 ξ h JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

12 Assignment rules of ELECTRE TRI-C Definition (Descending assignment rule) Choose a majority level λ (0.5 λ 1). Decrease h from (q + 1) until the first value such that ξ + h (a,λ) 0. If ξ + h (a,λ) ξ+ h+1 (a,λ), then assign action a to category C h. Otherwise, assign a to C h+1. Definition (Ascending assignment rule) Choose a majority level λ (0.5 λ 1). Increase h from 0 until the first value such that ξ h (a,λ) 0. If ξ h (a,λ) ξ h 1 (a,λ), then assign action a to category C h. Otherwise, assign a to C h 1. Example Comparing ELECTRE TRI-C rules ELECTRE TRI-B rules JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

13 Numerical example ELECTRE TRI-C assignment results Example Actions b 1 b 2 b 3 b 4 Descending Ascending a 1 R λ R λ C 2 C 4 a 2 R λ R λ R λ R λ C 1 C 4 a 3 C 3 C 3 a 4 C 2 C 3 a 5 I λ C 2 C 1 a 6 R λ R λ C 2 C 4 a 7 C 1 C 2 a 8 C 1 C 1 a 9 R λ R λ R λ R λ C 1 C 4 a 10 C 1 C 1 Note: λ = 0.70; Source: Data adapted from Merad et al., 2004 ELECTRE TRI-C rules Comparison ELECTRE TRI-C rules JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

14 Properties of the assignment rules I ELECTRE TRI-C Theorem a) The monotonicity, homogeneity, and stability requirements hold. b) If the strict separability condition is fulfilled, then the conformity requirement holds. Remark The properties of uniqueness and independence of the assignments are also fulfilled. JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

15 Basic assumptions Weak separability condition Definition (Dominance) The set of reference actions, B, fulfills the (strict) dominance relation if and only if j, Ω j (b h+1, b h ) 0 and j, Ω j (b h+1, b h ) > 0; h = 0,...,q. Definition (Weak separability condition) The set of reference actions, B, fulfills the weak separability condition if and only if j, Ω j (b h+1, b h ) 0 and j, Ω j (b h+1, b h ) > p j ; h = 0,...,q. Strict JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

16 Properties of the assignment rules II ELECTRE TRI-C Theorem If the weak separability condition is fulfilled, then there exists a compatible majority level, λ c, for which the conformity requirement holds, whenever the chosen majority level λ λ c, such that λ c = { } max σ(b h, b h+1 ) h = 0,..., q Example JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

17 Numerical example Compatible majority level Example b 1 b 2 b 3 b 4 b b b b Compatible majority level: λ c = max {σ(b h, b h+1 )} = 0.69 h=0,...,4 Source: Data adapted from Merad et al., 2004 Theorem JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

18 An overview of ELECTRE TRI-B Basic assumptions Ĉ = {Ĉ1,..., Ĉh,..., Ĉq} is the set of pre-defined and ordered categories, where Ĉ1 is the worst category and Ĉq the best one, with q 2. Each category Ĉh is defined by a lower profile limit, ˆb h 1, and an upper profile limit, ˆb h, such that ˆb h F ˆbh 1, h = 1,...,q. B = {ˆb 0, ˆb 1,..., ˆb h,..., ˆb q 1, ˆb q } is the set of the (q + 1) profile limits. ˆb 0 and ˆb q play a similar role as b 0 and b q+1 when using central reference actions. JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

19 An overview of ELECTRE TRI-B The most well-known assignment rules Definition (Pseudo-conjunctive assignment rule) Choose a majority level λ (0.5 λ 1). Decrease h from q until the first value such that ξ + h 1 (a,λ) 0. Assign action a to category Ĉh. Definition (Pseudo-disjunctive assignment rule) Choose a majority level λ (0.5 λ 1). Increase h from 0 until the first value such that ξ h (a,λ) 0 and ξ+ h (a,λ) < 0. Assign action a to category Ĉh. Comparing ELECTRE TRI-B rules ELECTRE TRI-C rules JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

20 Comparing results I ELECTRE TRI-C and ELECTRE TRI-B Theorem Consider (q + 2) reference actions defined to apply ELECTRE TRI-C with q categories. When such reference actions are used as profile limits of the (q + 1) categories in ELECTRE TRI-B, a) if an action a is assigned to C h by the ELECTRE TRI-C descending rule, then a is assigned to Ĉh or Ĉh+1 by the ELECTRE TRI-B pseudo-conjunctive rule. b) if an action a is assigned to C t by the ELECTRE TRI-C ascending rule, then a is assigned to Ĉk, with k t, by the ELECTRE TRI-B pseudo-disjunctive rule. Example Comparing results II JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

21 Numerical example Comparing assignment results Example ELECTRE TRI-C ELECTRE TRI-B Actions Descending Ascending Pseudo-conjunctive Pseudo-disjunctive a 1 C 2 C 4 C3 b C5 b a 2 C 1 C 4 C1 b C5 b a 3 C 3 C 3 C3 b C3 b a 4 C 2 C 3 C3 b C3 b a 5 C 2 C 1 C2 b C2 b a 6 C 2 C 4 C3 b C5 b a 7 C 1 C 2 C2 b C2 b a 8 C 1 C 2 C1 b C1 b a 9 C 1 C 4 C1 b C5 b a 10 C 1 C 1 C1 b C1 b Note: λ = 0.70; Source: Data adapted from Merad et al., 2004 JAD, Theorem JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

22 Conclusions In ELECTRE TRI-C the categories are defined through central reference actions instead of profile limits. Central reference actions and profile limits are two alternative ways of defining ordered categories. ELECTRE TRI-C fulfills the properties of uniqueness, independence, conformity, monotonicity, homogeneity, and stability. When the set of reference actions does not fulfill the strict separability condition, but only a weak separability condition, then a compatible majority level is required. A comparison with ELECTRE TRI-B shows the main similarities of the two methods. JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

23 References Almeida Dias, J.; Figueira, J.R. and Roy, B. (2008). ELECTRE TRI-C: A Multiple Criteria Sorting Method Based on Central Reference Actions. Technical Report 5/2008, CEG-IST, Tecnical University of Lisbon, Lisboa, Portugal. Merad, M.; Verdel, T.; Roy, B. and Kouniali, S. (2004). Use of multi-criteria decision-aids for risk zoning and management of large area subjected to mining-induced hazards. Tunnelling and Underground Space Technology, 19: Roy, B. (1996). Multicriteria Methodology for Decision Aiding. Kluwer Academic Publishers, Dordrecht. Roy, B. and Bouyssou, D., (1993). Aide Multicritère à la Décision : Méthodes et Cas. Economica, Paris, France. Yu, W. (1992). Aide Multicritère à la Décision dans le Cadre de la Problématique du Tri : Concepts, Méthodes et Applications. Thèse de Doctorat. LAMSADE, Université Paris-Dauphine, Paris, France. JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

24 Appendix 7 Appendix: Additional results Pseudo-criterion model Credibility index Binary relations Comparing ELECTRE TRI-C assignment rules Comparing ELECTRE TRI-B assignment rules Comparing results II ELECTRE TRI-C particular results JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

25 Pseudo-criterion model Definition (Pseudo-criterion) A pseudo-criterion is a function g j associated with two threshold functions, q j ( ) and p j ( ), satisfying the following condition: for all actions a in the sets of actions, g j (a) + p j and g j (a) + q j are non-decreasing monotone functions of g j (a). (Roy, 1996) Key concepts Remark Consider an ordered pair of actions (a, a ), and the two thresholds associated to the pseudo-criterion model, a) a P j a Ω j (a, a ) > p j ( ) b) a Q j a q j ( ) < Ω j (a, a ) p j ( ) c) a I j a q j ( ) Ω j (a, a ) q j ( ) JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

26 Key concepts Credibility index of ELECTRE methods Definition (Credibility index) The credibility of the comprehensive outranking of an action a over a, which means that a may be judged at least as good as a when taking all the criteria from F into account, is defined by aggregating the comprehensive concordance index, c(a, a ), and the partial discordance indices, d j (a, a ), as follows: σ(a, a ) = c(a, a ) n T j (a, a ) j=1 where, T j (a, a ) = { 1 dj (a,a ) 1 c(a,a ) if d j (a, a ) > c(a, a ) 1 otherwise Key concepts JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

27 Binary relations Definition (λ-binary relations) 1 λ-outranking: as λ a σ(a, a ) λ 0 2 λ-indifference: ai λ a σ(a, a ) λ 0 σ(a, a) λ 0 3 λ-incomparability: ar λ a σ(a, a ) λ < 0 σ(a, a) λ < 0 4 λ-preference: a a σ(a, a ) λ 0 σ(a, a) λ < 0 JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

28 Comparing assignment rules ELECTRE TRI-C Theorem a) If an action a is λ-indifferent to at least one reference action, then a is assigned by the descending rule to a category at least as good as the one a is assigned to when using the ascending rule. b) If an action a is λ-incomparable to at least one reference action, then a is assigned by the descending rule to a category at most as good as the one a is assigned to when using the ascending rule. c) Otherwise, both rules assign the action a to the same category or to two different but consecutive categories. Example ELECTRE TRI-C rules JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

29 Comparing assignment rules ELECTRE TRI-B Roy and Bouyssou, 1993, p. 395 If an action a is assigned to category Ĉk by the pseudo-conjunctive rule and to Ĉh by the pseudo-disjunctive rule, then k h. Furthermore, the two assignment rules provide the same results if and only if there is no t such that ξ + t (a,λ) < 0 and ξ t (a,λ) < 0 or there is at most one t such that ξ + t (a,λ) 0 and ξ t (a,λ) 0. ELECTRE TRI-B rules JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

30 Comparing results II ELECTRE TRI-C and ELECTRE TRI-B Theorem Consider (q + 1) profile limits defined to apply ELECTRE TRI-B with q categories. When such profile limits are used as reference actions of the (q 1) categories in ELECTRE TRI-C, a) if an action a is assigned to Ĉh by the ELECTRE TRI-B pseudo-conjunctive rule, then a is assigned to C h or C h 1 by the ELECTRE TRI-C descending rule. b) if an action a is assigned to Ĉt by the ELECTRE TRI-B pseudo-disjunctive rule, then a is assigned to C k, with k t by the ELECTRE TRI-C ascending rule. Comparing results I JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

31 ELECTRE TRI-C particular results Proposition a) If a λ-outranks b h, then a is assigned at least to C h by the descending rule. b) If b h λ-outranks a, then a is assigned at most to C h by the ascending rule. c) If a is λ-preferred to b h, then a is assigned at least to C h by both rules. d) If b h is λ-preferred to a, then a is assigned at most to C h by both rules. JAD, JRF, BR (LAMSADE & CEG-IST) ELECTRE TRI-C April, / 39

32 ELECTRE methods with interaction between criteria: An extension of the concordance index JOSÉ RUI FIGUEIRA 1, SALVATORE GRECO 2, BERNARD ROY 3 1 CEG-IST, Instituto Superior Técnico, Portugal 2 Faculty of Economics, University of Catania, Italy 3 LAMSADE, Université Paris-Dauphine, France Cost Action 0602, Troina, Sicily (11-16 April 2008) Figueira et al. (CEG-IST) Interaction Between Criteria / 41

33 Contents 1 Introduction 2 Illustrative Examples 3 Concepts: Definitions and notation 4 Types of interactions considered 5 Extensions of the concordance index 6 Conclusions 7 References Figueira et al. (CEG-IST) Interaction Between Criteria / 41

34 Introduction This presentation is devoted to an extension of the comprehensive concordance index of ELECTRE methods. Such an extension have been considered to take into account the interaction between criteria. Three types of interaction effects has been considered, mutual strengthening, mutual weakening, and antagonistic. In real-world decision-making situations is reasonable to consider the interaction between a small number of pairs of criteria. Various conditions, boundary, monotonicity, and continuity have been imposed. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

35 Introduction This presentation is devoted to an extension of the comprehensive concordance index of ELECTRE methods. Such an extension have been considered to take into account the interaction between criteria. Three types of interaction effects has been considered, mutual strengthening, mutual weakening, and antagonistic. In real-world decision-making situations is reasonable to consider the interaction between a small number of pairs of criteria. Various conditions, boundary, monotonicity, and continuity have been imposed. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

36 Introduction This presentation is devoted to an extension of the comprehensive concordance index of ELECTRE methods. Such an extension have been considered to take into account the interaction between criteria. Three types of interaction effects has been considered, mutual strengthening, mutual weakening, and antagonistic. In real-world decision-making situations is reasonable to consider the interaction between a small number of pairs of criteria. Various conditions, boundary, monotonicity, and continuity have been imposed. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

37 Introduction This presentation is devoted to an extension of the comprehensive concordance index of ELECTRE methods. Such an extension have been considered to take into account the interaction between criteria. Three types of interaction effects has been considered, mutual strengthening, mutual weakening, and antagonistic. In real-world decision-making situations is reasonable to consider the interaction between a small number of pairs of criteria. Various conditions, boundary, monotonicity, and continuity have been imposed. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

38 Introduction This presentation is devoted to an extension of the comprehensive concordance index of ELECTRE methods. Such an extension have been considered to take into account the interaction between criteria. Three types of interaction effects has been considered, mutual strengthening, mutual weakening, and antagonistic. In real-world decision-making situations is reasonable to consider the interaction between a small number of pairs of criteria. Various conditions, boundary, monotonicity, and continuity have been imposed. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

39 Introduction... It allows for the representation of some very important preference information, one that could not be modeled by the existing MCDA methodologies. This crucial preference information, is the interaction between criteria expressing preferences of the same sign (synergy and redundancy), or opposite sign (the power of the opposing criteria).... in Greco and Figueira (2003) Figueira et al. (CEG-IST) Interaction Between Criteria / 41

40 Introduction... It allows for the representation of some very important preference information, one that could not be modeled by the existing MCDA methodologies. This crucial preference information, is the interaction between criteria expressing preferences of the same sign (synergy and redundancy), or opposite sign (the power of the opposing criteria).... in Greco and Figueira (2003) Figueira et al. (CEG-IST) Interaction Between Criteria / 41

41 Introduction... It allows for the representation of some very important preference information, one that could not be modeled by the existing MCDA methodologies. This crucial preference information, is the interaction between criteria expressing preferences of the same sign (synergy and redundancy), or opposite sign (the power of the opposing criteria).... in Greco and Figueira (2003) Figueira et al. (CEG-IST) Interaction Between Criteria / 41

42 Introduction B. Roy. A propos de la signification des dépendances entre critères : Quelle place et quels modèles de prise en compte pour laide à la décision? Cahier du LAMSADE N. 244, Figueira et al. (CEG-IST) Interaction Between Criteria / 41

43 Introduction Application areas: Environmental problems Constructions of indices (in this case several pairs... of interaction criteria) Figueira et al. (CEG-IST) Interaction Between Criteria / 41

44 Introduction Application areas: Environmental problems Constructions of indices (in this case several pairs... of interaction criteria) Figueira et al. (CEG-IST) Interaction Between Criteria / 41

45 Introduction Application areas: Environmental problems Constructions of indices (in this case several pairs... of interaction criteria) Figueira et al. (CEG-IST) Interaction Between Criteria / 41

46 Illustrative Example 1 CHOOSING THE SITE FOR CONSTRUCTING A NEW HOTEL Criteria: g 1 : land purchasing and construction costs (investment costs) [min]; g 2 : annual operating costs (annual costs) [min]; g 3 : personnel recruitment possibilities (recruitment) [max]; g 4 : target client perceptions of the city district (image) [max]; g 5 : facility of access for the target clients (access) [max]. Mutual strengthening between criteria: investment and annual costs. Mutual weakening between criteria: image and access. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

47 Illustrative Example 1 CHOOSING THE SITE FOR CONSTRUCTING A NEW HOTEL Criteria: g 1 : land purchasing and construction costs (investment costs) [min]; g 2 : annual operating costs (annual costs) [min]; g 3 : personnel recruitment possibilities (recruitment) [max]; g 4 : target client perceptions of the city district (image) [max]; g 5 : facility of access for the target clients (access) [max]. Mutual strengthening between criteria: investment and annual costs. Mutual weakening between criteria: image and access. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

48 Illustrative Example 1 CHOOSING THE SITE FOR CONSTRUCTING A NEW HOTEL Criteria: g 1 : land purchasing and construction costs (investment costs) [min]; g 2 : annual operating costs (annual costs) [min]; g 3 : personnel recruitment possibilities (recruitment) [max]; g 4 : target client perceptions of the city district (image) [max]; g 5 : facility of access for the target clients (access) [max]. Mutual strengthening between criteria: investment and annual costs. Mutual weakening between criteria: image and access. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

49 Illustrative Example 1 CHOOSING THE SITE FOR CONSTRUCTING A NEW HOTEL Criteria: g 1 : land purchasing and construction costs (investment costs) [min]; g 2 : annual operating costs (annual costs) [min]; g 3 : personnel recruitment possibilities (recruitment) [max]; g 4 : target client perceptions of the city district (image) [max]; g 5 : facility of access for the target clients (access) [max]. Mutual strengthening between criteria: investment and annual costs. Mutual weakening between criteria: image and access. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

50 Illustrative Example 1 The comparison of site a with sites b, c, and d, in terms of the two financial criteria g 1 and g 2 b c d g 1 a is better than b a is worse than c a is better than d g 2 a is worse than b a is better than c a is better than d According to the classic definition of the concordance index, the role that g 1 and g 2 should have for supporting the answer to the assertion a is at least as good as b (or c or d) is characterized by the following weights, k 1 in the comparison with b, k 2 in the comparison with c, k 1 + k 2 in the comparison with d. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

51 Illustrative Example 1 The comparison of site a with sites b, c, and d, in terms of the two financial criteria g 1 and g 2 b c d g 1 a is better than b a is worse than c a is better than d g 2 a is worse than b a is better than c a is better than d According to the classic definition of the concordance index, the role that g 1 and g 2 should have for supporting the answer to the assertion a is at least as good as b (or c or d) is characterized by the following weights, k 1 in the comparison with b, k 2 in the comparison with c, k 1 + k 2 in the comparison with d. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

52 Illustrative Example 1 The comparison of site a with sites b, c, and d, in terms of the two financial criteria g 1 and g 2 b c d g 1 a is better than b a is worse than c a is better than d g 2 a is worse than b a is better than c a is better than d According to the classic definition of the concordance index, the role that g 1 and g 2 should have for supporting the answer to the assertion a is at least as good as b (or c or d) is characterized by the following weights, k 1 in the comparison with b, k 2 in the comparison with c, k 1 + k 2 in the comparison with d. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

53 Illustrative Example 1 The comparison of site a with sites b, c, and d, in terms of the two financial criteria g 1 and g 2 b c d g 1 a is better than b a is worse than c a is better than d g 2 a is worse than b a is better than c a is better than d According to the classic definition of the concordance index, the role that g 1 and g 2 should have for supporting the answer to the assertion a is at least as good as b (or c or d) is characterized by the following weights, k 1 in the comparison with b, k 2 in the comparison with c, k 1 + k 2 in the comparison with d. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

54 Illustrative Example 1 DMR considers the weights k 1 and k 2 appropriate, when only one criterion, supports a decision that one action is better than another one. However, he/she judges that the sum k 1 + k 2 is not sufficient to characterize the role of this criteria pair when both supports the decision Because in this case each criterion is strengthened by the other given the degree of complementarity between them. If one action is better than another one with respect to criteria g 1 and g 2 conjointly, it would be interesting to be able to take this mutual strengthening effect into account. This effect can be taken into account by increasing the weights k 1 and k 2 in the concordance index. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

55 Illustrative Example 1 DMR considers the weights k 1 and k 2 appropriate, when only one criterion, supports a decision that one action is better than another one. However, he/she judges that the sum k 1 + k 2 is not sufficient to characterize the role of this criteria pair when both supports the decision Because in this case each criterion is strengthened by the other given the degree of complementarity between them. If one action is better than another one with respect to criteria g 1 and g 2 conjointly, it would be interesting to be able to take this mutual strengthening effect into account. This effect can be taken into account by increasing the weights k 1 and k 2 in the concordance index. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

56 Illustrative Example 1 DMR considers the weights k 1 and k 2 appropriate, when only one criterion, supports a decision that one action is better than another one. However, he/she judges that the sum k 1 + k 2 is not sufficient to characterize the role of this criteria pair when both supports the decision Because in this case each criterion is strengthened by the other given the degree of complementarity between them. If one action is better than another one with respect to criteria g 1 and g 2 conjointly, it would be interesting to be able to take this mutual strengthening effect into account. This effect can be taken into account by increasing the weights k 1 and k 2 in the concordance index. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

57 Illustrative Example 1 DMR considers the weights k 1 and k 2 appropriate, when only one criterion, supports a decision that one action is better than another one. However, he/she judges that the sum k 1 + k 2 is not sufficient to characterize the role of this criteria pair when both supports the decision Because in this case each criterion is strengthened by the other given the degree of complementarity between them. If one action is better than another one with respect to criteria g 1 and g 2 conjointly, it would be interesting to be able to take this mutual strengthening effect into account. This effect can be taken into account by increasing the weights k 1 and k 2 in the concordance index. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

58 Illustrative Example 1 DMR considers the weights k 1 and k 2 appropriate, when only one criterion, supports a decision that one action is better than another one. However, he/she judges that the sum k 1 + k 2 is not sufficient to characterize the role of this criteria pair when both supports the decision Because in this case each criterion is strengthened by the other given the degree of complementarity between them. If one action is better than another one with respect to criteria g 1 and g 2 conjointly, it would be interesting to be able to take this mutual strengthening effect into account. This effect can be taken into account by increasing the weights k 1 and k 2 in the concordance index. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

59 Illustrative Example 1 The comparisons of site a with sites b, c, d in terms of the two purely ordinal criteria, g 4 and g 5 b c d g 4 a is better than b a is worse than c a is better than d g 5 a is worse than b a is better than c a is better than d The DMR judges that the sum k 4 + k 5 is too high to characterize the role of this criteria pair when both supports the decision that one action is better than another one, because in this case each criterion is weakened by the other due to the degree of redundancy between them. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

60 Illustrative Example 1 The comparisons of site a with sites b, c, d in terms of the two purely ordinal criteria, g 4 and g 5 b c d g 4 a is better than b a is worse than c a is better than d g 5 a is worse than b a is better than c a is better than d The DMR judges that the sum k 4 + k 5 is too high to characterize the role of this criteria pair when both supports the decision that one action is better than another one, because in this case each criterion is weakened by the other due to the degree of redundancy between them. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

61 Illustrative Example 1 The comparisons of site a with sites b, c, d in terms of the two purely ordinal criteria, g 4 and g 5 b c d g 4 a is better than b a is worse than c a is better than d g 5 a is worse than b a is better than c a is better than d The DMR judges that the sum k 4 + k 5 is too high to characterize the role of this criteria pair when both supports the decision that one action is better than another one, because in this case each criterion is weakened by the other due to the degree of redundancy between them. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

62 Illustrative Example 2 LAUNCHING A NEW DIGITAL CAMERA MODEL Criteria: g 1 : purchasing costs (cost) [min]; g 2 : weaknesses (fragility) [min]; g 3 : user friendliness of the controls (workability) [max]; g 4 : image quality (image) [max]; g 5 : aesthetics [max]; g 6 : volume [min]; g 7 : weight [min]. Antagonistic effect between criteria: cost and fragility. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

63 Illustrative Example 2 LAUNCHING A NEW DIGITAL CAMERA MODEL Criteria: g 1 : purchasing costs (cost) [min]; g 2 : weaknesses (fragility) [min]; g 3 : user friendliness of the controls (workability) [max]; g 4 : image quality (image) [max]; g 5 : aesthetics [max]; g 6 : volume [min]; g 7 : weight [min]. Antagonistic effect between criteria: cost and fragility. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

64 Illustrative Example 2 LAUNCHING A NEW DIGITAL CAMERA MODEL Criteria: g 1 : purchasing costs (cost) [min]; g 2 : weaknesses (fragility) [min]; g 3 : user friendliness of the controls (workability) [max]; g 4 : image quality (image) [max]; g 5 : aesthetics [max]; g 6 : volume [min]; g 7 : weight [min]. Antagonistic effect between criteria: cost and fragility. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

65 Illustrative Example 2 Comparisons of digital camera model a with models b, c, and d, according to criteria g 1 and g 2 : b c d g 1 a is better than b a at least as good as c a is better than d g 2 a is better than b a is better than c a is worse than d According to the classic definition of the concordance index the role these criteria should play in supporting the assertion model a is at least as good as model b (or c or d) is characterized by the following weights, k 1 + k 2 in the comparison with b, k 2 in the comparison with c, k 1 in the comparison with d. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

66 Illustrative Example 2 Comparisons of digital camera model a with models b, c, and d, according to criteria g 1 and g 2 : b c d g 1 a is better than b a at least as good as c a is better than d g 2 a is better than b a is better than c a is worse than d According to the classic definition of the concordance index the role these criteria should play in supporting the assertion model a is at least as good as model b (or c or d) is characterized by the following weights, k 1 + k 2 in the comparison with b, k 2 in the comparison with c, k 1 in the comparison with d. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

67 Illustrative Example 2 Comparisons of digital camera model a with models b, c, and d, according to criteria g 1 and g 2 : b c d g 1 a is better than b a at least as good as c a is better than d g 2 a is better than b a is better than c a is worse than d According to the classic definition of the concordance index the role these criteria should play in supporting the assertion model a is at least as good as model b (or c or d) is characterized by the following weights, k 1 + k 2 in the comparison with b, k 2 in the comparison with c, k 1 in the comparison with d. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

68 Illustrative Example 2 DMR considers that weights k 1 and k 2 adequately characterize the role these two criteria should play when comparing a with b and a with c However, he/she considers that the same is not true when comparing a with d Based on a customer survey, it seems that when one model is less fragile than another, the benefit derived from the lower cost is partially masked by the fact the model is less fragile. This phenomenon can be modeled by decreasing the weight of criterion g 1 in the concordance index of the assertion a is at least as good as b. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

69 Illustrative Example 2 DMR considers that weights k 1 and k 2 adequately characterize the role these two criteria should play when comparing a with b and a with c However, he/she considers that the same is not true when comparing a with d Based on a customer survey, it seems that when one model is less fragile than another, the benefit derived from the lower cost is partially masked by the fact the model is less fragile. This phenomenon can be modeled by decreasing the weight of criterion g 1 in the concordance index of the assertion a is at least as good as b. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

70 Illustrative Example 2 DMR considers that weights k 1 and k 2 adequately characterize the role these two criteria should play when comparing a with b and a with c However, he/she considers that the same is not true when comparing a with d Based on a customer survey, it seems that when one model is less fragile than another, the benefit derived from the lower cost is partially masked by the fact the model is less fragile. This phenomenon can be modeled by decreasing the weight of criterion g 1 in the concordance index of the assertion a is at least as good as b. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

71 Illustrative Example 2 DMR considers that weights k 1 and k 2 adequately characterize the role these two criteria should play when comparing a with b and a with c However, he/she considers that the same is not true when comparing a with d Based on a customer survey, it seems that when one model is less fragile than another, the benefit derived from the lower cost is partially masked by the fact the model is less fragile. This phenomenon can be modeled by decreasing the weight of criterion g 1 in the concordance index of the assertion a is at least as good as b. Figueira et al. (CEG-IST) Interaction Between Criteria / 41

72 Concepts: Definitions and notation Notation Pseudo criterion The criteria weights and the concordance index Partial concordance index c i (a, b) Properties of the comprehensive concordance index c(a, b) Figueira et al. (CEG-IST) Interaction Between Criteria / 41

73 Concepts: Definitions and notation Notation Pseudo criterion The criteria weights and the concordance index Partial concordance index c i (a, b) Properties of the comprehensive concordance index c(a, b) Figueira et al. (CEG-IST) Interaction Between Criteria / 41

74 Concepts: Definitions and notation Notation Pseudo criterion The criteria weights and the concordance index Partial concordance index c i (a, b) Properties of the comprehensive concordance index c(a, b) Figueira et al. (CEG-IST) Interaction Between Criteria / 41

75 Concepts: Definitions and notation Notation Pseudo criterion The criteria weights and the concordance index Partial concordance index c i (a, b) Properties of the comprehensive concordance index c(a, b) Figueira et al. (CEG-IST) Interaction Between Criteria / 41

76 Concepts: Definitions and notation Notation Pseudo criterion The criteria weights and the concordance index Partial concordance index c i (a, b) Properties of the comprehensive concordance index c(a, b) Figueira et al. (CEG-IST) Interaction Between Criteria / 41

77 Concepts: Definitions and notation Notation Pseudo criterion The criteria weights and the concordance index Partial concordance index c i (a, b) Properties of the comprehensive concordance index c(a, b) Figueira et al. (CEG-IST) Interaction Between Criteria / 41

78 Notation - F = {g 1, g 2,..., g i,..., g n } denote a coherent set or family of criteria; for the sake of simplicity we shall use also F as the set of criteria indices (the same will apply later on for subsets of F ); - A = {a, b, c,... } denote a finite set of actions with cardinality m; - g i (a) E i denote the evaluation of action a on criterion g i, for all a A and i F, where E i is the scale associated to criterion g i (no restriction is imposed to the scale type). - k i is the relative importance or weight of criterion g i. - C(aTb) represents the coalition of criteria in favor of the assertion atb, where T {P, Q, S} (introduced later). - C(aTb) denote the complement of C(aTb). Figueira et al. (CEG-IST) Interaction Between Criteria / 41

79 Notation - F = {g 1, g 2,..., g i,..., g n } denote a coherent set or family of criteria; for the sake of simplicity we shall use also F as the set of criteria indices (the same will apply later on for subsets of F ); - A = {a, b, c,... } denote a finite set of actions with cardinality m; - g i (a) E i denote the evaluation of action a on criterion g i, for all a A and i F, where E i is the scale associated to criterion g i (no restriction is imposed to the scale type). - k i is the relative importance or weight of criterion g i. - C(aTb) represents the coalition of criteria in favor of the assertion atb, where T {P, Q, S} (introduced later). - C(aTb) denote the complement of C(aTb). Figueira et al. (CEG-IST) Interaction Between Criteria / 41

80 Notation - F = {g 1, g 2,..., g i,..., g n } denote a coherent set or family of criteria; for the sake of simplicity we shall use also F as the set of criteria indices (the same will apply later on for subsets of F ); - A = {a, b, c,... } denote a finite set of actions with cardinality m; - g i (a) E i denote the evaluation of action a on criterion g i, for all a A and i F, where E i is the scale associated to criterion g i (no restriction is imposed to the scale type). - k i is the relative importance or weight of criterion g i. - C(aTb) represents the coalition of criteria in favor of the assertion atb, where T {P, Q, S} (introduced later). - C(aTb) denote the complement of C(aTb). Figueira et al. (CEG-IST) Interaction Between Criteria / 41

81 Notation - F = {g 1, g 2,..., g i,..., g n } denote a coherent set or family of criteria; for the sake of simplicity we shall use also F as the set of criteria indices (the same will apply later on for subsets of F ); - A = {a, b, c,... } denote a finite set of actions with cardinality m; - g i (a) E i denote the evaluation of action a on criterion g i, for all a A and i F, where E i is the scale associated to criterion g i (no restriction is imposed to the scale type). - k i is the relative importance or weight of criterion g i. - C(aTb) represents the coalition of criteria in favor of the assertion atb, where T {P, Q, S} (introduced later). - C(aTb) denote the complement of C(aTb). Figueira et al. (CEG-IST) Interaction Between Criteria / 41

82 Notation - F = {g 1, g 2,..., g i,..., g n } denote a coherent set or family of criteria; for the sake of simplicity we shall use also F as the set of criteria indices (the same will apply later on for subsets of F ); - A = {a, b, c,... } denote a finite set of actions with cardinality m; - g i (a) E i denote the evaluation of action a on criterion g i, for all a A and i F, where E i is the scale associated to criterion g i (no restriction is imposed to the scale type). - k i is the relative importance or weight of criterion g i. - C(aTb) represents the coalition of criteria in favor of the assertion atb, where T {P, Q, S} (introduced later). - C(aTb) denote the complement of C(aTb). Figueira et al. (CEG-IST) Interaction Between Criteria / 41

83 Notation - F = {g 1, g 2,..., g i,..., g n } denote a coherent set or family of criteria; for the sake of simplicity we shall use also F as the set of criteria indices (the same will apply later on for subsets of F ); - A = {a, b, c,... } denote a finite set of actions with cardinality m; - g i (a) E i denote the evaluation of action a on criterion g i, for all a A and i F, where E i is the scale associated to criterion g i (no restriction is imposed to the scale type). - k i is the relative importance or weight of criterion g i. - C(aTb) represents the coalition of criteria in favor of the assertion atb, where T {P, Q, S} (introduced later). - C(aTb) denote the complement of C(aTb). Figueira et al. (CEG-IST) Interaction Between Criteria / 41

84 Pseudo criterion A pseudo criterion is a function g i associated with the two threshold functions q i (g i (a)) and p i (g i (a)) satisfying the following condition, for all a A (Roy, 1991, 1996): g i (a) + p i (g i (a)) and g i (a) + q i (g i (a)) are non-decreasing monotone functions of g i (a). By definition, for all pairs (a, b) A A with g i (a) g i (b) and q i (g i (a)) p i (g i (a)), ai i b aq i b ap i b g i (a) g i (b) + q i (g i (b)); g i (b) + q i (g i (b)) < g i (a) g i (b) + p i (g i (b)); g i (b) + p i (g i (b)) < g i (a). If, q i (g i (a)) = p i (g i (a)), for all a A, then g i is called a quasi criterion. For a quasi criterion there is no ambiguity zone, that is, weak preference Q i Figueira et al. (CEG-IST) Interaction Between Criteria / 41

85 Pseudo criterion A pseudo criterion is a function g i associated with the two threshold functions q i (g i (a)) and p i (g i (a)) satisfying the following condition, for all a A (Roy, 1991, 1996): g i (a) + p i (g i (a)) and g i (a) + q i (g i (a)) are non-decreasing monotone functions of g i (a). By definition, for all pairs (a, b) A A with g i (a) g i (b) and q i (g i (a)) p i (g i (a)), ai i b aq i b ap i b g i (a) g i (b) + q i (g i (b)); g i (b) + q i (g i (b)) < g i (a) g i (b) + p i (g i (b)); g i (b) + p i (g i (b)) < g i (a). If, q i (g i (a)) = p i (g i (a)), for all a A, then g i is called a quasi criterion. For a quasi criterion there is no ambiguity zone, that is, weak preference Q i Figueira et al. (CEG-IST) Interaction Between Criteria / 41

86 Pseudo criterion A pseudo criterion is a function g i associated with the two threshold functions q i (g i (a)) and p i (g i (a)) satisfying the following condition, for all a A (Roy, 1991, 1996): g i (a) + p i (g i (a)) and g i (a) + q i (g i (a)) are non-decreasing monotone functions of g i (a). By definition, for all pairs (a, b) A A with g i (a) g i (b) and q i (g i (a)) p i (g i (a)), ai i b aq i b ap i b g i (a) g i (b) + q i (g i (b)); g i (b) + q i (g i (b)) < g i (a) g i (b) + p i (g i (b)); g i (b) + p i (g i (b)) < g i (a). If, q i (g i (a)) = p i (g i (a)), for all a A, then g i is called a quasi criterion. For a quasi criterion there is no ambiguity zone, that is, weak preference Q i Figueira et al. (CEG-IST) Interaction Between Criteria / 41

87 Pseudo criterion A pseudo criterion is a function g i associated with the two threshold functions q i (g i (a)) and p i (g i (a)) satisfying the following condition, for all a A (Roy, 1991, 1996): g i (a) + p i (g i (a)) and g i (a) + q i (g i (a)) are non-decreasing monotone functions of g i (a). By definition, for all pairs (a, b) A A with g i (a) g i (b) and q i (g i (a)) p i (g i (a)), ai i b aq i b ap i b g i (a) g i (b) + q i (g i (b)); g i (b) + q i (g i (b)) < g i (a) g i (b) + p i (g i (b)); g i (b) + p i (g i (b)) < g i (a). If, q i (g i (a)) = p i (g i (a)), for all a A, then g i is called a quasi criterion. For a quasi criterion there is no ambiguity zone, that is, weak preference Q i Figueira et al. (CEG-IST) Interaction Between Criteria / 41

88 The criteria weights and the concordance index The concordance index can be defined as follows, where, 8 c(a, b) = i F k i K c i(a, b), with K = i F 1, if g i (a) + q i (g i (a)) g i (b), (as i b), k i >< c i (a, b) = g i (a) + p i `gi (a) g i (b) p i `gi (a) q i `gi (a), if g i(a) + q i (g i (a)) < g i (b) g i (a) + p i (g i (a)), (bq i a), >: 0, if g i (a) + p i (g i (a)) < g i (b), (bp i a). When F is composed of quasi-criteria, c(a, b) = i C(aSb) k i K Figueira et al. (CEG-IST) Interaction Between Criteria / 41

89 The criteria weights and the concordance index The concordance index can be defined as follows, where, 8 c(a, b) = i F k i K c i(a, b), with K = i F 1, if g i (a) + q i (g i (a)) g i (b), (as i b), k i >< c i (a, b) = g i (a) + p i `gi (a) g i (b) p i `gi (a) q i `gi (a), if g i(a) + q i (g i (a)) < g i (b) g i (a) + p i (g i (a)), (bq i a), >: 0, if g i (a) + p i (g i (a)) < g i (b), (bp i a). When F is composed of quasi-criteria, c(a, b) = i C(aSb) k i K Figueira et al. (CEG-IST) Interaction Between Criteria / 41

90 Partial concordance index, c i (a, b) c i (.,.) 1 c i (b, a) c i (a, b) 0 c i (a, b) c i (b, a) g i (b) p i (b) g i (b) q i (b) g i (b) g i (b) + q i (b) g i (b) + p i (b) g i (a) Figueira et al. (CEG-IST) Interaction Between Criteria / 41

91 Partial concordance index, c i (a, b) c i (.,.) 1 c i (b, a) c i (a, b) 0 c i (a, b) c i (b, a) g i (b) p i (b) g i (b) q i (b) g i (b) g i (b) + q i (b) g i (b) + p i (b) g i (a) Figueira et al. (CEG-IST) Interaction Between Criteria / 41