Applying MCHP, ROR and the SMAA methodology to the ELECTRE III method with interaction between criteria

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1 Applying MCHP, ROR and the SMAA methodology to the ELECTRE III method with interaction between criteria Salvatore Corrente 1, José Rui Figueira 2, Salvatore Greco 1,3, Roman S lowiński 4 1 Department of Economics and Business, University of Catania, Italy 2 Instituto Superior Técnico, Universidade de Lisboa, Portugal 3 Portsmouth Business School, UK 4 Poznań University of Technology and Polish Academy of Sciences, Poland 83 rd EWG-MCDA, Barcelona, Spain, March 31-April 2, April / 47

2 We deal with... The ELECTRE III outranking method, An imprecise version of the SRF method, Possible interactions between criteria, SMAA and ROR to explore the whole set of parameters (weights and interaction coefficients). 1 April / 47

3 We deal with... The ELECTRE III outranking method, An imprecise version of the SRF method, Possible interactions between criteria, SMAA and ROR to explore the whole set of parameters (weights and interaction coefficients). 1 April / 47

4 We deal with... The ELECTRE III outranking method, An imprecise version of the SRF method, Possible interactions between criteria, SMAA and ROR to explore the whole set of parameters (weights and interaction coefficients). 1 April / 47

5 We deal with... The ELECTRE III outranking method, An imprecise version of the SRF method, Possible interactions between criteria, SMAA and ROR to explore the whole set of parameters (weights and interaction coefficients). 1 April / 47

6 We deal with... The ELECTRE III outranking method, An imprecise version of the SRF method, Possible interactions between criteria, SMAA and ROR to explore the whole set of parameters (weights and interaction coefficients). 1 April / 47

7 Plan The Multiple Criteria Hierarchy Process (MCHP), The ELECTRE III method and the MCHP, The imprecise SRF method, Interactions between criteria, SMAA and ROR applied to the hierarchical ELECTRE III method with interactions: The proposed model, A real world decision making problem: Universities ranking, Conclusions 1 April / 47

8 Multiple Criteria Hierarchy Process 1 Basic concepts: G set of criteria of all levels of the hierarchy, I G set of indices of all criteria in the different levels, G r G with r = (i 1,...,i h ) I G, criterion of the level h in the hierarchy, EL set of indices of elementary subcriteria (i.e. set of criteria in the leaves of the tree), E(G r ) set of indices of elementary subcriteria descending from criterion G r. 1 Corrente, Greco, S lowiński (2012) 1 April / 47

9 Multiple Criteria Hierarchy Process 1 Basic concepts: G set of criteria of all levels of the hierarchy, I G set of indices of all criteria in the different levels, G r G with r = (i 1,...,i h ) I G, criterion of the level h in the hierarchy, EL set of indices of elementary subcriteria (i.e. set of criteria in the leaves of the tree), E(G r ) set of indices of elementary subcriteria descending from criterion G r. 1 Corrente, Greco, S lowiński (2012) 1 April / 47

10 Multiple Criteria Hierarchy Process 1 Basic concepts: G set of criteria of all levels of the hierarchy, I G set of indices of all criteria in the different levels, G r G with r = (i 1,...,i h ) I G, criterion of the level h in the hierarchy, EL set of indices of elementary subcriteria (i.e. set of criteria in the leaves of the tree), E(G r ) set of indices of elementary subcriteria descending from criterion G r. 1 Corrente, Greco, S lowiński (2012) 1 April / 47

11 Multiple Criteria Hierarchy Process 1 Basic concepts: G set of criteria of all levels of the hierarchy, I G set of indices of all criteria in the different levels, G r G with r = (i 1,...,i h ) I G, criterion of the level h in the hierarchy, EL set of indices of elementary subcriteria (i.e. set of criteria in the leaves of the tree), E(G r ) set of indices of elementary subcriteria descending from criterion G r. 1 Corrente, Greco, S lowiński (2012) 1 April / 47

12 Multiple Criteria Hierarchy Process 1 Basic concepts: G set of criteria of all levels of the hierarchy, I G set of indices of all criteria in the different levels, G r G with r = (i 1,...,i h ) I G, criterion of the level h in the hierarchy, EL set of indices of elementary subcriteria (i.e. set of criteria in the leaves of the tree), E(G r ) set of indices of elementary subcriteria descending from criterion G r. 1 Corrente, Greco, S lowiński (2012) 1 April / 47

13 Multiple Criteria Hierarchy Process 1 Basic concepts: G set of criteria of all levels of the hierarchy, I G set of indices of all criteria in the different levels, G r G with r = (i 1,...,i h ) I G, criterion of the level h in the hierarchy, EL set of indices of elementary subcriteria (i.e. set of criteria in the leaves of the tree), E(G r ) set of indices of elementary subcriteria descending from criterion G r. 1 Corrente, Greco, S lowiński (2012) 1 April / 47

14 Elementary and partial concordance indices in MCHP-ELECTRE 2 For each elementary criterion g t, we define 1, if g t (b) g t (a) q t p ϕ t (a,b) = t [g t(b) g t(a)] p t q t, if q t < g t (b) g t (a) < p t 0, if g t (b) g t (a) p t while for each non-elementary criterion G r, that is, r I G \EL, the partial concordance index is: where: C r (a,b) = t E(G r) w t ϕ t (a,b) w t, importance of elementary criterion g t, q t and p t are the indifference and the preference thresholds of g t. 2 Corrente, Greco, S lowiński (2013) 1 April / 47

15 Elementary and partial concordance indices in MCHP-ELECTRE 2 For each elementary criterion g t, we define 1, if g t (b) g t (a) q t p ϕ t (a,b) = t [g t(b) g t(a)] p t q t, if q t < g t (b) g t (a) < p t 0, if g t (b) g t (a) p t while for each non-elementary criterion G r, that is, r I G \EL, the partial concordance index is: where: C r (a,b) = t E(G r) w t ϕ t (a,b) w t, importance of elementary criterion g t, q t and p t are the indifference and the preference thresholds of g t. 2 Corrente, Greco, S lowiński (2013) 1 April / 47

16 Elementary discordance index and partial credibility index of outranking For each elementary criterion g t we can define the partial discordance index: 0 if g t (b) g t (a) p t, [g d t (a,b) = t(b) g t(a)] p t v t p t if p t < g t (b) g t (a) < v t, 1 if g t (b) g t (a) v t. where v t is the veto threshold defined for g t. For each non-elementary criterion G r, we define also the partial credibility index: σ r (a,b) = C r (a,b) t E(G r):d t(a,b)>c r(a,b) 1 d t (a,b) 1 C r (a,b) 1 April / 47

17 Elementary discordance index and partial credibility index of outranking For each elementary criterion g t we can define the partial discordance index: 0 if g t (b) g t (a) p t, [g d t (a,b) = t(b) g t(a)] p t v t p t if p t < g t (b) g t (a) < v t, 1 if g t (b) g t (a) v t. where v t is the veto threshold defined for g t. For each non-elementary criterion G r, we define also the partial credibility index: σ r (a,b) = C r (a,b) t E(G r):d t(a,b)>c r(a,b) 1 d t (a,b) 1 C r (a,b) 1 April / 47

18 The descending and the ascending distillations... The ELECTRE III method 3 provides a partial ranking of the considered alternatives as the intersection of the two complete preorders obtained from two distillations: in the descending distillation, we order the alternatives from the best to the worst, {C 1,...,C k }, in the ascending distillation, we order the alternatives from the worst to the best. {F 1,...,F k }. 3 Roy (1978) 1 April / 47

19 The descending and the ascending distillations... The ELECTRE III method 3 provides a partial ranking of the considered alternatives as the intersection of the two complete preorders obtained from two distillations: in the descending distillation, we order the alternatives from the best to the worst, {C 1,...,C k }, in the ascending distillation, we order the alternatives from the worst to the best. {F 1,...,F k }. 3 Roy (1978) 1 April / 47

20 The descending and the ascending distillations... The ELECTRE III method 3 provides a partial ranking of the considered alternatives as the intersection of the two complete preorders obtained from two distillations: in the descending distillation, we order the alternatives from the best to the worst, {C 1,...,C k }, in the ascending distillation, we order the alternatives from the worst to the best. {F 1,...,F k }. 3 Roy (1978) 1 April / 47

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23 The final preorder of the ELECTRE-III method Given a,b A and a non-elementary criterion G r, a is preferred to b if a is in a class not worse than b in both distillations and in a better class than b in at least one distillation, a and b are indifferent if they belong to the same class in both distillations, a and b are incomparable if a is in a better class than b in one distillation and in a worse class than b in the other one. 1 April / 47

24 The final preorder of the ELECTRE-III method Given a,b A and a non-elementary criterion G r, a is preferred to b if a is in a class not worse than b in both distillations and in a better class than b in at least one distillation, a and b are indifferent if they belong to the same class in both distillations, a and b are incomparable if a is in a better class than b in one distillation and in a worse class than b in the other one. 1 April / 47

25 The final preorder of the ELECTRE-III method Given a,b A and a non-elementary criterion G r, a is preferred to b if a is in a class not worse than b in both distillations and in a better class than b in at least one distillation, a and b are indifferent if they belong to the same class in both distillations, a and b are incomparable if a is in a better class than b in one distillation and in a worse class than b in the other one. 1 April / 47

26 The final preorder of the ELECTRE-III method Given a,b A and a non-elementary criterion G r, a is preferred to b if a is in a class not worse than b in both distillations and in a better class than b in at least one distillation, a and b are indifferent if they belong to the same class in both distillations, a and b are incomparable if a is in a better class than b in one distillation and in a worse class than b in the other one. 1 April / 47

27 The SRF method 4 In the classical SRF method the DM is asked to provide the following preference information: 1 Rank the criteria from the most important to: L 1 = {g 2,g 4 }, L 2 = {g 1,g 3 }, L 3 = {g 5,g 7 } L 4 = {g 6 } 2 Insert some white cards between two successive sets of criteria to increase the difference of importance between the criteria in these sets: L 1 = {g 2,g 4 },{WC}, L 2 = {g 1,g 3 }, {WC,WC}, L 3 = {g 5,g 7 }, L 4 = {g 6 } 3 Provide the ratio z between the weights of the most important criteria and the weights of the least important ones: 4 Figueira, Roy (2002) z = w L1 /w L4 1 April / 47

28 The SRF method 4 In the classical SRF method the DM is asked to provide the following preference information: 1 Rank the criteria from the most important to: L 1 = {g 2,g 4 }, L 2 = {g 1,g 3 }, L 3 = {g 5,g 7 } L 4 = {g 6 } 2 Insert some white cards between two successive sets of criteria to increase the difference of importance between the criteria in these sets: L 1 = {g 2,g 4 },{WC}, L 2 = {g 1,g 3 }, {WC,WC}, L 3 = {g 5,g 7 }, L 4 = {g 6 } 3 Provide the ratio z between the weights of the most important criteria and the weights of the least important ones: 4 Figueira, Roy (2002) z = w L1 /w L4 1 April / 47

29 The SRF method 4 In the classical SRF method the DM is asked to provide the following preference information: 1 Rank the criteria from the most important to: L 1 = {g 2,g 4 }, L 2 = {g 1,g 3 }, L 3 = {g 5,g 7 } L 4 = {g 6 } 2 Insert some white cards between two successive sets of criteria to increase the difference of importance between the criteria in these sets: L 1 = {g 2,g 4 },{WC}, L 2 = {g 1,g 3 }, {WC,WC}, L 3 = {g 5,g 7 }, L 4 = {g 6 } 3 Provide the ratio z between the weights of the most important criteria and the weights of the least important ones: 4 Figueira, Roy (2002) z = w L1 /w L4 1 April / 47

30 The SRF method 4 In the classical SRF method the DM is asked to provide the following preference information: 1 Rank the criteria from the most important to: L 1 = {g 2,g 4 }, L 2 = {g 1,g 3 }, L 3 = {g 5,g 7 } L 4 = {g 6 } 2 Insert some white cards between two successive sets of criteria to increase the difference of importance between the criteria in these sets: L 1 = {g 2,g 4 },{WC}, L 2 = {g 1,g 3 }, {WC,WC}, L 3 = {g 5,g 7 }, L 4 = {g 6 } 3 Provide the ratio z between the weights of the most important criteria and the weights of the least important ones: 4 Figueira, Roy (2002) z = w L1 /w L4 1 April / 47

31 Some questions Considering robustness concerns the two following questions are very natural 5 : What about the possibility for the DM to express an uncertainty on the number of white cards between two successive sets of criteria? What about the possibility for the DM to express an uncertainty on the ratio z between the weights of the most important criteria and the weights of the least important ones? 5 Bottero et al April / 47

32 Some questions Considering robustness concerns the two following questions are very natural 5 : What about the possibility for the DM to express an uncertainty on the number of white cards between two successive sets of criteria? What about the possibility for the DM to express an uncertainty on the ratio z between the weights of the most important criteria and the weights of the least important ones? 5 Bottero et al April / 47

33 Some questions Considering robustness concerns the two following questions are very natural 5 : What about the possibility for the DM to express an uncertainty on the number of white cards between two successive sets of criteria? What about the possibility for the DM to express an uncertainty on the ratio z between the weights of the most important criteria and the weights of the least important ones? 5 Bottero et al April / 47

34 The imprecise SRF method In our proposal, the DM is asked to: 1 Rank the criteria from the most important to the least important with the possibilities of some ex-aequo, L 1 = {g 2,g 4 }, L 2 = {g 1,g 3 }, L 3 = {g 5,g 7 }, L 4 = {g 6 } 2 Insert an imprecise number of white cards between two successive sets of criteria to increase the difference of importance between the criteria in these sets: L 1 = {g 2,g 4 },[2,4], L 2 = {g 1,g 3 }, {WC}, L 3 = {g 5,g 7 }, [1,3], L 4 = {g 6 } 3 Provide an interval of variation for the ratio z between the weights of the most important criteria and the weights of the least important criteria: 2 w L1 /w L April / 47

35 The imprecise SRF method In our proposal, the DM is asked to: 1 Rank the criteria from the most important to the least important with the possibilities of some ex-aequo, L 1 = {g 2,g 4 }, L 2 = {g 1,g 3 }, L 3 = {g 5,g 7 }, L 4 = {g 6 } 2 Insert an imprecise number of white cards between two successive sets of criteria to increase the difference of importance between the criteria in these sets: L 1 = {g 2,g 4 },[2,4], L 2 = {g 1,g 3 }, {WC}, L 3 = {g 5,g 7 }, [1,3], L 4 = {g 6 } 3 Provide an interval of variation for the ratio z between the weights of the most important criteria and the weights of the least important criteria: 2 w L1 /w L April / 47

36 The imprecise SRF method In our proposal, the DM is asked to: 1 Rank the criteria from the most important to the least important with the possibilities of some ex-aequo, L 1 = {g 2,g 4 }, L 2 = {g 1,g 3 }, L 3 = {g 5,g 7 }, L 4 = {g 6 } 2 Insert an imprecise number of white cards between two successive sets of criteria to increase the difference of importance between the criteria in these sets: L 1 = {g 2,g 4 },[2,4], L 2 = {g 1,g 3 }, {WC}, L 3 = {g 5,g 7 }, [1,3], L 4 = {g 6 } 3 Provide an interval of variation for the ratio z between the weights of the most important criteria and the weights of the least important criteria: 2 w L1 /w L April / 47

37 The imprecise SRF method In our proposal, the DM is asked to: 1 Rank the criteria from the most important to the least important with the possibilities of some ex-aequo, L 1 = {g 2,g 4 }, L 2 = {g 1,g 3 }, L 3 = {g 5,g 7 }, L 4 = {g 6 } 2 Insert an imprecise number of white cards between two successive sets of criteria to increase the difference of importance between the criteria in these sets: L 1 = {g 2,g 4 },[2,4], L 2 = {g 1,g 3 }, {WC}, L 3 = {g 5,g 7 }, [1,3], L 4 = {g 6 } 3 Provide an interval of variation for the ratio z between the weights of the most important criteria and the weights of the least important criteria: 2 w L1 /w L April / 47

38 The imprecise SRF method From a formal point of view, let us denote by 1 L 1,...,L v the sets of criteria ordered from the most important to the least important, 2 e s the number of white cards included between L s and L s+1, s = 1,...,v 1, 3 z the ratio between the weights of criteria in L 1 and the weights of criteria in L v. 1 April / 47

39 The imprecise SRF method If the DM states that e s [low s,upp s ], then this information is translated to the following constraints w Ls+1 w Ls +(low s +1) C, w Ls+1 w Ls +(upp s +1) C, C > 0. where C is the importance of a white card. If the DM states that z [z low,z upp ], then this information is translated to the constraint z low w L 1 w Lv z upp. 1 April / 47

40 The imprecise SRF method The complete set of constraints translating the preference information provided by the DM is therefore the following: w Ls+1 w Ls +(low s +1) C, w Ls+1 w Ls +(upp s +1) C, for all s = 1,...,v 1, C > 0, z low w Lv w L1 0, w L1 z upp w Lv 0, w Lv > 0. 1 April / 47

41 The imprecise SRF method and the MCHP The imprecise SRF method can be easily applied in case of criteria structured in a hierarchical way since the DM has to apply the SRF method to the set of criteria {G (r,1),...,g (r,n(r)) } for each non-elementary criterion G r. 1 The constraints derived from the application of the imprecise SRF method can be expressed as function of the weights of elementary criteria only, since W r = w t, for all G r,r I G \EL, t E(G r) 2 The importance of the white cards has not to be the same for all criteria in the hierarchy. 1 April / 47

42 Interactions between criteria 6 Sometimes, the criteria are not independent. They present a certain degree of interaction: g t1 and g t2 present a mutual strengthening effect if the importance of {g t1,g t2 } is greater than the sum of the importance of the two criteria taken alone. A value w t1t 2 > 0 has to be added to w t1 +w t2 ; g t1 and g t2 present a mutual weakening effect if the importance of {g t1,g t2 } is lower than the sum of the importance of the two criteria taken alone. A value w t1t 2 > 0 has to be subtracted from w t1 +w t2 ; g t2 presents an antagonistic effect over g t1 if the importance assigned to g t1 has to be lowered in consequence of the presence of g t2. A value w t 1t 2 > 0 has to be subtracted from w t1. 6 Figueira, Greco, Roy (2009) 1 April / 47

43 Interactions between criteria 6 Sometimes, the criteria are not independent. They present a certain degree of interaction: g t1 and g t2 present a mutual strengthening effect if the importance of {g t1,g t2 } is greater than the sum of the importance of the two criteria taken alone. A value w t1t 2 > 0 has to be added to w t1 +w t2 ; g t1 and g t2 present a mutual weakening effect if the importance of {g t1,g t2 } is lower than the sum of the importance of the two criteria taken alone. A value w t1t 2 > 0 has to be subtracted from w t1 +w t2 ; g t2 presents an antagonistic effect over g t1 if the importance assigned to g t1 has to be lowered in consequence of the presence of g t2. A value w t 1t 2 > 0 has to be subtracted from w t1. 6 Figueira, Greco, Roy (2009) 1 April / 47

44 Interactions between criteria 6 Sometimes, the criteria are not independent. They present a certain degree of interaction: g t1 and g t2 present a mutual strengthening effect if the importance of {g t1,g t2 } is greater than the sum of the importance of the two criteria taken alone. A value w t1t 2 > 0 has to be added to w t1 +w t2 ; g t1 and g t2 present a mutual weakening effect if the importance of {g t1,g t2 } is lower than the sum of the importance of the two criteria taken alone. A value w t1t 2 > 0 has to be subtracted from w t1 +w t2 ; g t2 presents an antagonistic effect over g t1 if the importance assigned to g t1 has to be lowered in consequence of the presence of g t2. A value w t 1t 2 > 0 has to be subtracted from w t1. 6 Figueira, Greco, Roy (2009) 1 April / 47

45 Interactions between criteria 6 Sometimes, the criteria are not independent. They present a certain degree of interaction: g t1 and g t2 present a mutual strengthening effect if the importance of {g t1,g t2 } is greater than the sum of the importance of the two criteria taken alone. A value w t1t 2 > 0 has to be added to w t1 +w t2 ; g t1 and g t2 present a mutual weakening effect if the importance of {g t1,g t2 } is lower than the sum of the importance of the two criteria taken alone. A value w t1t 2 > 0 has to be subtracted from w t1 +w t2 ; g t2 presents an antagonistic effect over g t1 if the importance assigned to g t1 has to be lowered in consequence of the presence of g t2. A value w t 1t 2 > 0 has to be subtracted from w t1. 6 Figueira, Greco, Roy (2009) 1 April / 47

46 Interactions, MCHP and the ELECTRE III method For each elementary criterion g t, the following condition has to be satisfied w t1 w t1t 2 + > 0. t 2 EL: w t1 t 2 <0 t 3 EL w t 1t 3 For each (a,b) A A and for each non-elementary criterion G r, the partial concordance index is, therefore, redefined as follow: 1 ( ) Cr(a,b) = Wr(a, b) wt 1 ϕt (a,b) + 1 wt 1 t Z 2 ϕt 1 (a,b),ϕt (a,b) 2 t 1 C(b,a) E(G r) t 1,t 2 C(a,b) E(Gr) w ( ) t 1 t Z 2 ϕt 1 (a,b),ϕt (a,b) 2 t 1 C(b,a) E(Gr), t 2 C(b,a) E(Gr) (1) where Wr(a,b) = wt + wt 1 t 2 Z(ϕt 1 (a,b),ϕt (a,b)) 2 t E(Gr) t 1,t 2 C(a,b) E(Gr) t 1 C(b,a) E(Gr), t 2 C(b,a) E(Gr) w t 1 t 2 Z(ϕt 1 (a,b),ϕt (b,a)) 2 1 April / 47

47 The whole set of constraints Denoting by E r the set of constraints translating the information provided by the application of the imprecise SRF method to the set of criteria {G (r,1),...,g (r,n(r)) }, the whole set of constraints translating the information provided by the DM is the following: E [1] r IG\ELE r, w t1,t 2 > 0 if g t1 and g t2 are positively interacting, [2] w t1,t 2 < 0 if g t1 and g t2 are negatively interacting, w t 1,t 2 > 0 if g t2 presents an antagonistic effect over g t1, [3] w t1 w t1t 2 + > 0, for all t 1 EL, [4] {t 1,t 2} EL: w t1 t 2 >0 t 2 EL: w t1 t 2 <0 w t1t 2 t ELw t t 3 EL w t 1t 3 1 April / 47

48 The plurality of compatible weights and parameters... more than one set of weights and parameters is compatible with the preference information provided by the DM, we propose to apply the Stochastic Multiobjective Acceptability Analysis (SMAA) and the Robust Ordinal Regression (ROR). Both of them explore the whole set of weights and parameters even if in two different ways. 1 April / 47

49 SMAA 7 Given a,b A and a non-elementary criterion G r, we can computes: Pre r (a,b) : the frequency with which a is preferred to b on G r, Ind r (a,b) : the frequency with which a is indifferent to b on G r, Inc r (a,b) : the frequency with which a and b are incomparable on G r, Weak r (a,b) : the frequency with which a is weakly preferred to b on G r. Weak r (a,b) = Pre r (a,b)+ind r (a,b). Given a A and a non-elementary criterion G r, we can compute: the mean number of alternatives being weakly preferred to a on G r, the mean number of alternatives to which a is weakly preferred on G r. 7 Lahdelma, Hokkanen, Salminen (1998) 1 April / 47

50 SMAA 7 Given a,b A and a non-elementary criterion G r, we can computes: Pre r (a,b) : the frequency with which a is preferred to b on G r, Ind r (a,b) : the frequency with which a is indifferent to b on G r, Inc r (a,b) : the frequency with which a and b are incomparable on G r, Weak r (a,b) : the frequency with which a is weakly preferred to b on G r. Weak r (a,b) = Pre r (a,b)+ind r (a,b). Given a A and a non-elementary criterion G r, we can compute: the mean number of alternatives being weakly preferred to a on G r, the mean number of alternatives to which a is weakly preferred on G r. 7 Lahdelma, Hokkanen, Salminen (1998) 1 April / 47

51 ROR 8 Given a,b A and a non-elementary criterion G r, ROR computes: as N r b, a is necessarily weakly preferred to b on G r, if a is weakly preferred to b on G r for all compatible set of weights and parameters, as P r b: a is possibly weakly preferred to b on G r, if a is weakly preferred to b on G r for at least one compatible set of weights and parameters, as cn r b: a is not necessarily weakly preferred to b on G r, if a is not weakly preferred to b on G r for all compatible set of weights and parameters, as cp r b: a is not possibly weakly preferred to b on G r, if a is not weakly preferred to b on G r for at least one compatible set of weights and parameters. 8 Greco, Mousseau, S lowiński (2008) and Corrente, Greco, Kadziński, S lowiński (2013) 1 April / 47

52 The link between ROR and SMAA 9 Given a,b A and a non-elementary criterion G r, if Weak r (a,b) > 0%, then as P r b, if as N r b, then Weak r (a,b) = 100%, if as cn r b, then Weak r (a,b) = 0%, if Weak r (a,b) < 100%, then as cp r b. 9 Kadziński and Tervonen (2013a), (2013b) 1 April / 47

53 Flow chart of the proposed method 1 April / 47

54 The decision making problem Elementary subriterion Description Masters Graduation Rate (MGR) The percentage of new entrants that successfully completed their master programs Masters Graduating on Time (MGOT) The percentage of graduates that graduated within the time expected (normative time) for their masters programs Number of Research Publications (NRP) The number of research publications indexed in the Web of Science database, where at least one author is affiliated to the university (relative to the number of students) Citation Rate (CR) The average number of times that the university department s research publications (over the period ) get cited in other research, adjusted (normalized) at the global level to take into account differences in publication years and to allow for differences Proportion of Top Cited Publications (PTCP) The proportion of the university s research publications that, compared to other publications in the same field and in the same year, belong to the top 10% most frequently cited Number of Patents Awarded (NPA) The number of patents assigned to (inventors working in) the university (over the period ) Number of Spin-Offs (NSO) The number of spin-offs (i.e. firms established on the basis of a formal knowledge transfer arrangement between the institution and the firm) recently created by the institution (per 1,000 fte academic staff) Research and Knowledge Transfer Revenues (RKTR) Research revenues and knowledge transfer revenues from private sources (incl. not-for profit organizations), excluding tuition fees. Measured in e1,000s using Purchasing Power Parities. Expressed per fte academic staff. 1 April / 47

55 Performances of the universities on the considered criteria TL (G(1)) R (G(2)) KT (G(3)) University Country MGR (g(1,1)) MGOT (g(1,2)) NRP (g(2,1)) CR (g(2,2)) PTCP (g(2,3)) NPA (g(3,1)) NS0 (g(3,2)) RKTR (g(3,3)) U A Coruna (U1) Spain UAS Aalen (U3) Germany U Agder (U6) Norway U Alcalá (U7) Spain Ca Foscari U Venice (U37) Italy U Cantabria (U39) Spain Czech Tech U Prague (U58) Czech Republic U Ferrara (U72) Italy U Osijek (U144) Hungary Reutlingen UAS (U162) Germany G(0) 1 April / 47

56 Preference information Application of the imprecise SRF method: {R}, [1,2], {KT}, [0,1], {TL}, z [5,7] {MGOT}, [1,2], {MGR}, z [2,3] {CR}, [1,2], {PTCP}, [0,1], {NRP}, z [4,5] {RKTR}, [0,1], {NSO}, [1,2], {NPA}, z [4,5] Interactions between criteria: Citation Rate and Proportion of Top Cited Publications are negatively interacting, Number of Research Publications and Research and Knowledge Transfer Revenues are positively interacting, Proportion of Top Cited Publications presents an antagonistic effect over Masters Graduation Rate. 1 April / 47

57 Preference relation at comprehensive level U1 U3 U6 U7 U37 U39 U58 U72 U144 U162 U1 : UA Coruna U3 : UAS Aalen U6 : U Agder U7 : U Alcalá U37 : Cá Foscari U Venice U39 : U Cantabria U58 : Czech Tech U Prague U72 : U Ferrara U144 : U Osijek U162 : Reutlingen UAS April / 47

58 Incomparability relation at comprehensive level U1 U3 U6 U7 U37 U39 U58 U72 U144 U162 U1 : UA Coruna U3 : UAS Aalen U6 : U Agder U7 : U Alcalá U37 : Cá Foscari U Venice U39 : U Cantabria U58 : Czech Tech U Prague U72 : U Ferrara U144 : U Osijek U162 : Reutlingen UAS April / 47

59 Indifference relation at comprehensive level U1 U3 U6 U7 U37 U39 U58 U72 U144 U162 U1 : UA Coruna U3 : UAS Aalen U6 : U Agder U7 : U Alcalá U37 : Cá Foscari U Venice U39 : U Cantabria U58 : Czech Tech U Prague U72 : U Ferrara U144 : U Osijek U162 : Reutlingen UAS April / 47

60 Preference relation on Teaching and Learning U1 U3 U6 U7 U37 U39 U58 U72 U144 U162 U1 : UA Coruna U3 : UAS Aalen U6 : U Agder U7 : U Alcalá U37 : Cá Foscari U Venice U39 : U Cantabria U58 : Czech Tech U Prague U72 : U Ferrara U144 : U Osijek U162 : Reutlingen UAS April / 47

61 Indifference relation on Teaching and Learning U1 U3 U6 U7 U37 U39 U58 U72 U144 U162 U1 : UA Coruna U3 : UAS Aalen U6 : U Agder U7 : U Alcalá U37 : Cá Foscari U Venice U39 : U Cantabria U58 : Czech Tech U Prague U72 : U Ferrara U144 : U Osijek U162 : Reutlingen UAS April / 47

62 Preference relation on Research U1 U3 U6 U7 U37 U39 U58 U72 U144 U162 U1 : UA Coruna U3 : UAS Aalen U6 : U Agder U7 : U Alcalá U37 : Cá Foscari U Venice U39 : U Cantabria U58 : Czech Tech U Prague U72 : U Ferrara U144 : U Osijek U162 : Reutlingen UAS April / 47

63 Incomparability relation on Research U1 U3 U6 U7 U37 U39 U58 U72 U144 U162 U1 : UA Coruna U3 : UAS Aalen U6 : U Agder U7 : U Alcalá U37 : Cá Foscari U Venice U39 : U Cantabria U58 : Czech Tech U Prague U72 : U Ferrara U144 : U Osijek U162 : Reutlingen UAS April / 47

64 Indifference relation on Research U1 U3 U6 U7 U37 U39 U58 U72 U144 U162 U1 : UA Coruna U3 : UAS Aalen U6 : U Agder U7 : U Alcalá U37 : Cá Foscari U Venice U39 : U Cantabria U58 : Czech Tech U Prague U72 : U Ferrara U144 : U Osijek U162 : Reutlingen UAS April / 47

65 Preference relation on Knowledge Transfer U1 U3 U6 U7 U37 U39 U58 U72 U144 U162 U1 : UA Coruna U3 : UAS Aalen U6 : U Agder U7 : U Alcalá U37 : Cá Foscari U Venice U39 : U Cantabria U58 : Czech Tech U Prague U72 : U Ferrara U144 : U Osijek U162 : Reutlingen UAS April / 47

66 Indifference relation on Knowledge Transfer U1 U3 U6 U7 U37 U39 U58 U72 U144 U162 U1 : UA Coruna U3 : UAS Aalen U6 : U Agder U7 : U Alcalá U37 : Cá Foscari U Venice U39 : U Cantabria U58 : Czech Tech U Prague U72 : U Ferrara U144 : U Osijek U162 : Reutlingen UAS April / 47

67 The advantage of MCHP... TL (G(1)) R (G(2)) KT (G(3)) University Country MGR (g(1,1)) MGOT (g(1,2)) NRP (g(2,1)) CR (g(2,2)) PTCP (g(2,3)) NPA (g(3,1)) NS0 (g(3,2)) RKTR (g(3,3)) Norwegian Sch. Economics (U139) Norway Warsaw U Tech (U213) Poland G(0) 1 April / 47

68 Mean number of universities to which each university is weakly preferred 1 April / 47

69 Mean number of universities being weakly preferred to each university 1 April / 47

70 Difference between the previous indicators.. Comprehensive TL R KT U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U April / 47

71 Conclusions We have proposed an extension of the ELECTRE III method to the Multiple Criteria Hierarchy Process (MCHP), MCHP deals with problems in which evaluation criteria are organized in a hierarchical way, We can take into account also interaction between criteria and antagonistic effects between criteria, A new version of the SRF has been proposed to consider imprecise preference information provided by the DM SMAA and ROR have been used to take into account the plurality of instances of the preference model compatible with the provided information 1 April / 47

72 Conclusions We have proposed an extension of the ELECTRE III method to the Multiple Criteria Hierarchy Process (MCHP), MCHP deals with problems in which evaluation criteria are organized in a hierarchical way, We can take into account also interaction between criteria and antagonistic effects between criteria, A new version of the SRF has been proposed to consider imprecise preference information provided by the DM SMAA and ROR have been used to take into account the plurality of instances of the preference model compatible with the provided information 1 April / 47

73 Conclusions We have proposed an extension of the ELECTRE III method to the Multiple Criteria Hierarchy Process (MCHP), MCHP deals with problems in which evaluation criteria are organized in a hierarchical way, We can take into account also interaction between criteria and antagonistic effects between criteria, A new version of the SRF has been proposed to consider imprecise preference information provided by the DM SMAA and ROR have been used to take into account the plurality of instances of the preference model compatible with the provided information 1 April / 47

74 Conclusions We have proposed an extension of the ELECTRE III method to the Multiple Criteria Hierarchy Process (MCHP), MCHP deals with problems in which evaluation criteria are organized in a hierarchical way, We can take into account also interaction between criteria and antagonistic effects between criteria, A new version of the SRF has been proposed to consider imprecise preference information provided by the DM SMAA and ROR have been used to take into account the plurality of instances of the preference model compatible with the provided information 1 April / 47

75 Conclusions We have proposed an extension of the ELECTRE III method to the Multiple Criteria Hierarchy Process (MCHP), MCHP deals with problems in which evaluation criteria are organized in a hierarchical way, We can take into account also interaction between criteria and antagonistic effects between criteria, A new version of the SRF has been proposed to consider imprecise preference information provided by the DM SMAA and ROR have been used to take into account the plurality of instances of the preference model compatible with the provided information 1 April / 47

76 THANKS FOR YOUR ATTENTION 1 April / 47

77 References (1) M. Bottero, V. Ferretti, J.R. Figueira, S. Greco, and B. Roy. Dealing with a multiple criteria environmental problem with interaction effects between criteria through an extension of the Electre III method.. European Journal of Operational Research, 245(3): , S. Corrente, S. Greco, M. Kadziński, and R. S lowiński. Robust ordinal regression in preference learning and ranking. Machine Learning, 93: , S. Corrente, S. Greco, and R. S lowiński. Multiple Criteria Hierarchy Process in Robust Ordinal Regression. Decision Support Systems, 53(3): , S. Corrente, S. Greco, and R. S lowiński. Multiple Criteria Hierarchy Process with ELECTRE and PROMETHEE. Omega, 41: , April / 47

78 References (2) J.R. Figueira, S. Greco, and B. Roy. ELECTRE methods with interaction between criteria. An extension of the concordance index. European Journal of Operational Research, 199: , J.R. Figueira, and B. Roy. Determining the weights of criteria in the ELECTRE type methods with a revised Simos procedure. European Journal of Operational Research, 139: , S. Greco, V. Mousseau, and R. S lowiński. Ordinal regression revisited: multiple criteria ranking using a set of additive value functions. European Journal of Operational Research, 191(2): , April / 47

79 References (3) M. Kadziński, and T. Tervonen. Robust multi-criteria ranking with additive value models and holistic pair-wise preference statements. European Journal of Operational Research, 228: , M. Kadziński, and T. Tervonen. Stochastic ordinal regression for multiple criteria sorting problems. Decision Support Systems, 55:55 66, R. Lahdelma, J. Hokkanen, and P. Salminen. SMAA - stochastic multiobjective acceptability analysis. European Journal of Operational Research, 106(1): , B. Roy. ELECTRE III: Un algorithme de classements fondé sur une représentation floue des préférences en présence de critères multiples. Cahiers du CERO, 20(1):3 24, April / 47