ELECTRE METHODS (PART I)

Transcription

1 (PART I) José Rui FIGUEIRA Technical University of Lisbon 10th MCDM Summer School, Paris, France

2 Contents 1 2

3 Main references for this talk: 1 Figueira, J., B. Roy, and V. Mousseau (2005). methods. In J. Figueira, S. Greco, and M. Ehrgott (Eds.), Multiple Criteria Decision Analysis: State of the Art Surveys, pp New York, U.S.A.: Springer Science + Business Media, Inc. 2 Figueira, J.R., S. Greco, B. Roy, and R. Słowiński, (2010). methods: Main and recent developments. In C. Zopounidis and P. Pardalos (Eds.), Handbook of Multicriteria Analysis, Chapter 4, New York, USA: Springer. 3 Greco, S., R. Słowiński,, and V. Mousseau (2010). Robust ordinal regression. In M. Ehrgott, J.R. Figueira, and S. Greco (Eds.), Trends in Multiple Criteria Decision Analysis, pp New York, U.S.A.: Springer Science + Business Media, Inc.

4 1.1. methods were designed according to a constructivist conception of MCDA: A decision aiding situation (Roy, 2009). A decision aiding situation 1 Imagine that in a company or institution, a CEO is confronted with a certain decision aiding situation and has to make a decision. 2 The CEO needs the help of an analyst (an in-house operational service, a consultant, or a university research team). 3 Two key elements in a decision aiding situation are: The Analyst and the Decision Maker (DM). The latter is here represented by the CEO.

5 1.1. methods were designed according to a constructivist conception of MCDA: The fundamental pillars (Roy, 2009). The decision aiding activity is based on three fundamental pillars: 1 The actions (formal definition of the possible actions or alternatives). 2 The consequences (aspects, attributes, characteristics,... of the actions that allow to compare them). 3 The of a preference system (it consists of an implicit or explicit process, that for each pair of actions envisioned, assigns one and only one of the three possibilities: indifference, preference, or incomparability).

6 methods were designed according to a constructivist conception of MCDA (Roy, 2009). Based on the above three pillars: 1. The analyst should try to obtain a coherent structured set of results in order to guide the decision aiding process and facilitate the communications about the decisions. 2. The analyst must follow an approach that leads or aims to produce knowledge from a certain number working hypotheses defined a priori. 3. This approach should be based on models that are, at least co-constructed interactively with the DM.

7 1.1. methods were designed according to a constructivist conception of MCDA (Roy, 2009). Based on the above three pillars: 4. During the co-construction process, that takes into account the values of the DM, contradictory judgements or ambiguities may occur. 5. The analyst must admit that the novelty of these questions can bring (the DM) or the person this questioned to revise certain pre-existing preferences momentarily and locally.

8 1.2. Notation: Basic data. Basic data 1 A = {a 1, a 2,..., a i,...,a m } is the set of m potential actions. This set can be partially known a priori (it is frequent in sorting problems). 2 F = {g 1, g 2,...,g j,..., g n } is a coherent family of criteria, with n 3. 3 g j (a i ) is the performance of action a i on criterion g j, for all a i A and g j F. A performance matrix M can thus be built. 4 Assume w.l.g. that the higher the performance g j (a) is, the better for the DM (increasing direction of preference).

9 s. Four main comprehensive preference 1 I (Indifference) 2 P (strict preference) 3 Q (hesitation : weak preference) 4 R (incomparability). (For more details see Figueira et al., 2010)

10 through outranking relations: The concept of pseudo-criterion (Roy, 1996). 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 ordered pairs of actions (a, a ) A A such that g j (a) g j (a ), g j (a) + p j (g j (a )) and g j (a) + q j (g j (a )) are non-decreasing monotone functions of g j (a ), such that p j (g j (a )) q j (g j (a )) 0, for all a A.

11 through outranking relations: Partial binary relations. Partial binary relations (1) 1 g j (a) g j (a ) > p j (g j (a )) ap j a, 2 q j (g j (a )) < g j (a) g j (a ) p j (g j (a )) aq j a, 3 q j (g j (a)) g j (a) g j (a ) q j (g j (a )) ai j a.

12 through outranking relations: Partial binary relations. Partial binary relations (2) 1 S j = P j Q j I j 2 as j a means that a is at least as good as a on criterion g j. 3 When as j a the voting power of criterion g j, denoted by w j is taken in total (assume w.l.g. that w 1 + w w n = 1).

13 through outranking relations: Comprehensive outranking. Let S = P Q I, whose meaning is at least as good as. Comprehensive outranking Consider two actions, a and a and the relation = P Q. Four may occur: 1 asa and not(a Sa), i.e., a a (a is preferred in a broader sense to a ). 2 a Sa and not(asa ), i.e., a a (a is preferred in a broader sense to a). 3 asa and a Sa, i.e., aia (a is indifferent to a ). 4 not(asa ) and not(a Sa), i.e., ara (a is incomparable to a ).

14 : Concordance. Concordance 1 Concordance. To validate asa, a sufficient majority of criteria in favor of this assertion must occur. 2 The comprehensive concordance index c(a, a ) for each pair of actions (a, a ) A A, for all g j F is fundamental to all the methods in order to compute a concordance matrix C. c(a, a ) = w j + w j ϕ j {j g j C(a{P,Q,I}a }) {j g j C(a Qa)} where ϕ j = g j(a) g j (a ) + p j p j q j

15 : Voting power. Voting power This index comprises the summation of the voting power of the criteria that clearly are in favor of the assertion asa, plus the summation of the fraction, ϕ j, of the voting power for those criteria included in the hesitation group.

16 . : Graphical representation. ϕ j 1 0 g j(a ) p j(g j(a)) g j(a ) g j(a ) + p j(g j(a )) g j(a) g j(a ) q j(g j(a)) g j(a ) + q j(g j(a )) Figure: Variation of ϕ j for a given g j (a ) and variable g j (a)

17 2.3.Concordance : Discordance (1) Discordance 1 Discordance. The assertion asa cannot be validated if a minority of criteria is strongly against this assertions. 2 The concept of veto threshold, v j, gives the possibility to the criterion g j to impose its veto power. It means that g j (a ) is so much better than g j (a), that is not possible to allow that asa 3 The computation of the partial discordance indices leads to the construction of a discordance matrix, D. 4 The application of both types of indices is related to a specific method. For, in TRI they are combined with c(a, a ) to define a degree of credibility of the assertion asa (fuzzy relation).

18 : Discordance (2) Partial discordance index 8 >< 1 if g j (a) g j (a ) < v j (g j (a)), d j (a, a g ) = j (a) g j (a )+p j (g j (a)) if v p >: j (g j (a)) v j (g j (a)) j (g j (a)) g j (a) g j (a ) < p j (g j (a)), 0 if g j (a) g j (a ) p j (g j (a)).

19 2.4. Reminder and additional notation Reminder and additional notation 1 We use k j as the non-normalized weights for each criterion) - C(aSa ) is the coalition of criteria in favor of the assertion asa. - W {C(aSa }) = w j is the weight or power of the coalition C(aSa ). {j : g j C(aSa )} 2 q j ( ) is the indifference threshold of criterion g j. 3 p j ( ) is the preference threshold of criterion g j. 4 v j ( ) is a veto threshold of criterion g j.

20 2.4. Location of a new hotel (Figueira et al., 2009) (1) Location of a new hotel 1 Matrix below presents the performances of the five sites - a, b, c, d, and e - according to the five criteria. 2 The performances of criterion g 1 (investment costs) are expressed in thousands of e, designated Ke. 3 The indifference and the preference thresholds assigned to this criterion are q 1 (g 1 (x)) = g 1 (x) Ke and p 1 (g 1 (x)) = g 1 (x) Ke, respectively, where x is the worst of the two actions. 4 The performances of criterion g 2 (annual costs) are also expressed in Ke; the thresholds assigned to this criterion are q 2 (g 1 (x)) = g 1 (x) Ke and p 2 (g 1 (x)) = g 1 (x) Ke, respectively.

21 2.4. Location of a new hotel (Figueira et al., 2009) (2) Location of a new hotel 1 The performances of criteria g 3 (recruitment), g 4 (image), and g 5 (access) are expressed on the following seven-level qualitative scale: very bad (1), bad (2), rather bad (3), average (5), rather good (5), good (6), and very good (7). The values between parenthesis can be used in methods to code the different verbal statements. 2 Other ways of coding the verbal scale through the use of numerical values could be used by adjusting the thresholds values (see Martel and Roy, 2006). 3 The indifference threshold for each criterion has been set at one on the seven-level scale and the preference threshold at two levels. 4 In this there is no veto.

22 2.4. Performances matrix (Figueira et al., 2009) Performances matrix 1 Quantitative criteria: g 1 (investment costs) and g 2 (annual costs) 2 Qualitative criteria: g 3 (recruitment), g 4 (image), and g 5 (access) g 1 [min] g 2 [min] g 3 [max] g 4 [max] g 5 [max] a Ke Ke Average Average Average b Ke Ke Good Bad Very Good c Ke Ke Good Good Very Bad d Ke Ke Good Good Good e Ke Ke Good Very Bad Bad k j

23 2.4. Pairwise comparison Pairwise comparison 1 Does a outrank d, asd? For the moment we cannot answer this question. 2 The coalition of criteria in favor of asd: C(aSd) = {g 1, g 2 } 3 The power of this coalition: W {C(aSd)} = = 0.5 (normalized) 4 What about dsa?

24 The structure of methods. Each method comprises two main procedures: Two procedures 1 The first procedure is a Multiple Criteria Aggregation Procedure (MCAP) that builds one or possibly several outranking relations aim to compare, in a comprehensive way, each ordered pair of actions. 2 The second procedure, called Exploitation Procedure (EP) is used to obtain adequate results from which recommendations can be derived. 3 The nature of the results depends of the specific problematique.

25 Example: MCAP of III MCAP of III It is modeled through a credibility index i.e. a fuzzy measure denoted by σ(a, a ) [0, 1], which combines c(a, a ) and d j (a, a ): σ(a, a ) = c(a, a ) j J (a,a ) 1 d j (a, a ) 1 c(a, a ), where j J (a, a ) if and only if d j (a, a ) c(a, a ).

26 2.5. The nature of the results: Choosing (Mousseau, 1993; Roy, 2002). Choosing: Selecting a restricted number as small as possible of potential actions, which justify to eliminating others.

27 2.5. The nature of the results: Choosing (Mousseau, 1993; Roy, 2002). Choosing: Selecting a restricted number as small as possible of potential actions, which justify to eliminating all others.

28 2.5. The nature of the results: Choosing (Mousseau, 1993; Roy, 2002). Choosing: Selecting a restricted number as small as possible of potential actions, which justify to eliminating all others. Choice set Actions rejected

29 2.5. The nature of the results: Ranking (Mousseau, 1993; Roy, 2002). Ranking: Ranking of actions from the best to the worst, with the of ties (ex aequo) and incomparabilities.

30 Ranking: Ranking of actions from the best to the worst, with the of ties (ex aequo) and incomparabilities.

31 2.5. The nature of the results: Sorting (Mousseau, 1993; Roy, 2002). Ordinal classification or sorting: Assigning each potential action to one of the categories among those of a family previously defined; the categories are ordered, in general, from the worst to the best one.

32 2.5. The nature of the results: Sorting (Mousseau, 1993; Roy, 2002). Ordinal classification or sorting: Assigning each potential action to one of the categories among those of a family previously defined; the categories are ordered, in general, from the worst to the best one. Category 1 Category 2. Category k

33 2.5. The nature of the results: Sorting (Mousseau, 1993; Roy, 2002). Ordinal classification or sorting: Assigning each potential action to one of the categories among those of a family previously defined; the categories are ordered, in general, from the worst to the best one. Cat. 1 Cat. 2. Cat. k

34 2.5. The nature of the results: Absolute versus relative evaluation (Roy, 1996). In sorting problems there is an 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. As for the remaining problematiques the actions are compared against each other and thus there exists a relative evaluation instead of an absolute evaluation as for the previous case.

35 Software (Figueira et al., 2005). Choosing: I, IV, and IS. Ranking: II, III, and IV. Ordinal classification or sorting: TRI. New software (see later on).

36 2.6. Strong 2.7. Weaknesses (PART II) José Rui FIGUEIRA Technical University of Lisbon 10th MCDM Summer School, Paris, France

37 Contents 2.6. Strong 2.7. Weaknesses Strong 2.7. Weaknesses

38 Introduction 2.6. Strong 2.7. Weaknesses Summary 1 The qualitative nature of some criteria 2 The heterogeneity of scales 3 The non-relevance of compensatory effects 4 The imperfect knowledge and arbitrariness 5 The reasons for and reasons against and outranking

39 2.6. Some strong of methods 2.6. Strong 2.7. Weaknesses Strong 1. They have the possibility of taking into account the qualitative nature of some criteria. They allow thus to consider the original data. 2. They can deal with very heterogeneous scales to model noisy, delay, aesthetics, cost,... Whatever the nature of scales, every procedure can run by preserving the original performances of the actions. 3. The compensatory effects are not pertinent. This is due to the fact that the weights cannot be interpreted as substitution rates. Contrarily to other methods there is no need in methods to use, from the starting point of their application, identical and commensurable scales.

40 2.6. Some strong of methods 2.6. Strong 2.7. Weaknesses Strong Consider the following with 4 criteria and only 2 actions (scales: [0,10]). The weighted-sum model was chosen, i.e, V(a) = n j=1 w jg j (a). In the considered, the weights, w j, are equal for all criteria: g 1 g 2 g 3 g 4 a a V(a 1 ) = > V(a 2 ) = This shows, in an obvious way, the possibility that a big preference difference not favorable to a 1 on one of the criteria (g 4 ) can be compensated by 3 differences of weak amplitude on the remaining criteria, in such a way that a 1 becomes finally preferred to a 2. In methods this effect does not occurs in a systematic way.

41 2.6. Some strong of methods Strong 2.6. Strong 2.7. Weaknesses 4. They are adequate to take the imperfect knowledge of the data and the arbitrariness related to the construction of the criteria. This is modeled through the indifference and preference thresholds. Consider the same with the following (constant) discrimination thresholds: g 1 g 2 g 3 g 4 a a q j p j

42 2.6. Some strong of methods 2.6. Strong 2.7. Weaknesses Strong If on criterion g 3 we change the performance from 7.3 to 7.1, the score moves from to (V(a 1 ) V(a 2 ) = 0.050). Consequently there is a reinforcement of the preference in favor of a 1. On the other hand, with c(a 1, a 2 ) and c(a 2, a 1 ) remain unchanged. Now, if we consider 7.5 instead of 7.3, then V(a 2 ) = 8.150, and consequently a 2 Pa 1. Again this small variation is too small. When adding the discrimination thresholds and using methods, c(a 1, a 2 ) = = 0.75 and c(a 2, a 1 ) = = 0.8. Thus, a 2 Pa 1.

43 2.6. Some strong of methods 2.6. Strong 2.7. Weaknesses Strong 5. They are based in a certain sense in the reasons for and the reasons against of an outranking between two actions (concordance and discordance). Consider the same and that a veto threshold should v j = 3, for all j = 1,..., 4. g 1 g 2 g 3 g 4 a a q j p j v j If s = 0.8 then a 2 Sa 1 and not(a 2 Sa 1 ). But, if s = 0.7, a 1 Ia 2. Since d 4 (a 2, a 1 ) = 1, g 4 imposes a veto, for whatever the chosen s. We get allays not(a 2 Sa 1 ).

44 2.7. Weaknesses Introduction 2.6. Strong 2.7. Weaknesses Summary 1 Scoring actions 2 The quantitative nature of family of criteria 3 The independence with respect to irrelevant alternatives 4 The intransitivities

45 2.7. Some weaknesses of methods 2.6. Strong 2.7. Weaknesses Some weaknesses 1. Scoring the actions. In certain contexts it is required to assign a score to each action. When the decision makers require each action should appear associated with a score, the methods are not adequate for such a purpose and the scoring based methods should be applied instead. The decision makers should be, however, aware that they cannot provide information that leads, for, to intransitivities or to incomparabilities between certain pairs of actions. Indeed, this score is very fragile.

46 2.7. Some weaknesses of methods 2.6. Strong 2.7. Weaknesses Some weaknesses 2. The quantitative nature of the family of criteria. When all the criteria are quantitative it is better to use other methods. But, if we want to take into account a completely or even a partial noncompensatory method, the reasons for and against, or the imperfect character of at least one criterion, even under such conditions, we can use the methods.

47 2.7. Some weaknesses of methods 2.6. Strong 2.7. Weaknesses Some weaknesses 3. The independence with respect to irrelevant alternatives. Except TRI-B, TRI-C, the remaining methods does not fulfill the independence w.r.t. irrelevant alternatives (Roy, 1973). In 1973, B. Roy shows that rank reversal may occur and consequently the property of independence with respect to irrelevant alternatives can be violated when dealing with outranking relations. Notice that rank reversal may occur only when the set of potential actions is subject to evolve, which is quite a natural assumption, but one that is not present in many hard decision-aiding processes where the number of alternatives is rather small and easily identified.

48 2.7. Some weaknesses of methods 2.6. Strong 2.7. Weaknesses Some weaknesses 4. Intransitivities may also occur in methods (Roy, 1973). It is also well-known that methods using outranking relations (not only the methods) do not need to fulfill the transitivity property. This aspect represents only a weakness if we impose a priori that preferences should be transitive. There are, however, some raisons that lead us to do not impose transitivity.

49 3. Some recent developments (>2000) (PART III) 3.1. Methodological 3.2. New approaches 3.3. Axiomatic and meaningfulness analysis 3.4. Other aspects 4. Applications 4.1. Some applications areas 4.2. Real-world applications José Rui FIGUEIRA Technical University of Lisbon 10th MCDM Summer School, Paris, France 5. Concluding remarks

50 Contents 3. Some recent developments (>2000) 3.1. Methodological 3.2. New approaches 3.3. Axiomatic and meaningfulness analysis 1 3. Some recent developments (>2000) 3.1. Methodological 3.2. New approaches 3.3. Axiomatic and meaningfulness analysis 3.4. Other aspects 3.4. Other aspects 4. Applications 4.1. Some applications areas 4.2. Real-world applications 5. Concluding remarks 2 4. Applications 4.1. Some applications areas 4.2. Real-world applications 3 5. Concluding remarks

51 3. Some recent developments (>2000) 3.1. Methodological (1) Recent developments 3. Some recent developments (>2000) 1 Pure inference based approaches after the work by Mousseau and Słowiński (1998) (Software: TRI-Assistant): 3.1. Methodological 3.2. New approaches 3.3. Axiomatic and meaningfulness analysis 3.4. Other aspects 4. Applications 4.1. Some applications areas 4.2. Real-world applications 5. Concluding remarks inferring only the weights (Mousseau et al, 2001); inferring veto (Mousseau and Dias, 2006); and, inferring category bounds (Ngo The and Mousseau, 2002). Some manageable disaggregation procedures for valued outranking relations (Mouuseau and Dias, 2006); Inconsistent judgements (Mousseau et al., 2006a; Mousseau et al., 2006b) or an inadequate preference model (Figueira, 2009).

52 3. Some recent developments (>2000) 3.1. Methodological (2) Recent developments 3. Some recent developments (>2000) 3.1. Methodological 3.2. New approaches 3.3. Axiomatic and meaningfulness analysis 3.4. Other aspects 4. Applications 4.1. Some applications areas 4.2. Real-world applications 5. Concluding remarks 2 The inference-robustness based approach for inferring weights and derive robust conclusions in sorting problems (Dias et al., 2002). Software: IRIS. 3 The pseudo-robustness based approach dealing with simulation methods mainly for ranking and sorting problems (Tervonen et al., 2008, 2009). Software: SMAA-III, SMAA-TRI. 4 New robustness analysis concepts (Aissi and Roy, 2009; Roy, 2009). These papers are more general, but some techniques can be applied to methods.

53 3. Some recent developments (>2000) 3.2. New approaches New approaches 3. Some recent developments (>2000) 3.1. Methodological 3.2. New approaches 3.3. Axiomatic and meaningfulness analysis 3.4. Other aspects 4. Applications 4.1. Some applications areas 4.2. Real-world applications 5. Concluding remarks 1 Bi-polar outranking relations implemented in RUBIS software (Bisdorff et al., 2007, 2008). 2 The weights of the interaction coefficients and the modifications in the concordance index (Figueira et al., 2009). 3 Handling with the reinforced preference and the counter-veto effects (Roy and Słowiński, 2009). 4 TRI-C, TRIN, NC (Almeida-Dias et al., 2010a, 2010b). 5 The possible and the necessary approach for methods (-GKMS) by Greco et al., (2009, 2010).

54 3. Some recent developments (>2000) 3.3 Axiomatic and meaningfulness Axiomatic and meaningfulness 3. Some recent developments (>2000) 3.1. Methodological 3.2. New approaches 3.3. Axiomatic and meaningfulness analysis 3.4. Other aspects 4. Applications 4.1. Some applications areas 4.2. Real-world applications 5. Concluding remarks 1 Axiomatic analysis of I method by using conjoint measurement theory (Greco et al., 2001). 2 Representing preferences through conjoint measure and the decision rule approach (Greco et al., 2002). 3 An axiomatic analysis based on a general conjoint measure framework with application to a variant of TRI (Bouyssou and Marchant, 2007a,b). 4 An axiomatic analysis of the concordance-discordance relations (Bouyssou and Pirlot, 2009). 5 Representing preferences by decision rules (Greco et al., 2002). 6 The meaningfulness of methods (Martel and Roy, 2006).

55 3. Some recent developments (>2000). 3.4 Other aspects 3. Some recent developments (>2000) 3.1. Methodological 3.2. New approaches 3.3. Axiomatic and meaningfulness analysis 3.4. Other aspects 4. Applications 4.1. Some applications areas 4.2. Real-world applications 5. Concluding remarks Other aspects 1 The relative importance of criteria (Figueira and Roy, 2002). 2 Concordant outranking with criteria of ordinal significance (Bisdorff, 2004). 3 Evolutionary approaches (Leyva-López et al., 2008; Doumpos et al., 2009). 4 The EPISSURE method for the assessment of non-financial performances (André and Roy, 2007; André, 2009). 5 Group decision aiding (Damart et al., 2007; Greco et al., 2009, 2010).

56 4. Applications 4.1. Some applications areas 3. Some recent developments (>2000) 3.1. Methodological 3.2. New approaches 3.3. Axiomatic and meaningfulness analysis 3.4. Other aspects 4. Applications 4.1. Some applications areas 4.2. Real-world applications 5. Concluding remarks Areas 1 Agriculture and Forest Management. 2 Energy. 3 Environment and Water Management. 4 Finance. 5 Medicine. 6 Military. 7 Project selection (call for tenders). 8 Transportation. 9...

57 4. Applications 4.2. Concrete cases (1) 3. Some recent developments (>2000) 3.1. Methodological 3.2. New approaches 3.3. Axiomatic and meaningfulness analysis 3.4. Other aspects 4. Applications 4.1. Some applications areas 4.2. Real-world applications 5. Concluding remarks Areas Sorting cropping systems (Arondel and Girardin, 2000). Land-use suitability assessment (Joerin et al., 2001). Greenhouse gases emission reduction (Georgopoulou, 2003). Risk zoning of an area subjected to mining-inducing hazards (Merad et al., 2004). Participatory decision-making on the localization of waste-treatment plants (Norese, 2006).

58 4. Applications 4.2. Concrete cases (1) 3. Some recent developments (>2000) 3.1. Methodological 3.2. New approaches 3.3. Axiomatic and meaningfulness analysis 3.4. Other aspects 4. Applications 4.1. Some applications areas 4.2. Real-world applications 5. Concluding remarks Areas Material selection of bipolar plates for polymer electrolyte membrane fuel cell (Shanian and Savadogo). Assisted reproductive technology (Matias, 2008). Promotion of social and economic development (Autran-Gomes et al., 2009). Sustainable demolition waste management strategy (Roussat et al., 2009). Assessing the risk of nano-materials (Tervonen et al., 2009).

59 5. Concluding remarks Concluding remarks 3. Some recent developments (>2000) 3.1. Methodological 3.2. New approaches 3.3. Axiomatic and meaningfulness analysis 3.4. Other aspects 4. Applications 4.1. Some applications areas 4.2. Real-world applications 5. Concluding remarks 1 methods have a long history of successful real-world applications with impact on the life of populations (see Figueira et al., 2005)). 2 When applying methods analysts should pay attention to the characteristics of the context and also to the (theoretical) weaknesses of these methods. Note that all the MCDA methods have theoretical limitations. 3 Software implementations of high quality along with friendly interfaces render possible the application to a vast range of applications. 4 Research on methods is not a death field. It stills evolving and rapidly, namely over of the first years of this new millennium.

60 3. Some recent developments (>2000) 3.1. Methodological 3.2. New approaches 3.3. Axiomatic and meaningfulness analysis 3.4. Other aspects 4. Applications 4.1. Some applications areas Thank You! (very much for your attention) 4.2. Real-world applications 5. Concluding remarks