Chapter 1 ELECTRE METHODS - Lamsade
Contents
1. V Mousseau and R Slowinski Inferring an ELECTRE TRI model from as signment examples Journal of Global Optimization 12 2 157 174 1998 V Mousseau R Slowinski and P Zielniewicz ELECTRE TRI 2 0a Method ological guide and user s documentation Document du LAMSADE 111 Uni versit Paris Dauphine 1999 A Ngo The and V Mousseau Using assignment examples to infer category limits for the ELECTRE TRI method Journal of Multi Criteria Decision Analysis 11 1 29 43 2002 M F Norese and S Viale A multi profile sorting procedure in the public administration European Journal of Operational Research 138 365 379 2002 E Parent and P Schnabele Le choix d un am nagement aquacole Exem ple d utilisation de la m thode ELECTRE III et comparaison avec d autres m thodes multicrit res d aide a la d cision Cahier du LAMSADE 47 Univer sit Paris Dauphine 1988 J Pomerol and S Barba Romero Multicriterion Decision Making in Manage ment Kluwer Academic Publishers Dordrecht 2000 K Raju and C Pillai Multicriterion decision making in river basin planning and development European Journal of Operational Research 112 2 249 257 1999 J Richard Aide la d cision strat gique en PME In E Jacquet Lagr ze and Y Siskos editors M thode de d cision multicrit re pages 119 143 Editions Hommes et Techniques Paris 1983 D Rios Insua and S French A framework for sensitivity analysis in dis2. IRIS infers parameter values and allows to draw robust conclusions by indicating the range of assignments for each action that do not contradict any constraint If it is not possible to fulfill all of the constraints IRIS tells the user where is the source of inconsistency It was developed with Delphi Borland and runs on Windows 98 Me 2000 NT and XP SFR was designed to determine the relative importance coefficients for ELECTRE family methods It is based on a very simple pro cedure the pack of cards technique created by J Simos and try to assess these coefficients by questioning the DM in an indirect way It was developed with the Delphi Borland 3 0 and runs on Windows 98 Me 2000 and XP The software ELECTRE IS II IV TRI and TRI Assistant were developed under a collaborative project between researchers from the Institute of Computing Science of the Technical University of Poznan Poland and LAMSADE Universit Paris Dauphine France while IRIS and SRF result from a collaborative project between researchers from LAMSADE and the Faculty of Economics of the University of Coimbra INESC Coimbra Portugal ELECTRE methods were successful applied in many areas 1 2 w CON DD Ot A Agriculture and Forest Management 4 31 62 118 128 130 131 Energy 8 9 19 39 40 54 108 126 Environment and Water Management 12 40 41 44 59 78 80 85 86 118 121 124 125 131 132 77 88 58 Finance 3
3. 30 46 47 48 56 61 143 144 145 146 Military 6 36 140 Project selection call for tenders 13 21 24 65 107 137 Transportation 11 12 23 38 45 73 111 112 110 114 116 Varia 33 34 81 83 129 107 Acknowledgements 23 6 Conclusion Since their first appearance in 1965 see 10 ELECTRE methods on one side had a strong impact on the Operational Research community mainly in Europe and provoked the development of other outranking methods see for example Chapters and as well as other com plementary multiple criteria methodologies see for example Chapters and Most importantly the development of ELECTRE methods is strongly connected with the birth of the European Working Group of Multiple Criteria Decision Aiding www inescc pt ewgmcda On the other side ELECTRE methods experienced a widespread and large use in real world situations Despite their almost four decades of existence research stills active in this field We can also mention some of recent developments and avenues for future research generalization of the concordance and non discordance methods 133 robustness analysis 26 27 29 parameters elicitation techniques 74 interaction between criteria 66 42 multiple DMs and social interaction 22 Acknowledgments This work has benefited from the Luso French grant n 500B4 ICCTI Ambassade de France au Portugal The first author was also supported by
4. LAMSADE some of these papers were cited in the introduction The books 64 95 100 122 123 are also good references in the area ELECTRE software manuals also contain much material both on theoretical and pedagogical issues 2 43 75 117 134 142 Finally several other works deserve to be mentioned because they include information concerning ELECTRE methods 5 15 16 20 37 52 79 87 125 In what follows we will only summarize the elementary concepts un derlying ELECTRE methods details will be omitted More sophisti cated presentations can however be found in the references cited above Description of methods is presented in problematic and chronological order 3 1 Choice Problematic Let us remind the purpose of choice problematic before presenting meth ods The objective of this problematic consists of aiding DMs in selecting a subset of actions as small as possible in such a way that a single action may finally be chosen The order in which methods will be presented permit us to understand the historical introduction of the two fundamental concepts in multiple criteria decision aiding veto thresholds and pseudo criteria 3 1 1 ELECTRE I The purpose underlying the description of this method is rather theoretical and pedagogical The method does not have a significant practical interest given the very nature of real world applications having usually a vast spectrum of quantitative and qualitative elementary con
5. Ren Decartes 1977 J Moscarola and B Roy Proc dure automatique d examen de dossiers fond e sur une segmentation trichotomique en pr sence de crit res multiples RAIRO Recherche Op rationnelle 11 2 145 173 1977 V Mousseau Probl mes li s l valuation de l importance relative des crit res en aide multicrit re la d cision R flexions th oriques exp rimentations et impl mentations informatiques PhD thesis Universit Paris Dauphine 1993 30 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 V Mousseau Eliciting information concerning the relative importance of cri teria In P Pardalos Y Siskos and C Zopounidis editors Advances in Multi criteria Analysis Nonconvex Optimization and its Applications pages 17 43 Kluwer Academic Publishers Dordrecht 1995 V Mousseau and L Dias Valued outranking relations in ELECTRE provid ing manageable disaggregation procedures European Journal of Operational Research in press 2004 V Mousseau J Figueira and J Naux Unsing assignment examples to infer weights for ELECTRE TRI method Some experimental results European Journal of Operational Research 130 2 263 275 2001 V Mousseau B Roy and I Sommerlatt Elaboration d un outil d aide a la d cision en vue de l volution de la tarification des transports publics en Ile de France Journal of Decision Systems 9 2 289 315 2000
6. de crit res perme ttant une int gration des divers aspects li s au temps dans l aide la d cision en mati re de transports 1re phase 1975 2e phase 1977 D l gation G n rale la Recherche Scientifique et Technique DGRST SAEI Action Th matique Programm e ATP Socio Economie des Transports 1977 References 33 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 B Roy and V Mousseau A theoretical framework for analysing the notion of relative importance of criteria Journal of Multi Criteria Decision Analysis 5 145 159 1996 B Roy M Pr sent and D Silhol A programming method for determining which Paris metro stations should be renovated European Journal of Opera tional Research 24 318 334 1986 B Roy and J Skalka ELECTRE IS Aspects m thodologiques et guide d utilisation Document du LAMSADE 30 Universit Paris Dauphine 1984 B Roy R Slowinski and W Treichel Multicriteria programming of water resources supply systems for rural areas Water Resources Bulletin 28 1 13 31 1992 B Roy and D Vanderpooten The European school of MCDA Emergence basic features and current works Journal of Multi Criteria Decision Analysis 5 1 22 37 1996 T Saaty The Analytical Hierarchy Process McGraw Hill New York 1980 P Salminen J Hokkanen and R Lahdelma Comparing multicriteria methods in the context of environmental
7. imperfect knowledge only could be taken into account in ELEC TRE methods through the use of probabilistic distributions and expected utility criterion Clearly more work needed to be done Research in this area was still in its initial stages Another way to cope with uncer tain imprecision and ill determination has been introduced the thresh old approach 19 49 50 114 For more details and a comprehensive treatment of this issue see 14 96 97 Just a few years later a new method for ranking actions was devised ELECTRE III electre three 93 116 The main new ideas introduced by this method were the use of pseudo criteria see 92 and fuzzy binary outranking relations An other ELECTRE method known as ELECTRE IV electre four arose from a new real world problem related to the Paris subway network 38 45 110 111 113 It now became possible to rank actions without using the relative criteria importance coefficients this is the only ELEC TRE method which does not make use of such coefficients In addition the new method was equipped with an embedded outranking relations framework Methods created up to this point were particularly designed to help decision making in choosing and ranking actions However in the late seventies a new technique of sorting actions into predefined and or dered categories was proposed i e the trichotomy procedure 67 68 94 This is a decision tree based approach Several years later in order
8. problems European Journal of Operational Research 104 3 485 496 1998 A Scharlig D cider sur Plusieurs Crit res Panorama de l Aide la D cision Multicrit re Presses Polytechniques et Universitaires Romandes Lausanne 1985 A Sch rlig Pratiquer ELECTRE et PROMETHEE Un Compl ment D cider sur Plusieurs Crit res Presses Polytechniques et Universitaires Ro mandes Lausanne 1996 R Silva Lopez Une technique syst matique de mod lisation multicrit re pour l valuation d alternatives de lutte contre la pollution industrielle Technolo gies propres ou curatives Document du LAMSADE 60 Universit Paris Dauphine 1990 J Simos Gestion des D chets Solides Urbains Genevois Les Faits le Traite ment l Analyse Presses Polytechniques et Universitaires Romandes Lau sanne 1990 Y Siskos and Ph Hubert Multi criteria analysis of the impacts of energy alternatives survey and a new comparative approach European Journal of Operational Research 13 278 299 1983 Y Siskos G Wascher and H Winkels A bibliography on outranking ap proaches 1966 1982 Cahier du LAMSADE 31 Universit Paris Dauphine 1983 R Srinivasa L Duckstein and C Arondel Multicriterion analysis for sus tainable water resources planning A case study in Spain Water Resources Management 14 435 456 2000 Y Tanguy and N Leclere Une application des m thodes de choix multi crit res l tude de l ad q
9. situations may occur a aSb and not bSa i e aPb a is strictly preferred to b a bSa and not aSb i e bPa b is strictly preferred to a a aSb and bSa i e alb a is indifferent to b a Not aSb and not bSa i e aRb a is incomparable to b ELECTRE methods build one or several crispy fuzzy or embedded outranking relations Note that using outranking relations to model preferences introduces a new preference relation R incomparability This relation is useful to account for situations in which the DM and or the analyst are not able to compare two actions The construction of an outranking relation is based on two major concepts 1 Concordance For an outranking aSb to be validated a sufficient majority of criteria should be in favor of this assertion 2 Non discordance When the concordance condition holds none of the criteria in the minority should oppose too strongly to the assertion aSb These two conditions must be fulfilled for validating the assertion aSb Given a binary relation on set A it is extremely helpful to build a graph G V U where V is the set of vertices and U the set of arcs For each action a A we associate a vertex i V and for each pair of actions a b A the arc i l exists either if aPb or alb An action a outranks b if and only if the arc i exists If there is no arc between vertices 7 and l it means that a and b are incomparable if two reversal arcs exist there is an
10. the construction of cardinal value functions International Transactions in Operational Research 1 489 500 1994 O Barda J Dupuis and P Lencioni Multicriteria location of thermal power plants European Journal of Operational Research 45 332 346 1990 M Beccali M Cellura and D Ardente Decision making in energy plan ning The ELECTRE multicriteria analysis approach compared to a fuzzy sets methodology Energy Conversion and Management 39 16 18 1869 1881 1998 R Benayoun B Roy and B Sussman ELECTRE Une m thode pour guider le choix en pr sence de points de vue multiples Note de travail 49 SEMA METRA International Direction Scientifique 1966 P Bertier G Gagey J de Montgolfier and B Roy Choix de trac s au toroutiers en milieu suburbain Faut il vraiment endommager des sites ur bains et ou forestiers lesquels Communication aux journ es de formation transports et agglom rations Nice 1972 25 26 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Y B tolaud and R F vrier Conservation des for ts suburbaines et passage des autoroutes Revue Foresti re Fran aise pages 197 200 1973 P Blondeau M Sp randio and F Allard Multicriteria analysis of ventilation in summer period Building and Environment 37 2 165 176 2002 D Bouyssou Modelling inaccurate determination uncertainty imprecision using multip
11. to help decision making in a large banking company which faced to the problem of accepting or refusing credits requested by firms a specific method ELECTRE A was devised and applied in 10 sectors of activ ity This should have remained confidential The most recent sorting method ELECTRE TRI electre tree was greatly inspired by these earlier works It removed everything they had of specific given their context of application Indeed this new method is at the same time both simpler and more general 141 142 ELECTRE methods are still evolving Section 4 presents recent de velopments on the topic and avenues for future research 2 Main Features of ELECTRE Methods This section presents a set of key issues concerning ELECTRE methods the context in which they are relevant modelling with an outranking relation their structure the role of criteria and how to account for imperfect knowledge 2 1 In What Context Are ELECTRE Methods Relevant ELECTRE methods are relevant when facing decision situations with the following characteristics see 99 109 122 1 The decision maker DM wants to include in the model at least three criteria However aggregation procedures are more adapted in situations when decision models include more than five criteria up to twelve or thirteen And at least one of the following situations must be verified 2 Actions are evaluated for at least one criterion on an ordinal scale see 8
12. 4 or on a weakly interval scale see 63 These scales are not suitable for the comparison of differences Hence it is difficult and or artificial to define a coding that makes sense in terms of gj a g b ha g5 c 95 evaluation of action x on criterion g preference differences of the ratios where g x is the ELECTRE Methods 5 3 A strong heterogeneity related with the nature of evaluations exists among criteria e g duration noise distance security cultural sites monuments This makes it difficult to aggregate all the criteria in a unique and common scale 4 Compensation of the loss on a given criterion by a gain on another one may not be acceptable for the DM Therefore such situations require the use of noncompensatory aggregation procedures see Chapter 5 For at least one criterion the following holds true small differences of evaluations are not significant in terms of preferences while the accumulation of several small differences may become significant This requires the introduction of discrimination thresholds indif ference and preference which leads to a preference structure with a comprehensive intransitive indifference binary relation see Chap ter 2 2 Modelling Preferences Using an Outranking Relation Preferences in ELECTRE methods are modelled by using binary out ranking relations S whose meaning is at least as good as Considering two actions a and b four
13. Chapter 1 ELECTRE METHODS Jos Figueira Faculdade de Economia and INESC Coimbra Universidade de Coimbra Av Dias da Silva 165 8004 512 Coimbra Portugal figueira fe uc pt Vincent Mousseau Bernard Roy LAMSADE Universit Paris Dauphine Place du Mar chal De Lattre de Tassigny 75775 Paris Cedex 16 France mousseau roy lamsade dauphine fr Abstract Keywords Over the last three decades a large body of research in the field of ELEC TRE family methods appeared This research has been conducted by several researchers mainly in Europe The purpose of this chapter is to present a survey of the ELECTRE methods since their first appearance in mid sixties when ELECTRE I was proposed by Bernard Roy and his colleagues at SEMA consultancy company The chapter is organized in five sections The first section presents a brief history of ELECTRE methods The second section is devoted to the main features of ELEC TRE methods The third section describes the different ELECTRE methods existing in the literature according to the three main problem atics choosing ranking and sorting The fourth section presents the recent developments and future issues on ELECTRE methods Finally the fifth section is devoted to the software and applications An ex tensive and up to date bibliography is also provided in the end of this chapter Multiple criteria decision aiding Outranking approaches ELECTRE methods 2 1 Introductio
14. Readings in Multiple Criteria Decision Aid pages 155 183 Springer Verlag Berlin 1990 B Roy The outranking approach and the foundations of ELECTRE methods Theory and Decision 31 49 73 1991 B Roy Multicriteria Methodology for Decision Aiding volume 12 of Nonconvex Optimization and its Applications Kluwer Academic Publishers Dordrecht 1996 32 101 102 103 104 105 106 107 108 109 110 111 112 113 114 B Roy missing link in OR AD Robustness analysis Foundations of Com puting and Decision Sciences 23 3 141 160 1998 B Roy Decision aiding today What should we expect In T Gal T Stewart and T Hanne editors Advances in MultCriteria Decision Making MCDM Models Algorithms Theory and Applications pages 1 1 1 35 Kluwer Aca demic Publishers Boston 1999 B Roy A French English decision aiding glossary In Newsletter of the EWG on MCDA Supplement 1 volume Series 3 1 Faculty of Economics INESC Coimbra Portugal 2000 B Roy Optimisation et analyse multicrit re In C Jessua C Labrousse and D Vitry editors Dictionnaire des Sciences Economiques pages 640 643 Presses Universitaires de Frances Paris 2001 B Roy Robustesse de quoi et vis vis de quoi mais aussi robustesse pourquoi en aide la d cision In C Henggeler Antunes J Figueira and J Climaco editors Proceedings of the 56th Meeting of the European Worki
15. Zopounidis Operational Tools in the Management of Financial Risks Kluwer Academic Publishers Dordrecht 1998 145 C Zopounidis and A Dimitras Multicriteria Decision Aid Methods for the Prediction of Business Failure Kluwer Academic Publishers Dordrecht 1998 146 C Zopounidis and C Hurson Portfolio selection and multicriteria analysis In C Floudas and P Pardalos editors Encyclopedia of Optimization Vol IV pages 310 322 Kluwer Academic Publishers Dordrecht 2001
16. at preferences increase with the value on each criterion After determining the credibility index we should introduce a A cutting level of the fuzzy relation in order to obtain a crisp outranking relation This level can be defined as the credibility index smallest value compatible with the assertion aSb Let gt denote the preference J denote the indifference relation and R denote the incomparability binary relations The action a and the profile b may be related to each other as follows a albp iff aSby and by Sa b a gt bn iff aSby and not bn Sa c bn gt a iff not aSby and bn Sa d aRby iff not aSby and not bp Sa The objective of the exploitation procedure is to exploit the above bi nary relations The role of this exploitation is to propose an assignment This assignment can be grounded on two well known logics 1 The conjunctive logic in which an action can be assigned to a category when its evaluation on each criterion is at least as good as the lower limit which has been defined on the criterion to be in this category The action is hence assigned to the highest category fulfilling this condition 2 The disjunctive logic in which an action can be assigned to a cat egory if it has on at least one criterion an evaluation at least as good as the lower limit which has been defined on the criterion to be in this category The action is hence assigned to the highest category fulfilling this condition With this dis
17. ation Both the strong and weak relations are built thanks to the definition of two concordance levels sl gt s where s s 0 5 1 minjezw Now the concordance condition with the assertion a outranks b can be de fined as follows c aSb gt s and c aSb gt c bSa forr 1 2 The exploiting procedure is a four step algorithm ELECTRE Methods 13 1 Partioning the set A First let us consider the relation S over A In a similar way like in ELECTRE I this relation may define on A one or several cycles If all the actions belonging to each maximal cycle are grouped together into a single class a partition on A will be obtained Let A denote this partition When each class of A is not a singleton the actions belonging to that class will be considered as ex quo For the purpose of comparison between elements of A a preference relation will be used This relation has the same meaning as the relation gt for ELECTRE I 2 Building a complete pre order Z on A After obtaining A the procedure identifies a subset B of classes of A following the rule no other is preferred to them according to the relation gt t After removing B from A and applying the same rule to A B a subset B will be found The procedure iterates in the same way till define the final partition on A B B Now on the basis of S we may define a rough version of the complete pre order Z1 while placing in the head of this p
18. ciences Economiques et Sociales Dunod Paris 1969 volume 1 1970 volume 2 B Roy Crit res multiples et mod lisation des pr f rences l apport des rela tions de surclassement Revue d Economie Politique 1 1 44 1974 B Roy From optimization to multicriteria decision aid Three main opera tional attitudes In H Thiriez and S Zionts editors Multiple Criteria Decision Making volume 130 of Lecture Notes in Economics and Mathematical Systems pages 1 34 Springer Verlag Berlin 1976 B Roy ELECTRE III Un algorithme de classements fond sur une repr sentation floue des pr f rences en pr sence de crit res multiples Cahiers du CERO 20 1 3 24 1978 B Roy A multicriteria analysis for trichotomic segmentation problems In P Nijkamp and J Spronk editors Multiple Criteria Analysis Operational Methods pages 245 257 Gower Press Farnborough 1981 B Roy M thodologie Multicrit re d aide la D cision Economica Paris 1985 B Roy Main sources of inaccurate determination uncertainty and impreci sion in decision models In B Munier and M Shakun editors Compromise Negotiation and Group Decision pages 43 62 Reidel Publishing Company Dordrecht 1988 B Roy Main sources of inaccurate determination uncertainty and imprecision Mathematical and Computer Modelling 12 10 11 1245 1254 1989 B Roy The outranking approach and the foundations of ELECTRE methods In C A Bana e Costa editor
19. crete multi objective decision making European Journal of Operational Research 54 176 190 1991 E Roba B Sussmann and M Theys Les M thodes de Choix Multicrit re Appliqu es la S lection du Personnel Models of Manpower The English University Press London 1970 F Roberts Measurement Theory with Applications to Decision Making Util ity and the Social Sciences Addison Wesley New York 1979 References 31 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 M Rogers and M Bruen Choosing realistic values of indifference preference and veto thresholds for use with environmental criteria within ELECTRE European Journal of Operational Research 107 3 542 551 1998 M Rogers and M Bruen A new system for weighting environmental criteria for use within ELECTRE III European Journal of Operational Research 107 3 552 563 1998 M Rogers M Bruen and L Maystre ELECTRE and Decision Support Kluwer Academic Publishers London 2000 J Rohmer and J Veret Choix d une strat gie commerciale dans la distribution succursaliste In E Jacquet Lagr ze and Y Siskos editors M thode de d cision multicrit re pages 91 118 Editions Hommes et Techniques Paris 1983 B Roy Classement et choix en pr sence de points de vue multiples la m thode ELECTRE RIRO 8 57 75 1968 B Roy Alg bre Moderne et Th orie des Graphes Orient es vers les S
20. decades on a broad range of real life applications In the current description of ELECTRE III we will omit several for mulae details The novelty of this method is the introduction of pseudo criteria instead of true criteria In ELECTRE III the outranking relation can be interpreted as a fuzzy relation The construction of this relation requires the definition of a credibility index which characterizes the credibility of the assertion a outranks b aSb let p aSb denote this index It is defined by using both the concordance index as determined in ELECTRE IS c aSb and a discordance index for each criterion gj in F that is d aSb The discordance of a criterion g aims at taking into account the fact that this criterion is more or less discordant with the assertion aSb The discordance index reaches its maximal value when criterion g puts its veto to the outranking relation it is minimal when the criterion g is not discordant with that relation To define the value of the discordance index on the intermediate zone we simply admitted that this value grows in proportion to the difference g b g a This index can now be presented as follows 1 if gj b gt g a v gj a d aSb 9 if gj b lt gj a p g a 95 b 93 a p3 95 v g a p 5 g5 a otherwise The credibility index is defined as follows 1 d aSb p aSb c aSb Il T ca55 JEJ dj aSb gt c aSb Notice that
21. elected Such a subset A may be determined with the help of the graph kernel concept Kg The justifi cation of the use of this concept can be found in 109 When the graph contains no direct cycles there exists always a unique kernel otherwise the graph contains no kernels or several But let us point out that a graph G may contain direct cycles If that is the case a preprocessing step must take place where maximal direct cycles are reduced to single ton elements forming thus a partition on A Let A denote that partition Each class on A Aj Ao is now composed of a set of considered equivalent actions It should be noticed that a new preference relation is defined on A Ay gt Aq amp Ja and Ib such that aSb for A In ELECTRE I all the actions which form a cycle are considered indifferent which may be criticized ELECTRE IS was designed to mitigate this inconvenient see Section 3 1 3 3 1 2 ELECTRE Iv The name ELECTRE Iv was an un official name created for designating ELECTRE I with veto threshold 64 This method is equipped with a different but extremely useful tool The new tool made possible for analysts and DMs to overcome the difficulties related to the heterogeneity of scales Whichever the scales type this method is always able to select the best compromise action or a subset of actions to be analyzed by DMs The new tool introduced was the veto threshold v that can be a
22. for choice problematic The acronym ELECTRE stands for 10 95 ELimination Et Choix Traduisant la REalit ELimination and Choice Expressing the REality and was cited for commercial reasons At the time it seemed adequate and served well to promote the new tool Nevertheless the develop ments in ELECTRE methods over the last three decades the way in which we consider the tool today and the methodological foundations of multiple criteria decision aiding have made the meaning of the acronym unsatisfactory An atypical ELECTRE method was also created to deal with the problem of highway layout in the Ile de France region it was called the meaningful compensation method 11 12 25 91 109 This approach was based on substitution rates These rates were ill defined stakehold ers views about their values strongly differed it was only possible to fix a minimum and maximum value for each one On such a basis a set of embedded fuzzy relations has been defined In the late sixties a different real world decision making situation arose in media planning concerning the definition of an advertising plan For such a purpose the question was how to establish an ad equate system of ranking for periodicals magazines newspapers This led to the birth of ELECTRE II electre two a method for deal ing with the problem of ranking actions from the best option to the worst 1 43 106 107 However in a world where perfect knowledge is rare
23. ging to a cycle are no longer considered as indifferent as in the previous versions of ELECTRE for choice problems Now we take into account the concept of degree of robustness of a outranks b It is a reinforcement of veto effect and allow us to build true classes of ex quo ties and thus define an acycle graph over these classes In such conditions there is always a single kernel 3 2 Ranking Problematic In ranking problematic we are concerned with the ranking of all the actions belonging to a given set of actions from the best to the worst possibly with ex quo There are three different ELECTRE methods to deal with this problematic 3 2 1 ELECTRE II From an historical and pedagogical point of view it is interesting to present ELECTRE II This method was the first of ELECTRE methods especially designed to deal with ranking problems Without going into further detail it is important to point out that ELECTRE II was also the first method to use a technique based on the construction of an embedded outranking relations sequence The construction procedure is very closed to ELECTRE Iv in the sense that it is also a true criteria based procedure Hence it is not surprising that the no veto condition remains the same However con cordance condition is modified in order to take into account the notion of embedded outranking relations There are two embedded relations a strong outranking relation followed by a weak outranking rel
24. higher 2 ELECTRE III IV is a software which implements ELECTRE III and ELECTRE IV methods described in Section 3 2 It runs on Windows 3 1 95 98 2000 Millennium and XP 3 ELECTRE TRIis a multiple criteria decision aiding tool designed to deal with sorting problems This software implements ELEC TRE TRI method described in Section 3 3 The ELECTRE TRI software versions 2 x were developed with the C programming language and runs on Microsoft Windows 3 1 95 98 Me 2000 XP and NT This software integrates ELECTRE TRI Assistant which enables the user to define the weights indirectly i e fixing the model parameters by giving some assignment examples cor responding to desired assignments or past decisions The weights are thus inferred through a certain form of regression Hence ELECTRE TRI Assistant reduces the cognitive effort required from the DM to elicit the preference parameters 22 4 SA IRIS Interactive Robustness analysis and parameters Inference for multiple criteria Sorting problems This DSS has been built to support the assignment of actions described by their evaluation on multiple criteria to a set of predefined ordered categories using a variant of ELECTRE TRI Rather than demanding precise values for the model s parameters IRIS allows to enter constraints on these values namely assignment examples that it tries to restore When the constraints are compatible with multiple assignments for the actions
25. ified to assign it by means of com parisons to profiles bounds limits norms or references Results are expressed using the absolute notion of assigned or not assigned to a category similar or not similar to a reference profile adequate or not adequate to some norms The sorting problematic refers thus to absolute judgements It consists of assigning each action to one of the pre defined categories which are defined by norms or typical elements of the categories The assignment of an action a to a specific category does not influence the category to which another action b should be assigned 3 3 1 ELECTRE TRI ELECTRE TRI is designed to assign a set of actions objects or items to categories In ELECTRE TRI cat egories are ordered let us assume from the worst C1 to the best Cx Each category must be characterized by a lower and an upper profile Let C Ci Ch Cy denote the set of categories The assignment of a given action a to a certain category Cp results from the comparison ELECTRE Methods 17 of a to the profiles defining the lower and upper limits of the categories bn being the upper limit of category Cp and the lower limit of category Ch 1 for all h 1 k For a given category limit bp this comparison rely on the credibility of the assertions aSby and b Sa This credibility index is defined as in ELECTRE III In what follows we will assume without any loss of generality th
26. ights do not depend neither on the ranges nor the encoding of the scales Let us point out that these parameters can not be interpreted as substitution rates as in compensatory aggregation procedures AHP 120 MACBETH 7 and MAUT 55 Veto thresholds express the power attributed to a given criterion to be against the assertion a outranks b when the difference of the evalua tion between g b and g a is greater than this threshold These thresh olds can be constant along a scale or it can also vary ELECTRE Methods 7 A large quantity of works have been published on the topic of relative importance of criteria The following list is not exhaustive 35 64 69 86 87 115 116 125 136 2 5 Discrimination Thresholds To take into account the imperfect character of the evaluation of actions see Chapter ELECTRE methods make use of discrimination in difference and preference thresholds This leads to a pseudo criterion model on each criterion see Chapter Discrimination thresholds account for the imperfect nature of the eval uations and are used for modelling situations in which the difference between evaluations associated with two different actions on a given cri terion may either a justify the preference in favor of one of the two actions preference threshold p be compatible with indifference between the two actions indiffer ence thresholds q be interpreted as an hesitation between opting f
27. indifference between both a and b An outranking relation is not necessarily transitive Preference in transitivities come from two different situations Condorcet effect see Chapter and incomparabilities between actions This requires an exploitation procedure to derive from such a relation results that fit the problematic see Chapter 2 3 Structure of ELECTRE Methods ELECTRE methods comprise two main procedures construction of one or several outranking relation s followed by an exploitation procedure The construction of one or several outranking relation s aims at com paring in a comprehensive way each pair of actions The exploitation procedure is used to elaborate recommendations from the results ob tained in the first phase The nature of the recommendations depends on the problematic choosing ranking or sorting Hence each method is characterized by its construction and its exploitation procedures For more details the reader may consult the following references 70 98 99 109 135 138 2 4 About the Relative Importance of Criteria The relative role attached to criteria in ELECTRE methods is defined by two distinct sets of parameters the importance coefficients and the veto thresholds The importance coefficients in ELECTRE methods refer to intrinsic weights For a given criterion the weight wj reflects its voting power when it contributes to the majority which is in favor of an outranking The we
28. junctive rule the assignment of an action is generally higher than with the conjunctive rule This is why the conjunctive rule is usually interpreted as pessimistic while the disjunctive rule is inter preted as optimistic This interpretation optimistic pessimistic can be permuted according to the semantic attached to the outranking relation When no incomparability occurs in the comparison of an action a to the limits of categories a is assigned to the same category by both the 18 optimistic and the pessimistic procedures When a is assigned to differ ent categories by the optimistic and pessimistic rules a is incomparable to all intermediate limits within the highest and lowest assignment categories ELECTRE TRI is a generalization of the two above mentioned rules The generalization is the following in the conjunctive rule replace in the condition on each crite rion by on a sufficient majority of criteria and in the absence of veto a in the disjunctive rule replace the condition on at least one cri terion by on a sufficient minority of criteria and in the absence of veto The two procedures can be stated as follows 1 Pessimistic rule An action a will be assigned to the highest cate gory Cp such that aSbp_1 a Compare a successively with br r k 1 k 2 0 b The limit bp is the first encountered profile such that aSbp Assign a to category Ch11 2 Optimistic rule An acti
29. king examples European Journal of Operational Research to appear 2004 L Dias V Mousseau J Figueira and J Climaco An aggregation disaggregation approach to obtain robust conclusions with ELECTRE TRI European Journal of Operational Research 138 332 348 2002 A I Dimitras C Zopounidis and C Hurson Assessing financial risks using a multicriteria sorting procedure The case of country risk asessment Founda tions of Computing and Decision Sciences 29 97 109 2001 L Duckstein and M Gershon Multicriterion analysis of a vegetation manage ment problem using ELECTRE II Applied Mathematical Modelling 7 4 254 261 1983 J Evans Sensitivity analysis in decision theory Decision Sciences 15 239 247 1984 D Ferrand and J Martel Le choix multicrit re des items d une chelle de mesure Math matiques et Sciences Humaines 89 35 59 1985 J Fichefet and J Leclercq A multicriteria microcomputer assisted system for the identification of bacteria European Journal of Operational Research 24 12 22 1986 J Figueira and B Roy Determining the weights of criteria in the ELECTRE type methods with a revised Simos procedure European Journal of Opera tional Research 139 317 326 2002 V Gabrel M thodologie pour la planification de production de syst me d observation de la terre par satellites Aspects algorithmiques et multicrit res PhD thesis Universit Paris Dauphine 1994 T Gal T Stewar
30. l circumstances we are not able we do not want or we do not know how to assign a value to those coefficients It does not mean that we would be satisfied with the pre order obtained when applying ELECTRE III with the same value for all the coefficients w Another approach we could take would be determining a pre order which takes into account all the pre orders obtained from the application of several combinations of the weights Obviously this situation will be unman ageable ELECTRE IV is also a procedure based on the construction of a set of embedded outranking relations There are five different relations S1 9 The St relation r 1 2 3 4 accepts an outranking in a less credible circumstances than the relation S It means while remaining on a merely ordinal basis the assignment of a value p for the credibility index p aSb to the assertion aSb The chosen values must be such that pr gt pr41 Furthermore the movement from one credibility value p to another p 1 must be perceived as a considerable loss The ELECTRE IV exploiting procedure is the same as in ELECTRE III 3 3 Sorting Problematic A set of categories must be a priori defined The definition of a category is based on the fact that all potential actions which are assigned to it will be considered further in the same way In sorting problematic each action is considered independently from the others in order to determine the categories to which it seems just
31. le criteria In A Lockett and G Islei editors Improving Deci sion Making in Organisations volume 335 of Lecture Notes in Economics and Mathematical Systems pages 78 87 Springer Verlag Berlin 1989 D Bouyssou E Jacquet Lagr ze P Perny R Slowinski D Vanderpooten and Ph Vincke editors Aiding Decisions with Multiple Criteria Essays in Honour of Bernard Roy Kluwer Academic Publishers Dordrecht 2001 D Bouyssou T Marchant M Pirlot P Perny A Tsouki s and Ph Vincke Evaluation and Decision Model A Critical Perspective Kluwer Academic Pub lishers Dordrecht 2000 D Bouyssou and B Roy La notion de seuils de discrimination en analyse multicrit re INFOR 25 4 302 313 1987 P Buffet J Gr my M Marc and B Sussmann Peut on choisir en tenant compte de crit res multiples Une m thode ELECTRE et trois applications Revue METRA VI 283 316 1967 A Charpentier and E Jacquet Lagr ze La promotion de l lectricit dans l entreprise et l utilisation des m thodes multicrit res In H Thiriez and S Zionts editors Multiple Criteria Decision Making volume 130 of Lecture Notes in Economics and Mathematical Systems pages 364 377 Springer Ver lag Berlin 1976 A Colorni M Paruccini and B Roy editors A MCD A Aide Multi Crit re la D cision Multiple Criteria Decision Aiding The European Commission Joint Research Center Luxemburg 2001 G Colson The OR s prize winner a
32. ls of variation for the preference parameters Definition 2 A preference elicitation process proceeds through an in teraction between DMs and analysts in which DMs express information about their preferences within a specific aggregation procedure It is possible to distinguish among direct and indirect elicitation tech niques 20 4 2 1 Direct Elicitation Techniques In direct elicitation procedures DMs should provide information directly on the values of the preference parameters A major drawback of such techniques is that it is difficult to understand the precise meaning of the assertions of the DMs This is why ELECTRE methods are usually implemented by using indirect elicitation procedures 4 2 2 Indirect Elicitation Techniques Indirect elicitation techniques do not require from DMs to provide answers to questions related to the values of the preference parameters On the contrary these techniques proceeds indirectly by posing questions whose answers can be interpreted through the aggregation procedure Such techniques make use of the disaggregation paradigm 51 60 For instance DIVAPIME 70 and SRF 35 elicitation techniques make it possible to determine the vector of the relative importance coefficients from pairwise comparisons of fictitious actions or a relative importance ranking of criteria Recent developments concerning elicitation techniques have been pro posed for the ELECTRE TRI method Inference procedures have bee
33. n A brief History How far back in history should we go to discover the origins of ELECTRE methods Some years ago B Roy and D Vanderpooten 119 published an article The European School of MCDA Emergence Basic Features and Current Works Journal of Multi Criteria Decision Analysis on this very topic This introduction is largely based on their paper but additional material has been included to define the origins more precisely and to look more deeply into the history of ELECTRE methods We have also benefited from an old but nonetheless excellent bibliography containing a lot of references collated by Y Siskos G Wascher and H Winkels 127 The latter only covers the period 1966 1982 but contains many valuable references The origins of ELECTRE methods go back to 1965 at the European consultancy company SEMA which is still active today At that time a research team from SEMA worked on a concrete multiple criteria real world problem regarding decisions dealing with the development of new activities in firms For solving this problem a general multi ple criteria method MARSAN M thode d Analyse de Recherche et de S lection d Activit s Nouvelles was built The analysts used a weighted sum based technique included in the MARSAN method for the selection of the new activities 57 When using the method the engineers from SEMA noticed serious drawbacks in the application of such a technique B Roy was thus consul
34. n developed to elicit the parameters values from assignment examples i e an assignment that is imposed by DMs on specific actions It is possible to infer all the preference parameters simultaneously 74 we will refer to such a case by complete inference The induced mathematical program ming model to be solved is however non linear Thus its resolution is computationally difficult for real world problems In such cases it is possible to infer a subset of parameters only see Figure 1 1 Concordant coalition parameters weights and cutting level 72 a Discordance related parameters veto thresholds 28 Category limits 76 ELECTRE Methods 21 ELECTRE TRI Inferring from examples Direct elicitation Partial inference Complete inference Inferring weights Inferring veto Inferring category limits Figure 1 1 Inferring parameter values for ELECTRE TRI 5 Software and Applications The implementation of ELECTRE methods in real world decision prob lems involving DMs requires software packages Some of them are widely used in large firms and universities in particular ELECTRE IS ELEC TRE II IV ELECTRE TRI and IRIS Among the software available at LAMSADE are http www lamsade dauphine fr english sof tware html 1 ELECTRE IS is a generalization of ELECTRE I It is an imple mentation of ELECTRE IS described in Section 3 1 This software runs on a IBM compatible computer on Windows 98 and
35. nd the software ARGOS How a multijudge and multicriteria ranking GDSS helps a jury to attribute a scientific award Computers amp Operations Research 27 7 8 741 755 2000 S Damart Aide la d cision et processus de d cision collectifs La contribu tion des outils d aide la d cision aux d marches de concertation PhD thesis Universit Paris Dauphine Paris 2003 S Damart V Mousseau and I Sommerlatt Du mode d implication d acteurs multiples dans le cadre de l utilisation d un mod le d affectation multicrit re Analyse au regard d une application la tarification des transports publics INFOR 40 3 199 222 2002 N Danila M thodologie d aide a la d cision dans le cas d investissements fort d pendants PhD thesis Universit Paris Dauphine 1980 J de Montgolfier and P Bertier Approche Multicrit re des Probl mes de D cision Editions Hommes et Techniques Paris 1978 References 27 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 L Dias and J Climaco On computing ELECTRE s credibility indices under partial information Journal of Multi Criteria Decision Analysis 8 2 74 92 1999 L Dias and J Climaco ELECTRE TRI for groups with imprecise information on parameter values Group Decision and Negotiation 9 5 355 377 2000 L Dias and V Mousseau Inferring ELECTRE s veto related parameters from outran
36. ne on thresholds and semi orders may how ever illuminate the issue of inaccurate data and permit to build a more general procedure the so called ELECTRE IS method The main novelty of ELECTRE IS is the use of pseudo criteria instead of true criteria This method is an extension of the previous one aiming at taking into account a double objective primarily the use of possible no nil indifference and preference thresholds for certain criteria belonging to F and correlatively a backing up reinforcement of the veto effect when the importance of the concordant coalition decreases Both concordance and no veto conditions change Let us present separately the formulae for each one of theses conditions e Concordance condition Let us start by building the following two indices sets 1 concerning the coalition of criteria in which aSb i 5 ET gj a q g la 95 2 concerning the coalition of criteria in which bQa I ET g a q lg la lt gjla lt 95 b P 95 b The concordance condition will be c aSb X wy X pu s jeT je where _ gj a pi gi a 9 b 7 p g5 a a5 9 the coefficient p decreases linearly from 1 to 0 when g describes the range g a gz g a gta p5 g5 a 12 e no veto condition The no veto condition can be stated as follows gjla vj gj gt 95 b a5 95 b n where _ 1 e aSb wj 7 l s wj Nj In the exploitation procedure actions belon
37. nen and P Salminen Choosing a solid waste management system us ing multicriteria decision analysis European Journal of Operational Research 98 1 19 36 1997 J Hugonnard and B Roy Le plan d extension du m tro en banlieue parisienne un cas type d application de l analyse multicrit re Les Cahiers Scientifiques de la Revue Transports 6 77 108 1982 C Hurson and C Zopounidis On the use of multicriteria decision aid methods to portfolio selection The Journal of Euro Asian Management 1 2 69 94 1995 C Hurson and C Zopounidis Gestion de Portefeuille et Analyse Multicrit re Economica Paris 1997 C Hurson and C Zopounidis On the use of multicriteria decision aid methods to portfolio selection In J Climaco editor Multicriteria Analysis pages 496 507 Springer Verlag Berlin 1997 E Jacquet Lagreze Le probleme de l agr gation des pr f rences Une classe de proc dures seuil Math matiques et Sciences Humaines 43 29 37 1973 E Jacquet Lagr ze How we can use the notion of semi orders to build out ranking relations in multicriteria decision making In E Wendt and C Vleck editors Utility Subjective Probability and Human Decision Making pages 87 112 Reidel Dordrecht 1975 E Jacquet Lagr ze and Y Siskos Assessing a set of additive utility functions for multicriteria decision making The UTA method European Journal of Operational Research 10 151 164 1982 E Jacquet Lagr
38. ness of such recommendations They strongly depend on the set of central values attributed to the parameters Should the analyst analyze the influence of a variation of each parameter considered sep arately on the results And then enumerate those parameters which provoke a strong impact on the results when their values vary from the central positions This is a frequent way to proceed in classical opera tions research methods and it is called sensitivity analysis 32 53 79 82 But this kind of analyzes has rather a theoretical interest than a practi cal one Analysts are most often interested in building recommendations which remain acceptable for a large range of the parameters values Such recommendations should be elaborated from what we call the robust con clusions Chapter 101 105 109 Definition 1 A conclusion C is said to be robust with respect to a domain Q of possible values for the preference and technical parame ters if there is no a particular set of parameters w Q which clearly invalidates the conclusion C A robustness concern consists of all the possible ways that contribute to build synthetic recommendations based on the robust conclusions Possible ways to deal with robustness concerns in ELECTRE methods are illustrated for example in 26 27 29 109 116 Chapters and 4 2 Elicitation of Parameter Values Implementing ELECTRE methods requires to determine values or in terva
39. ng Group Multi ple Criteria Decision Aiding Coimbra Portugal 3 5 October 2002 Commiss o de Coordena o da Regi o Centro Coimbra Portugal to appear 2004 B Roy and P Bertier La m thode ELECTRE II Note de travail 142 SEMA METRA Metra International 1971 B Roy and P Bertier La m thode ELECTRE II Une application au m dia planning In M Ross editor OR 72 pages 291 302 North Holland Publishing Company Amsterdam 1973 B Roy and D Bouyssou Comparison of two decision aid models applied to a nuclear plant siting example European Journal of Operational Research 25 200 215 1986 B Roy and D Bouyssou Aide Multicrit re la D cision M thodes et Cas Economica Paris 1993 B Roy and J Hugonnard Classement des prolongements de lignes de m tro en banlieue parisienne pr sentation d une m thode multicrit re originale Cahiers du CERO 24 2 3 4 153 171 1982 B Roy and J Hugonnard Le plan d extension du m tro en banlieue parisienne un cas type de l analyse multicrit re Les Cahiers Scientifiques de la Revue Transports 6 77 108 1982 B Roy and J Hugonnard Ranking of suburban line extension projects on the Paris metro system by a multicriteria method Transportation Research 16A 4 301 312 1982 B Roy and J Hugonnard R ponse Monsieur Gargaillo Les Cahiers Scien tifiques de la Revue Transports 7 58 64 1982 B Roy E Jacquet Lagr ze and M Blancher Elaboration
40. on a will be assigned to the lowest category Cy such that by gt a a Compare a successively with b r 1 2 k 1 b The limit b is the first encountered profile such that b gt a Assign a to category Cp 4 Recent Developments and Future Issues Although several decades past since the birth of the first ELECTRE method research on ELECTRE family method stills active today Some of the recent developments are shortly described in this Section 4 1 Robustness Concerns When dealing with real world decision problems DMs and analysts are often facing with several sources of imperfect knowledge regarding the available data This leads to the assignment of arbitrary values to certain variables In addition modelling activity frequently requires to choose between some technical options introducing thus an additional source of arbitrariness to the problem For these reasons analysts hesitate ELECTRE Methods 19 when assigning values to the preference parameters weights thresholds categories lower and upper limits and the technical parameters dis cordance and concordance indices A cutting level of ELECTRE methods In practice it is frequent to define a reference system built from the assignment of central values to these two types of parameters Then an exploitation procedure should be applied in order to obtain outputs which are used to elaborate recommendations But what about the meaningful
41. or a preference or an indifference between the two actions These thresholds can be constant or vary along the scale When they are variable we must distinguish between direct the evaluation of the best action is taken into account and inverse when they are computed by using the worst evaluation How to assign values to such thresholds There are several techniques which can be used some of them come directly from the definition of threshold and other ask for the concept of dispersion threshold see Sec tion 4 2 A dispersion threshold allow us to take into account the concept of probable value and the notion of optimistic and pessimistic values It translates the plausible difference due to over or under estimations which affect the evaluation of a consequence or of a performance level It should be noticed that there are no true values for thresholds Therefore the values chosen to assign to the thresholds are the most convenient the best adapted for expressing the imperfect character of the knowledge For more details about thresholds see 2 17 95 100 102 103 104 109 8 3 A Short Description of ELECTRE Methods A comprehensive treatment of ELECTRE methods may be found in the books by B Roy and D Bouyssou 109 and Ph Vincke 139 Much of the theory developed on this field is presented in these books This theory however was foreshadowed in earlier papers namely by B Roy and his colleagues at SEMA and later at
42. re 15 5 471 478 2000 J March Ambiguity and the engineering of choice Bell Journal of Economics 9 587 608 1978 J Martel N Khoury and M Bergeron An application of a multicriteria ap proach to portfolio comparisons Journal of the Operational Research Society 39 617 628 1988 J Martel and A Rousseau Cadre de r f rence d une d marche multicrit re de gestion int gr e des ressources en milieu forestier Document technique 93 11 Minist re de environment et de la faune Qu bec 1993 J Martel and B Roy Analyse de la signifiance de diverses proc dures d agr gation multicrit re Annales du LAMSADE 1 Universit Paris Dauphine 2002 L Maystre J Pictet and J Simos Les M thodes Multicrit res ELECTRE Presses Polytechniques et Universitaires Romandes Lausanne 1994 J Michel La s lection des projects du programme architecture nouvelle In Actes du congr s Association Francaise pour la Cybern tique Economique et Technique AFCET Aide la D cision Paris May 16 17 1974 pages 111 121 1974 L Montano Guzm n Fuzzy measures and integrals in the MCDA sorting prob lematic Methodology and application to the diagnosis of firms PhD thesis Universit Libre de Bruxelles Brussels Belgium 2002 J Moscarola Aide la d cision en pr sence de crit res multiples fond e sur une proc dure trichotomique M thodologie et application Th se de doctorat de 3e cycle Universit Paris V
43. re order and in an ex quo position all classes of B1 then those of B and so forth In order to define Z1 in a more accurate way we examine if it is possible to refine this pre order on the basis of the relation S This refinement consists of using the information that brings this less believable outranking to decide between the various classes of a subset B when it contains several classes This refinement of the rough version is obtained while using S to define over BP a complete pre order that takes place between B and BPt 3 Determining a complete pre order Z on A The procedure to obtain this pre order is quite similar to the above one only two modifications are needed e apply the rule they are not preferred to any other instead of no other is preferred to them let Bt B denote the partition thus obtained e define the rough version of the complete pre order Z by putting it in the queue of this pre order and in an ex quo position all classes of B then those of B and so forth 4 Defining the partial pre order Z The partial pre order Z is an intersection of Z and Zo Z Z N Z2 and it is defined in the following way aZb aZ b and aZ b 14 3 2 2 ELECTRE III ELECTRE III was designed to improve ELECTRE II and thus deal with inaccurate imprecise uncertain or ill determination of data This purpose was actually achieved and ELEC TRE III was applied with success during the last two
44. sequences leading to the construction of a contradictory and very heterogeneous set of criteria with both numerical and ordinal scales associated with them In addition a certain degree of imprecision uncertainty or ill determination is always attached to the knowledge collected from real world problems The method is very simple and it should be applied only when all the criteria have been coded in numerical scales with identical ranges In such a situation we can assert that an action a outranks b that is a is at least as good as b denoted by aSb only when two conditions hold ELECTRE Methods 9 On the one hand the strength of the concordant coalition must be powerful enough to support the above assertion By strength of the concordant coalition we mean the sum of the weights associated to the criteria forming that coalition It can be defined by the following concordance index assuming for the sake of formulae simplicity that Die Wj 1 where J is the set of the indices of the criteria c aSb 5 Wj j gj a gt g b where j gj a gt g b is the set of indices for all the criteria belonging to the concordant coalition with the outranking relation aSb In other words the value of the concordance index must be greater than or equal to a given concordance level s whose value generally falls within the range 0 5 1 minjez wj i e c aSb gt s On the other hand no discordance against the as
45. sertion a is at least as good as 6 may occur The discordance is measured by a discordance level defined as follows OO adn a 85 as a This level measures in some way the power of the discordant coalition meaning that if its value surpasses a given level v the assertion is no longer valid Discordant coalition exerts no power whenever d aSb lt v Both concordance and discordance indices have to be computed for every pair of actions a b in the set A where a b It is easy to see that such a computing procedure leads to a binary relation in comprehensive terms taking into account the whole set of criteria on the set A Hence for each pair of actions a b only one of the following situations may occur aSb and not bSa i e aPb a is strictly preferred to b a bSa and not asb i e bPa b is strictly preferred to a a aSb and bSa i e alb a is indifferent to b Not aSb and not bSa i e aRb a is incomparable to b This preference indifference framework with the possibility to resort to incomparability says nothing about how to select the best compromise action or a subset of actions the DM will focus his attention on In the construction procedure of ELECTRE I method only one outranking relation S is matter of fact 10 The second procedure consists of exploiting this outranking relation in order to identify a small as possible subset of actions from which the best compromise action could be s
46. t tributed to certain criteria gj belonging to the family of criteria F The concept of veto threshold is related in some way to the definition of an upper bound beyond which the discordance about the assertion a outranks b can not surpass and allow an outranking In practice the idea of threshold is however quite different from the idea of the dis concordance level like in ELECTRE I Indeed while discordance level is related to the scale of criterion g in absolute terms for an action a from A threshold veto is related to the preference differences between g a and g b In terms of structure and formulae little changes occur when moving from ELECTRE I to ELECTRE Iv The only difference being the dis cordance condition now called no veto condition which may be stated as follows g a vj gi a Z g Yj ET ELECTRE Methods 11 To validate the assertion a outranks b it is necessary that among the minority of criteria that are opposed to this assertion none of them puts its veto ELECTRE Iv uses the same exploitation procedure as ELECTRE I But this method is by no means complete the problem of imperfect knowledge remains 3 1 3 ELECTRE IS How general an ELECTRE method can be when applied to choice decision making problems Is it possible to take into account simultaneously the heterogeneity of criteria scales and imperfect knowledge about real world decision making situations Pre vious theoretical research do
47. t and T Hanne editors Advances in MultCriteria Deci sion Making MCDM Models Algorithms Theory and Applications Kluwer Academic Publishers Boston 1999 L Gargaillo R ponse l article Le plan d extension du m tro en banlieue parisienne un cas type de l analyse multicrit re Les Cahiers Scientifiques de la Revue Transports 7 52 57 1982 E Georgopoulou D Lalas and L Papagiannakis A multicriteria decision aid approach for energy planning problems The case of renewable energy option European Journal of Operational Research 103 1 38 54 1997 E Georgopoulou Y Sarafidis S Mirasgedis S Zaimi and D Lalas A multi ple criteria decision aid approach in defining national priorities for greenhouse 28 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 gases emissions reduction in the energy sector European Journal of Opera tional Research 146 199 215 2003 M Gershon L Duckstein and R McAniff Multiobjective river basin planning with qualitative criteria Water Resources Research 18 193 202 1982 S Greco and J Figueira Dealing with interaction between bi polar multiple criteria preferences in outranking methods Research Report 11 2003 INESC Coimbra Portugal 2003 J Grolleau and J Tergny Manuel de r f rence du programme ELECTRE II Document de travail 24 SEMA METRA International Direction Scientifique 1971 J Hokka
48. ted and soon tried to find a new method to over come the limitations of MARSAN The ELECTRE method for choosing the best action s from a given set of actions was thus devised in 1965 and was later referred to as ELECTRE I electre one In that same year July 1965 the new multiple criteria outranking method was presented for the first time at a conference les journ es d tudes sur les m thodes de calcul dans les sciences de l homme in Rome Italy Nevertheless the original ideas of ELECTRE methods were first merely published as a research report in 1966 the notorious Note de Travail 49 de la SEMA 10 Shortly after its appearance ELECTRE I was found to be suc cessful when applied to a vast range of fields 18 but the method did not become widely known until 1968 when it was published in RIRO la Revue d Informatique et de Recherche Op rationnelle 89 This article presents a comprehensive description of ELECTRE and the foundations of the outranking approach the reader may also consult the graph theory book by B Roy 90 The method has since evolved and given rise to an unofficial version ELECTRE Iv electre one vee This version took into account the notion of a veto threshold A further version known as ELECTRE IS electre one esse appeared subsequently see 117 and ELECTRE Methods 3 was used for modelling situations in which the data was imperfect see below This is the current version of ELECTRE methods
49. the grant SFRH BDP 6800 2001 Funda o para a Ci ncia e Tec nologia Portugal and gratefully acknowledges DIMACS Research Cen ter at Rutgers University for the welcome during his sabbatical leave August 2002 May 2003 The authors are thankful to Dominique Champ Brunet for the many valuable references she provided References 10 11 R Abgueguen La S lection des Supports de Presse Robert Lafont Paris 1971 A Ait Younes R Azibi and B Roy ELECTRE IS Manuel d utilisation Document du LAMSADE 118 and 118bis Universit Paris Dauphine 2000 A Andenmatten Evaluation du Risque de D faillance des Emetteurs d Obligations Une Approche par l Aide Multicrit re la D cision Presses Polytechniques et Universitaires Romandes Lausanne Suisse 1995 C Arondel and P Girardin Sorting cropping systems on the basis of their impact on groundwater quality European Journal of Operational Research 127 467 482 2000 C Bana e Costa editor Readings in Multiple Criteria Decision Aid Springer Verlag Heidelberg 1990 C Bana e Costa and C Dinis das Neves Describing and formalizing the eval uation process of Portuguese navy officiers In A Lockett and G Islei editors Improving Decision Making in Organisations volume 335 of Lecture Notes in Economics and Mathematical Systems pages 355 369 Springer Verlag Berlin 1989 C Bana e Costa and J Vansnick MACBETH An interactive path towards
50. tial pre order Z is defined as a partition on the set A into q ordered classes Biase Br B where B1 is the head class in Z1 Each class Bp is composed of ex quo elements according to Z1 The complete pre order Z is determined in a similar way where A is partitioned into u ordered classes B Bn Bu Bu being the head class Each one of these classes is obtained as a final distilled of a distillation procedure The procedure designed to compute Z starts first distillation by defining an initial set Do A it leads to the first final distilled By After getting By in the distillation 1 the procedure sets Do A BiU UB According to Z1 the actions in class B are preferable to those of class B 41 for this reason distillations that lead to these classes will be called as descending top down The procedure leading to Z2 is quite identic but now the actions in Bn41 are preferred to those in class Bp these distillations will be called ascending bottom up The partial pre order Z will be computed as the intersection of Z and Z2 A complete pre order is finally suggested taking into account the par tial pre orders and some additional considerations The way the incom parabilities which remain in the pre order are treated is nevertheless subject to criticism 16 3 2 3 ELECTRE IV In Section 2 4 we pointed out the diff culty to define the relative importance coefficients of criteria However in severa
51. tome 1 Guide d utilisation tome 2 Document du LAM SADE 85 et 85bis Universit Paris Dauphine 1994 D Vanderpooten The construction of prescriptions in outranking methods In C Bana e Costa editor Readings in Multiple Criteria Decision Aid pages 184 216 Springer Verlag Heidelberg 1990 J Vansnick On the problem of weight in multiple criteria decision making the non compensatory approach European Journal of Operational Research 24 288 294 1986 M Ventura Evry 1 Concours d am nagement urbain Cahier 31 Institut d Amenagement et d Urbanisme de la R gion Parisienne IAURP Paris 1973 Ph Vincke Exploitation of a crisp relation in a ranking problem Theory and Decision 32 3 221 240 1992 Ph Vincke Multicriteria Decision Aid John Wiley amp Sons New York 1992 D Wijnmalen A case study in military decision making with mission oriented multiple objectives In J Climaco editor Multicriteria Analysis pages 549 560 Springer Verlag Berlin 1997 W Yu Aide multicrit re la d cision dans le cadre de la probl matique du tri Concepts m thodes et applications PhD thesis Universit Paris Dauphine 1992 W Yu ELECTRE TRI Aspects m thodologiques et manuel d utilisation Document du LAMSADE 74 Universit Paris Dauphine 1992 C Zopounidis Evaluation du Risque de D faillance de l Entreprise M thodes et Cas d Application Economica Paris 1995 References 35 144 C
52. uation des personnels aux postes de travail In 34 130 131 132 133 134 135 136 137 138 139 140 141 142 143 Actes du congr s Association Fran aise pour la Cybern tique Economique et Technique AFCET Aide la D cision Paris May 16 17 1974 pages 139 147 1974 A Tecle and L Duckstein A procedure for selecting MCDM techniques for forest resources management In A Goicoechea L Duckstein and S Zionts editors Multiple Criteria Decision Making and Support at the Interface of Industry Business and Finance pages 19 32 Springer Verlag New York 1992 A Tecle M Fogel and L Duckstein Multicriterion analysis of forest wa tershed management alternatives Water Resources Bulletin 24 1169 1178 1988 J Teng and G Tzeng Multicriteria evaluation for strategies of improving and controlling air quality in the super city case study of Taipei city Journal of Environmental Management 40 3 213 229 1994 A Tsouki s P Perny and Ph Vincke From concordance discordance to the modelling of positive and negative reasons in decision aiding In D Bouyssou E Jacquet Lagr ze P Perny R Stowiriski D Vanderpooten and Ph Vincke editors Aiding Decisions with Multiple Criteria Essays in Honour of Bernard Roy pages 147 174 Kluwer Academic Publishers Dordrecht 2002 D Vall e and P Zielniewicz ELECTRE III IV version 3 x Aspects M thodologiques
53. when d aSb 1 it implies that p aSb 0 since c aSb lt 1 The definition of p aSb is thus based on the following main ideas a When there is no discordant criterion the credibility of the out ranking relation is equal to the comprehensive concordance index b When a discordant criterion activates its veto power the assertion is not credible at all thus the index is null ELECTRE Methods 15 c For the remaining situations in which the comprehensive concor dance index is strictly lower than the discordance index on the discordant criterion the credibility index becomes lower than the comprehensive concordance index because of the opposition effect on this criterion The index p aSb corresponds to the index c aSb weakened by pos sible veto effects In 71 a modification of the valued outranking relation used in the ELECTRE III and ELECTRE TRI was proposed The modification requires the implementation of the discordance concept Such a modi fication is shown to preserve the original discordance concept the new outranking relation makes it easier to solve inference programs The exploitation procedure starts by deriving from the fuzzy relation two complete pre orders as in ELECTRE II A final partial pre order Z is then built as the intersection of the two complete pre orders Z1 and Z2 which are obtained according to two variants of the same principle both acting in an antagonistic way on the floating actions The par
54. ze and Y Siskos editors M thode de D cision Multicrit re Editions Hommes et Techniques Paris 1983 R Kampke Sensitivity analysis for assessing preferentially independent order relations Computers amp Operations Research 23 12 1119 1130 1996 A Karagiannidis and N Moussiopoulos Application of ELECTRE III for the integrated management of municipal solid wastes in the greater Athens area European Journal of Operational Research 97 3 439 449 1997 R Keeney and H Raiffa Decisions with Multiple Objectives Preferences and Value Tradeoffs John Wiley amp Sons New York 1976 References 29 56 57 58 59 60 61 62 63 64 65 66 67 68 69 N Khoury J Martel and P Yourougou A multicriterion approach to country selection for global index founds Document de travail FSA 91 13 Facult des Sciences de l Administration Universit Laval Qu bec Canada 1991 R Laffy La m thode MARSAN pour la recherche de produits nouveaux Presented at the ESOMAR congress Copenhagen September 1966 R Lahdelma P Salminen and J Hokkanen Locating a waste treatment facility by using stochastic multicriteria acceptability analysis with ordinal criteria European Journal of Operational Research 142 345 356 2002 M Mahmoud and L Garcia Comparison of different multicriteria evalua tion methods for the Red Bluff diversion dam Environmental Modelling and Softwa
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