Modelling Bipolar Multicriteria Decision Making J. Tinguaro Rodr´ıguez, Bego˜na Vitoriano, Javier Montero, Daniel G´omez Abstract— In this paper we revisit some classical multicriteria decision making aid models in order to stress the presence of dual concepts, which will be consistent with Bipolar Fuzzy Sets (sometimes called Atanassov’s Intuitionistic Fuzzy Sets). In addition, we point out how such a dual approach is a non necessary binary heritage, so we can conclude how relevant in practice are decision aid models based in linguistic terms.
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I. I NTRODUCTION
ULTICRITERIA DECISION MAKING (MCDM) is a main field of research that implies quite a number of mathematical models. Being decision making modelling mainly devoted to complex problems, such a complexity use to come with several individuals that express conflicting views about the same problem and suggest different alternatives, being these alternatives valuated from different perspectives On the one hand, we have to formalize the information that each individual has in mind. Of course, the basis for such an information can be data, i.e., a measurable set of observations obtained perhaps by means of a previously designed experiment. But it should be acknowledged that the information being actually managed by each individual is not a rough data set, but an elaborated product where a particular logic (and other cultural heritage influences) may play a key role in order to understand data, find a meaning for data, and even the design of the experiment to be run (see, e.g., [22]). Such an elaborated information that each decision maker has in mind should be again processed according to a compatible and reliable procedure, so we can reach a common consistent formal representation. On the other hand, information from different decision makers should be in some way amalgamated and decomposed to produce another processed information that everybody can at least comprehend and check, so a true epistemological dialog can be initiated and eventually allow some kind of decision making. It should be stressed with [24] that the general objective of mathematical modelling should not be the decision itself, but only a better understanding of the problem, so a solution (most probably not yet an alternative) can be grasped (see, e.g., [29], [32]). We should not forget that Medicine tells us that the part of the human brain making a decision is associated to emotions, which is different from the part of the brain in charge of the rational analysis [4]. In this sense, some J. Tinguaro Rodr´ıguez, Bego˜na Vitoriano and Javier Montero are with the Faculty of Mathematics, Complutense University of Madrid, Spain (email: {jtrodrig,bvitoriano,monty}@mat.ucm.es). Daniel G´omez is with the Faculty of Statistics, Complutense University of Madrid, Spain (email:
[email protected]). This work was partially supported by Grant TIN2006-06190 from the Government of Spain.
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of the tremendous theoretical effort in order to formalize rationality may be misleading meanwhile it is focused on acts. A practical approach to the concept of rationality should focuss mainly on the argument supporting the decision. Rationality should be mainly applied to the discourse supporting each decision, and such a consistency is obviously graded (see [9] for a valued approach to rationality). Fuzziness plays a natural role within this context [22]. For example, we can not accept in real life that the most tiny mistake changes the pretended discourse into an absolutely inconsistent discourse (some consequences of this argument are pointed out in [20], [8], [21]). We should search for the rationality behind our acts, where our true human decision has been made. In fact, there is only one way of being consistent in our acts: to become ourselves extreme radical persons (the relevance of the discourse above contradictory acts is quite obvious if we are talking about political parties in democracy, for example). Moreover, as pointed out in [17], [18], acknowledging complexity implies also an effort to avoid the linear ordering as the unique consistent basic element for decomposing and amalgamating. In this sense, Atanassov’s Intuitionistic Fuzzy Sets (AIFS) [1] (or Bipolar Fuzzy Sets, see [5], [19], [35]) indeed represent an alternative approach, once some modeling issues are corrected). In this paper we shall always view these sets from the perspective addressed in [23], where among other things (including the possibility of more informational structures) duality is considered instead of negation, and non-determinacy is specifically associated to ignorance in terms of a particular state valuation structure, to be defined in each particular problem. In this paper we shall show how this intuitionistic (in the improper sense of Atanassov, see [1], [2]) was already present is some classical multicriteria models, meanwhile a strict decision making approach is being pursued. We must acknowledge that the term intuitionistic is considered misleading by most of the scientific community, as stated in [13] (see also the whole volume 23, issue 8, of the International Journal of Intelligent Systems). In particular, in this paper we are concerned with decision problems where positive and negative evaluations for each criteria are taken into account. Hence, the term bipolar is indeed more appropriate to our context, and AIFS should be better called ”Bipolar Fuzzy Sets”, as claimed in [13]. Notice anyway that such bipolar models can be generalized following the approach of [23]. II. A N INTRODUCTION TO STANDARD MULTICRITERIA ANALYSIS
A standard approach to a multicriteria decision making problem is to assume that basic decision making is given in
terms of a binary preference relation, i.e., a direct comparison between every possible pair of alternatives. It is not the objective of this paper to discuss about the obvious difficulties to obtain from decision makers a complete answer to a big amount of comparisons, how to assure they will be consistent, how such a consistency should be understood, or even how those alternatives are defined. As pointed out in [24], decision making use to imply a process from quite abstract alternatives that at each step produces a more specific alternative till decision maker considers he can jump into reality leaving details in some kind of randomness (last moment adjustments perhaps out of control). But note that these comments apply both to crisp and fuzzy frameworks. Given a finite family of crisp alternatives X, a fuzzy weak preference relation is a mapping µ : X × X → [0, 1] that assigns to each pair of alternatives (x, y) ∈ X × X a value µ(x, y) the degree to which alternative x is better or indifferent to y. From this information we can build up (see [26]) the complete preference structure relation, defining the strict preference, indifference and incomparability relations (see [14] but also [20] to see an application within group decision making). Nevertheless, it is very relevant for the purpose of this paper to point out with [23] that mathematical entities are never defined because of their names, but the other way round: the name should follow from the mathematical properties. Hence, it is a terrible mistake to treat preference intensities as numbers without taking into account which preference we are referring to, without taking into account the relative position of such a number within the particular preference structure. As shown in [23], similar mathematical entities model different real objects if we change the relationship between them. In particular, neither individuals or criteria and alternatives should be analyzed without taking into account their structure in each particular problem we address. And such structures are in no way unique in general. This was a key argument in [23] to distinguish between Atanassov’s intuitionistic fuzzy sets [2] and interval valued fuzzy sets (see, e.g., [7]). In the framework of aggregation functions, for example, we can not accept [15] that such an aggregation is described by means of a unique binary operation that does not take into account the situation of each data (see [12], [10] for a solution to linear data). And we can neither accept that an aggregation function must be defined in terms of an un-structured sequence of a unique binary operation. III. B IPOLARITY IN SOME STANDARD MULTICRITERIA MODELS
The classical ELECTRE model [29], for example, assumes basic data in terms of a family of fuzzy preference relations for each criteria. Being C a finite family of criteria describing the main features of each alternative x ∈ X, for each criteria i ∈ C
it is assumed in [29] (see also [33], [28]) the existence of a fuzzy preference relation µi : X × X → [0, 1] such that µi (x, y) represents the degree of intensity to which alternative x is weakly better than alternative y whenever we restrict comparison to criteria i. Again, it is not the purpose of this paper to discuss how we can reach to those criteria, which in fact should be searched quite often from the direct preferences between alternatives, looking for a meaningful and tractable decomposition (see [34] and [17], [18], [16] for interesting attempts within a crisp and a fuzzy framework). Based on the above information about the decision maker preferences decomposed by criteria, an outranking relation [29] is defined over the cartesian product of all alternatives. Together with such an outranking relation, another veto fuzzy set is obtained. And from both intensities a fuzzy set of non-dominated alternatives is defined, hopefully containing a non-empty set of un-dominated alternatives [27], [25]. It is interesting to notice that the meaning of such an outranking relation seems to be close in meaning to a strict preference of each alternative x over each alternative y, and that veto meaning seems to be related to the opposite strict preference. But in no way outranking is the negation of veto. Again, it is not the objective of this paper to offer a formal overview of ELECTRE’s model. Let us here just point out that Roy [29] realizes the practical need of introducing dual concepts in order to analyze how good a property is. It is not simply the strict preference of x over y in opposition to the strict preference of alternative y over x, but how an alternative x is supported in opposition to the intensity to which an alternative is unacceptable. We can not make a decision without taking into account pro arguments and contra arguments at the same time (and a contra argument is not simply the logic negation of a pro argument). Our main thesis here is that many standard multicriteria models build up dual structures like the one underlying ELECTRE [29] model with his outranking versus veto analysis. This duality looks in nature the same duality that appears in Atanassov’s model, once we accept [23] explanation. Notice also that of course, and consistently with [23], outranking and veto structures are in [29] far away from defining a fuzzy partition in the sense of Ruspini [30] (see [11]). Following a main argument of [11], [23], a basic dual classification (plus the initial ignorance state) is in general unnatural, since standard linguistic scales requires five or seven levels. If many authors model preferences from basic two-level approach is just because decision making plays an excessive role: at the end, it looks like an alternative should be chosen or not. But this should not be the case if we accept that final decision making is out of focuss and that our only objective it to help the true decision maker to get a better picture of the problem and the consequences of the actual alternatives under consideration. Perhaps from a good analysis the decision maker can start a search for a different
alternative still not at hand (the key claims of [29] and [32] are still valid today). IV. C ONCLUSIONS In this paper we stress that the idea of taking into account pro and contra arguments relative to each alternative in order to make a decision has been a quite standard approach to decision making, and that such dual view directly connects with bipolar ([5], [19], [35]) or AIFS model [2], at least if we accept the alternative approach offered in [23]. In addition, we also point out that the naive approach of Atanassov not only brings conceptual doubts, as shown in [13] (see also [3]), but it is too focused on decision making, which in fact suggests a bipolar close-to-binary framework (see [24]). Accepting the natural generalization presented in [23], more sophisticated valuation spaces should be developed (see also [6]) Aknowledgements: We deeply thank one of the referees for a careful lecture and suggestions that indeed improved this paper. Some comment in this paper has been taken from this referee. R EFERENCES [1] K.T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems vol. 20, 87-96, 1986. [2] K.T. Atanassov, Intuitionistic Fuzzy sets, Physica-Verlag., Heidelberg, New York, 1999. [3] K.T. Atanassov, “A personal view on intuitionistic fuzzy sets,” in: H. Bustince, F. Herrera and J. Montero (Eds.), Fuzzy Sets and Their Extensions: Representation, Aggregation and Models, pp. 23-43, Springer Verlag, Berlin, 2008. [4] A. Bechara, D. Tranel and H. Damasio, “Characterization of the decision-making deficit of patients with ventromedial prefrontal cortex lesions,” Brain vol. 123, 2189-2202, 2000. [5] S. Benferhat, D. Dubois, S. Kaci and H. Prade, “Bipolar possibility theory in preference modelling: representation, fusion and optimal solutions,” Information Fusion vol. 7, 135-150, 2006. [6] H. Bustince, F. Herrera and J. Montero (Eds.), Fuzzy Sets and Their Extensions: Representation, Aggregation and Models, Springer Verlag, Berlin, 2008. [7] H. Bustince, J. Montero, M. Pagola, E. Barrenechea and D. G´omez, “A survey on interval-valued fuzzy sets,” in: W. Pedrycz, A. Skowron and V. Kreinovichedrycz (Eds.), Handbook of Granular Computing, pp. 491-515, John Wiley and Sons, 2008. [8] V. Cutello and J. Montero, “A characterization of rational amalgamation operations,” International Journal of Approximate Reasoning, vol. 8, 325-344, 1993. [9] V. Cutello and J. Montero, “Fuzzy rationality measures,”, Fuzzy Sets and Systems, vol. 62, 39-44, 1994. [10] V. Cutello and J. Montero, “Recursive connective rules,” Int. J. Intelligent Systems, vol. 14, pp. 3-20, 1999. [11] A. Del Amo, J. Montero, G. Biging and V. Cutello, “Fuzzy classification systems,” European Journal of Operational Research, vol. 156, pp. 459-507, 2004. [12] A. Del Amo, J. Montero and E. Molina, “Representation of consistent recursive rules,” European Journal of Operational Research, vol. 130, pp. 29-53, 2001. [13] D. Dubois, S. Gottwald, P. Hajek, J. Kacprzyk and H. Prade, “Terminological difficulties in fuzzy set theory - the case of intuitionistic fuzzy sets,” Fuzzy Sets and Systems vol. 156, 485-491, 2005. [14] J. Fodor and M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer Academic Publishers, Dordrecht, 1994. [15] D. G´omez and J. Montero, “A discussion on aggregation operators,” Kybernetika, vol. 40, pp. 107-120, 2004. [16] D. G´omez, J. Montero and J. Y´a nez, “An algorithmic approach to preference representation,” International Journal of Uncertainty, Fuzzyness and Knowledge-Based Systems vol. 16, 1-18, 2008.
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