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. I. I NTRODUCTION M 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 conflict- ing 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 decom- posed to produce another processed information that ev- erybody 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: dagomez@estad.ucm.es). This work was partially supported by Grant TIN2006-06190 from the Government of Spain. 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. Ra- tionality 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. AN INTRODUCTION TO STANDARD MULTICRITERIA ANALYSIS A standard approach to a multicriteria decision making problem is to assume that basic decision making is given in 978-1-4244-2764-2/09/$25.00 ©2009 IEEE