Fuzzy information representation for decision aiding J. Montero Faculty of Mathematics Complutense University Madrid 28040 (Spain) monty@mat.ucm.es D. Gómez Faculty of Statistics Complutense University Madrid 28040 (Spain) dagomez@estad.ucm.es S. Muñoz Faculty of Mathematics Complutense University Madrid 28040 (Spain) smunoz@estad.ucm.es Abstract In this paper we want to stress the relevance of decision aid procedures in complex decision making prob- lems and claim for an extra effort in order to develop appropriate rep- resentation tools when fuzzy criteria or objectives are present. In particu- lar, we point out how some painting algorithms may help decision mak- ers to understand problems subject to fuzziness based upon a graphical first approach, like Statistics use to do. Moreover, we point out that al- though the standard communication tool with machines are either data or words, we should also consider cer- tain families of graphics for such a role, mainly for the output. Keywords: Multicriteria decision making, decision aid, criteria repre- sentation, fuzzy sets. 1 Introduction When Zadeh [34] introduced fuzzy sets he was postulating the existence of certain uncer- tainty decision makers use to deal with which needed a more appropriate and efficient rep- resentation than the only existing model for uncertainty, i.e., probability theory. A big sci- entific dispute was declared with Bayesian ap- proach to probability, since they claim fuzzi- ness and any other kind of uncertainty can be consistently represented according to their model (see, e.g., [5, 6, 13, 22, 23, 27, 32] but also [8, 9, 20, 35] and a personal view in [26]). No matter if we accept or not the axioms of the Bayesian approach, leading towards the existence of a unique representation of uncer- tainty, the true fact is that most students and even many researchers do not know about any alternative representation for uncertainty. This situation is a consequence of a more im- portant issue: most people do not know about any other formal alternative to binary logic which represents, together with observation, a key pilar of present Science [25]. We can recall here philosopher Francis Ba- con when he wrote that ”those who have han- dled the sciences have been either empiricists or dogmatists. Empiricists are like ants, who only collect things and make use of them. Ra- tionalists are like spiders, who weave webs out of their own bodies. But the bee has a middle policy: it extracts material from the flowers of the gardens and meadows, and digests and transforms it by its own powers.” The problem we want to stress first in this paper is that experiments still need a logical structure behind, so it is not so clear that experiments are in one side and logic is on the other side. Experiments are usually built in order to answer certain questions provided within a particular logic, and the way we ob- serve and understand reality depends on a logic previously defined. Then we shall stress that any decision aid tool needs a protocol for communication with hu- man being. Quite often this protocol is as- L. Magdalena, M. Ojeda-Aciego, J.L. Verdegay (eds): Proceedings of IPMU’08, pp. 1425–1430 Torremolinos (Málaga), June 22–27, 2008 sumed within a linguistic framework, but we may be forgetting that part of the success of Statistics is based upon the great explanatory potential some simple graphic have. 2 Experiments and probability Perhaps the main success of Statistics is quite concentrated in two fields mathematicians do not care too much about: the design of exper- iments and what is usually called Descriptive Statistics. The first one represents the first stage of any statistical study, and the second one appears once those experiments have been run. Design of experiments tells how reality must be observed in order to maximize information, at the same time that cost is being reduced (this is actually done by means of a sometimes quite boring mathematical reckoning). But Descriptive Statistics pretends a simpli- fied global view of data, of course subject to manipulation and frequent errors. It is sur- prising how some complex situations are eas- ily explained with a simple standard statisti- cal graphic. A key argument to be pointed out is how Probability Theory is fully consistent with bi- nary logic: uncertainty is about observability of certain event, but not about the event itself (we may not know if such an event happened or not, but for sure either the event hap- pens or the event does not happens). Hence, those experiments Statistics talk about are conceived and designed according to such a binary logic. At this point, it is interesting to note that most introductory books to Probability The- ory assume that the experimental space Ω has been someway given (see, for example, [11]), so events (that define a Boolean algebra ac- cording to binary operators and, or and no) can be represented in terms of subsets of that experimental space, which looks like a regular set but it is defined as a set of possible results during an experiment (so, it is not a regular set). On the contrary, a consistent approach to Kol- mogorov [21] can be based upon events, sim- ply assuming the existence of a set representa- tion (which can be assured as soon the struc- ture of events is being assumed to follow the Boolean structure, see [33]). But even in this case, an appropriate exper- iment should be built in such a way that we can have an answer to every question we make, does event A holds when I get result ω during the experiment? And according to the assumed binary logic, only one of two answers is allowed: yes or no. Once binary logic is being assumed, events have to be crisp in the above sense. Defining a different kind of experiments, subject to an alternative non-binary logic, should be a main objective of fuzzy researchers in order to build the new Science being claimed by those soft sciences where most information is given in linguistic terms rather than Set Theory terms. 3 Complex decision making problems and decision aiding Complex decision making problems are deeply related to the above discussion, once they are not confused with difficult decision making problems. By a difficult decision making problem we mean here those problems that have been or can be modelled in terms of a classical op- timization problem, where basic information fits into a real space: we may find of course that the optimal solution does not exist, per- haps because there is an inner conflict be- tween criteria or objectives. But obtaining an informative output about the true situa- tion hinges simply on our reckoning capacity. We just need faster algorithms. By a complex decision making problem we mean here a problem where basic informa- tion is subject to deep modelling uncertain- ties. Criteria or objectives are poorly defined or simply not provided, and they have to be estimated from the available information, per- haps linguistic preferences. Any classical ap- proximation to such this kind of problems is subject to an essential criticism, most often 1426 Proceedings of IPMU’08 because a linguistic term is forced to fit into a crisp representation. Complexity in our context refers to modelling, no matter if the proposed model is difficult or not to solve. Within such a modelling complexity in this paper we stress the relevance of two problems, respectively located at both sides of any deci- sion aid procedure, in between such a proce- dure and decision makers: • Estimation of criteria or objectives from preferences, for a better knowledge of de- cision maker’s mind. • Graphical representation of results, for a better knowledge of possible conse- quences of decision maker’s alternatives. 4 Estimation of criteria or objectives from preferences A typical complex situation is when the basic information is given in terms of preferences. Decision maker compares a set of given al- ternatives, which can be poorly defined (as pointed out in [25], the most important hu- man alternatives are strategic and therefore poorly defined, so details are fixed in the very last moment). Since in a complex problem it is always desir- able that our model allows the possibility of creating new alternatives, initially not taken into account (see [28, 31]), understanding the space were these alternatives move is essen- tial. In this sense, we can try some kind of decom- position or representation in terms of some possible underlying criteria or objectives. Pursuing this objective, an interesting ap- proach has been recently proposed by some of the authors [18, 19] by generalizing the dimen- sion theory restricted in [10] to partial order sets. Although more efficient algorithms and approaches are being investigated, we should stress that from this approach an alternative to Saaty’s importance weighting [30] can be developed (see [15]). The relevance of this approach is to suggest decision makers with some hints in order to understand decision maker’s own mind. Hav- ing a hint about possible underlying criteria or objectives should help in the search of new alternatives. In Statistics we find tools for reducing data dimension that can be complementary. 5 Producing graphical representations Another key issue is the output decision maker will get from our decision aid tool. First of all, we must acknowledge that no deci- sion maker will accept black boxes, i.e., a ma- chine telling decision makers what to do, un- less the proposed model fully fits their mind. The true objective in a complex decision mak- ing problem is to help decision makers to un- derstand the problem they are facing to (see again [28, 31]). We should then realize that too often mathe- matical models use to assume that input and output are the same kind of information. But this is not true in most cases, where we give simple information (the one we as decision makers have) and we expect some help about a problem we cannot face, at least directly. So, we are acknowledging that such a prob- lem is more complex than the available in- formation. Why then should we expect that the output will be similar to the input? If our data are given in terms of words, we should be trying a more complex representation frame- work for the output, rather than the linguistic one. A natural more sophisticated framework for communication is the graphical framework (which may contain words, at least in their written form). Of course, a certain restriction to some singu- lar kind of graphics is needed, in order to build up a proper logic within a structured family of graphical symbols (see [24] for an interesting discussion within the language structure). In this context, an interesting approach for Proceedings of IPMU’08 1427 classification is being developed by some of the authors in [17, 16] generalizing Ruspini’s [29] fuzzy partitions [1, 2] in order to produce representative paintings where colors repre- senting classes are subject to gradation. The final objective is to be able to produce similar graphics to those used in statistics, but allow- ing color gradation (first results are being ap- plied in a remote sensing framework, see [14]). We find here obvious reckoning difficulties, so a great effort is being made in order to reduce computing time keeping meaningful outputs. Improvements are being tested against stan- dard libraries of figures together with real re- mote sensing images. For example, the algo- rithm proposed in [17] can be significatively improved in time together with the number of classes under consideration, so pictures will be more easily manageable. There is a lot of work ahead under this ap- proach, deserving in our opinion more inter- est than too sophisticated tools difficult to be managed by regular decision makers. A key characteristic for a revolutionary tool should be its simplicity. 6 Final comments In this paper we stress the relevance of aiding tools for decision making, specifically multi- criteria problems. The development of aiding tools represents a key issue in complex decision making prob- lems, where the key characteristics and their relationship use to be difficult to be captured, measured or described. The amount and the structure of data (initial or processed) may represent themselves serious difficulties for a direct intuition of the problem, not allowing neither an easy a priori modelling or an easy a posteriori explanation). In fact, quite often the deep objective problem in a multicrite- ria decision making is to find out those crite- ria which explain present preferences and will help future decisions. Needless to say, such criteria may not be representable in the real line but, on the contrary, they can be poorly defined [34]. In particular, two key decision aid tools are considered in this paper. On one hand, those procedures allowing a better understanding of decision maker’s mind (pursuing an accu- rate, significative and suggesting representa- tion model of the problem). On the other hand, those procedures allowing a simplified, significative and suggesting view of possible results, so that decision maker can understand the consequences of each decision and even find some hints about possible new alterna- tives. Those two key procedures address the first and the last stage of any decision aid method- ology, i.e., the communication between ma- chine and decision maker. In this sense, we point out that meanwhile the linguistic sup- port may be the standard way in which hu- man beings give information, graphical infor- mation can be the standard way in which in- formation is given back to the decision maker. Certainly, there are additional important paradigms in complex decision making that have not being addressed in this paper. For example, aggregation operators do play a key role in any summarizing process (see, e.g., [12] and [4]). In fact, we usually get from the set of rough data certain significative indices, which will be quite often the true base for the pos- terior treatment or representation. Aggrega- tion operators taking into account operative reckoning (see, e.g., [3, 7]) and the underlying structure of data should become a key issue in the next future (see, e.g., [24]). But at the end, a picture can be more illustra- tive than many words, and sometimes we can- not control the affective pressure words may represent in decision maker’s mind. Acknowledgements This research has been partially supported by the Government of Spain, grant TIN2006- 06190. References [1] A. Amo, D. Gómez, J. Montero and G. Biging (2001): Relevance and re- dundancy in fuzzy classification systems. 1428 Proceedings of IPMU’08 Mathware and Soft Computing 8:203– 216. [2] A. Amo, J. Montero, G. Biging and V. Cutello (2004): Fuzzy classification sys- tems. European Journal of Operational Research 156:459–507. [3] A. Amo, J. Montero and E. Molina (2001): Representation of consistent re- cursive rules. European Journal of Oper- ational Research 130:29–53. [4] T. Calvo, G. Mayor and R. Mesiar, Eds. (2002): Aggregation Operators New Trends and Applications (Physica- Verlag, Heidelberg). [5] P. Cheeseman (1986): Probabilistic ver- sus fuzzy reasoning. In Uncertainty in Artificial Intelligence, L.N. Kanal and J.F. Lemmer (eds.), Elsevier Science Pub., pages 85–102. [6] P. Cheeseman (1988): An inquiry into computer understanding (with discus- sion). Computational Intelligence 4:58– 142. [7] V. Cutello and J. Montero (1999): Recur- sive connective rules. Int. J. Intelligent Systems 14, 3–20. [8] D. Dubois and H. Prade (1989): Fuzzy sets, probability and measurement. Eu- ropean Journal of Operational Research 40:135–154. [9] D. Dubois and H. Prade (1993): Fuzzy sets and probability, misunderstandings, bridges and gaps. In Second IEEE In- ternational Conference on Fuzzy Systems volume 2, IEEE Press, pages 1059–1068. [10] B. Dushnik and E.W. Miller (1941): Par- tially ordered sets. American Journal of Mathematics 63:600–610. [11] W. Feller (1968): An Introduction to Probability Theory and Its Applications, volume 1, Wiley. [12] J. Fodor and M. Roubens (1994): Fuzzy Preference Modelling and Multicriteria Decision Support. Theory and Decision Library (Kluwer Academic Publishers, Dordrecht). [13] S. French (1984): Fuzzy decision analy- sis: some criticisms. TIMS/Studies in the Management Sciences 20:29–44. [14] D. Gómez, G. Biging and J. Montero (in press): Accuracy statistics for judging soft classification. International Journal of Remote Sensing. [15] D. Gómez, J. Montero and J. Yáñez (2006): Measuring criteria weights by means of dimension theory. Mathware and Soft Computing 13:173-188. [16] D. Gómez, J. Montero and J. Yánez (2006): A coloring algorithm for im- age classification. Information Sciences 176:3645-3657. [17] D. Gómez, J. Montero, J. Yáñez and C. Poidomani (2007): A graph coloring al- gorithm approach for image segmenta- tion. Omega 35:173–183. [18] J. González-Pachón, D. Gómez, J. Mon- tero and J. Yáñez (2003): Searching for the dimension of binary valued prefer- ence relations. Int. J. Approximate Rea- soning 33:133–157. [19] J. González-Pachón, D. Gómez, J. Mon- tero and J. Yáñez (2003): Soft dimension theory. Fuzzy Sets and Systems 137:137– 149. [20] G.J. Klir (1989): Is there more to un- certainty than some probability theorists might have us believe?. International Journal of General Systems 15:347–378. [21] A.N. Kolmogorov (1956): Foundations of the theory of probability, Chelsea Publish- ing. [22] M. Laviolette and J. Seaman (1994): The efficacy of fuzzy representations of uncer- tainty (with discussion). IEEE Transac- tion on Fuzzy Systems 2:4–42. Proceedings of IPMU’08 1429 [23] D.V. Lindley (1982): Scoring rules and the inevitability of probability. Interna- tional Statistical Review 50 , 1–26. [24] J. Montero, D. Gómez and H. Bustince (2007): On the relevance of some fami- lies of fuzzy sets. Fuzzy Sets and Systems 158:2429-2442. [25] J. Montero, V. López and D. Gómez (2007): The role of fuzziness in decision making. In D. Ruan et al. (eds.), Fuzzy Logic: an spectrum of applied and theo- retical issues, Springer), pages 337–349. [26] J. Montero and M. Mendel (1998): Crisp acts, fuzzy decisions. In S. Barro et al. (eds.), Advances in Fuzzy Logic, Univer- sidad de Santiago de Compostela, pages 219–238. [27] B. Natvig (1983): Possibility versus probability. Fuzzy Sets and Systems 10:31–36. [28] B. Roy (1993): Decision science or decision-aid science. European Journal of Operational Research 66:184–203. [29] E.H. Ruspini (1969): A new approach to clustering. Information and Control 15:22–32. [30] T.L. Saaty (1978): Exploring the inter- face between hierarchies, multiple objec- tives and fuzzy sets. Fuzzy Sets and Sys- tems 1:57–68. [31] G. Shafer (1986): Savage revisited (with discussion). Statistical Science 1:463– 501. [32] W. Stallings (1977): Fuzzy set theory versus bayesian statistics. IEEE Trans- action on Systems, Man and Cybernetics 7:216–219. [33] M.H. Stone (1936): The Theory of Rep- resentations of Boolean Algebras. Trans- actions of the American Mathematical Society 40:37–111. [34] L.A. Zadeh (1965): Fuzzy sets. Informa- tion and Control 8:338–353. [35] L.A. Zadeh (1986): Is probability the- ory sufficient for dealing with uncertainty in AI, a negative view. In Uncertainty in Artificial Intelligence, L.N. Kanal and J.F. Lemmer (eds.), Elsevier, pages 103– 116. 1430 Proceedings of IPMU’08