Email: {dubois, prade}@irit.fr. ABSTRACT ... which maps the attribute values describing the situation to evaluate into a discrete domain (classification), or a.
Fuzzy Criteria and Fuzzy Rules in Subjective Evaluation —A General Discussion— Didier Dubois and Henri Prade Institut de Recherche en Informatique de Toulouse (IRIT) – CNRS Université Paul Sabatier, 118 route de Narbonne 31062 Toulouse Cedex 4, France Email: {dubois, prade}@irit.fr
ABSTRACT This position paper emphasizes the interest of fuzzy set-based methods in subjective evaluation problems, where complex situations, or objects, have to be classified into categories, or have to be assessed under the form of global estimates. More particularly, different types of approaches to these problems, such as multiple-criteria aggregation, rule-based inference, case-based reasoning, are reviewed and put in perspective. Important issues in subjective evaluation, such as the expressibility of an approach with respect to the description of the evaluation process, are pointed out. The paper illustrates this general overview by means of some examples taken from the literature. 1 - SUBJECTIVE EVALUATION A subjective evaluation problem aims at providing a global assessment of a (complex) situation or object. The situation or the object under consideration is supposed to be described in terms of various attributes, and the assessment may be made either in a discrete way by classifying the situation according to given categories, or in an absolute way by computing a global rate, taking its value in a continuous scale. It may also correspond to the extrapolation of the value of some parameter attached to the situation. A daylife example of subjective evaluation problem is the pricing of a house from the knowledge of its surface, of its location w.r.t. downtown, its age, and some other attributes. This problem generally amounts to the computation of an estimated price, but viewing it as a discrete evaluation problem, it would mean classifying the price of the house with respect to context-dependent categories like "cheap", "moderate", "expensive", etc. Subjective evaluation problems can be encountered in a great number of areas. Let us, for instance, mention the evaluation of complex notions such as comfort (Levrat et al., 1997), creditworthiness of a customer in a bank (Zimmermann and Zysno, 1983), the assessment of damages, of defects in quality control (Ménage, 1996), the design of attractive products (e.g., Grabisch et al., 1997, in cosmetics), the selection of candidates from their profiles, the evaluation of recommended amounts of calories in dietetics in specific situations,… Fuzzy set methods have been advocated for a long time in subjective evaluation for representing the result of the evaluation (by means of fuzzy estimates, or fuzzy classes), and/or for describing the evaluation process itself (even if all the attributes describing the situation or the object are precisely known). A general introduction to fuzzy set methods in subjective evaluation can be found in the recent book (Club CRIN Logique Floue, 1997). More generally, subjective evaluation problems have an important place in Fuzzy Information Engineering (Dubois et al., 1997) in relation with decision issues. Formally speaking, the subjective evaluation problem can be viewed as the synthesis, the identification of a function which maps the attribute values describing the situation to evaluate into a discrete domain (classification), or a continuous one (absolute evaluation). More generally, we may look for the degree of membership of the situation to a category, or have a function yielding a fuzzy evaluation. This function is in general not available as such, but is implicitly, and partially, described in terms of criteria, or by means of expert rules, or through some fuzzy algorithm. It may also happen that the function is only partially known by exemplification through prototypical examples of situations for which the evaluation is available. We might also think of a neural net approach for synthesizing the function from a set of training examples, but then, we would not take advantage of the available knowledge on the evaluation process if any, and there will be no basis for explaining the evaluation results to a user. In the following, we briefly discuss the different knowledge-based approaches that have been just mentioned and the role of fuzzy set methods in each of them, before concluding on the interest of a unified view of the different approaches. 2 - MULTICRITERIA VIEW The evaluation function, say f, which relates the attribute values x1 , …, xn (assumed to be precisely known for simplicity) describing the situation or the object under consideration to its estimation y, is supposed in the multiplecriteria view, to be obtained by an appropriate aggregation of the evaluations of the attribute values x i by means of criteria Ci, for i = 1,n. Namely, f can be decomposed into local evaluation functions of each xi :
f(x 1 , …, xn ) = ϕ(C1 (x1), …, Cn (xn)). In the above expression, ϕ is not necessarily an associative aggregation operation. Different specific aggregation operations may be used for combining evaluations over subsets of the criteria (triangular-norms, averages, etc.), or ϕ may involve even more sophisticated functions taking into account some interaction between the criteria. Such an approach raises several questions about – the choice of proper scales (what kind? qualitative or numerical scale?), their commensurability, and the meaningfulness of the aggregation operations w.r.t. the scale; – the practical elicitation of the membership functions, and of the appropriate operations (compensatory, or purely logical conjunctions, for instance); see (Dubois and Prade, 1988) on this latter point, where the elicitation of aggregation operations is based on the knowledge of the decision's maker's behavior in well-contrasted situations; – the modelling of the importance (by means of weights or thresholds) of the criteria, and more generally of the interaction between criteria (Carlsson and Fullér, 1994; Grabisch, 1997). A pioneering example of such an approach is the assessment of the credit-worthiness of a bank customer (Zimmermann and Zysno, 1983; see also Zimmermann, 1997). This approach has been used in many other applications, including financial analysis. Another worth mentioning example of this type of approach to a completely different application, is the cutting of long woods in vine pruning (Tisseyre et al., 1996, 1997). In such a problem, the professional pruner for deciding what long wood to keep for next year, takes into account six main criteria (reflecting experts' preferences), the diameter, the length, the direction, the linearity of the cane and its X and Y positions with respect to the axis of the vine trunk and the wire respectively. Tisseyre et al. (1996) have succeeded in identifying an aggregation function for combining all these criteria and obtaining evaluation results which are well in agreement with expert opinions, in standard situations. In special cases where one or several of the important criteria are not satisfied at all, the way the expert selects the long wood which will not be cut, largely differs from his approach to standard situations. Tisseyre et al. (1996, 1997) use a fuzzy rule-based approach for describing the long wood selection in such special cases. It raises the issue of integrating a multiple-criteria view and a rule-based approach in the same framework. Conditional prioritized requirements are a particular instance of rule-based expressions which can be captured in the framework of a multiple-criteria aggregation approach. This type of problem has been encountered in database querying systems by Lacroix and Lavency (1987) who deal with requirements of the form "P1 should be satisfied, and among the solutions to P 1 (if any) the ones satisfying P2 are preferred, and among those satisfying both P 1 and P2 , those satisfying P 3 are preferred, and so on", where P 1, P2 , P3 …, are binary constraints for simplicity. It should be understood in the following way: satisfying P 2 if P1 is not satisfied is of no interest; satisfying P3 if P2 is not satisfied is of no use even if P 1 is satisfied. Thus, there is a hierarchy between the requirements. A request looking for candidates such that "if they are not graduated they should have professional experience, and if they have professional experience, they should preferably have communication abilities", is an example where only conditional constraints, organized in a hierarchical way, take place. It can be represented by an expression of the form min[max(Prof.exp.(d), Grad.(d)), max(Com.ab.(d), 1 – min(1 – Grad.(d), Prof.exp.(d), α)] so that if d has professional experience and communication abilities, d completely satisfies the request, as well as if d is graduated; d satisfies the request to the degree 1 – α if d is not graduated and has professional experience only. d does not satisfy the request at all if d is neither graduated nor has professional experience (even if d has communication abilities). See (Dubois and Prade, 1996b) for details on this approach, where a conditional requirement of the form "if P is true then d should be C" is represented (and estimated) by max(C(d), 1 – P(d)) where P(d) = 1 if the choice d satisfies the context P and is 0 otherwise, and where C(d) rates d according to criterion C (outside context P, C(d) is not taken into account). 3. RULE-BASED VIEW In this approach, the assessment of the evaluation is described by means of a set of rules which relate conditions pertaining to the attribute values describing the situation or the object under consideration to the global evaluation through intermediary variables usually. A toy illustration of this kind of approach is given in Figure 1 which summarizes the dependency structure of a set of rules which relate the mentioned attributes. In the example of Figure 1, one or several rules relate the values of the indicated attributes (through their location w.r.t. to partitions of attribute domains) to the values of more abstract concepts. This example is dealt with in Dubois and Prade (1988). A basic issue in such an approach is the appropriate modelling of the fuzzy rules which are used for describing the evaluation or the classification process. Different kinds of rules may be encountered (Dubois and Prade, 1996a). Namely
– gradual rules: which are of the form "the more X is A, the more Y is B"; – certainty rules: which are of the form "the more X is A, the more certain Y is in B"; – possibility rules: which are of the form "the more X is A, the more possible the values in B for Y". Note that rules with conclusions pervaded with uncertainty may appear in the description of the evaluation process, as well as gradual rules which provide a rough qualitative summarization of the relationships between variables. As it may be suggested by the example of Figure 1, the borderline between a multicriteria and a rule-based view of an evaluation process is rather vaguely located. This state of fact was already appearing in maybe, the very first example of fuzzy evaluation provided by Zadeh (1973) (see Figure 2), where he described the concept 'oval' through a set of fuzzy instructions, with fuzzy conditional statements very similar to fuzzy rules. But, in his final analysis of the algorithm, he pointed out that it is "approximately equivalent to" a conjunctive aggregation of fuzzy criteria. A detailed discussion of the relative merits of the multiple-criteria view vs. the rule-based view can be found in (Dubois and Prade, 1994a). The rule-based approach is however clearly better if explanations have to be given to the user about the obtained evaluations. Besides, note that fuzzy subjective evaluation algorithms, apart from fuzzy it-then rules, may also include some fuzzy analytical statement of a function using fuzzy arithmetic operations, whose application may also be fuzzily conditioned (e.g., "if x and y are close enough, make their average", where x and y may be fuzzy valued). Quality of the applicant Special skills
Basic skills
Education
Test scores IQ test
Work experience
Manual skills
Artistic skills
Sports skills
Examples of rules (without the assessment of the uncertainty of the conclusion):
French test Oral test
Adaptability
Written test
• If she/he has received an appropriate education and her/his tests scores is acceptable then she/he definitely has the basic skills for the job • If her/his score in the IQ test is approximately between 10 and 14 and her/his score in the French test is approximately between 6 and 9 then her/his tests scores are considered as rather poor
• If her/his score in the IQ test is approximately greater than 10 and 14 and her/his score in the French test is approximately between 6 and 9 then her/his tests scores are considered as good • If her/his score in the oral test is approximately between 6 and 9 and her/his score in the written French test is acceptable then her/his score in the French test is considered as rather poor
• If she/he has some special skills then it may increase her/his qualification for the job • If she/he has artistic skills then she/he has some special skills
Figure 1: Example of a rule-based evaluation 4. CONCLUDING REMARKS A last worth noticing way of describing an evaluation process is through prototypical examples. Then, the evaluation of a case under consideration can be made by means of a fuzzy case-based reasoning machinery, as, e.g. the one presented in (Dubois et al., 1997). Such an inference machinery can be based on a meta-principle expressing that the more similar two situations are, the more (possible) the similarity of the corresponding classification (or estimates). This metaprinciple, represented by means of gradual or possibility rules, is the basis for the implementation of an interpolation mechanism yielding the evaluation of the current case. This general discussion suggests the interest of making a careful study of the merits of the different ways of describing an evaluation (or classification) process (criteria-based, rule-based, case-based, and neural net-based methodologies) from the point of view – of the assumptions pertaining to the information required by each methodology; – of the elicitation and validation of the knowledge; – of the expressivity; – of the explanation capabilities;
– of the computationality; – of the joint use of several methodologies. Lastly, it should be pointed out that subjective evaluation problems are different from data fusion problems. In both situations, pieces of information have to be aggregated. However, in data fusion, information comes from multiple sources and we are more interested in properly sorting the information, taking into account the uncertainty and getting rid of conflicts between sources (see Dubois and Prade (1994b) for an approach to data fusion in the fuzzy setting). In subjective evaluation, the problem is often one of trading off between several points of view. However, the output of data fusion procedures may sometime provide input information for subjective evaluation procedures (when providing the imprecise evaluation of some attribute, for instance). "As a very simple example of a fuzzy definitional algorithm, we shall consider the fuzzy concept oval. It should be emphasized again that the oversimplified definition that will be given is intended only for illustrative purposes and has no pretense at being an accurate definition of the concept oval. The instructions comprising the algorithm OVAL are listed here. The symbol T in these instructions stands for the object under test. The term CALL CONVEX represents a call on a subalgorithm labeled CONVEX, which is a definitional algorithm for testing whether or not T is convex. An instruction of the form IF A THEN B should be interpreted as IF A THEN B ELSE go to next instruction. Algorithm OVAL: 1) 2) 3) 4) 5) 6)
IF T is not closed THEN T is not oval; stop IF T is self-intersecting THEN T is not oval; stop IF T is not CALL CONVEX THEN T is not oval; stop IF T does not have two more or less orthogonal axes of symmetry THEN T is not oval; stop IF the major axis of T is not much longer than the minor axis THEN T is not oval; stop T is oval; stop.
Subalgorithm CONVEX: Basically, this subalgorithm involves a check on whether the curvature of T at each point maintains the same sign as one moves along T in some initially chosen direction. [The detailed algorithm is not reproduced here.] In this connection, it should be noted that […] the algorithm OVAL as stated is approximately equivalent to the expression oval = closed ∩ non-self-intersecting ∩ convex ∩ more or less orthogonal axes of symmetry ∩ major axis much larger than minor axis
(*)
which defines the fuzzy set oval as the intersection of the fuzzy and nonfuzzy sets whose labels appear on the right-hand side of (*). However, one significant difference is that the algorithm not only defines the right-hand side of (*), but also specifies the order in which the computations implicit in (*) are to be performed."
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