Gisella Facchinetti

Note on ranking fuzzy triangular numbers

This paper deals with the problem of ranking a set of alternatives, represented by triangular fuzzy numbers, in decision‐making situations. Three new methods are proposed, and a notion of preference between alternatives is suggested. A comparison with other methods is provided in the concluding table.

A characterization of a general class of ranking functions on triangular fuzzy numbers

This paper aims to the discussion of properties for ranking fuzzy numbers. An approach of axiomatic type is followed providing the definition of two groups of requirements which characterize a ranking function through its behaviour on triangular fuzzy numbers. Particularly, the problem of making the evaluations on triangular fuzzy numbers sensitive to their spread has been deeply analyzed.

Evaluations of fuzzy quantities

This paper deals with the problem of evaluating a fuzzy quantity. We call fuzzy quantity any non-normal and non-convex fuzzy set, defined as the union of two, or more, non-normal fuzzy numbers. In order to introduce either ranking or defuzzifing procedures, we propose a definition which arises from crisp set theory: it is based on a particular fuzzy number evaluation, weighted average value (WAV) that works on @a-cuts levels and depends on two parameters, a real number @l and an additive measure S; @l is connected with the optimistic or pessimistic point of view of the decision maker, S allows the decision maker to choose evaluations in particular subsets of the fuzzy number support, according to his preference.

Ranking Functions Induced by Weighted Average of Fuzzy Numbers

In this paper we present two definitions of possibilistic weighted average of fuzzy numbers, and by them we introduce two different rankings on the set of real fuzzy numbers. The two methods are dependent on several parameters. In the first case, the parameter is constant and the results generalize what Carlsson and Fuller have obtained in (2001). In the second case, the parameter is a function, not fixed a priori by the decision maker, but it depends on the position of the interval on the real axe. In all the two cases we call the parameter degree of risk, which takes into account of a risk-tendency or aversion of the decision maker.

A Fuzzy Expert System for Solving Real-Option Decision Processes

This paper presents a new approach to real options. The current options-based models have provided new insights into capital-budgeting decisions. Unfortunately they are not widely used by corporate managers and practitioners as they are formally complex, rather difficult to understand and rest on strong implicit assumptions that considerably limit their scope of application. We propose a possible alternative by using a fuzzy expert system, on the basis of Mastroleo, Facchinetti and Magni (2001). We draw up a decision tree with multiple uncertain variables affecting the value of an investment opportunity, consisting of a defer option, a growth option, an abandonment option. Some simulations are conducted to test the economic soundness of the model as well as its consistency with the current models in the literature. A rather refined study can be accomplished by showing how inputs and outputs of the model interrelate one another.

Vagueness evaluation of the crisp output in a fuzzy inference system

Fuzzy models generally provide an output characterized by vagueness, which is expressed through a solution fuzzy set. In many applications, the response of the model is transformed in a crisp value through some defuzzification methods for solution fuzzy region, thus losing its fuzziness. Only to preserve a few indications of its vagueness, some indices summarizing the spread of the output membership function could be used to associate them with the crisp output, such as its standard deviation, the quartile deviation, the coefficients of skewness and kurtosis. The behaviour of such indices is examined in a large number of possible, though unlikely, output solutions and in an application of a fuzzy inference system for evaluating university teaching activity. The results seem to suggest that the 20-80 mid-percentile range could be a good measure of the vagueness dispersion, while the coefficient of skewness could provide a useful indication about the asymmetry of the solutions shape. Moreover, a rough estimate of dispersion was obtained from a triangle approximating the solution fuzzy region because the results were straightforwardly deduced from formulae involving the abscissae of its vertices. The results generally appear to underestimate the true values of the standard deviations; the 15-85 mid-percentile range of the approximating triangle seemed to be a more suitable rough appraisal of fuzzy output dispersion.

International market selection for small firms: a fuzzy-based decision process

Purpose – This study aims to build and test an International Market Selection (IMS) decision process method that is able to capture, within a small firm’s risk-averse setting, the entrepreneurs experience, reduce cognitive biases, and preserve the flexibility of the decision, by combining the advantages of systematic and behavioural-based international market selection approaches. Design/methodology/approach – The unit of analysis is the IMS decision process of a small firm venturing abroad. We adopt a ranking approach based on three-step screening. We assess the markets through a multi-criteria approach with a wider set of variables aggregated within a tree-shaped model. To obtain the ranking, we use a Fuzzy Expert System (FES) as an evaluative tool. Findings – The results show that the proposed decision method is consistent with the entrepreneur’s strategic orientation and experience, while preserving the flexibility requested for decision-making in small firms. Unlike traditional behavioural IMS appro...

The Total Variation of Bounded Variation Functions to Evaluate and Rank Fuzzy Quantities

In this paper, we present a different approach to introduce evaluation and ranking of fuzzy quantities. These general fuzzy sets are obtained by the union of several fuzzy sets. They are neither normal nor convex. The idea we have followed is to use the total variation and the bounded variation function definitions applied to the membership function of a fuzzy set to introduce its evaluation. This approach has produced that the well‐known method of area compensation, introduced by Fortemps and Roubens only in a geometrical framework, is now presented in a general contest and useful for any fuzzy set. Moreover, this new representation formula provides an α‐cut view. This aspect, absent in Fortemps and Roubens paper, offers an evaluation by a weighted average of alfa‐cuts values, where the weights are connected with the number of subintervals that produce every α‐cut. Following the same idea, we have introduced the ambiguity definition of a general fuzzy set. By this new definition of evaluation and the consequent ambiguity, we present a way to rank fuzzy quantities.

Evaluation and interval approximation of fuzzy quantities

In this paper we present a general framework to face the problem of evaluate fuzzy quantities. A fuzzy quantity is a fuzzy set that may be non normal and/or non convex. This new formulation contains as particular cases the ones proposed by Fortemps and Roubens [7], Yager and Filev [12, 13] and follows a completely different approach. It starts with idea of “interval approximation of a fuzzy number” proposed, e.g., in [4, 8, 9].

A fuzzy approach to the geography of industrial districts

Building on Istat [5], Brusco and Paba [4] developed a mathematical algorithm which tries to identify the number and the importance of Italian industrial districts. Unfortunately, this algorithm is too rigid and it rules out some important small-firm, specialized local industries. To solve this problem, we propose a Fuzzy System. We obtain two main results. ( l ) With the fuzzy approach, most of the industrial districts identified in the current literature are still included, but it is now possible to recover some important local production systems ruled out by the crisp approach. (2) The score obtained allows us to rank, sector by sector, all the industrial districts according to their quantitative importance and to their adherence to the characteristics emphasized in the literature, whereas the crisp approach says only yes or not.