Handbook of Granular Computing

The arithmetical and topological structures of fuzzy numbers have been developed in the 1980s and this enabled to design the elements of fuzzy calculus (see [6, 7]); Dubois and Prade stated the exact analytical fuzzy mathematics and introduced the well-known LR model and the corresponding formulas for the fuzzy operations. For the basic concepts see, e.g., [8–12]. More recently, the literature on fuzzy numbers has grown in terms of contributions to fuzzy arithmetic operations and to the use of simple formulas to approximate them; an extensive recent survey and bibliography on fuzzy intervals is in [13]. Zadeh’s extension principle (with some generalizations) plays a very important role in fuzzy set theory as it is a quite natural and reasonable principle to extend the operators and the mapping from classical set theory, as well as its structures and properties, into the operators and the mappings in fuzzy set theory ([14, 15]). In general, the arithmetic operations on fuzzy numbers can be approached either by the direct use of the membership function (by Zadeh’s extension principle) or by the equivalent use of the α-cuts representation. The arithmetic operations and more general fuzzy calculations are natural when dealing with fuzzy reasoning and systems, where variables and information are described by fuzzy numbers and sets; in particular, procedures and algorithms have to take into account the existing dependencies (and constraints) relating all the operands involved and their meaning. The essential uncertainties are generally modeled in the preliminary definitions of the variables, but it is very important to pay great attention to how they propagate during the calculations. A solid result in fuzzy theory and practice is that calculations cannot be performed by using the same rules as in arithmetic with real numbers and in fact fuzzy calculus will not always satisfy the same properties (e.g., distributivity, invertibility, and others). If not performed by taking into account existing dependencies between the data, fuzzy calculations will produce excessive propagation of initial uncertainties (see [16–19]). As we will see, the application of Zadeh’s extension principle to the calculation of fuzzy expressions requires to solve simultaneously global (constrained) minimization and maximization problems and they have typically a combinatorial structure; the task is not easy, except for particular cases. For this reason, general algorithms have been proposed (the vertex method and its variants) but also specific methods based on the exploitation of the problem at hand to produce exact solutions or generate approximated subproblems to be solved more efficiently than the original ones. By the α-cuts approach, it is possible to define a parametric representation of fuzzy numbers that allow a large variety of possible shapes and is very simple to implement, with the advantage of obtaining a much wider family of fuzzy numbers than for standard LR model (see [20–22]). This representation has the relevant advantage of being applied to the same [0, 1] interval for all the fuzzy numbers involved in the computations. In many fields of different sciences (physics, engineering, economics, social, and political sciences) and disciplines, where fuzzy sets and fuzzy logic are applied (e.g., approximate reasoning, image processing, fuzzy systems modeling and control, fuzzy decision making, statistics, operations research and optimization, computational engineering, artificial intelligence, and fuzzy finance and business) fuzzy numbers and arithmetic play a central role and are frequently and increasingly the main instruments (see [1, 9, 11, 12, 17, 19, 23, 24]). A significant research activity has been devoted to the approximation of fuzzy numbers and fuzzy arithmetic operations, by following essentially two approaches: the first is based on approximating the non-linearities introduced by the operations, e.g., multiplication and division (see [20, 21] and references therein); the other consists in producing trapezoidal (linear) approximations based on the minimization of appropriate distance measures to obtain preservation of desired elements like expected intervals, values, ambiguities, correlation, and properties such as ordering, invariancy to translation, and scale transformation (see [25–29]). An advantage of the second approach is that, in general, the shape representations are simplified, but possibly uncontrolled errors are introduced by forcing linearization; on the other hand, the first approach has the advantage of better approximating the shape of the fuzzy numbers and this allows in most cases to control and reduce the errors but with a computational cost associated with the handling of non-linearities. A difficulty in the adoption of fuzzy modeling is related to the fact that, from a mathematical and a practical view, fuzzy numbers do not have the same algebraic properties common to the algebra of real numbers (e.g., a group algebraic structure) as, for example, the lack of inverses in fuzzy arithmetic (see [30]). It follows that modeling fuzzy numbers and performing fuzzy calculations has many facets and possible solutions have to balance simple representations and approximated calculations with a sufficient control in error propagation. The organization of the chapter is the following: Section 12.2 contains an introduction to the fuzzy numbers in the unidimensional and multidimensional cases; Section 12.3 introduces some simple and flexible representations of the fuzzy numbers, based on shape-function modeling; in Section 12.4 the fundamental elements of the fuzzy operations and calculus are given; in Sections 12.5 and 12.6 we describe the procedures and detail some algorithms for the fuzzy arithmetic operations; and in Section 12.7 we illustrate some extensions to fuzzy mathematics (integration and differentiation of fuzzy-valued functions, fuzzy differential equations). The final Section 12.8 contains a brief account of recent applications and some concluding remarks.