Extended gradual interval (EGI) arithmetic and its application to gradual weighted averages

Abstract

We combine the concepts of gradual numbers and Kaucher arithmetic on extended intervals to define extended gradual interval (EGI) arithmetic in which subtraction and division operators are the inverse operators of addition and multiplication, respectively. Use of the proposed EGI operators can lead to non-monotonic gradual intervals that are not fuzzy subsets and cannot be represented by fuzzy intervals. In this context and when fuzzy representation results are desired, an approximation strategy is proposed to determine the nearest fuzzy interval of the non-monotonic gradual interval obtained. This approximation is viewed as an interval regression problem according to an optimization procedure. The EGI operators are applied to the common fuzzy weighted average (FWA) leading to a gradual weighted average (GWA).

Introduction

In many practical situations, fuzzy numbers and fuzzy intervals [14], [20], [23], [42], [49], [64] are frequently used to represent uncertain information, especially in decision-making, control theory, and approximate reasoning problems [5], [6], [7], [13], [35], [47]. Fortin et al. introduced the concept of gradual numbers [24], which provide a new outlook on fuzzy intervals [21], [22]. A gradual number is defined by an assignment function that can represent the essence of graduality. A fuzzy interval can be represented as a pair of gradual numbers. These lower and upper bounds are called left and right profiles. This approach allows extension of standard interval computation methods to fuzzy intervals. Indeed, it is possible to exploit fuzzy arithmetic directly by using conventional interval analysis in a space of functions (gradual bounds) [19], [24].

More generally, a real interval becomes a gradual interval when its bounds are gradual numbers. It is then represented by the ordered pair corresponding to its two gradual bounds. Moreover, in a gradual context, no monotonicity assumption is imposed on the gradual interval boundaries [19], [24], [76]. Thus, the concept of a gradual interval is more general than that of a fuzzy interval. More details on gradual numbers and their relationships with fuzzy intervals [19], [24] and an extensive survey and bibliography on fuzzy intervals [20] can be found in the literature.

Extending the usual arithmetic operations on real numbers to intervals is not a new problem for either the crisp case [60], [61], [73] or the fuzzy case [18], [58], [59], [80]. Zadeh's extension principle is computationally expensive owing to the requirement to solve a nonlinear programming problem [17], [65]. Considering fuzzy numbers or intervals as a collection of α-cuts is a conventional approach [4], [26], [27], [29], [41].

Fuzzy and interval arithmetic operations give results that are usually too large or imprecise [42], [43], [60], [61]. Indeed, conventional interval arithmetic operations may produce some counterintuitive results in which addition and subtraction (multiplication and division) are not inverse operations. This problem has been of long interest. In this context, additional constraints can be added to variables to obtain exact inverse operators [42], [53], [66]. For instance, in [42], [66] requisite constraints are proposed. The idea consists in doing arithmetic with constraints dictated by the context of the problem. In practice, constraints are achieved by assuming that the α-cuts from two variables are the same. The approach is efficient since it avoids overestimation due to the occurrence of interactive variables. However, the calculus of fuzzy quantities is still pessimistic about precision.

An interesting constraint method for fuzzy arithmetic was proposed to obtain inverse operations [53]. In this method, constraint interval arithmetic where an interval is viewed as a single-valued function that ranges in $[0,1]$ is performed on each α-cut. The results obtained using this technique [53] are strictly equivalent to those given by gradual arithmetic, whereby the arithmetic on functions is used on the left and right profiles [24]. Stefanini proposed exact subtraction and division operators (named gH-difference and g-division) between fuzzy intervals according to a generalization of the Hukuhara difference [72]. Another approach was developed in the 1970s by Ortlof and then Kaucher. They proposed a new type of interval arithmetic in which the order condition between the lower and upper bounds of a conventional interval is no longer required [40]. This is known as Kaucher arithmetic or extended interval arithmetic. Since then, it has been investigated by several authors either theoretically or at the application level [16], [28], [56], [57], [67], [68], [77].

In this study we merge the gradual interval approach and Kaucher arithmetic. This leads to extended gradual intervals (EGIs) [5], [8] and associated arithmetic operators that provide a means for computing exact inverse operations.

In this framework, regardless of the method used, inverse computations can lead to non-gradual intervals for which the interval boundaries are not well ordered. Thus, if the proposed EGI arithmetic gives results strictly equivalent to those obtained by constraint fuzzy arithmetic [53] or the use of exact operators [72], it yields compact existence conditions for obtaining proper gradual interval (PGI) results. The proposed method is used in a purely gradual context, so it is possible to check a priori if the operations obtained give a PGI or not. These conditions are very useful in many engineering applications such as model regression inversion, diagnosis problems, and inverse controller design [5], [6] for which real implementations have to be developed. Computed EGI results can sometimes be unrealistic, for example, when improper intervals are obtained. An analogy can be made with complex numbers when they are used for computation but real numbers are required for the result. Therefore, an important issue is to find conditions that can guarantee that the result obtained belongs to the PGI set. Thus, for many practical situations an existence condition is required to determine admissible trajectories for the inverse problem a priori. For instance, when considering a linear or nonlinear system, from an interval control point of view, being able to express conditions to obtain PGI results leads to the answer to the following question: is it possible to determine sufficient conditions to guarantee the existence of the inverse controller? In other words, what are the desired trajectories that can be accepted by the controlled system?

It is advisable then to study and analyze a posteriori if the gradual results obtained are interpretable as fuzzy intervals. In this case, only monotonic gradual numbers are useful in defining fuzzy intervals [24]. However, some fuzzy interval computations can lead to non-monotonic gradual intervals that are not fuzzy subsets and cannot be represented by fuzzy intervals since the interval boundaries are not monotonic [76]. In this context and when a fuzzy interval is not reached, an approximation strategy can be used to determine the nearest parametric fuzzy representation. Many parametric approximation methods have been proposed in the literature. The main objective of these approximations is to obtain from a given fuzzy interval some simpler shapes that are easy to handle and have natural interpretations. In this context, trapezoidal and triangular fuzzy intervals are the shapes most used in current applications [12], [32], [33], [55]. For instance, Ma et al. proposed a symmetric triangular approximation [55]. Chanas used the Hamming distance to derive the nearest approximation [12]. Grzegorzewski and Mrowka described a trapezoidal approximation of fuzzy numbers in which they derive an interesting and reasonable compromise between loss of information and a sophisticated approximation form [32]. Grzegorzewski also proposed algorithms for a trapezoidal approximation that preserve the expected interval value [33]. It is clear that the approximation chosen may lead to loss of information about fuzziness. To overcome this problem, most of the proposed methods rely on minimization of a measure of distance between the original fuzzy interval and its nearest approximation [31], [63]. Stefanini also presented a fuzzy approximation of the gH-difference [72]. Our proposed approach is inspired by previous studies where the objective is to identify the nearest fuzzy approximation of a non-monotonic gradual interval by minimizing a performance criterion. In this context, fuzzy interval identification is viewed as an interval regression problem for input–output data [3], [9], [15], [74], [75]. From a practical point of view and to illustrate the gradual computation and approximation concept, we use the common fuzzy weighted average (FWA) [11], [17], [25], [38], [46], [52].

We only view the weighted average as a potential application of the proposed gradual operators. Motivation for our weighted average choice is threefold. Our aims were:

• To compare the performance obtained with recent FWA results reported by Liu et al. [52];

• To propose a gradual weighted average (GWA), which can be viewed as an optimistic counterpart to the FWA according to Kaucher arithmetics; and

• To compute the weighted average using a sequence of elementary EGI operations between gradual intervals in which the idempotence property of the weighted average is conserved.

In Section 2 we introduce endpoint (EP) and midpoint–radius (MR) representations. Section 3 describes gradual intervals in detail. Section 4 presents the arithmetic operations associated with PGIs and EGIs. We apply the GWA and its approximation methodology in Section 5. Simulation examples are detailed in Section 6. We present concluding remarks in Section 7.