Execution of Arithmetic Operations Involving the Second-Order Fuzzy Numbers

The need to improve the adequacy of conventional models of the source data uncertainty in order to undertake research using fuzzy mathematics methods has led to the development of natural improvement in the analytical description of the fuzzy numbers' membership functions. Given this, in particular, in order to describe the membership functions of the three-parametric fuzzy numbers of the (L-R)-type, the modification implies the following. It is accepted that these functions' parameters (a modal value, the left and right fuzzy factors) are not set clearly by their membership functions. The numbers obtained in this way are termed the second-order fuzzy numbers (bi-fuzzy). The issue, in this case, is that there are no rules for operating on such fuzzy numbers. This paper has proposed and substantiated a system of operating rules for a widely used and effective class of fuzzy numbers of the (L-R)-type whose membership functions' parameters are not clearly defined. These rules have been built as a result of the generalization of known rules for operating on regular fuzzy numbers. We have derived analytical ratios to compute the numerical values of the membership functions of the fuzzy results from executing arithmetic operations (addition, subtraction, multiplication, division) over the second-order fuzzy numbers. It is noted that the resulting system of rules is generalized for the case when the numbers-operands' fuzziness order exceeds the second order. The examples of operations execution over the second-order fuzzy numbers of the (L-R)-type have been given.


Introduction
The bi-fuzzy numbers (the second-order fuzzy numbers) were introduced in [1] as a natural generalization of the regular fuzzy numbers. The fundamental novelty of the technology that formally defines these numbers is that the parameters of their membership functions are themselves fuzzy with their own membership functions. An important positive feature of the emerging complexity of the structure of mathematical notation of fuzzy numbers is the arising possibility to radically improve the adequacy of the uncertainty models of objects in the external world and within their functioning environment. The resulting formalism significantly expands the space of the mathematical models of real-world objects operating under conditions of hierarchical uncertainty. The two-stage character of the analytical description of the membership functions' parameters of the second-order fuzzy numbers makes it possible to more accurately and strictly determine the real inaccuracy of the conventional descriptions of objects and their functioning environment. However, the practical usefulness and effectiveness of using this new field of fuzzy mathematics are largely limited by the lack of the algebra of operations on the corresponding fuzzy numbers, which requires further research.

Literature review and problem statement
Following the pioneering work [1], papers [2,3], which addressed the theory of fuzzy sets, focused on the prospect of expanding the fuzzy mathematics arsenal by using the sec-ond-order fuzzy numbers and the relevance of the study in this area. The issues of building the second-order fuzzy numbers' membership functions and the need to form the axiomatics for these numbers are discussed in [4,5]. Papers [5,6] provide examples of building the second-order fuzzy numbers' membership functions and discuss the emerging opportunities to improve the adequacy of system models under the conditions of uncertainty. Studies [7][8][9] address the use of the second-order fuzzy numbers in solving fuzzy logic problems. Contemporary works [10,11] discuss the issues associated with the possible application of the second-order fuzzy numbers in regression analysis. In recent years, papers [12][13][14][15] have explored the possibilities of practical use of second-order fuzzy numbers with an interval membership function. The authors note the simplicity to form appropriate membership functions, but it is noted that the models that arise are not adequate enough. In studies [16,17], the application area of interval numbers expands toward the development of models for the non-linear management tasks.
Thus, the following important first steps have been taken in known studies on the second-order fuzzy numbers. The idea of constructing fuzzy numbers above the first order has been formulated, and the methods of building the second-order fuzzy numbers' membership functions have been considered. Practical examples of numerous problems have been given, in which the use of these numbers significantly improves the adequacy of mathematical models. However, these problems are solved at the simplest possible interval level of the description of the membership function parameters. It is clear that the practical usefulness of the results obtained in this case is very relative.
A brief analysis of known works has made it possible to outline a problematic issuethe lack of general rules for operations involving the second-order fuzzy numbers. A solution to this problem in its most general form is hardly feasible. Below, this problem is solved for an important particular case where fuzzy numbers have a membership function of the (L-R)-type.
This choice is clear: the membership functions of such numbers are similarly and clearly defined by a set of their numerical characteristics. In particular, the membership function of a three-parametric number of the (L-R)-type is assigned by the set <m, α, β>, where m is the modal value of a fuzzy number, α and β are, respectively, the left and right fuzziness factors. If this number is bi-fuzzy, the parameters m, α, β are the fuzzy numbers, defined by the following sets: This natural and easy-to-interpret notation is hereafter used in the development of rules for the execution of operations on the second-order fuzzy numbers of the (L-R)-type.

The aim and objectives of the study
The aim of this study is to build a sound system of rules for the execution of arithmetic operations involving the second-order fuzzy numbers. Devising these rules would make it possible to move from a decorative description of problem statements to actually solve them in order to obtain specific results under the conditions of clearly defined source data. In this case, the computational solving process is implemented by executing a clearly described sequence of arithmetic operations.
To accomplish the aim, the following tasks have been set: -to build a system for forming the rules of executing arithmetic operations involving the second-order fuzzy numbers; to define the formula ratios for computing the parameters of the membership functions of the results from executing binary arithmetic operations involving the second-order fuzzy numbers.

Building a system of rules for executing arithmetic operations involving the second-order fuzzy numbers of the (L-R)-type
Let the membership functions of two fuzzy numbers of the (L-R)-type be set by the threesomes of their parameters' values: Introduce a binary operation 1 2 , , .
In accordance with the rules from [18] for the arithmetic of fuzzy numbers of the (L-R)-type, the formulae for calculating the parameters of the membership function of the fuzzy result of main operations take the following form: In these ratios, the parameters of the membership functions of the results of arithmetic operations are found from the computations based on formulae (1) to (4) involving regular numbers. The character of these ratios allows them to be summarized in the following way. If, according to the idea from [1], the fuzzy numbers' membership function parameters are the fuzzy numbers themselves, this can be used constructively. Assume that the parameters m 1 , α 1 , β 1 , m 2 , α 2 , β 2 of the membership functions of the fuzzy numbers x 1 and x 2 are the fuzzy numbers themselves with membership functions defined by the following sets Then the parameters of computation results according to (1) to (4) involving the fuzzy numbers described in (5), (6) become fuzzy numbers, and the results of these operations would determine the second-order fuzzy numbers. Thus, a system is proposed that forms the rules for the execution of arithmetic operations on fuzzy numbers whose order is higher than the first order. In this case, the parameters of the fuzzy numbers' membership functions of the assigned order in general ratios (1) to (4) are used to calculate the parameters of the membership functions of the next, higher order.
Calculate the parameters of the fuzzy numbers' membership functions resulting from the execution of x 1 +x 2 , x 1 -x 2 , x 1 •x 2 , x 1 /x 2 , operations by using (1) to (6). Define the following system of rules for the execution of arithmetic operations involving the bi-fuzzy numbers of the (L-R)-type.
We shall perform the appropriate operations.
The resulting ratios (7) assign the membership functions of the parameters m, α, β for the result of adding the fuzzy numbers x 1 and x 2 . The parameters of these membership functions are defined through the parameters of the membership functions of the fuzzy terms 1 set by ratios (5), (6). Thus, the result of the addition of two bi-fuzzy numbers x 1 and x 2 is also a bi-fuzzy number. Subtraction.
In accordance with (2), to calculate the parameters of the membership function of the fuzzy result from executing a subtraction operation, one needs to find the parameters of the membership functions of the fuzzy numbers m 1 -m 2 , α 1 +β 2 , β 1 +α 2 .
Perform the appropriate operations.

Multiplication.
Using (3), compute the parameters of the membership function of the fuzzy number x 1 x 2 taking into consideration the fuzziness of the parameters of the membership function of each of the multipliers defined from (5) and (6). To this end, one needs to consistently calculate the membership functions of the following products of the fuzzy numbers Perform the appropriate operations. Introduce Next, according to (3), one needs to find the parameters of the membership functions of the following fuzzy numbers: Thus, formulae (11), (16), (17) set parameters for the membership functions of the result of multiplying two bifuzzy numbers.
We shall compute the parameters of the membership function of the bi-fuzzy result from the addition Z=x 1 +x 2 in line with formulae (7).

Subtraction.
We shall compute the parameters of the membership function of the bi-fuzzy result from the subtraction Z=x 1 -x 2 in line with formula (8).    Solving is over. Thus, the results of the execution of arithmetic operations involving the second-order fuzzy numbers are again the fuzzy numbers of the same order.

Discussion of results of building a system of the rules for executing operations involving the bi-fuzzy numbers of the (L-R)-type
The natural generalization of the canonical definition of fuzzy numbers, which led to the emergence of the second-order fuzzy numbers, gave rise to the issue associated with the absence of algebra for the bi-fuzzy numbers. In this case, when studying the real objects of the physical world, only a superficial, qualitative analysis of them is at hand. Moreover, even if the object under study can be quantified, the absence of algebra for the fuzzy numbers whose values set the state of the object excludes the possibility to manage that state. This paper has examined the simplest part of this problemthe algebra of arithmetic operations involving bi-fuzzy numbers of the (L-R)-type. The choice of this very field of the algebra of bi-fuzzy numbers is associated with wide and effective use of them in solving many practical problems. This is a consequence of the simplicity of the (L-R)-type number description as a set of the values of their membership functions' parameters. We have considered a case where the membership functions' parameters of the fuzzy numbers of the (L-R) type are themselves set by fuzzy numbers of the (L-R)-type. This work has proposed the rules for executing the arithmetic operations involving the resulting second-order fuzzy numbers. The formulae have been derived to compute the parameters of the membership function of the results of the execution of addition (7), subtraction (8), multiplication (11), (16), (17), and division (18), (25), (26) operations involving the bifuzzy numbers of the (L-R)-type.
Note that the resulting system of rules for performing arithmetic operations involving numbers of the (L-R)-type is easy to summarize. Introduce the following definition. A fuzzy number of the n-th order denotes a fuzzy number whose membership function's parameters are the fuzzy numbers of the (n-1)-th order. According to this definition, a standard fuzzy number has a membership function whose parameters are standard numbers. Next, the fuzzy number of the second order has a membership function whose parameters are the fuzzy numbers of the first order; a fuzzy number of the third order has a membership function whose parameters are the second-order fuzzy numbers, etc. It is clear that the recurrent application of the proposed rules makes it possible to obtain an analytical description of the results of the execution of arithmetic operations involving fuzzy numbers of any order.
The possible areas of further research are related to the generalization of the proposed approach in order to build rules for the execution of other algebraic operations: raising a bi-fuzzy number to a bi-fuzzy power, taking the logarithm of a bi-fuzzy number on a bi-fuzzy base, etc. In addition, it is interesting to construct methods for solving specific problems (solving equations and systems of equations, optimization, etc.) for the case where the original data are bi-fuzzy, etc. The task of constructing the bi-fuzzy models of inaccurate numbers is also of great practical interest.

1.
A system of rules has been built for performing arithmetic operations (addition, subtraction, multiplication, division) for the fuzzy numbers of the (L-R)-type of the second order. This system is axiomatic and makes it possible to unambiguously obtain a complete and consistent set of rules for operating on fuzzy numbers whose order is higher than the first order.
2. The formulae have been proposed to compute the parameters of the membership functions of the results of executing arithmetic operations involving the second-order fuzzy numbers of the (L-R)-type. Using these rules makes it possible to solve in practice the numerous problems of analysis and synthesis of the systems for which the fuzziness of parameter setting exceeds the first order.