Performing Arithmetic Operations Over the (L–R)-Type Fuzzy Numbers

The issue of constructing a system of rules to perform binary operations over fuzzy numbers has been formulated and considered. The set problem has been solved regarding the (L–R)-type fuzzy numbers with a compact carrier. Such a problem statement is predetermined by the simplicity of the analytical notation of these numbers, thereby making it possible to unambiguously set a fuzzy number by a set of values of its parameters. This makes it possible, as regards the (L–R)-type numbers, to reduce the desired execution rules for fuzzy numbers to the rules for simple arithmetic operations over their parameters. It has been established that many cited works provide ratios that describe the rules for performing operations over the (L–R)-type fuzzy numbers that contain errors. In addition, there is no justification for these rules in all cases.<br><br>In order to build a correct system of fuzzy arithmetic rules, a set of metarules has been proposed, which determine the principles of construction and the structure of rules for operation execution. Using this set of metarules has enabled the development and description of the system of rules for performing basic arithmetic operations (addition, subtraction, multiplication, division). In this case, different rules are given for the multiplication and division rules, depending on the position of the number carriers involved in the operation, relative to zero. The proposed rule system makes it possible to correctly solve many practical problems whose raw data are not clearly defined. This system of rules for fuzzy numbers with a compact carrier has been expanded to the case involving a non-finite carrier. The relevant approach has been implemented by a two-step procedure. The advantages and drawbacks of this approach have been identified.


Introduction
Practical tasks related to the analysis and synthesis of systems, decision theory, management theory, are resolved under conditions of uncertainty. Methods from a probability theory or a fuzzy set theory are used to describe this uncertainty. Probability theory is a strict axiomatic theory. However, it is this circumstance that limits the scope of its application for many practical problems. The fact is that the fundamental concepts of this theory, such as the distribution of random probabilities, can be strictly defined if the mechanism of forming these random quantities does not change over multiple observations. In reality, the conditions for the formation of observed quantities can vary significantly, resulting in the inadequacy of the results derived from them. Less demanding is the description of uncertainty in terms of fuzzy set theory [1]. This theory is a step towards bringing together the impeccable accuracy of classical mathematics and the pervasive inaccuracy of the real world [2].

Literature review and problem statement
The objects of fuzzy mathematics are fuzzy numbers. As in any mathematical theory, the success of solving practical problems in terms of fuzzy mathematics is determined by the correctness of the rules for performing operations over the objects of this theory. These rules are introduced as follows [3,4].
Introduce the * symbol of an arbitrary binary arithmetic queue (addition, subtraction, multiplication, division). Any such operation assigns the numbers x 1 and x 2 a certain result z. We also introduce the «reverse» operation ⊗, which assigns its second element to the result of composition z and one of its elements (for example, x 1 ) its second element. Set the variants for a binary arithmetic operation: It is clear that the difficulty of performing computational operations in accordance with (8) depends on the form of the membership functions of the elements of composition x 1 and x 2 , in specific cases, it can be unacceptably high. Therefore, many works [9][10][11][12] consider simplifying the procedure for arithmetic operations on fuzzy numbers. In this case, fuzzy numbers of different types are proposed to be used as operands: sigmoidal [12], interval [13], trapezoidal [14,15] numbers. However, the low adequacy of these variants for describing uncertainty has led to the development of a more informative unified description of the membership functions of fuzzy numbers in the form of so-called (L-R)-type functions [1][2][3][4][5][6][7][8]. The membership function of such a fuzzy number x takes the form: where m is the mode of a fuzzy number x, α, β are the left and right coefficients of fuzziness α β > > ( ) the arbitrary functions that are not ascending over a set of non-negative numbers, and L R 0 0 1 ( )= ( )= .
According to (9), the (L-R)-type fuzzy number can be unequivocally defined by the three parameters < > m, , . α β Note that the asymmetric nature of the membership functions of the (L-R)-type fuzzy numbers provides a possibility of obtaining good-quality approximations for a wide class of specific membership functions. In this case, a universal and extremely simple way of formally describing the membership functions of the (L-R)-type numbers initiated the development of a specific technology for performing the simplest of operations (addition, subtraction, multiplication, division) over these numbers using only the parameters of these numbers. As a result, a large number of works have emerged that each has its own rules for performing operations over the (L-R)-type fuzzy numbers. Significant differences among them are due to the need to analyze known results and to construct a unified approach to the formation of the justified set of rules for implementing arithmetic operations over the (L-R)-type numbers. The relevant results are given below.
Let the (L-R)-type fuzzy numbers x 1 and x 2 be set by the membership functions: Thus, these two fuzzy numbers are defined by sets x m In accordance with this, the fuzzy number z, resulting from the operation z x x = * 1 2 , is described by the set < > m, , , α β where the numbers m, α, β are determined by the rules corresponding to the * operation. Consider the known rules for performing arithmetic operations (addition, subtraction, multiplication, and division). A large number of publications provide their own rules for performing operations on fuzzy numbers. We shall choose some of the most cited ones from those papers, containing the main results related to this issue.
Studies [16,17] propose the following set of rules for performing operations over the (L-R) fuzzy numbers.
Subtraction z x x = − 1 2 . Multiplication Ratio (13) (a subtraction operation) is wrong. It is clear that each fuzzy coefficient of the number z that sets the parameters for membership function (9) should depend simultaneously on the degree of blurriness in both the reduced and subtracted. Ratio (14) also produces an inaccurate result. For simple reasons, it is clear that the values of the fuzzy coefficients α and β in the result of the multiplication of fuzzy numbers should be higher than the fuzzy coefficients of the efficients. However, if the modal values of the efficients are small, the fuzzy result obtained according to (14) may be inexplicably small. In particular, if the modal values m 1 and m 2 are close to zero, then the result, even with the high vagueness of the efficients, would be an almost clear number, which, of course, is not the case.
Work [18] reports the following rules for operation execution. The rule of addition repeats (12). Subtraction The proposed rule (15) of the subtraction operation makes the erroneous ratio (13) absurd. According to this formula, the blurring level of the result of the subtraction for any two blurred, to any degree (but almost equally), reduced and subtracted would be almost zero, which contradicts common sense.
In work [19], the addition rule repeats (12). Division Rule (17) makes an unsuccessful attempt to improve the consideration of importance of the efficients blurriness. As is shown below, the reported formula is not accurate. The proposed ratio (18) to describe the results of division is also not accurate. The denominator of the third component in formula (18) should include a parameter that describes the fuzziness of the divider rather than the divisible in fraction x x 1 2 .
In the set of rules for the execution of operations, proposed in [20,21], the rules of addition and subtraction repeat the previous (12) and (16). Multiplication Division Ratio (19) repeats error (14). Rule (20) for the division of fuzzy numbers, as is shown below, contains inaccuracies in determining the left and right fuzzy coefficients.
Thus, a brief analysis of the known sets of rules for performing arithmetic operations over the (L-R)-type fuzzy numbers suggests that there is not any substantiated system of rules for operating on such numbers, which requires continued research.

The aim and objectives of the study
The aim of this study is to develop a sound system of fuzzy arithmetic rules.
To accomplish the aim, the following tasks have been set: -to devise a unified, scientifically-substantiated approach to the wording of rules for performing arithmetic operations on the (L-R)-type fuzzy numbers; -to implement the devised approach and to form with its use a system of rules for the execution of fuzzy arithmetic operations.

Rules for performing operations over the (L-R)-type fuzzy numbers
The desired approach to the formation of a system of rules for performing arithmetic operations over the (L-R)-type fuzzy numbers should be simple, naturally interpreted, and general in character, regardless of the type of operation. In accordance with this, we shall introduce a set of metarules for forming the rules for the execution of operations over the (L-R)-type fuzzy numbers.
The proposed metarules for compiling the rules for performing operations over the (L-R)-type fuzzy numbers structurally employ their following fundamental feature: the membership functions of such numbers are clearly determined by the numerical values of their parameters (a mode and the fuzzy coefficients). The rules for performing operations over the (L-R)-type fuzzy numbers should define algebra, that is, meet the generally accepted requirement: for any pair of fuzzy numbers from the (L-R)-type numbers class, no operation involving these numbers should exclude the operation result from this class. Let M LR be a set of the (L-R)-type fuzzy numbers with a finite carrier and a pair of fuzzy numbers x 1 and x 2 belong to this set, that is, x M LR 1 ∈ , x M LR 2 ∈ . Introduce *, an arbitrary operation on fuzzy numbers belonging to M LR . Then, the fuzzy number z, obtained as a result of this operation, is equal to describe in the standard way: x m Taking into consideration (21), (22), we shall introduce the following metarules for the formation of fuzzy arithmetic rules. Thus, the introduced metarules 1-3 clearly set a formal procedure for calculating the parameters of the membership function of the fuzzy number resulting from performing arithmetic operations over the (L-R)-type fuzzy numbers. We shall use these metarules to consistently describe the rules for performing basic operations over the (L-R)-type fuzzy numbers with a finite carrier. Introduce x m Next, in line with (23) to (25), write down the rules for performing arithmetic operations over numbers x 1 and x 2 , accompanying them with explanatory drawings. Fig. 1

. The membership functions of operands
Addition z x x = + The rules for multiplication and division operations vary depending on the position of the x 1 and x 2 number carriers relative to zero. We shall give these rules for different variants of ratios between m 1 , α 1 , β 2 and m 2 , α 2 , β 2 . Variant 1.
Multiplication z x x = ⋅ The division operation in this variant is impossible as the carrier of the divider covers zero.
Thus, we have devised the system of rules for performing arithmetic operations over the (L-R)-type fuzzy numbers.

Discussion of results obtained in the development of the fuzzy number arithmetic rule system
This paper proposes a strictly based system of rules for performing arithmetic operations on fuzzy numbers. These rules, when necessary, take into consideration the location of the carriers of fuzzy arguments of the operation performed relative to zero. The proposed analytical ratios have been obtained to execute rules over the (L-R)-type fuzzy numbers with a compact carrier. For other cases, if, for example, the membership functions of the fuzzy arguments are Gaussian, then these rules would produce an approximate result. The system of rules for performing operations over fuzzy numbers with an infinite carrier, corresponding to such a situation, can be obtained as follows. Operations are performed in two stages. In the first stage, for each of the arguments of the implemented rule, their membership functions are cut on a certain set of levels. To this end, for the appropriate pair of the membership functions: . The corresponding operation is performed according to the rules introduced above. The described steps are performed for each of the d-levels. The undeniable advantage of the described approach is the possibility to restore the operation over the original fuzzy numbers at any predefined accuracy. The ob vious drawback is the laboriousness of obtaining the result, associated with the need to multiply solve the set of equations (38), not necessarily easily solved, which naturally limits the scope of application of this approach.
Possible areas of further research include the development of an expanded system of rules for any binary algebraic operations in order to create a formal basis for problem solving, analysis, and fuzzy system optimization under uncertain conditions [22,23]; the construction of rules for performing operations over fuzzy numbers of the second type.

1.
We have devised a scientifically-substantiated approach to form a system of rules of fuzzy numbers arithmetic whose axiomatic basis is a set of metarules.
2. Using this approach, a strictly based system of rules has been built for performing arithmetic operations over the (L-R)-type fuzzy numbers with a compact carrier. The proposed rule system has been expanded to perform operations on fuzzy numbers with a non-finite carrier. In this case, the initial problem is reduced to a set of simple problems with a compact carrier.