Arithmetic Operations and Expected Values of Regular Interval Type-2 Fuzzy Variables

: High computation complexity restricts the application prospects of the interval type-2 fuzzy variable (IT2-FV), despite its high degree of freedom in representing uncertainty. Thus, this paper studies the fuzzy operations for the regular symmetric triangular IT2-FVs (RSTIT2-FVs)—the simplest IT2-FVs having the greatest membership degrees of 1. Firstly, by deﬁning the medium of an RSTIT2-FV, its membership function, credibility distribution


Introduction
There are various types of uncertainty in real life (e.g., fluctuations in the stock market, the costs and demands in the suppliers' market, etc.), forcing us to choose reasonable mathematical tools to describe the research objects. Therefore, the ordinary fuzzy set theory proposed by Zadeh [1] in 1965 has "deepened" over the past five decades. Among the vast amount of theories(e.g., the fuzzy random sets [2], the twofold fuzzy sets [3], the bifuzzy sets [4], the fuzzy soft sets [5], the interval-valued picture hesitant fuzzy set [6]), the type-2 fuzzy set (T2-FS) is an important branch attracting constant attention due to its superiority in representing the uncertainty of the membership function (MF).
The MF, which measures the degree to which an FV is likely to take a certain value, is usually provided by experts, based on their experiences or experiments under the circumstances where there are a lack of data, or where semantic ambiguity exists. In the ordinary fuzzy set theory, the MF of an FV should be uniquely determined. In reality, however, different people have their own understanding of the same fuzzy concept; that is, the determination process of the MF is subjective. Then, how could one guarantee that subjective judgments lead to objective results? There is no mature or effective method yet. Consequently, in order to reflect the inner-and inter-individual differences, Zadeh [7] proposed the concept of T2-FS in 1975, the MF (named primary MF) of which is also an FV.
For a general T2-FV, the fuzziness of its secondary MFs (the MFs of the primary MF) brings about both high computational complexity and visualization difficulty [8]. Although numerous scholars have made delicate research on type-reduction approaches, including the Karnik-Mendel (KM) algorithm-based iterative methods [9], the statistical sampling based geometric strategies [10], and the possibility measure based methods [11][12][13], the applicability of the T2-FVs is seriously restricted. Up until now, the studies on the application of T2-FVs have mainly focused on fuzzy logic systems, programmings, and game theory [14][15][16][17][18]. Readers may refer to [19][20][21] for related papers on T2-FS type-reduction and its application areas.
To the best of our knowledge, however, the existing studies focus on algorithm-based type-reductions and defuzzification methods, whereas the research for fuzzy arithmetic operations of the IT2-FVs are relatively limited. The first work is by Lee and Chen [29] who defined three arithmetic operations between two IT2-FVs, whose UMF and LMF are both r-polygonal shape, named addition, subtraction, and multiplication. Since then, some scholars have followed Lee and Chen's work while applying multi-attribute decision making approaches [30], such as the analytic hierarchy process [31,32] and TOPSIS [33], to make ranking and evaluation decisions [34][35][36]. Besides, for the trapezoidal interval type-2 fuzzy numbers, some other operators, such as the geometric Bonferroni mean operator [37] and Heronian mean operators [38] were proposed. Aguero and Vargas [39] is another work focusing on the operations between two IT2-FVs, where the vertex method, based on α−cut, was suggested. Nevertheless, these operators involve complex operations themselves, and thus are inadequate at solving many realistic problems concerning more complex operations. Hence, there is an urgent need to put forward a brand-new and convenient operational law for fuzzy arithmetic operations of the IT2-FVs.
On the other hand, within the framework of the traditional fuzzy set theory, Zhou et al. [40] proposed an operational law for calculating the inverse credibility distribution of the strictly monotone functions between mutually independent type-1 FVs (T1-FVs) and verified that their method can greatly simplify the fuzzy arithmetical operations. Therefore, learning from their ideas, we attempt to extend the inverse credibility distribution-based operational law to the type-2 fuzzy set theory. To demonstrate the efficiency of the proposed method, the simplest and most special type of IT2-FVs, i.e., the regular symmetric triangular IT2-FV (RSTIT2-FV), is studied in this paper, as an extension of the most commonly used symmetric triangular FV in the type-1 fuzzy set system. The RSTIT2-FV this paper focuses on has two characteristics: (1) the shapes of the UMF and LMF are both symmetric triangles; (2) the vertexes of the UMF and LMF overlap at the membership degree of 1.
The main contributions of this paper are as follows. First, we give a clear mathematical definition of the RSTIT2-FV and define the medium of an RSTIT2-FV as a type-reduction operator, and then set forth the possibility, necessity, and credibility measures, as well as the credibility distribution and inverse credibility distribution. Second, the operational law based on the proposed inverse credibility distribution for mutually independent RSTIT2-FVs is given, which can help calculate for the strictly monotone functions of the RSTIT2-FVs in a relatively easy way. Thirdly, an expected value operator via inverse credibility distribution for the RSTIT2-FV is defined, and its linearity is proved. Finally, some numerical examples are illustrated to verify the efficiency of the proposed methodology.
The rest of this paper is organized as follows. In Section 2, the related concepts are briefly introduced. Then, the RSTIT2-FV is defined in Section 3, as well as the credibility distribution function and inverse credibility distribution. In Section 4, the operation law for the proposed RSTIT2-FV is provided. Finally, the expected value and its property of linearity are introduced and proved in Section 5.

Preliminaries
To begin, some fundamental concepts of the fuzzy set theory will be briefly introduced in advance. Definition 1. (Zadeh [7]) A type-1 fuzzy set (T1-FS) B can be defined as where µ B (x) is the real-valued membership function, and X is the universe of x.
Definition 5. (Zhou et al. [40]) An LR-type FV η, which has a continuous and strictly increasing credibility distribution Φ η is called a regular LR-FV.
Definition 6. (Zadeh [7], Mendel and John [43]) A type-2 fuzzy set (T2-FS), denoted as S and characterized by a type MF µ S (x, u). And S can be expressed as where u ∈ J x ⊆ [0, 1], x ∈ X is called the primary MF of x, µ S (x, u) is called the secondary MF of u.
It can be easily seen from Definition 6 that T2-FS is defined on a three-dimensional space. To better characterize its features, Mendel and John [43] defined the information it mapped to the two-dimensional plane as the footprint of uncertainty (FOU).  [22]) For a T2-FS S who has an N-element discrete universe of discourse X and primary membership functions J x 1 , J x 2 , · · · , J x N , an embedded T1-FS S e has N elements, one each from J x i , i = 1, 2, · · · , N, namely u 1 , u 2 , · · · , u N , i.e.,

Remark 1.
The embedded T1-FS can be extended to the continuous situations and expressed as Obviously, there are uncountable numbers of embedded T1-FSs in infinite domains.

Remark 2.
In fact, for a given x = x , µ S (x ) in Definition 9 is exactly the secondary MF, µ S (x , u), in Definition 6. Therefore, J x in Definition 9 can also be regarded as the primary MF in Definition 6.
Definition 10. (Men at al. [44]) For any x ∈ X and u ∈ J x , if µ S (x, u) is identically equal to 1, then the T2-FS S is an interval T2-FV (IT2-FV).
According to Definition 10, the secondary membership of an IT2-FV is unique and equal everywhere, i.e., the uncertainty of the MF and the information of the third dimension for an IT2-FV can be ignored, and the IT2-FV can be uniquely determined by the region bounded by its UMF and LMF.

Symmetric Triangular Interval Type-2 Fuzzy Variable
In this section, a special kind of IT2-FV, named regular symmetric triangular IT2-FV, is introduced.

RSTIT2-FV and Its Medium
The specific definition of the IT2-FV studied in this paper is as follows.
Definition 11. An IT2-FV S is called a regular symmetric triangular IT2-FV (RSTIT2-FV) if its UMF and LMF are as in the following forms, and and can be denoted as c − l U c c + l U c − l L c c + l L , where the spreads of the UMF and LMF, l U and l L , satisfy l U > l L , and the peak of them, 1, are reached when x is equal to c.
An RSTIT2-FV is visualized in Figures 1 and 2, respectively, in a three-dimensional space and two-dimensional plane, respectively, and Figure 2 can be understood as a top view of Figure 1.  In order to facilitate the subsequent definitions of credibility distribution and operational law for the RSTIT2-FV, the medium of an RSTIT2-FV is defined in advance.

Definition 12.
Let S be an RSTIT2-FV, η be a T1-FV, and their MFs satisfy then η is called the medium of S.
Definition 12 endows equal weights to every possible value of the MFs for a certain x, it is a relatively evenhanded way to eliminate the uncertainty of MFs in an uncertain and complex environment. Following from Definition 12, the analytical expression of µ η (x) can be calculated as according to Equations (1) and (2). Figure 3 shows the medium η with the red polyline representing its MF, from which we can readily infer that η is a regular LR-FV. Moreover, µ η can be regarded as one of the embedded type-1 sets of S, according to Definition 8.

The Credibility Distribution of the RSTIT2-FVs
By means of the medium η, we can define the credibility distribution and inverse credibility distribution in light of Liu [41]'s definition of credibility measure for the T1-FVs. Beforehand, the definition of the credibility measure for an RSTIT2-FV is given. Definition 13. Let S be an RSTIT2-FV with the medium of η, and A be a fuzzy event from the universe, then the possibility, necessity, and credibility measures of A for S are, respectively, It can be easily proven that the credibility measure of an RSTIT2-FV is self-dual and satisfies the monotonicity. Theorem 1. For a fuzzy event A from the universe and the credibility measure of an RSTIT2-FV, Cr, we have Cr{A} + Cr{A c } = 1.
Proof. According to Definition 13, it can be deduced that Theorem 2. Suppose that C and D are two fuzzy events satisfying C ⊆ D, then for the credibility measure of the RSTIT2-FV, we have Cr{C} ≤ Cr{D}.
Proof. In light of Definition 13, it can be obtained that On the other hand, we have D c ⊆ C c since C is a subset of D. Thus, we can further obtain Again, on the basis of Definition 13, we have Accordingly, it can be derived that Cr{C} ≤ Cr{D} in view of Equations (4) and (5).
Based on Definition 13 and Theorems 1 and measure-mono, the possibility, necessity, and credibility measures of a specific fuzzy event {S ≤ r} for an RSTIT2-FV S can be calculated by Pos{S ≤ r} = sup x≤r µ η (x), Next, we can define the credibility distribution of S as follows.

Definition 14.
Suppose that S is an RSTIT2-FV with a medium of η, then the credibility distribution Φ S of S is defined as Cr{S ≤ x}.
On the basis of Definition 13 and Equation (3), we can easily obtain that and Then by using Definition 14 and Equations (1) and (6) , the credibility distribution Φ S can be easily calculated as Figure 4 gives the visualization of Φ S . Obviously, Φ S is a continuous and strictly increasing function. Definition 15. Let S be an RSTIT2-FV, then its inverse credibility distribution is defined as the inverse function of Φ S (see Figure 5) and can be calculated by

Remark 3.
According to the definition of the credibility distribution [45] and inverse credibility distribution [40] of the TI-FV, are, respectively, the credibility distribution and inverse credibility distribution of the medium η.

Example 1.
Denote an RSTIT2-FV as S 1 = 1 4 7 3 4 5 , then according to Definition 11, we have l U = 3 and l L = 1, and the UMF and LMF are, respectively, Afterwards, based on Equation (3) in Definition 12, the MF of the medium η 1 for S 1 is Consequently, on the basis of Definition 14, the possibility of the fuzzy event S ≤ x is 1, x ∈ (4, +∞), and the necessity of it is 1, x ∈ (7, +∞).
Finally, with a view to Equation (9) in Definition 15, we have Example 2. Given that S 2 = 0 4 8 2 4 6 is an RSTIT2-FV, then the MF of its medium η 2 , the credibility distribution and inverse credibility distribution are as follows, following from Equations (3), (8) and (9).
In the previous subsection, the credibility distribution and the inverse credibility distribution for a single RSTIT2-FV had been given. Now, we consider a situation where more than one RSTIT2-FV are involved in an arithmetic operation.

Operational Law
This section introduces the operational law for fuzzy arithmetic operations regarding RSTIT2-FVs, which can help efficiently calculate Φ S in Definition 16. As a prerequisite for its application, the definition of independence is introduced at first.

Independence
Although Liu and Liu [11] has given the definition of independence for T2-FVs based on the possibility measure, it is somewhat more complex and is not that practicable. Therefore, a new definition of independence based on the credibility measure and the concept of medium in Definition 13 is proposed in this section. Definition 17. The RSTIT2-FVs S i , i = 1, 2, · · · , n. S i , i = 1, 2, · · · , n are said to be mutually independent if Cr{S i ∈ B i , i = 1, 2, · · · , n} = min 1≤i≤n Cr{S i ∈ B i } for any subsets B 1 , B 2 , · · · , B n . [46]) Let η i , i = 1, 2, · · · , n be n T1-FVs, they are said to be mutually independent if Cr{η i ∈ B i , i = 1, 2, · · · , n} = min 1≤i≤n Cr{η i ∈ B i } for any subsets B 1 , B 2 , · · · , B n .
Example 3. Suppose that S 1 in Example 1 and S 2 in Example 2 are mutually independent RSTIT2-FVs, and f i (S 1 , S 2 ), i = 1, 2, 3 are three functions of them. Then the inverse credibility distributions of f i (S 1 , S 2 ) can be given directly by Equation (12).
It is obviously seen that f 1 is strictly increasing with respect to S 1 and S 2 . Hence, on the basis of Theorem 4 and Equations (10) and (11), we have Case 2: Set f 2 (S 1 , S 2 ) = S 1 − S 2 . In this case, f 2 is strictly increasing with respect to S 1 , and strictly decreasingm with respect to S 2 . Following from Equation (11), we have Then, with a view to Theorem 4 and Equations (10) and (14), we can readily obtain that which can be visualized in Figure 7. Case 3: Set f 3 (S 1 , S 2 ) = S 2 /S 1 . Similarly, f 3 is strictly increasing with respect to S 2 , and strictly decreasing with respect to S 1 . According to Equation (10) Therefore, it can be obtained that following from Theorem 4 and Equations (11) and (16), as shown in Figure 8. Comparatively, if we calculate the inverse credibility function of S 1 + S 2 by means of Zadeh's extension principle [1], we should first determine the turning points of the MF of the medium f i (η 1 , η 2 ) for f i (S 1 , S 2 ) by identifying the key values of µ η 1 (x) and µ η 2 (x), and then calculate the MF µ f (η 1 ,η 2 ) (x) and credibility distribution Φ f i (S 1 ,S 2 ) (x).Due to space limitations, the solution process is placed in Appendix A, from which we can see that the operational law proposed in this paper can simplify the fuzzy operations without computing the intermediate procedures and possess the correctness.

Expected Value
The expected value operator is a crucial numerical characteristic of fuzzy variables; however, to the best of our knowledge, there is not yet a clear expected value definition for the IT2-FVs in the existing literature. Therefore, we define the expected value of the IT2-FVs as well as their strictly monotone functions via the credibility distribution in this section.

Definition 20.
Suppose that S is an RSTIT2-FV or an FV derived from multiple RSTIT2-FVs operations, then the expected value of S can be defined as Obviously, both integrals in Equation (18) are finite.

Theorem 5.
Given that S is an RSTIT2-FV or obtained from multiple RSTIT2-FVs operations, then its expected value can be simplified as the following form Proof. In light of Definitions 5 and 20, we have where Φ −1 S (α) is the inverse credibility distribution of S, which can be calculated by Equation (9). The results show that the expected value operator we defined is an unbiased estimation. (13), the expected values of f 1 = S 1 + S 2 in Example 3 can be calculated as Proof. When k > 0, it is easy to know from Definition 4 that the inverse credibility distribution of kS is kΦ −1 S (α). Then, by using Theorem 5, it can be deduced that

Example 5. Following from Theorem 5 and Equation
when k < 0, the inverse credibility distribution of kS is kΦ −1 S (1 − α), then we have Theorem 7. Let S 1 and S 2 be two mutually independent RSTIT2-FVs, then the linearity of the expected value operator concerning the two RSTIT2-FVs can be expressed in the following form, Proof. When k 1 > 0 and k 2 > 0, (k 1 S 1 + k 2 S 2 ) is strictly increasing monotone with respect to S 1 and S 2 and, thus, the inverse credibility distribution of (k 1 S 1 + k 2 S 2 ) can be easily obtained as in the views of Definition 4. Following that, and based on Theorems 5 and 6, we have when k 1 > 0 and k 2 < 0, (k 1 S 1 + k 2 S 2 ) is strictly increasing with respect to S 1 but strictly decreasing with respect to S 2 . It follows immediately from Theorems 5 and 6 that In the other two cases (i.e., k 1 < 0 and k 2 > 0, k 1 < 0 and k 2 < 0), it is easy to verify that Theorem 7 still holds. Due to space limitations, the details of the proofs are not provided in this paper. Remark 6. Definition 7 can be extended to the general case as where S 1 , S 2 , · · · , S n are mutually independent RSTIT2-FVs and k 1 , k 2 , · · · , k n are constants.
Theorem 7 and Remark 6 demonstrate that the introduced expected value operator is consistent with the property of linearity, which can help calculate the expected values of the functions of mutually independent RSTIT2-FVs without calculating the inverse credibility distribution of these RSTIT2-FVs. Example 6. On the basis of Theorem 7 and Example 4, the expected value of the RSTIT2-FV f 1 = S 1 + S 2 in Example 3 can be easily calculated by which is equal to the result figured out by the integral form in Example 5.

Conclusions
In this paper, we defined a special kind of TIT2-FV based on the most typical and simplest symmetric triangular type-1 FV, and proposed a novel operational law to calculate the arithmetic operations for the strictly monotone functions of the mutually independent RSTIT2-FVs as well as the expected value operator via the inverse credibility distribution. The comparative results of the numerical examples verified that the operational law is conducive to simplify the calculation processes, and the expected value operator is an unbiased estimator possessing the property of linearity. Thus, it can be applied to many areas involving complex fuzzy operations.
It is noteworthy that the methodology presented in this paper are not confined to the symmetric and triangular types, but could also be extended to the asymmetric cases and many other types of type-2 fuzzy variables, such as trapezoidal and normal TIT2-FVs, and the regularity of the TIT2-FVs can be extended to more generalized cases, which would be analyzed in the near future. In other words, the proposed method is a universal methodology, providing new insight for future research of many scholars, in both theory and practice, regarding the type-2 fuzzy set theory.

Acknowledgments:
The authors especially thank the editors and anonymous reviewers for their kind reviews and helpful comments. Any remaining errors are ours.

Conflicts of Interest:
We declare that we have no relevant or material financial interest that relate to the research described in this paper. The manuscript has neither been published before, nor has it been submitted for consideration of publication in another journal.