The NOWA weighted sampling type-reduction method for interval type-2 fuzzy sets and its application

Abstract

In order to aggregate all uncertain information of the footprint of uncertainty (FOU) and improve the speed of defuzzification for interval type-2 fuzzy sets (IT2 FS), the normal ordered weighted averaging (NOWA) weighted sampling method is proposed to transform an IT2 FS into a type-1 fuzzy set (T1 FS), whose defuzzified values closely approximate those of the expected values of all embedded T1 FSs. Compared with the existing methods, the proposed method is more simple and practical, and the result is mainly determined by the numbers and weights of random samples in each vertical slice of FOU. Finally, based on the NOWA weighted sampling method, a new possibility degree is introduced to solve multi-attribute decision-making problems in which the attribute values take the form of interval type-2 fuzzy numbers.

Keywords

Interval type-2 fuzzy set type-reduction normal OWA weight sampling

1 Introduction

In 1975, Zadeh [12] initially introduced the concept of the type-2 fuzzy set (T2 FS), which is regarded as an extension of the type-1 fuzzy set (T1 FS). The advantage of T2 FSs over T1 FSs is theis capability for model second-order uncertainties [2]. The interval type-2 fuzzy set (IT2 FS) is the most widely used T2 FS, and it has been widely applied to many practical fields [3 , 7–9]. Because the computational complexity is very high in an IT2 FS, the IT2 FS logic system needs two parts to complete the defuzzification process: type reduction and defuzzification proper. Type reduction is the procedure by which an IT2 FS is converted to a T1 FS (i.e. ordinary fuzzy set). Karmik and Mendel [15] introduced a centroid type reduction method to convert IT2 FSs to T1 FSs. However, it is very difficult to find the centroid of a large number of T1 FSs (embedded T1 FS).

From a computational viewpoint, IT2 FS is more difficult to deal with than T1 FS because we cannot determine the membership of an element in a set as 0 or 1. In fact, the membership of an IT2 FS may be between 0 and 1. To avoid this difficulty, some defuzzification approaches have been proposed in the literature for dealing with T2 FSs [1 , 19]. Among these methods, Nie and Tan [14] proposed a simple type reduction method (N-T method), which is based on the vertical-slice representation. Greenfield et al. [20] showed the method of sampling embedded sets and proved the sampling method was a good type-reduction method both theoretically and practically. However, the N-T method relied only on the lower and upper bounds of its FOU. The sampling method does not consider the weight of each random sample value. The goal of the N-T method and the sampling method is to make the defuzzification calculation simpler. In real world situations, some individuals may be too high or too low to assign preferences to their like or dislike objects. Therefore, inspired by this idea, the purpose of this paper is to reduce the order and aggregate all uncertain information of a vertical slice that comprises n discrete points for IT2 FS by using normal OWA (NOWA) weighted random sampling in the upper and lower membership function. The NOWA weighted sampling method takes into account not only the random sampling but also the weight of the sample.

The remainder of the paper is structured as follows. Section 2 introduces some concepts essential for interval type-2 fuzzy sets. In Section 3, we introduce the normal OWA operator weights and its properties. In Section 4, the NOWA weighted sampling method is presented and two numerical examples are given to illustrate the effectiveness of this proposed method. In Section 5, a confidence interval possibility degree is proposed to compare interval type-2 fuzzy numbers and applied to solve a multi-attribute decision making (MADM) problem. Finally, Section 6 is the conclusion.

2 Fundamental concepts of IT2 FS

The basic concepts of IT2 FS are introduced below to facilitate the following discussions. Interval type-2 fuzzy sets, which are characterized by fuzzy low and upper membership functions, are an extension of type-1 fuzzy sets [20].

Definition 1. Given a universal set X, an interval type-2 fuzzy set $\tilde{A}$ is defined as $\tilde{A} = \int_{x \in X} \int_{u \in J_{X}} 1 / - (x, u)$ (1) where J_X ⊆ [0, 1] is called the primary membership of x.

The concept of an IT2 FS can also be defined by the footprint of uncertainty (FOU). IT2 FS use lower and upper membership functions to limit the scope of FOU. Mathematically, the FOU may be expressed as the union of all the primary memberships, i.e. $FOU (\tilde{A}) = \underset{x \in X}{\cup} J_{x}$ (2)

A FOU is bounded by a lower membership function (LMF) ${\underline{μ}}_{\tilde{A}} (x)$ and an upper membership function (UMF) ${\bar{μ}}_{\tilde{A}} (x)$ , which are associated with the lower and upper bounds of (x, u) respectively. Using the LMF and UMF, the interval representation of FOU is $FOU (\tilde{A}) = \underset{x \in X}{\cup} [{\underline{μ}}_{\tilde{A}} (x), {\bar{μ}}_{\tilde{A}} (x)]$ (3)

For discrete universes of discourse X = {x₁, x₂, …, x_N} and discrete J_x, an embedded T1 FS A_e has N elements, one each from J_{x
₁}, J_{x
₂}, …, J_{x
_N}, namely u₁, u₂, …, u_N, i.e. $\begin{matrix} {\tilde{A}}_{e} = \sum_{i = 1}^{N} u_{i} / x_{i}, & u_{i} \in J_{x_{i}} \subseteq [0, 1] \end{matrix}$ (4)

Its graphical representation is shown in Fig. 1.

3 The normal OWA operation weights

An ordered weighted aggregation (OWA) operator was introduced by Yager [18]. This operator has been investigated in many documents and used in an astonishingly wide range of applications [22].

Definition 2. Suppose w = (w₁, w₂, …, w_n) to be the weight vector of the OWA operator; the OWA operator can be defined as follows: ${OWA}_{w} (a_{1}, a_{2}, \dots, a_{n}) = \sum_{j = 1}^{n} w_{j} b_{j}$ (5) where b_j is the jth largest element of the collection of the all aggregated objects a₁, a₂, …, a_n.

One important issue in the OWA operator is to determine its associated weights. The normal distribution is one of the most common distributions in nature, that is, many phenomena in nature obey the normal distribution.

Definition 3. [22] Suppose w = (w₁, w₂, …, w_n) to be the weight vector of the OWA operator; the normal OWA operator weights are defined by the following formula: $w_{i} = \frac{1}{\sqrt{2 π σ_{n}}} e^{- [(i - μ_{n})^{2} / 2 σ_{n}^{2}]}, i = 1, 2, \dots, n$ (6) where μ_n and σ_n are the mean and standard deviation of the natural sequence 1, 2, …, n. Obviously, μ_n and σ_n can be obtained by the following formulas, respectively:

$\begin{matrix} μ_{n} & = & \frac{1}{n} \frac{n (1 + n)}{2} = \frac{1 + n}{2}, \\ σ_{n} & = & \sqrt{\frac{1}{n} \sum_{i = 1}^{n} (i - μ_{n})^{2}} \end{matrix}$ (7)

Consider that w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Thus, by Equation (6), we have

$\begin{matrix} w_{i} & = & \frac{\frac{1}{\sqrt{2 π σ_{n}}} e^{- [(i - μ_{n})^{2} / 2 σ_{n}^{2}]}}{\sum_{j = 1}^{n} \frac{1}{\sqrt{2 π σ_{n}}} e^{- [(j - μ_{n})^{2} / 2 σ_{n}^{2}]}} \\ = & \frac{e^{- [(i - μ_{n})^{2} / 2 σ_{n}^{2}]}}{\sum_{j = 1}^{n} e^{- [(j - μ_{n})^{2} / 2 σ_{n}^{2}]}} \end{matrix}$ (8)

Because the mean of the natural sequence 1, 2, …, n, is (1 + n)/2, then Equation (8) can be rewritten as $w_{i} = \frac{e^{- [(i - (1 + n) / 2)^{2} / 2 σ_{n}^{2}]}}{\sum_{j = 1}^{n} e^{- [(j - (1 + n) / 2)^{2} / 2 σ_{n}^{2}]}}, i = 1, 2, \dots, n$ (9)

4 The NOWA weighted sampling method of defuzzification

The type-reduction of a discretised type-2 set may be characterized as the union of its embedded T1 FS ${\tilde{A}}_{e}$ , i.e. $\tilde{A} = \sum_{i = 1}^{N} {\tilde{A}}_{e}^{i}$ , where N is the number of vertical slices and ${\tilde{A}}_{e}^{i}$ is the number of elements on the ith slice [20]. The traditional type-reduction method (K-M method) requires every embedded T1 FS to be processed. In order to overcome the computational bottleneck, the sampling defuzzification method is a feasible method [20].

Suppose the continuous vertical-slice is discretized into n points. Then for each vertical-slice value x_i, the secondary domain value lies within the interval $[{\underline{μ}}_{\tilde{A}} (x_{i}), {\bar{μ}}_{\tilde{A}} (x_{i})]$ (as shown in Fig. 1). Because the enumeration of all the possible embedded sets is not practical, a process of random sampling is feasible to select a sample of them.

Although the sampling method is simpler computationally than the exhaustive method, it is not reasonable to use the same probability to select the secondary domain values. Clearly, the value in the secondary domain interval $[{\underline{μ}}_{\tilde{A}} (x_{i}), {\bar{μ}}_{\tilde{A}} (x_{i})]$ is more reliable, and should be given greater weight. The closer to the value $\frac{1}{2} ({\underline{μ}}_{\tilde{A}} (x_{i}) + {\bar{μ}}_{\tilde{A}} (x_{i}))$ , the greater the weight, i.e., it satisfies the NOWA operator weights.

For an IT2 FS with discretized (or discrete) universes of discourse X, let a set of sequences {x₁, x₂, …, x_N} satisfy x₁ ≤ x₂ ≤ … ≤ x_N, x_i ∈ X. Then, the type reduction of $\tilde{A}$ is decided by the footprint of uncertainty, i.e., it is decided by the interval $[{\underline{μ}}_{\tilde{A}} (x_{i}), {\bar{μ}}_{\tilde{A}} (x_{i})], i = 1, 2, \dots, N$ . The NOWA weighted sampling defuzzification algorithm can be shown as follows:

Step 1. For any x_i ∈ X, in accordance with the principle of uniform distribution, randomly select m points on the intervals $({\underline{μ}}_{\tilde{A}} (x_{i}), \frac{{\underline{μ}}_{\tilde{A}} (x_{i}) + {\bar{μ}}_{\tilde{A}} (x_{i})}{2})$ and $(\frac{{\underline{μ}}_{\tilde{A}} (x_{i}) + {\bar{μ}}_{\tilde{A}} (x_{i})}{2}, {\bar{μ}}_{\tilde{A}} (x_{i}))$ , respectively.

Step 2. In each vertical slice x_i ∈ X, these 2m random sampling points and three endpoints ${\underline{μ}}_{\tilde{A}} (x_{i}), \frac{{\underline{μ}}_{\tilde{A}} (x_{i}) + {\bar{μ}}_{\tilde{A}} (x_{i})}{2}$ and ${\bar{μ}}_{\tilde{A}} (x_{i})$ constitute a new set of sequences {μ_{x
_i} (j)} , (j = 1, 2, …, 2m + 3), which satisfies μ_{x
_i} (1) ≤ μ_{x
_i} (2) ≤ … ≤ μ_{x
_i} (2m + 3).

Step 3. Match the normal OWA weights w_k with their respective ${\bar{μ}}_{x_{i}} (k), (k = 1, 2, \dots, 2 m + 3)$ . Then, using Equation (5), obtain the expected membership degree value of ${\tilde{A}}_{e} (x_{i})$

$\begin{matrix} μ_{{\tilde{A}}_{E}} (x_{i}) & = & {TR}_{NOWA} (\tilde{A} (x_{i})) \\ = & \sum_{k = 1}^{2 m + 3} w_{k} μ_{x_{i}} (k), x_{i} = 1, 2, \dots, N \end{matrix}$ (10)

i.e.,

$\begin{matrix} μ_{{\tilde{A}}_{E}} (x_{i}) & = & {TR}_{NOWA} (\tilde{A} (x_{i})) \\ = & \frac{\sum_{k = 1}^{2 m + 3} μ_{x_{i}} (k) e^{- [(k - (2 + m))^{2} / 2 σ_{2 m + 3}^{2}]}}{\sum_{j = 1}^{2 m + 3} e^{- [(j - (2 + m))^{2} / 2 σ_{2 m + 3}^{2}]}} \end{matrix}$ (11)

Step 4. Hence, the centroid (or defuzzified value) of IT2 FS can be expressed as $c (\tilde{A}) = \frac{\sum_{i = 1}^{N} x_{i} μ_{{\tilde{A}}_{E}} (x_{i})}{\sum_{i = 1}^{N} μ_{{\tilde{A}}_{E}} (x_{i})}$ (12)

For the user of the NOWA weighted sampling defuzzification algorithm, there is one decision to be made: the number of embedded sets (m). Theoretically, the larger the number value m, the more accurate the results. The following example shows that in practical application, as long as a small value m is selected, we can get very good accuracy results.

If m = 0, we can easily find $μ_{{\tilde{A}}_{E}} (x_{i}) = \frac{{\underline{μ}}_{\tilde{A}} (x_{i}) + {\bar{μ}}_{\tilde{A}} (x_{i})}{2}$ (13)

That is consistent with the Nie-Tan method (N-T method); the N-T method is actually a special case of the NOWA weighted sampling method.

Example 1. Let us consider the case of a triangular interval type-2 fuzzy number $\tilde{A} = ((0, 0.4, 0.9; 1), (0.3, 0.4, 0.5; 1))$ , defuzzifying the triangular test set at a degree of discretisation of 0.1, and a sample size of m = 0,1,2,3.

By setting the sampling value, both the expected value (i.e. defuzzified value) and centroid are calculated in Table 1.

In order to better understand how the NOWA weighted sampling type-reducer method compared to the K-M method, the centroid of IT2 FS calculated using Equation (12) is compared against that of the K-M iterative algorithm. The results of the K-M type-reducer are obtained by first calculating the generalized centroid [c_l, c_r] of an IT2 FS, and then taking the centroid average x_c,KM = (c_l + c_r)/2, where the interval [c_l, c_r] is the lower and upper bound of the type-reduced set.

Example 2.Table 2 for the following four IT2 FSs, which are asymmetrical triangular membership functions about x = 5, gives the centroids obtained using the K-M type reducer by [14]. Under coarse discretisation (e.g., the degree of discretisation is 0.05), exhaustive defuzzification is practicable, enabling corroboration of the NOWA weighted sampling method. The corresponding results for the IT2 FS with triangular membership functions are tabulated in Table 2, with the sample numbers m = 0, 2, 4, 6, 8, 10 in each vertical slice.

By the asymmetric test sets, Table 2 shows that the results of the NOWA weighted sampling method are very close to the K-M method and to those produced by exhaustive defuzzification. The accuracy of the NOWA weighted method is ideal, and improves with the increase of sampling numbers. Our experiments on asymmetric test sets of known defuzzified values have shown that through the NOWA weighted sampling method, great improvement in efficiency may be achieved without losing the accuracy of the premise. In the following section, we will use the method to solve a decision problem.

5 Application to multi-criteria decision making

In this section, we present a new possibility degree method for calculating the ranking values of interval type-2 fuzzy sets. Wu and Mendel [6] defined the variance $V (\tilde{A})$ of an IT2 FS $\tilde{A}$ based on the embedded T1 FS ${\tilde{A}}_{e}$ as $V (\tilde{A}) = \frac{\sum_{i = 1}^{N} [x_{i} - c (\tilde{A})]^{2} μ_{A_{E}} (x_{i})}{\sum_{i = 1}^{N} μ_{A_{E}} (x_{i})}$ (14)

We use the “3σ” principle in statistics, where the confidence intervals of the two IT2 FSs $\tilde{A}$ and $\tilde{B}$ are ${\tilde{A}}_{CI} = [c (\tilde{A}) - 3 \sqrt{V (\tilde{A})}, c (\tilde{A}) + 3 \sqrt{V (\tilde{A})}]$ and ${\tilde{B}}_{CI} = [c (\tilde{B}) - 3 \sqrt{V (\tilde{B})}, c (\tilde{B}) + 3 \sqrt{V (\tilde{B})}]$ , respectively. Then, a new possibility degree based on the confidence interval of the two IT2 FSs is defined as follows: $p (\tilde{A} \geq \tilde{B}) = min {max (\frac{c (\tilde{A}) + 3 \sqrt{V (\tilde{A})} - c (\tilde{B}) + 3 \sqrt{V (\tilde{B})}}{6 \sqrt{V (\tilde{A})} + 6 \sqrt{V (\tilde{B})}}, 0), 1}$ (15)

Theorem 1. The confidence interval possibility degree $p ({\tilde{A}}_{1} \geq {\tilde{A}}_{2})$ of two IT2 FSs ${\tilde{A}}_{1}$ and ${\tilde{A}}_{2}$ has the following properties:

$0 \leq p ({\tilde{A}}_{1} ≻ {\tilde{A}}_{2}) \leq 1$ , $0 \leq p ({\tilde{A}}_{2} ≻ {\tilde{A}}_{1}) \leq 1$ .

For any two IT2 FSs ${\tilde{A}}_{1}$ and ${\tilde{A}}_{2}$ , $p ({\tilde{A}}_{1} ≻ {\tilde{A}}_{2}) + p ({\tilde{A}}_{2} ≻ {\tilde{A}}_{1}) = 1$ .

$p ({\tilde{A}}_{1} ≻ {\tilde{A}}_{2}) \geq 0.5$ if and only if $c ({\tilde{A}}_{1}) \geq c ({\tilde{A}}_{2})$ .

For IT2 FSs ${\tilde{A}}_{1}$ , ${\tilde{A}}_{2}$ and ${\tilde{A}}_{3}$ , if $p ({\tilde{A}}_{1} ≻ {\tilde{A}}_{2}) \geq 0.5$ and $p ({\tilde{A}}_{2} ≻ {\tilde{A}}_{3}) \geq 0.5$ then $p ({\tilde{A}}_{1} ≻ {\tilde{A}}_{2}) + p ({\tilde{A}}_{2} ≻ {\tilde{A}}_{3}) \geq p ({\tilde{A}}_{1} ≻ {\tilde{A}}_{3})$ .

In order to pairwise compare to n interval type-2 fuzzy numbers, then the confidence interval possibility degree matrices can be constructed as follows: $P = (p_{ij})_{n \times n} = (p ({\tilde{A}}_{i} \geq {\tilde{A}}_{j}))_{n \times n}$ (16)

Then, the ranking value $Rank ({\tilde{A}}_{i})$ of the interval type-2 fuzzy set can be obtained as follows [23]: $Rank ({\tilde{A}}_{i}) = \frac{1}{n (n - 1)} (\sum_{k = 1}^{n} p ({\tilde{A}}_{i} ≻ {\tilde{A}}_{k}) + \frac{n}{2} - 1)$ (17) where $\sum_{i = 1}^{n} Rank ({\tilde{A}}_{i}) = 1$ . The greater the ranking value $Rank ({\tilde{A}}_{i})$ , the greater the IT2 FS ${\tilde{A}}_{i}$ (i = 1, 2, …, n).

In this section, we present a new method for handling a MADM problem based on the proposed possibility degree method under an interval type-2 fuzzy environment. An example from Hu et al. [11] is used to illustrate the procedures and feasibility of the proposed possibility degree method under an interval type-2 fuzzy environment. They are a Chinese company mainly engaged in producing and selling nonferrous metals. In order to expand the market, the company needs to go abroad to seek minerals to promote future development.

Now, the company’s overseas investment department needs to make decisions on global minerals investment. Recently, the overseas investment department of this company has established an investment decision-making team composed of seven experts and executive managers to investigate the seven overseas companies.

Seven countries (alternatives) are represented by A₁, A₂, A₃, A₄, A₅, A₆, and A₇. In the process of evaluation, the main considerations are four attributes: “Resources”, “Politics and Policy”, “Economy”, and “Infrastructure”.

Aiming at this problem, Hu et al. proposed a new possibility degree method based on the lower and upper membership functions of the IT2 FS. Based on a TIT2-WAA operator, the overall values Z (A_i) of alternatives A_i are shown as follows: $\begin{matrix} Z (A_{1}) & = & ((0 . 60, 0 . 77, 0 . 89; 1), \\ (0 . 655, 0 . 77, 0 . 86; 0 . 9)) \\ Z (A_{2}) & = & ((0 . 56, 0 . 73, 0 . 84; 1), \\ (0 . 63, 0 . 73, 0 . 785; 0 . 9)) \\ Z (A_{3}) & = & ((0 . 66, 0 . 86, 0 . 98; 1), \\ (0 . 76, 0 . 86, 0 . 92; 0 . 9)) \\ Z (A_{4}) & = & ((0 . 68, 0 . 85, 0 . 96; 1), \\ (0 . 765, 0 . 85, 0 . 905; 0 . 9)) \\ Z (A_{5}) & = & ((0 . 72, 0 . 89, 0 . 98; 1), \\ (0 . 805, 0 . 89, 0 . 935; 0 . 9)) \\ Z (A_{6}) & = & ((0 . 62, 0 . 80, 0 . 91; 1), \\ (0 . 71, 0 . 80, 0 . 85; 0 . 9)) \\ Z (A_{7}) & = & ((0 . 66, 0 . 80, 0 . 88; 1), \\ (0 . 73, 0 . 80, 0 . 84; 0 . 9)) \end{matrix}$

The above overall values are interval type-2 fuzzy numbers, so they cannot be directly compared. Under discretisation (e.g. a degree of discretisation 0.01), and the sample number m = 10 of each vertical slice, the centroid and variance of the overall values are calculated in Table 3 based on the NOWA weighted sampling method and Equation (14).

A possibility degree matrix P is determined based on Equation (16) as follows: $p = (\begin{matrix} 0.5 & 0.5721 & 0.3771 & 0.3881 & 0.3138 & 0.4625 & 0.4516 \\ 0.4279 & 0.5 & 0.3047 & 0.3142 & 0.2317 & 0.3887 & 0.3676 \\ 0.6229 & 0.6953 & 0.5 & 0.5135 & 0.4519 & 0.5881 & 0.5930 \\ 0.6119 & 0.6858 & 0.4865 & 0.5 & 0.4357 & 0.5760 & 0.5797 \\ 0.6862 & 0.7683 & 0.5481 & 0.5643 & 0.5 & 0.6482 & 0.6621 \\ 0.5375 & 0.6113 & 0.4119 & 0.4240 & 0.3518 & 0.5 & 0.4938 \\ 0.5484 & 0.6324 & 0.4070 & 0.4203 & 0.3379 & 0.5062 & 0.5 \end{matrix})$

Then, we can utilize the possibility degree ranking method to get the best alternative. The results are Rank (A₁) =0.1325, Rank (A₂) =0.1199, Rank (A₃) =0.1539, Rank (A₄) =0.1518, Rank (A₅) =0.1637, Rank (A₆) =0.1388, and Rank (A₇) =0.1393. Because Rank (A₅) > Rank (A₃) > Rank (A₄) > Rank (A₇) > Rank (A₆) > Rank (A₁) > Rank (A₂), the preference order of the alternatives is A₅ > A₃ > A₄ > A₇ > A₆ > A₁ > A₂. Therefore, the most desirable global mineral supplier is A₅.

Using the above example, we computed the possibility degree to compare those interval type-2 fuzzy numbers based on different definitions proposed by Chen and Lee [21], Hu et al. [11], and Equation (17) as developed by us. The calculated results are shown in Table 4.

As can be seen from Table 4, the best alternative obtained by the three possibility degree methods is consistent, but the overall ranking results from second to fifth are not the same. The reason is that Chen’s method only considers the boundaries of UMF and LMF, which leads to an error of A₄ > A₃ > A₆ > A₇ rather than A₃ > A₄ > A₇ > A₆. Compared with Hu’s method, our method not only considers the values of UMF and LMF, but also considers all the values of embedded type-1 MF. Therefore, the result is more accurate and reasonable. From Table 4, it can be observed that the overall order result of second and third is not the same. The reason is that the values of all embedded type-1 MF are not taken into consideration at the same time in Hu’s method, which leads to an error of A₄ > A₃ rather than A₃ > A₄.

Thus, a comparison of the results shows that our proposed possibility degree approach performs better than the other two possibility degree methods.

6 Conclusion

Although there are different methods for computing the type-reduction of an IT2 FS, this paper focuses on the weighted sampling for all the vertical-slice values of the universe of discourse X. The example shows that the difference between the weighted sampling algorithm and K-M method is relatively small. The accuracy is mainly determined by the numbers and weights of random samples in each vertical slice, even if the degree of discretisation is coarse. The advantage of the proposed method is that it requires less computation, while the calculation is simple and the accuracy is good.

Because the results from the studies reported herein appear promising, future plans include extending the interval representative embedded set method to generalized type-2 fuzzy sets. On the other hand, we will also study whether the degrees of discretisation and sample numbers affect the accuracy of the resultant defuzzified value and analysis to better understand how it differs from existing type-reducers. Finally, the presented NOWA sampling method can be used to represent the centroid and variance of any IT2 FS, so its application to fuzzy ranking and decision-making problems has a wide potential.

Footnotes

Acknowledgments

This work was supported in part by the Natural Science Foundation of Jiangsu Province of China (No. BK20130242) and the Fundamental Research Funds for the Central Universities (No. 2015B28014).

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