*Article* **Normalized Weighted Bonferroni Harmonic Mean-Based Intuitionistic Fuzzy Operators and Their Application to the Sustainable Selection of Search and Rescue Robots**

#### **Jinming Zhou 1, Tomas Baležentis 2,\* and Dalia Streimikiene <sup>2</sup>**


Received: 11 January 2019; Accepted: 3 February 2019; Published: 13 February 2019

**Abstract:** In this paper, Normalized Weighted Bonferroni Mean (NWBM) and Normalized Weighted Bonferroni Harmonic Mean (NWBHM) aggregation operators are proposed. Besides, we check the properties thereof, which include idempotency, monotonicity, commutativity, and boundedness. As the intuitionistic fuzzy numbers are used as a basis for the decision making to effectively handle the real-life uncertainty, we extend the NWBM and NWBHM operators into the intuitionistic fuzzy environment. By further modifying the NWBHM, we propose additional aggregation operators, namely the Intuitionistic Fuzzy Normalized Weighted Bonferroni Harmonic Mean (IFNWBHM) and the Intuitionistic Fuzzy Ordered Normalized Weighted Bonferroni Harmonic Mean (IFNONWBHM). The paper winds up with an empirical example of multi-attribute group decision making (MAGDM) based on triangular intuitionistic fuzzy numbers. To serve this end, we apply the IFNWBHM aggregation operator.

**Keywords:** Bonferroni harmonic mean; aggregation operator; intuitionistic fuzzy set; multiple attribute group decision making; search and rescue robots

#### **1. Introduction**

Decision making seeks to pick the best-performing option (alternative) among the feasible ones in order to satisfy a certain objective represented by an attribute. In practice, many decisions require considering more than one objective and, hence, more than one attribute. This being the case, one faces a multi-attribute decision making (MADM) problem. Basically, MADM is defined as the identification of the best-performing alternative among the feasible ones, taking multiple attributes into consideration. As multiple attributes are involved in the problem, the issue of aggregation of the decision information arises. The aggregation operators may be employed in order to summarize the decision information in MADM and, thus, consider multiple objectives simultaneously. What is more, the aggregation operators can be adjusted to account for interrelations among the decision variables.

The theory and applications of aggregation operators have been developing due to an increasing prevalence of the MADM problem in different domains [1–4]. There have been some aggregation operators available for handling MADM problems involving intuitionistic fuzzy (IF) sets [5–8]. In order to exploit multiple desirable properties of the IF sets, different types of intuitionistic fuzzy numbers (IFNs) have been established and employed for various empirical applications [9–12]. The theory of the aggregation operators has also been extended in regards to different types of IFNs. For instance, the triangular intuitionistic fuzzy numbers (TIFNs) were introduced [13,14] and applied for information

aggregation by offering the corresponding extension of averaging operators, namely the intuitionistic fuzzy weighted arithmetic aggregation operator.

Yet another example regarding the aggregation operators for the IFNs was proposed by Wan and Dong [15], who developed the ordered weighted aggregation operator along with the hybrid weighted aggregation operator. The latter approach was based on the use of the measures of the expectation and expectant score determined by the position of the center of gravity of IFNs considered in the analysis. Wu and Cao [16] proposed a family of intuitionistic trapezoidal fuzzy operators weighted geometric operators (including the ordered, induced ordered, and hybrid ones).

The earlier literature has mostly opted for treating the IF information used for aggregation as showing no interdependency relations. As a result, the possible existing intercorrelation among the arguments has not been accounted for. One of the possible means for accounting for interdependence existing among the arguments of the MADM problems is the Bonferroni mean (BM) operator [17]. Yager showed that the BM may be obtained as a sum the products of arguments to be aggregated and the average value of all the arguments save the one under consideration. What is more, the arithmetic average may be replaced with the other types of means [18] including, for instance, the Choquet integral [19] or ordered weighted average operator.

Further modifications of the BM methodology were offered by Beliakov et al. [20], who developed the generalized BM. The concept of the BM has been extended for the intuitionistic fuzzy information by Xu and Yager [21] to handle the intercorrelation among the arguments throughout the aggregation. Dutta and Guha [22] proposed substituting the aggregation operators for the inner and outer means in the calculations.

While seeking to aggregate the uncertain information, the uncertain BM operator along with its ordered and Choquet integral versions were developed [23]. The generalized weighted BM operator and its intuitionistic fuzzy counterpart were introduced by Xia et al. [24]. The latter operators included expert assessments in order to improve the robustness of the aggregation. An additional technique for aggregating the IFNs—the intuitionistic fuzzy weighted power harmonic mean (IFWPHM) operator—was proposed by Das and Guha [25]. The harmonic aggregation operators for the MADM problems based upon the fuzzy information were proposed by Xu [26]. The latter group of fuzzy weighted harmonic operators includes mean, ordered mean, and hybrid mean operators. Wei [27] suggested using the order-inducing variables in the process of aggregation of the fuzzy information and devised the fuzzy induced ordered weighted harmonic mean operator. The use of the BM in the fuzzy MADM was furthered in [28] by developing the fuzzy Bonferroni harmonic mean operator and the ordered counterpart.

In the existing literature, applications of the BM operators have mostly been limited to cases where information was represented by the intuitionistic fuzzy sets established with respect to a finite universe of discourse [29–31]. However, the methods available for handling the intuitionistic fuzzy numbers, e.g., triangular intuitionistic fuzzy numbers (TIFNs), as arguments of the aggregation operators, are rather scarce in the literature. In order to extend the domain for application of the intuitionistic fuzzy information in MADM, we propose the normalized weighted triangular intuitionistic fuzzy Bonferroni harmonic mean (NWTIFBHM) operator, which is capable of aggregating the triangular intuitionistic fuzzy information. The proposed approach relies on the Bonferroni mean (BM). More specifically, we exploit the normalized weighted Bonferroni mean (NWBM) and establish the intuitionistic fuzzy normalized weighted Bonferroni harmonic mean (IFNWBHM). The proposed approach is then tested by solving a multi-attribute group decision making (MAGDM) problem involving the IFNWBHM.

The remainder of this paper unfolds as follows. Section 2 discusses the preliminary concepts and operations. Section 3 proposes the normalized weighted triangular intuitionistic fuzzy Bonferroni harmonic mean along with several important results. Section 4 presents application to MAGDM with triangular intuitionistic fuzzy information. Finally, an illustrative example is implemented with a comparative analysis of several prevalent aggregation operators with the proposed approach.

#### **2. Preliminaries**

In this section, we discuss the information carriers used for MADM, namely TIFNs. We further discuss the means for aggregations of TIFNs, which allow the utilities for the alternatives comprising the MADM problem to be derived. As the outcomes of such aggregations are also TIFNs, the ranking procedure is outlined.

#### *2.1. TIFNs and the Associated Arithmetic Operations*

Oftentimes, decision making cannot rely on precise information delivered in the form of exact (real) numbers. However, uncertain estimates can be provided regarding a certain phenomenon. Such being the case, one can embark by using the fuzzy numbers rather than crisp ones. Among different types of representation of the fuzzy information, the intuitionistic fuzzy numbers can be perceived as a generalization of the fuzzy numbers. Further on, a TIFN can be defined as an intuitionistic fuzzy set (defined in terms of a fuzzy membership function and a fuzzy non-membership function) attached to a certain real value. Mathematically, the membership and non-membership functions for a certain TIFN A are defined as [32]:

$$\mu\_A = \begin{cases} \frac{\chi - a}{b - a} \omega\_{A'} & a \le x \le b \\ \omega\_{A'} & x = b \\ \frac{c - \chi}{c - b} \omega\_{A'} & b \le x \le c \\ 0, & c!s \end{cases} \tag{1}$$

and

$$\nu\_A = \begin{cases} \frac{b - x + u\_A(x - a)}{b - a}, & a \le x \le b \\ u\_{A'} & x = b \\ \frac{x - b + u\_A(c - x)}{c - b}, & b \le x \le c \\ 1, & c \text{else} \end{cases} \tag{2}$$

where parameters *ω<sup>A</sup>* and *uA* represent the upper limit of the value of the membership function and the minimum level of the non-membership function, respectively, with restrictions on their individual value and sum thereof given by 0 ≤ *ω<sup>A</sup>* ≤ 1, 0 ≤ *uA* ≤ 1 and 0 ≤ *ω<sup>A</sup>* + *uA* ≤ 1. The values of the membership and non-membership functions comprise the "core" of the degree of dependency of *x* to *A*, whereas the "uncertain" part is given by the hesitancy function *πA*(*x*) = 1 − *μA*(*x*) − *νA*(*x*), which is related to the constrains on the two functions discussed above. This definition is different from that of triangular fuzzy numbers as the latter does not involve the "uncertain part ".

In order to successfully apply the TIFNs for the MADM, the operational laws for TIFNs need to be established [32]. Let us consider the two TIFNs defined as *A*<sup>1</sup> = ([*a*1, *b*1, *c*1]; *ωA*<sup>1</sup> , *uA*<sup>1</sup> ) and *A*<sup>2</sup> = ([*a*2, *b*2, *c*2]; *ωA*<sup>2</sup> , *uA*<sup>2</sup> ), and assume that there exists a real number *λ* > 0,. Given the aforementioned variables, the following calculations serve as the operational laws for the TIFNs:


The operational laws feature the following properties [32]:


**Proof.** The commutativity, distributivity, and associativity are implied by the definition of operational laws as follows:

*A*<sup>2</sup> ⊕ *A*<sup>1</sup> = ([*a*<sup>2</sup> + *a*1, *b*<sup>2</sup> + *b*1, *c*<sup>2</sup> + *c*1]; *ωA*<sup>2</sup> ∧ *ωA*<sup>1</sup> , *uA*<sup>2</sup> ∨ *uA*<sup>1</sup> ) = ([*a*<sup>1</sup> + *a*2, *b*<sup>1</sup> + *b*2, *c*<sup>1</sup> + *c*2]; *ωA*<sup>1</sup> ∧ *ωA*<sup>2</sup> , *uA*<sup>1</sup> ∨ *uA*<sup>2</sup> ) = *A*<sup>1</sup> ⊕ *A*2, ∴ *A*1⊕ *A*<sup>2</sup> = *A*<sup>2</sup> ⊕ *A*1. *A*<sup>2</sup> ⊗ *A*<sup>1</sup> = ([*a*2*a*1, *b*2*b*1, *c*2*c*1]; *ωA*<sup>2</sup> ∧ *ωA*<sup>1</sup> , *uA*<sup>2</sup> ∨ *uA*<sup>1</sup> ) = ([*a*1*a*2, *b*1*b*2, *c*1*c*2]; *ωA*<sup>1</sup> ∧ *ωA*<sup>2</sup> , *uA*<sup>1</sup> ∨ *uA*<sup>2</sup> ) = *A*<sup>1</sup> ⊗ *A*2; ∴ *A*1⊗ *A*<sup>2</sup> = *A*<sup>2</sup> ⊗ *A*1. *λ*(*A*<sup>1</sup> ⊕ *A*2) = ([*λ*(*a*<sup>1</sup> + *a*2), *λ*(*b*<sup>1</sup> + *b*2), *λ*(*c*<sup>1</sup> + *c*2)]; *ωA*<sup>1</sup> ∧ *ωA*<sup>2</sup> , *uA*<sup>1</sup> ∨ *uA*<sup>2</sup> ) = ([*λa*<sup>1</sup> + *λa*2, *λb*<sup>1</sup> + *λb*2, *λc*<sup>1</sup> + *λc*2]; *ωA*<sup>1</sup> ∧ *ωA*<sup>2</sup> , *uA*<sup>1</sup> ∨ *uA*<sup>2</sup> ) = ([*λa*1, *λb*1, *λc*1]; *ωA*<sup>1</sup> , *uA*<sup>1</sup> ) + ([*λa*2, *λb*2, *λc*2]; *ωA*<sup>2</sup> , *uA*<sup>2</sup> ) = *λ*([*a*1, *b*1, *c*1]; *ωA*<sup>1</sup> , *uA*<sup>1</sup> ) + *λ*([*a*2, *b*2, *c*2]; *ωA*<sup>2</sup> , *uA*<sup>2</sup> ) = *λA*<sup>1</sup> ⊕ *λA*<sup>2</sup> ∴ *λ*(*A*1⊕ *A*2) = *λA*<sup>1</sup> ⊕ *λA*<sup>2</sup> *λ*(*A*<sup>1</sup> ⊗ *A*2) = *λ*([(*a*1*a*2),(*b*1*b*2),(*c*1*c*2)]; *ωA*<sup>1</sup> ∧ *ωA*<sup>2</sup> , *uA*<sup>1</sup> ∨ *uA*<sup>2</sup> ) = ([(*λa*1)*a*2,(*λb*1)*b*2,(*λc*1)*c*2]; *ωA*<sup>1</sup> ∧ *ωA*<sup>2</sup> , *uA*<sup>1</sup> ∨ *uA*<sup>2</sup> ) = ([*a*1(*λa*2), *b*1(*λb*2), *c*1(*λc*2)]; *ωA*<sup>1</sup> ∧ *ωA*<sup>2</sup> , *uA*<sup>1</sup> ∨ *uA*<sup>2</sup> ) ([(*λa*1)*a*2,(*λb*1)*b*2,(*λc*1)*c*2]; *ωA*<sup>1</sup> ∧ *ωA*<sup>2</sup> , *uA*<sup>1</sup> ∨ *uA*<sup>2</sup> ) = ([*λa*1, *λb*1, *λc*1]; *ωA*<sup>1</sup> , *uA*<sup>1</sup> ) ⊗ ([*a*2, *b*2, *c*2]; *ωA*<sup>2</sup> , *uA*<sup>2</sup> ) = *λ*([*a*1, *b*1, *c*1]; *ωA*<sup>1</sup> , *uA*<sup>1</sup> ) ⊗ ([*a*2, *b*2, *c*2]; *ωA*<sup>2</sup> , *uA*<sup>2</sup> )=(*λA*1) ⊗ *A*<sup>2</sup> ([*a*1(*λa*2), *b*1(*λb*2), *c*1(*λc*2)]; *ωA*<sup>1</sup> ∧ *ωA*<sup>2</sup> , *uA*<sup>1</sup> ∨ *uA*<sup>2</sup> ) = ([*a*1, *b*1, *c*1]; *ωA*<sup>1</sup> , *uA*<sup>1</sup> ) ⊗ ([*λa*2, *λb*2, *λc*2]; *ωA*<sup>2</sup> , *uA*<sup>2</sup> ) = ([*a*1, *b*1, *c*1]; *ωA*<sup>1</sup> , *uA*<sup>1</sup> ) ⊗ *λ*([*a*2, *b*2, *c*2]; *ωA*<sup>2</sup> , *uA*<sup>2</sup> ) = *A*<sup>1</sup> ⊗ (*λA*2) ∴ *λ*(*A*<sup>1</sup> ⊗ *A*2)=(*λA*1) ⊗ *A*<sup>2</sup> = *A*<sup>1</sup> ⊗ (*λA*2) *λ*1*A* = ([*λ*1*a*, *λ*1*b*, *λ*1*c*]; *ωA*, *uA*), *λ*2*A* = ([*λ*2*a*, *λ*2*b*, *λ*2*c*]; *ωA*, *uA*) ∴ *λ*1*A* + *λ*2*A* = ([(*λ*<sup>1</sup> + *λ*2)*a*,(*λ*<sup>1</sup> + *λ*2)*b*,(*λ*<sup>1</sup> + *λ*2)*c*]; *ωA*, *uA*) = (*λ*<sup>1</sup> + *λ*2)*A Aλ*<sup>1</sup> = - [*aλ*<sup>1</sup> , *bλ*<sup>1</sup> , *cλ*<sup>1</sup> ]; *ωA*, *uA* , *Aλ*<sup>2</sup> = - [*aλ*<sup>2</sup> , *bλ*<sup>2</sup> , *cλ*<sup>2</sup> ]; *ωA*, *uA* . ∴ *Aλ*1+*λ*<sup>2</sup> = - [*aλ*1+*λ*<sup>2</sup> , *bλ*1+*λ*<sup>2</sup> , *cλ*1+*λ*<sup>2</sup> ]; *ωA*, *uA* = - [*aλ*<sup>1</sup> *aλ*<sup>2</sup> , *bλ*<sup>1</sup> *aλ*<sup>2</sup> , *cλ*<sup>1</sup> *aλ*<sup>2</sup> ]; *ωA*, *uA* <sup>=</sup> *<sup>A</sup>λ*<sup>1</sup> <sup>⊗</sup> *<sup>A</sup>λ*<sup>2</sup>

The TIFNs (and fuzzy numbers in general) are rather complex structures associated with elements of the real line. Therefore, it is often useful to approximate the fuzzy numbers by assuming a certain level of the (non-)membership function and projecting the fuzzy numbers on a real line. The elements of the real set satisfying the requirements associated with the values of the (non-)membership functions are then treated as those belonging to the set approximating a certain fuzzy number (including a TIFN). The latter approach is referred to as cutting of the fuzzy numbers. An *α*-cut of a TIFN is a subset of crisp values which satisfy *A*(*α*) = {*x*|*μA*(*x*) ≥ *α* } [32], where the chosen lower level of the membership function is 0 ≤ *α* ≤ *ωA*. Given Equation (1), every *α*-cut is a closed interval, which is obtained as

$$\left[A^{L}(a), A^{L}(a)\right] = \left[a + \frac{a(b-a)}{\omega\_{A}}, c - \frac{a(c-b)}{\omega\_{A}}\right] \tag{3}$$

Similarly, a *β*-cut of TIFN *A* is defined as a subset of crisp values for which the non-membership function does not exceed the upper limit, i.e., *A*(*β*) = {*x*|*νA*(*x*) ≤ *β* }, where the upper limit of the non-membership function is given by 0 ≤ *uA* ≤ *β* ≤ 1. Given the properties stipulated by Equation (2), each *β*-cut of TIFN is a projection of a certain TIFN on the real line represented by a closed interval, as follows:

$$[A^L(\beta), A^{\text{ul}}(\beta)] = [\frac{(1-\beta)b + (\beta - \mu\_A)a}{1 - \mu\_A}, \frac{(1-\beta)b + (\beta - \mu\_A)c}{1 - \mu\_A}] \tag{4}$$

Thus, one can obtain the projections of a TIFN on a real line with respect to the shape of membership and non-membership functions and the desirable level of these functions. The obtained *α*-cut and *β*-cut of a certain TIFN can be further used in, e.g., comparing the underlying TIFNs.

#### *2.2. Bonferroni Mean*

This subsection discusses the properties of the Bonferroni mean and its relevance to decision making problems. There have been different aggregation operators established in the literature, serving a number of objectives with respect to the nature of the data aggregated, preferences of the decision makers, and the interaction among the arguments. One of the topical issues the users of the aggregation operators needs to consider is the possible interrelationships among the data. This is particularly important in such cases where some deviating inputs may distort the result of aggregation and thus render a less meaningful outcome of the MADM. The deviating inputs may occur either due to measurement errors or due to biased expert ratings (whether intentionally or unintentionally). In order to avoid such situations, there have been some aggregations operators controlling for the degree of interrelationships among the data.

The BM can be applied in order to ensure that the interlinkages existing among the data are taken into account during the analysis. The BM was introduced by [17]. Later on, the BM-based aggregation operator was presented in order to allow for effective decision making based on possible interrelated data by Yager [18]. Thus, the BM aggregation operator can be employed for MADM. Indeed, the BM generalizes a family of well-known means.

Let there be two non-negative parameters *p*, *q* ≥ 0 along with a set of n non-negative arguments *ai*, *i* = 1, 2, ··· , *n*. Then, if

$$BM^{p,q}(a\_1, a\_2, \cdots, a\_n) = \left(\frac{1}{n(n-1)} \sum\_{\substack{i,j=1 \\ i \neq j}}^n a\_i^p a\_j^q \right)^{\frac{1}{p+q}},\tag{5}$$

*BMp*,*<sup>q</sup>* is termed the Bonferroni Mean (*BM*). Indeed, the following characteristics can be attributed to the BM:


The different combinations of the parameters *p* and *q* result in special cases of the BM representing various types of means. Especially setting either of the parameters to zero results in the family of mean operators involving no interactions among the arguments. Thus, setting *q* = 0 and considering Equation (1), one arrives at the following kind of aggregation:

$$BM^{p,0}(a\_1, a\_2, \dots, a\_n) = \left(\frac{1}{n(n-1)} \sum\_{\substack{i,j=1 \\ i \neq j}}^n a\_i^p a\_j^0\right)^{\frac{1}{p+1}} = \left(\frac{1}{n} \sum\_{i=1}^n a\_i^p\right)^{\frac{1}{p}},\tag{6}$$

which represents a generalized mean operator outlined in [19]. In general, higher values of *p* for fixed *q* imply greater importance of the larger values. By further modifying the parameters governing the aggregation, one can obtain the special cases of the BM as follows:

• If one sets *p* = 2, *q* = 0, then the interactions are ignored and higher values of the arguments are additionally rewarded and Equation (6) becomes the square mean:

$$BM^{2,0}(a\_1, a\_2, \cdots, a\_{\text{ll}}) = \left(\frac{1}{n} \sum\_{i=1}^n a\_i^2\right)^{\frac{1}{2}}.\tag{7}$$

• If one assumes *p* = 1, *q* = 0, then interactions remain ignored and arguments do not benefit from showing higher values, with Equation (6) becoming the arithmetic average:

$$BM^{1,0}(a\_1, a\_2, \cdots, a\_n) = \frac{1}{n} \sum\_{i=1}^n a\_i. \tag{8}$$

• If one picks the boundary condition *p* → ∞, *q* = 0, then the interactions remain ignored, with the greatest importance put on the largest argument, i.e., Equation (6) boils down to the maximum operator:

$$\lim\_{p \to \infty} BM^{p,0}(a\_1, a\_2, \cdots, a\_n) = \max\_i \{a\_i\}. \tag{9}$$

• If the boundary condition is set with *p* → 0, *q* = 0, then the interactions among the arguments are ignored and the lowest values become the most important ones, with Equation (6) being reduced to the geometric mean operator:

$$\lim\_{p \to 0} BM^{p,0}(a\_1, a\_2, \cdots, a\_n) = \left(\prod\_{i=1}^n a\_i\right)^{\frac{1}{n}}.\tag{10}$$

In the case where one assumes positive values for both of the parameters, similar operators merge. However, they account for the interactions among the arguments in the latter case. Let *p* = 1, *q* = 1, then Equation (6) takes the following form:

$$BM^{1,1}(a\_1, a\_2, \cdots, a\_n) = \left(\frac{1}{n} \sum\_{i=1}^n a\_i \left(\frac{1}{n-1} \sum\_{\substack{j=1 \\ i \neq j}}^n a\_j\right)\right)^{\frac{1}{2}}\tag{11}$$

Up to now, we have not included the preferences of decision makers in the analysis. In order to reflect their taste, the weights can be introduced in the decision making. In order to handle this kind of information, we can further introduce an additional instance of the BM. Let there be two parameters *p*, *q* ≥ 0 and a vector of the arguments to be aggregated *ai* (the elements of the vector are non-negative and indexed over *i* = 1, 2, ··· , *n*). Furthermore, let there be vector weights *w* = (*w*1, *w*2, ··· , *wn*) *T*, such that the weights are non-negative *wi* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, 2, ··· , *<sup>n</sup>*, and normalized *<sup>n</sup>* ∑ *i*=1 *wi* = 1. If the aggregation of the argument vector is carried out in the following manner

$$\text{NNBM}^{p,q}(a\_1, a\_2, \cdots, a\_n) = \left(\sum\_{\substack{i=1 \\ i, j=1 \\ i \neq j}}^n w\_i a\_i^p \frac{w\_j}{1 - w\_i} a\_j^q \right)^{\frac{1}{p+q}}$$

then *NWBMp*,*<sup>q</sup>* is referred to as the normalized weighted Bonferroni mean (*NWBM*) [33]. Some particular cases of the NWBM can be obtained by imposing certain conditions on the weight vector. Indeed, assuming equal weighting, i.e., *wi* = <sup>1</sup> *<sup>n</sup>* , *i* = 1, 2, ··· , *n*, leads to the BM.

#### *2.3. Normalized Weighted Bonferroni Harmonic Mean*

The harmonic means are often used in the decision making due to their desirable properties. Thus, we can consider the harmonic mean in the context of the NWBM in order to improve the decision making process. Let there be two values of parameters *p*, *q* ≥ 0 and a vector of arguments (non-negative numbers) for the aggregation *ai*, *i* = 1, 2, ··· , *n*, and let there be the underlying vector of the argument weights *w* = (*w*1, *w*2, ··· , *wn*) *<sup>T</sup>*, satisfying the non-negativity condition *wi* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, 2, ··· , *<sup>n</sup>*, and the normalization condition *<sup>n</sup>* ∑ *i*=1 *wi* = 1. Given these premises, the following aggregation operator

$$\mathbb{N}^{\mathcal{W}B H M^{p,q}\_{\mathcal{A}}(a\_1, a\_2, \cdots, a\_n)} = \frac{1}{\left(\brace{\sum\_{\begin{subarray}{c} \text{in} \\ i, j = 1 \ \text{a}\_i^{\text{in}} \ (1 - w\_i) \text{a}\_j^{\text{q}} \\ \text{i } j = 1 \end{subarray}}^{\text{I}} \right)^{\frac{1}{p + q}}}$$

can be established and *NWBHMp*,*<sup>q</sup>* is referred to as the normalized weighted Bonferroni Harmonic Mean (NWBHM). The *NWBHMp*,*<sup>q</sup>* features similar properties to the BM; however, there are certain superiorities. In general, the NWBHM features idempotency, monotonicity, commutativity, and boundedness.

#### *2.4. A Ranking Approach for TIFNs*

As the prioritization of the alternatives remains the focus of the MADM, the ranking of fuzzy ratings is important in order to identify the most desirable decision. This can be achieved by applying certain ranking procedures for TIFNs in our case. Thus, this section presents a relatively new approach towards ranking the TIFNs. The ranking is based on the concept of the (*α*, *β*)-cut of the TIFNs. The TIFNs are represented by the interval numbers due to the applications of the (*α*, *β*)-cut, whereas the resulting interval numbers are ranked by applying the concept of the probability of dominance [34]. The ranking of the intervals representing the TIFNs allows one to draw conclusions on the ranking of the underlying TIFNs.

Let *a* = [*aL*, *aU*] and *b* = [*bL*, *bU*] be the two interval numbers, where the endpoints are represented by the ordered values so that *<sup>a</sup><sup>L</sup>* ≤ *<sup>a</sup><sup>U</sup>* and *<sup>b</sup><sup>L</sup>* ≤ *<sup>b</sup>U*. Note that if *<sup>a</sup><sup>L</sup>* = *<sup>a</sup>U*, then the interval number degenerates to a real number *a* .

Let a and b be any two real numbers, and then the probability of a > b is defined as follows:

$$p(a>b) = \begin{cases} 1, a>b; \\ 0.5, a=b; \\ 0, a$$

Let there be the two arbitrarily chosen interval numbers, *a* = [*aL, aU*] and *b* = [*bL, bU*]. For these two numbers, the probability of dominance of *a* over *b*, i.e., *a* ≥ *b*, can be calculated as follows:

$$p(a \ge b) = \frac{\max\left\{0, L(a) + L(b) - \max\left\{b^{II} - a^L, 0\right\}\right\}}{L(a) + L(b)}\tag{12}$$

where the width of the intervals is defined as *<sup>L</sup>*(*a*) = *<sup>a</sup><sup>U</sup>* − *<sup>a</sup><sup>L</sup>* and *<sup>L</sup>*(*b*) = *<sup>b</sup><sup>U</sup>* − *<sup>b</sup>L*. The resulting probability *p*(*a* ≥ *b*) features a number of properties [34]:


Up to now, we have focused on the case of two interval numbers. However, decision making often requires considering more than two interval numbers (e.g., comparison of more than two alternatives). We can, thus, extend the case of the two interval numbers to the general case of multiple interval numbers following [34]. Let there be *m* TIFNs defined in terms of the parameters of the membership and non-membership functions *Ai* = - [*ai*, *bi*, *ci*]; *ωAi* , *uAi* , *i* = 1, 2, ··· , *m*. The ranking of the TIFNs based on the probability of dominance can be carried out in the following manner:

**Step 1.** For each TIFN, compute the (*α*, *β*)-cut by using Equations (3) and (4), where parameters *α* and *β* are chosen with respect to the extreme values of the membership and non-membership functions for a given set of TIFNs so that 0 ≤ *<sup>α</sup>* ≤ ∧*<sup>m</sup> <sup>i</sup>*=1*ωAi* , ∨*<sup>m</sup> <sup>i</sup>*=1*uAi* ≤ *β* ≤ 1 and 0 ≤ *α* + *β* ≤ 1. The resulting interval numbers representing the TIFNs are given by:

$$A\_i(\alpha) = [A\_i^L(\alpha), A\_i^{\text{LI}}(\alpha)], \\ A\_i(\beta) = [A\_i^L(\beta), A\_i^{\text{LI}}(\beta)]$$

where the decision-maker sets the values of *α, β*.

**Step 2.** Calculate the composite interval capturing both the membership and non-membership functions for a certain TIFN:

$$\begin{aligned} A\_i(\lambda) &= [A\_i^L(\lambda), A\_i^{\vert L}(\lambda)] = \lambda A\_i(a) + (1 - \lambda) A\_i(\beta) \\ &= [\lambda A\_i^L(a) + (1 - \lambda) A\_i^L(\beta), \lambda A\_i^{\vert L}(a) + (1 - \lambda) A\_i^{\vert L}(\beta)], (i = 1, 2, \cdots, m) \end{aligned}$$

where *λ* ∈ [0, 1] represents the risk aversion of the decision maker as represented by the lower and upper values of the intervals covered by the membership and non-membership functions for the given levels of *α* and *β* (lower values of *λ* imply higher risk aversion of the decision maker).

**Step 3.** Establish the preference relations matrix representing pairwise comparisons among all the alternatives:

$$P = \left(p\_{ij}\right)\_{m \times m'} \tag{13}$$

where the elements of *P* are given as *pij* = *p*(*Ai* ≥ *Aj*) = *p*(*Ai*(*λ*) ≥ *Aj*(*λ*)) based on Equation (12) for 1 ≤ *i* ≤ *m*, 1 ≤ *j* ≤ *m*.

**Step 4.** Aggregate results of the pairwise comparisons for each alternative by calculating the ranking indicator *RI*(*Ai*) as follows [34]:

$$RI(A\_i) = \frac{1}{m(m-1)} \left(\frac{m}{2} - 1 + \sum\_{j=1}^{m} p\_{ij}\right) \tag{14}$$

**Step 5.** The TIFNs are ranked with respect to the associated values of the ranking indicator *RI*(*Ai*), *i* = 1,2, ..., *m,* so that higher values of the indicator imply higher ranking of the alternatives.

#### *2.5. Normalized Weighted Triangular Intuitionistic Fuzzy Bonferroni Harmonic Mean*

In Section 2.3, we presented the NWBHM operator for the real numbers. In order to process the TIFNs, we extend the NWBHM operator. Specifically, the NWTIFBHM operator is proposed. The proposed aggregation operator can be applied for decision making based upon the TIFNs.

For *p*, *q* ≥ 0, let there be a collection of the TIFNs *Ai* = ([*ai*, *bi*, *ci*], *ωAi* , *uAi* ), *i*=1,2,...,*n,* defined on the positive part of the real line along with the associated weight vector *w* = (*w*1, *w*2, ··· , *wn*) *T*, such that *wi* <sup>≥</sup> 0, for *<sup>i</sup>* = 1,2,...,*n*, and *<sup>n</sup>* ∑ *i*=1 *wi* = 1. If

$$\begin{aligned} \text{NNTIIFBHM}^{p,q}(A\_1, A\_2, \cdots, A\_n) &= \frac{1}{\left(\begin{array}{c} \\ \oplus^{\text{n}} \\ i, j = 1 \\ i \neq j \end{array}\right) \left(\begin{array}{c} \text{w}\_{i} \\ \left(\begin{array}{c} \text{w}\_{i} \\ (1 - \text{w}\_{i})A\_{i}^{\text{p}} \end{array}\right) \left(\begin{array}{c} \text{w}\_{j} \\ \text{A}\_{j}^{\text{q}} \end{array}\right) \end{aligned} \tag{15}$$

then *NWTIFBHMp*,*<sup>q</sup>* is termed the normalized weighted triangular intuitionistic fuzzy Bonferroni Harmonic mean (*NWTIFBHM*). We can derive the following results given the operational laws for the TIFNs stipulated in Equations (1)–(4).

Let there be *p*, *q* ≥ 0 and a collection of positive TIFNs to be aggregated, *Ai* = ([*ai*, *bi*, *ci*], *ωAi* , *uAi* ), *i* = 1, 2, ... , *n,* TIFNs, with weight vector *w* = (*w*1, *w*2, ··· , *wn*) *<sup>T</sup>*, such that *wi* <sup>≥</sup> 0, (*<sup>i</sup>* = 1,2,...,*n*) and *<sup>n</sup>* ∑ *i*=1 *wi* = 1. The given set of TIFNs can be aggregated by the NWTIFBHM operator and the result of aggregation is also a TIFN. Specifically, the result of the aggregation is defined as follows (Proof see Appendix A):

$$\begin{array}{c} \text{NWTIFBHM}^{p,q}(A\_1, A\_2, \dots, A\_n) = & ( [\underbrace{\frac{1}{(\underbrace{\sum\limits\_{i}}}{\sum\limits\_{i}} \frac{\mu\_{\frac{1}{\sqrt{n}}\mu\_{i}}{\sqrt{\frac{n}{n}}\mathforalli}\frac{\mu\_{\frac{1}{\sqrt{n}}\mu\_{i}}{\sqrt{\frac{n}{n}}}}}{\psi\_{i,j} = 1}, \overbrace{\begin{array}{c} \frac{\mu\_{\frac{1}{\sqrt{n}}\mu\_{i}}}{\sum\limits\_{i} \frac{\mu\_{\frac{1}{\sqrt{n}}\mu\_{i}}{\sqrt{\frac{n}{n}}\mathforalli}\frac{\mu\_{\frac{1}{\sqrt{n}}\mu\_{i}}{\sqrt{\frac{n}{n}}}}}^{\frac{1}{\sqrt{n}}} \quad \left( \underset{i,j}{\frac{\mu}{\sum\limits\_{i} \frac{\mu\_{i}\mu\_{j}}{\sqrt{\frac{n}{n}}\mathforalli}\frac{\mu\_{\frac{1}{\sqrt{n}}\mu\_{i}}{\sqrt{\frac{n}{n}}}}}{\psi\_{i,j} = 1} \right); \end{array} \tag{16}$$

The desirable properties of the *NWTIFBHM* operator can be proved by exploiting the relevant theorems. The main results are presented below.

**Idempotency.** If there exists a collection of TIFNs *Ai, i* = 1,2,...,*n,* where all the elements are equal to a certain value, i.e., *Ai* = *A* = ([*a*, *b*, *c*], *ωA*, *uA*), then the application of the NWTIFBHM operator results in that value:

$$NWTIFBHMM^{p,q}(A\_1, A\_2, \cdots, A\_n) = NWTIFBHHM^{p,q}(A, A\_i, \cdots, A) = A...$$

**Commutativity.** Let there be a set of positive TIFNs, *Ai* = ([*ai*, *bi*, *ci*], *ωAi* , *uAi* ), *i* = 1,2,...,*n,* and let there be a permutation of (*A*1, *<sup>A</sup>*2, ··· , *An*) denoted by (*A*1, *<sup>A</sup>*2, ··· , *<sup>A</sup>n*). Then, the following relationship holds:

$$NWTIFBHMM^{p,q}(\widetilde{A}\_1, \widetilde{A}\_2, \dots, \widetilde{A}\_n) = NWTIFBHHM^{p,q}(A\_1, A\_2, \dots, A\_n).$$

**Monotonicity.** Let there be the two sets of TIFNs, *Ai* = ([*ai*, *bi*, *ci*], *ωAi* , *uAi* ) and *Ai* = ([*ai*, *bi*, *ci*], *ωAi* , *uAi* ), with *<sup>i</sup>* = 1,2,...,*n*. If *ai* ≥ *ai*, *bi* ≥ *bi*, *ci* ≥ *ci*, *<sup>ω</sup>Ai* ≥ *<sup>ω</sup>Ai* and *uAi* ≥ *uAi* for all *i*. Then, the results of aggregation are also related in the same manner. Formally,

$$NWTIFBHM^{p,q}(A\_1, A\_2, \cdots, A\_n) \ge NWTIFBHM^{p,q}(\overline{A}\_1, \overline{A}\_2, \cdots, \overline{A}\_n)$$

**Boundedness.** Let there be a collection of TIFNs denoted by *Ai* = ([*ai*, *bi*, *ci*], *ωAi* , *uAi* ), *i* = 1, 2, ··· , *n*. Furthermore, let there be negative and positive ideal solutions associated with the set defined by *A*<sup>−</sup> = ([∧*iai*, ∧*ibi*, ∧*ici*], ∧*iωAi* , ∨*iuAi* ) and *<sup>A</sup>*<sup>+</sup> = ([∨*iai*, <sup>∨</sup>*ibi*, <sup>∨</sup>*ici*], <sup>∨</sup>*iωAi* , ∧*iuAi* ), respectively. Then, the result of aggregation by the NWTIFBHM is bounded by those two ideal solutions as follows:

$$A^{-} \le NWTIFBHIM^{p,q}(A\_1, A\_2, \cdots, A\_n) \le A^{+} $$

The ordered aggregation operators consider the position of the ordered arguments. Thus, the ordered NWTIFBHM (NWTIFOBHM) operator can be defined. Let there be *p*, *q* ≥ 0 and let there be a set of TIFNs denoted by *Ai* = ([*ai*, *bi*, *ci*], *ωAi* , *uAi* ), *i* = 1,2,...,n. Assume there are weights associated with the i-th largest value such that *wi* <sup>≥</sup> 0, *<sup>i</sup>* = 1,2,...,*n*, and *<sup>n</sup>* ∑ *i*=1 *wi* = 1. Then, the application of the NWTIFOBHM results in a TIFN as defined below:

*NWT IFOBHMp*,*q*(*A*1, *<sup>A</sup>*2, ··· , *An*) = <sup>1</sup> <sup>⎛</sup> ⎜⎜⎜⎜⎜⎜⎝ ⊕*n i*, *j* = 1 *i* = *j wi* (1−*wi*)*A<sup>p</sup> σ*(*i*) ⊗ *wj Aq σ*(*j*) ⎞ ⎟⎟⎟⎟⎟⎟⎠ 1 *p*+*q* = ([ <sup>1</sup> ( *<sup>n</sup>* ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*<sup>a</sup> p σ*(*i*) *a q σ*(*j*) ) 1 *p*+*q* , <sup>1</sup> ( *<sup>n</sup>* ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*<sup>b</sup> p σ*(*i*) *b q σ*(*j*) ) 1 *p*+*q* , <sup>1</sup> ( *<sup>n</sup>* ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*<sup>c</sup> p σ*(*i*) *c q σ*(*j*) ) 1 *p*+*q* ]; ∧*n <sup>i</sup>*=1*ωAi* , ∨*<sup>n</sup> <sup>i</sup>*=1*uAi* ) (17)

where the ordered arguments are denoted by *Aσ*(*i*) = ([*aσ*(*i*), *bσ*(*i*), *cσ*(*i*)], *ωAσ*(*i*) , *uAσ*(*i*) ), *i* = 1, 2, ··· , *n*, and (*σ*(1), *σ*(2), ··· , *σ*(*n*)) is a permutation of {1,2,...,*n*}, ensuring the ordering of the arguments, i.e., *Aσ*(*i*−1) ≥ *Aσ*(*i*) for *i* = 2, 3, ..., *n*.

#### **3. MAGDM Based on the Triangular Intuitionistic Fuzzy Information and the NWTIFBHM Operator**

This section presents the MAGDM approach based on the proposed aggregation indicators. An empirical example is provided. Finally, the comparative analysis is carried out in order to compare the proposed framework against the existing ones.

#### *3.1. MAGDM Framework*

The MAGDM problem can be solved by applying the NWTIFBHM operator to aggregate the decision information for the alternatives under consideration. This sub-section outlines the main stages of the MAGDM based upon the NWTIFBHM operator.

Let there be a finite set of *n* alternatives, *X* = {*X*1, *X*2, ··· , *Xn*}, and a finite set of *m* criteria, *C* = {*C*1, *C*2, ··· , *Cm*}. The MAGDM problem involves decision makers *Dt*, *t* = 1, 2, ... , *T*, with associated decision matrices *At* = (*Atij*) *<sup>n</sup>*×*m*, where elements thereof represent the ratings of each alternative against each criterion. The ratings provided by the experts are aggregated and the organized in the aggregate decision matrix *A* = (*Atij*) *<sup>n</sup>*×*m*.

**Step 1.** Establish the individual decision matrices *At*. The weights of criteria are arranged into vector *w*. Note that the weights can be established based on objective methods (e.g., entropy) or subjective ones (e.g., pair-wise comparisons).

**Step 2.** Aggregate the ratings provided by the decision makers for each alternative and criterion. The NWTIFBHM operator given by Equation (16) can be applied (assuming *p* = *q* = 1) for the aggregation. The resulting elements of the aggregate matrix are thus defined as:

$$A\_{t\_i} = \text{NWTIFBH}\text{HM}^{p,q}(A\_{t\_{ij}}) \\ j = 1,2,\cdots,m; \\ t = 1,2,\cdots,T.$$

**Step 3** Calculate the final fuzzy utility scores for each alternative considering all the criteria and experts respectively by exploiting Equation (16).

Calculate the ranking indicator defined by Equation (14) for each fuzzy utility score *At* representing the overall performance of alternative *Xi, i* = 1,2, ..., *n*.

**Step 4.** Rank the alternatives based on the values of the ranking indicator *RI*(*At*) by assigning the highest ranks to the alternatives featuring the highest values of *RI*(*At*).

#### *3.2. Application for the Case of Search and Rescue Robot Selection*

In order to illustrate the possibilities for application of the proposed framework for the MAGDM problem, this sub-section presents its application to the case of the selection of search and rescue robots. This particular illustration is important in the sense that the performance of search and rescue robots is rather crucial for handling emergencies [35]. Accordingly, the performance of search and rescue robots should be assessed in a comprehensive manner.

Given the suggestions provided by the earlier literature [35], we consider four criteria when evaluating the performance of search and rescue robots, including: (1) viability—*C*1, (2) athletic ability—*C*2, (3) working ability—*C*3, and (4) communication control capability—*C*4. Assume there are four search and rescue robots *Xi* (*i* = 1,2,3,4) to be evaluated. Furthermore, the evaluation relies on expert opinions (i.e., one needs to solve an MAGDM problem). The experts provide their ratings for each alternative against the four criteria. The resulting individual decision matrices are outlined in Tables 1–4. The group of experts is assumed not to be a completely homogenous one. Accordingly, the experts are assigned with different weights arranged into vector *η* = (0.20,0.30,0.35,0.15)T, where each element is associated with a corresponding expert *Dt* (*t* = 1,2,3,4).




**Table 2.** Decision matrix *A*<sup>2</sup> given by expert *D*2.

**Table 3.** Decision matrix *A*<sup>3</sup> given by expert *D*3.


**Table 4.** Decision matrix *A*<sup>4</sup> given by expert *D*4.


The decision matrices *At* are constructed and the decision making proceeds as follows:

**Step 1.** Provide decision matrices *At*, *t* = 1, 2, 3, 4, and the weight vector of criteria *w* = (0.22, 0.20, 0.28, 0.30) *T*.

Utilize the NWTIFBHM operator as defined by Equation (A1) with *p* = *q* = 1 on individual decision matrices to obtain the group ratings associated with each alternative under consideration given the assessments provided by the four experts. Table 5 presents the aggregate decision matrix.


**Table 5.** The overall performance value *Ati* ,(*i*, *t* = 1, 2, 3, 4) by decision makers.

**Step 2.** The overall utilities are obtained for the alternatives under consideration. Decision makers' rankings of all the alternatives are calculated and the weight vector *η* = (0.20, 0.30, 0.35, 0.15) *<sup>T</sup>* of decision makers and the aggregated value are given as follows:

*A*<sup>1</sup> = ( [ 0.2353 0.3605 0.4385];0.5000, 0.4000),*A*<sup>2</sup> = ([ 0.2758 0.3891 0.4810];0.6000, 0.4000 ), *A*<sup>3</sup> = ( [ 0.2433 0.4203 0.5393];0.5000, 0.4000), *A*<sup>4</sup> = ( [ 0.3213 0.5189 0.6381];0.5000, 0.4000 ).

**Step 3.** The overall utility scores are expressed in the TIFNs. Therefore, we further utilize the probabilistic ranking approach outlined in Section 2.4 The ranking indicators are obtained by assuming *α* = *β* = *λ* = 0.5. The following values of the ranking indicator are obtained for each alternative *Xi*:

$$RI(A\_1) = 0.1154, \ RI(A\_2) = 0.1923, \ RI(A\_3) = 0.2692, \ RI(A\_4) = 0.3462... $$

**Step 4.** Given the values of the ranking indicator, the following ranking is obtained: *RI*(*A*4) > *RI*(*A*3) > *RI*(*A*2) > *RI*(*A*1). *X*<sup>4</sup> is identified as the most preferable (in the sense of the underlying fuzzy utility) search and rescue robot, as evidenced by the associated ranking indicator *RI*(*A*4) showing the largest value among the alternatives.

#### *3.3. Comparative Analysis*

In order to test the performance of the proposed operator, we solve the problem of the selection of the search and rescue robots by applying various aggregation operators, i.e., the weighted power average (TIFWPA) operator [31], weighted power geometric (TIFWPG) operator [36], weighted geometric mean (TIFWGM) operator [16], weighted power harmonic mean (TIFWPHM) operator [25], and weighted arithmetic mean (TIFWAM) operator [37] extended for the TIFNs. The comparative analysis is proceeded by implementing the procedure outlined in Section 3.1 and replacing the NWTIFBHM operator with the abovementioned aggregation operators. This results in the rankings of the alternatives associated with different aggregation operators. The results are summarized in Table 6.

**Table 6.** The ranking order rendered by the different methods.


The results in Table 6 clearly indicate that the use of the aggregation indicators which are not capable of handling extreme deviations in the data (i.e., the TIFWGM [16] and TIFWAM [37] operators) render rather different results from the rest of the operators. At the other end of the spectrum, the operators capable of accounting for possibly biased ratings (i.e., the proposed TIFWPHM operator, the weighted power average operator [31], and the weighted power geometric operator [36]) rendered similar results. It can be noted that all the operators belonging to the latter group can address the issue of the outlying data, yet the approach is different. Specifically, both the TIFWPA operator [31] and TIFWPG operator [36] allow low weights to be assigned for the outlying data and, thus, minimize their influence indirectly. On the other hand, the TIFWPHM operator [25] (here, it is the degenerate form of TrIFWPHM in [25]) focuses directly (due to its harmonic nature) on the outlying data to reduce the influence thereof on the final results of the aggregation. The NWTIFBHM showed the same best alternative, yet the ranking X3 appeared to be better in this case (the NWTIFBHM showed the same best alternative, yet the ranking X4 appeared to be better in this case).

Therefore, the proposed NWTIFBHM operator is suitable for dealing with situations where different importance of the arguments should be established given possibly biased rankings and the resulting inter-relationship patterns.

We further analyze the performance of the proposed NWTIFBHM operator by adjusting the underlying parameters. Specifically, parameters *α* and *β* determine the degree of uncertainty when constructing the (*α*, *β*)-cuts representing the underlying TIFNs, whereas parameter *λ* reflects the risk version when comparing the TIFNs. We will test the impact of changes in the values of these parameters on the results of the aggregation and ranking of the alternatives.

First, we fix the values of the parameter *α* = *β* = 0.5 and allow *λ* to vary, i.e., *λ* ∈ [0, 1]. The ranking is repeated for several values of *λ* and the results are summarized in Table 7. As one can note, the resulting ranking order is stable based on NWTIFBHM with fixed (*α*, *β*). Figure 1 presents the results graphically and depicts the resulting ranking indicators for each alternative under

different parameter values. As it can be seen from Figure 1, as *α* = *β* = 0.5, given the changes of *λ* (within interval defined by *λ* ∈ [0, 1]), the stability of the ranking remains rather high.


**Table 7.** The ordering of different *λ* based on NWTIFBHM operator (*α* = *β* = 0.5). λ

**Figure 1.** Sensitivity analysis of NWTIFBHM evaluation results (*α* = *β* = 0.5).

Note: For the convenience of observation, the curves for *λ* = 0.1 and *λ* = 0.4 are shifted up and down by 0.01 units, respectively, and the curves for *λ* = 0.6 and *λ* = 0.9 are coincident; x-axis represents the alternatives under consideration.

Second, we allow parameters *α* or *β* to change with *λ* remaining fixed at 0.5 (either *β* or *α* remains fixed at 0.5 too). Since 0 ≤ *α* ≤ *wA*, 0 ≤ *uA* ≤ *β* ≤ 1, we consider *α* ∈ [0, 0.5] and *β* ∈ [0.4, 1] in the numerical example. The results are given in Tables 8 and 9. It is easy to see that the proposed approach is specific, with a rather high stability of the results.


**Table 8.** The ordering of different *α* based on NWTIFBHM operator (*λ* = *β* = 0.5).


**Table 9.** The ordering of different *β* based on NWTIFBHM operator (*λ* = *α* = 0.5).

Table 8 and Figure 2 present the results when parameter *α* varies for the fixed values of *β* and *λ*. As shown in Figure 2, as *λ* = *β* = 0.5, the changes in *α* within *α* ∈ (0, 0.5) that induce greater changes in the ranking indicator for robots *X*1, *X*3, *X*<sup>4</sup> are affected to a higher degree, but the overall stability, sorting results remain unchanged. Thus, the changes can be considered to be more quantitative than qualitative.

**Figure 2.** Sensitivity analysis of NWTIFBHM evaluation results (*λ* = *β* = 0.5).

Table 9 and Figure 3 deal with the case where *β* varies for fixed *α* and *λ*. As shown in Figure 3, as *λ* = *α* = 0.5, the values of the ranking indicator for robots *S*1, *S*<sup>3</sup> are more sensitive to changes in *β*, if opposed to the other alternatives. However, the overall ranking remains stable.

The analysis suggests that the proposed aggregation operator performs similarly to the other aggregation operators capable of accounting for the inter-relationships among the data. The changes in the parameters of the operator did not render significant changes in the rankings. Thus, the proposed model can be considered to be effective and stable.

**Figure 3.** Sensitivity analysis of NWTIFBHM evaluation results (*λ* = *α* = 0.5).

#### **4. Conclusions**

Based on the Bonferroni mean, we developed the Bonferroni harmonic mean, which addresses the inter-relationships among the data to be aggregated to a higher extent. Specifically, the outlying observations receive much lower significance without any additional processing. The normalized harmonic Bonferroni mean allows for incorporating the preferences of the decision makers regarding the importance of the arguments to be aggregated. These concepts were integrated with the triangular fuzzy numbers, allowing uncertain information in the decision making problems to be represented. As a result, we have proposed the NWTIFBHM operator.

The new operator was applied in an illustrative example on a MAGDM problem. The comparative analysis comprised two directions: comparison with the existing approaches and sensitivity to changes in the underlying parameters. The analysis showed that the proposed aggregation operator is effective and is not heavily impacted by the changes in the underlying parameters.

Future research can be directed towards extension of the proposed aggregation operator by applying the generalized normalized weighted Bonferroni mean [33], probabilistic averages [38–40], Pythagorean fuzzy sets [12], and Choquet integrals [41], along with combinations thereof [42,43]. Simulation studies can be carried out to check the performance of the proposed approach in different settings [44] and to relate it to databases for real-life situations [45,46]. From the empirical viewpoint, applications of the NWTIFNBH operator for decisions in real-life problems can be considered across different sectors.

#### **Author Contributions:** Writing—original draft, J.Z.; Writing—review & editing, T.B. and D.S.

**Funding:** The work was supported in part by the National Social Science Fund Project (No. 2017YYRW07), Anhui Province Natural Science in Universities of General Project (No. TSKJ2017B22), and the general project of teaching research project of Anhui Polytechnic University (No. 2018JYXM43).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

In this appendix, we provide the proof of Equation (16).

**Proof.** Utilizing the principles of the operational laws for the TIFNs, one can obtain

$$\left(\frac{w\_i}{(1-w\_i)A\_i^{\mathcal{V}}}\right)\otimes \left(\frac{w\_j}{A\_j^{\mathcal{V}}}\right) = \left([\frac{w\_i w\_j}{(1-w\_i)a\_i^p a\_j^{q'}}, \frac{w\_i w\_j}{(1-w\_i)b\_i^p b\_j^{q'}}, \frac{w\_i w\_j}{(1-w\_i)c\_i^p c\_j^{q'}}\right]; \omega\_{A\_i}\wedge\omega\_{A\_j}, \mu\_{A\_i}\vee\mu\_{A\_j}\right)$$

Initially, one can derive that

⊕*n i*, *j* = 1 *i* = *j wi* (1−*wi*)*A<sup>p</sup> i* ⊗ *wj Aq j* = ([ *<sup>n</sup>* ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*a p i a q j* , *<sup>n</sup>* ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*b p i b q j* , *<sup>n</sup>* ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*c p i c q j* ]; ∧*<sup>n</sup> <sup>i</sup>*=1*ωAi* , ∨*<sup>n</sup> <sup>i</sup>*=1*uAi* ) (A1)

By exploiting the principle of mathematical induction upon *n* in the following manner:

(1) when *n* = 2, given (15), we can show:

⊕2 *i*, *j* = 1 *i* = *j* (( *wi* (1−*wi*)*A<sup>p</sup> i* ) ⊗ ( *wj Aq j* )) = (( *<sup>w</sup>*<sup>1</sup> (1−*w*1)*A<sup>p</sup>* 1 ) <sup>⊗</sup> ( *<sup>w</sup>*<sup>2</sup> *Aq* 2 )) <sup>⊕</sup> (( *<sup>w</sup>*<sup>2</sup> (1−*w*2)*A<sup>p</sup>* 2 ) <sup>⊗</sup> ( *<sup>w</sup>*<sup>1</sup> *Aq* 1 )) = ([ *<sup>w</sup>*1*w*<sup>2</sup> (1−*w*1)*a p* 1 *a q* 2 + *<sup>w</sup>*2*w*<sup>1</sup> (1−*w*2)*a p* 2 *a q* 1 , *<sup>w</sup>*1*w*<sup>2</sup> (1−*w*1)*b p* 1 *b q* 2 + *<sup>w</sup>*2*w*<sup>1</sup> (1−*w*2)*b p* 2 *b q* 1 , *<sup>w</sup>*1*w*<sup>2</sup> (1−*w*1)*c p* 1 *c q* 2 + *<sup>w</sup>*2*w*<sup>1</sup> (1−*w*2)*c p* 2 *c q* 1 ]; *ωA*<sup>1</sup> ∧ *ωA*<sup>2</sup> , *uA*<sup>1</sup> ∨ *uA*<sup>2</sup> ) = ([ <sup>2</sup> ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*a p i a q j* , <sup>2</sup> ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*b p i b q j* , <sup>2</sup> ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*c p i c q j* ]; ∧<sup>2</sup> *<sup>i</sup>*=1*ωAi* , ∨<sup>2</sup> *<sup>i</sup>*=1*uAi* )

(2) assume that *n* = *k* and Equation (15) holds so that

$$\begin{aligned} \stackrel{\ominus^{k}}{i}\_{i,j} &= 1 \quad \left( \left( \frac{w\_{i}}{(1-w\_{i})A\_{i}^{q}} \right) \odot \left( \frac{w\_{j}}{A\_{j}^{q}} \right) \right) \\ \stackrel{\text{i}}{i} &\neq j \\ &= \left( [ \quad \sum\_{\begin{subarray}{c} 1 \ (1-w\_{i})A\_{i}^{p}a\_{j}^{q} \\ i,j=1 \end{subarray}} \frac{w\_{i}w\_{j}}{(1-w\_{i})a\_{i}^{p}a\_{j}^{q}}, \sum\_{\begin{subarray}{c} 1 \ (1-w\_{i})b\_{i}^{p}b\_{j}^{q} \\ i,j=1 \end{subarray}} \frac{w\_{i}w\_{j}}{(1-w\_{i})c\_{i}^{p}c\_{j}^{q}} \right); \stackrel{\text{i}}{\wedge} &\stackrel{\text{i}}{i}\_{i=1} \omega\_{A\_{i}}, \forall\_{i=1}^{k} \mu\_{A\_{i}} \text{/} \\ &\neq j \end{aligned} \tag{A2}$$

(3) subsequently, assume *n* = *k* + 1 and by the virtue of (15), get

$$\begin{aligned} \stackrel{\oplus}{i}\_{l,j}^{k+1} &= 1 \left( \left( \frac{w\_i}{(1-w\_i)A\_i^l} \right) \otimes \begin{pmatrix} w\_i \\ A\_j^l \end{pmatrix} \right) = \left( \stackrel{\oplus}{\oplus}\_{l,j}^k = 1 \left( \left( \frac{w\_i}{(1-w\_i)A\_i^l} \right) \otimes \begin{pmatrix} w\_i \\ A\_j^l \end{pmatrix} \right) \right) \odot\\ \stackrel{\oplus}{i}\_{l+1}^k &= \left( \stackrel{\oplus}{\oplus}\_{l+1}^k \begin{pmatrix} w\_i \\ A\_{k+1}^l \end{pmatrix} \right) \odot \oplus \left( \stackrel{\oplus}{\ominus}\_{j-1}^k \begin{pmatrix} \left( \frac{w\_{k+1}}{(1-w\_{k+1})A\_{k+1}^l} \right) \odot \begin{pmatrix} w\_i \\ A\_j^l \end{pmatrix} \right) \right) \end{aligned} \tag{A3}$$

We now prove that

$$\begin{split} \mathbb{E}\_{i=1}^{k} \left( \left( \frac{w\_{i}}{(1-w\_{i})A\_{i}^{p}} \right) \odot \left( \frac{w\_{k+1}}{A\_{k+1}^{q}} \right) \right) &= \left( \bigcup\_{i=1}^{k} \frac{w\_{i}w\_{i}}{(1-w\_{i})a\_{i}^{p}a\_{i}^{q}} \Big| \sum\_{i=1}^{k} \frac{w\_{i}w\_{i}}{(1-w\_{i})b\_{i}^{p}b\_{i}^{q}} \Big| \sum\_{i=1}^{k} \frac{w\_{i}w\_{i}}{(1-w\_{i})c\_{i}^{p}c\_{i}^{q}} \right); \\ \wedge\_{i=1}^{k} \left( \omega\_{A\_{i}} \wedge \omega\_{A\_{k+1}} \right) \vee \bigvee\_{i=1}^{k} \left( \mu\_{A\_{i}} \vee \mu\_{A\_{k+1}} \right) \end{split} \tag{A4}$$

By applying the principle of the mathematical induction upon *k*.

*Symmetry* **2019**, *11*, 218

(a) Let *k* = 2, and by the virtue of Equation (A4), one can show

⊕2 *i*=1 *wi* (1−*wi*)*A<sup>p</sup> i* ⊗ *wk*+<sup>1</sup> *Aq k*+1 <sup>=</sup> *<sup>w</sup>*<sup>1</sup> (1−*w*1)*A<sup>p</sup>* 1 ⊗ *wk*+<sup>1</sup> *Aq k*+1 <sup>⊕</sup> *<sup>w</sup>*<sup>2</sup> (1−*w*2)*A<sup>p</sup>* 2 ⊗ *wk*+<sup>1</sup> *Aq k*+1 = ([ *<sup>w</sup>*1*wk*+<sup>1</sup> (1−*w*1)*a p* 1 *a q k*+1 + *<sup>w</sup>*2*wk*+<sup>1</sup> (1−*w*2)*a p* 2 *a q k*+1 , *<sup>w</sup>*1*wk*+<sup>1</sup> (1−*w*1)*b p* 1 *b q k*+1 + *<sup>w</sup>*2*wk*+<sup>1</sup> (1−*w*2)*b p* 2 *b q k*+1 , *w*1*wk*+<sup>1</sup> (1−*w*1)*c p* 1 *c q k*+1 + *<sup>w</sup>*2*wk*+<sup>1</sup> (1−*w*2)*c p* 2 *c q k*+1 ];(*ωA*<sup>1</sup> ∧ *ωAk*+<sup>1</sup> ) ∧ (*ωA*<sup>2</sup> ∧ *ωAk*+<sup>1</sup> ),(*uA*<sup>1</sup> ∨ *uAk*<sup>+</sup><sup>1</sup> ) ∨ (*uA*<sup>1</sup> ∨ *uAk*<sup>+</sup><sup>1</sup> )) (A5) = ([ <sup>2</sup> ∑ *i*=1 *wiwj* (1−*wi*)*a p i a q j* , 2 ∑ *i*=1 *wiwj* (1−*wi*)*b p i b q j* , 2 ∑ *i*=1 *wiwj* (1−*wi*)*c p i c q j* ]; ∧<sup>2</sup> *<sup>i</sup>*=1(*ωAi* ∧ *<sup>ω</sup>Ak*+<sup>1</sup> ), ∨<sup>2</sup> *<sup>i</sup>*=1(*uAi* ∨ *uAk*<sup>+</sup><sup>1</sup> )) (A6)

(b) Assume Equation (A4) is valid for any given *k* = *k*<sup>0</sup>

$$\begin{split} \ominus\_{i=1}^{k\_0} \left( \left( \frac{w\_i}{(1-w\_i)A\_i^q} \right) \odot \left( \frac{w\_{k+1}}{A\_{k+1}^q} \right) \right) \\ = \left( [\sum\_{i=1}^{k\_0} \frac{w\_i w\_j}{(1-w\_i)a\_i^q a\_j^q}, \sum\_{i=1}^{k\_0} \frac{w\_i w\_j}{(1-w\_i)b\_i^q b\_j^q}, \sum\_{i=1}^{k\_0} \frac{w\_i w\_j}{(1-w\_i)c\_i^q c\_j^q} \right); \forall \begin{subarray}{c} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${}\_i\$} \text{\${$$

(c) Subsequently, we demonstrate that the following holds for any *k* = *k*<sup>0</sup> + 1:

⊕*k*0+<sup>1</sup> *i*=1 *wi* (1−*wi*)*A<sup>p</sup> i* ⊗ *wk*+<sup>1</sup> *Aq k*+1 <sup>=</sup> <sup>⊕</sup>*k*<sup>0</sup> *i*=1 *wi* (1−*wi*)*A<sup>p</sup> i* ⊗ *wk*+<sup>1</sup> *Aq k*+1 <sup>⊕</sup> *wk*0+<sup>1</sup> (1−*wk*0+1)*A<sup>p</sup> k*0+1 ⊗ *wk*+<sup>1</sup> *Aq k*+1 = ([ *<sup>k</sup>*<sup>0</sup> ∑ *i*=1 *wiwj* (1−*wi*)*a p i a q j* <sup>+</sup> *wk*0+1*wk*+<sup>1</sup> (1−*wk*0+1)*<sup>a</sup> p <sup>k</sup>*0+1*<sup>a</sup> q k*+1 , *k*0 ∑ *i*=1 *wiwj* (1−*wi*)*b p i b q j* <sup>+</sup> *wk*0+1*wk*+<sup>1</sup> (1−*wk*0+1)*<sup>b</sup> p <sup>k</sup>*0+1*<sup>b</sup> q k*+1 , *k*0 ∑ *i*=1 *wiwj* (1−*wi*)*c p i c q j* <sup>+</sup> *wk*0+1*wk*+<sup>1</sup> (1−*wk*0+1)*<sup>c</sup> p <sup>k</sup>*0+1*<sup>c</sup> q k*+1 ]; ∧*k*0 *<sup>i</sup>*=1(*ωAi* <sup>∧</sup> *<sup>ω</sup>Ak*+<sup>1</sup> ) <sup>∧</sup> (*ωAk*0+<sup>1</sup> <sup>∧</sup> *<sup>ω</sup>Ak*+<sup>1</sup> ), <sup>∨</sup>*k*<sup>0</sup> *<sup>i</sup>*=1(*uAi* <sup>∨</sup> *uAk*<sup>+</sup><sup>1</sup> ) <sup>∨</sup> (*uAk*0+<sup>1</sup> <sup>∨</sup> *uAk*<sup>+</sup><sup>1</sup> ))

Clearly,

$$\wedge\_{i=1}^{k} (\omega\_{A\_i} \wedge \omega\_{A\_{k+1}}) = \wedge\_{i=1}^{k+1} \omega\_{A\_i} \vee\_{i=1}^{k} (\omega\_{A\_i} \vee \omega\_{A\_{k+1}}) = \vee\_{i=1}^{k+1} \omega\_{A\_i}$$

Hence,

$$\begin{split} \ominus\_{i=1}^{k} \left( \left( \frac{w\_{i}}{(1-w\_{i})A\_{i}^{\prime}} \right) \otimes \left( \frac{w\_{k+1}}{A\_{k+1}^{\prime}} \right) \right) \\ = \left( [\sum\_{i=1}^{k} \frac{w\_{i}w\_{k+1}}{(1-w\_{i})a\_{i}^{\prime}a\_{i+1}^{\prime}}, \sum\_{i=1}^{k} \frac{w\_{i}w\_{k+1}}{(1-w\_{i})b\_{i}^{\prime}b\_{k+1}^{\prime}}, \sum\_{i=1}^{k} \frac{w\_{i}w\_{k+1}}{(1-w\_{i})c\_{i}^{\prime}c\_{k+1}^{\prime}} \right); \wedge\_{i=1}^{k+1} \omega\_{A\_{i}\prime} \vee\_{i=1}^{k+1} \mu\_{A\_{i}} \right) \end{split} \tag{A8}$$

Similarly,

$$\begin{split} \bigoplus\_{j=1}^{k} \left( \left( \frac{w\_{k+1}}{(1-w\_{k+1})A\_{k+1}^{p}} \right) \odot \left( \frac{w\_{j}}{A\_{j}^{q}} \right) \right) \\ = \left( \bigcup\_{j=1}^{k} \frac{w\_{k+1}w\_{j}}{(1-w\_{k+1})a\_{k+1}^{p}a\_{j}^{q}}, \sum\_{j=1}^{k} \frac{w\_{k+1}w\_{j}}{(1-w\_{k+1})b\_{k+1}^{p}a\_{j}^{q}}, \sum\_{j=1}^{k} \frac{w\_{k+1}w\_{j}}{(1-w\_{k+1})c\_{k+1}^{p}a\_{j}^{q}} \right); \wedge\_{j=1}^{k+1} \omega\_{A\_{j}r} \vee\_{j=1}^{k+1} u\_{A\_{j}} \end{split} \tag{A9}$$

From Equations (A3), (A8) and (A9), we get

<sup>⊕</sup>*k*+<sup>1</sup> *i*, *j* = 1 *i* = *j wi* (1−*wi*)*A<sup>p</sup> i* ⊗ *wj Aq j* = ([ *k* ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*a p i a q j* , *<sup>k</sup>* ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*b p i b q j* , *<sup>k</sup>* ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*c p i c q j* ]; ∧*<sup>k</sup> <sup>i</sup>*=1*ωAi* , ∨*<sup>k</sup> <sup>i</sup>*=1*uAi* )

⊕([ *k* ∑ *i*=1 *wiwk*+<sup>1</sup> (1−*wi*)*a p i a q k*+1 , *k* ∑ *i*=1 *wiwk*+<sup>1</sup> (1−*wi*)*b p i b q k*+1 , *k* ∑ *i*=1 *wiwk*+<sup>1</sup> (1−*wi*)*c p i c q k*+1 ]; <sup>∧</sup>*k*+<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *ωAi* , <sup>∨</sup>*k*+<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *uAi* ) ⊕([ *k* ∑ *j*=1 *wk*+1*wj* (1−*wk*+1)*a p <sup>k</sup>*+1*a q j* , *k* ∑ *j*=1 *wk*+1*wj* (1−*wk*+1)*b p <sup>k</sup>*+1*b q j* , *k* ∑ *j*=1 *wk*+1*wj* (1−*wk*+1)*c p <sup>k</sup>*+1*c q j* ]; <sup>∧</sup>*k*+<sup>1</sup> *<sup>j</sup>*=<sup>1</sup> *ωAj* , <sup>∨</sup>*k*+<sup>1</sup> *<sup>j</sup>*=<sup>1</sup> *uAj* ) = ([ *k*+1 ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*a p i a q j* , *<sup>k</sup>*+<sup>1</sup> ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*b p i b q j* , *<sup>k</sup>*+<sup>1</sup> ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*c p i c q j* ]; <sup>∧</sup>*k*+<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *ωAi* , <sup>∨</sup>*k*+<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *uAi* )

As a result, Equation (A1) is valid for *n* = *k* + 1. Therefore, Equation (A1) is valid for any *n*. Considering Equation (A1) alongside operational law (3)

1 ⎛ ⎜⎜⎜⎜⎜⎜⎝ <sup>⊕</sup>*k*+<sup>1</sup> *i*, *j* = 1 *i* = *j wi* (1−*wi*)*A<sup>p</sup> i* ⊗ *wj Aq j* ⎞ ⎟⎟⎟⎟⎟⎟⎠ 1 *p*+*q* = ([ <sup>1</sup> ( *<sup>k</sup>*+<sup>1</sup> ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*<sup>a</sup> p i a q j* ) 1 *p*+*q* , <sup>1</sup> ( *<sup>k</sup>*+<sup>1</sup> ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*<sup>b</sup> p i b q j* ) 1 *p*+*q* , <sup>1</sup> ( *<sup>k</sup>*+<sup>1</sup> ∑ *i*, *j* = 1 *i* = *j wiwj* (1−*wi*)*<sup>c</sup> p i c q j* ) 1 *p*+*q* ]; <sup>∧</sup>*k*+<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *ωAi* , <sup>∨</sup>*k*+<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *uAi* ) (A10)

Exploiting Equation (A10) as well as operational law (4), one can show that

$$\begin{array}{c} ( [\underbrace{1}\_{\begin{subarray}{c}\sum\limits\_{i}\frac{\boldsymbol{w}\_{i}\boldsymbol{w}\_{j}}{\|\boldsymbol{\uppi}-\boldsymbol{w}\_{i}\|^{2}\boldsymbol{\uppi}\end{subarray}}},\frac{\text{NWTIFBH}\mathbf{H}\mathbf{P}^{\boldsymbol{\upbeta}}(\boldsymbol{A}\_{1},\boldsymbol{A}\_{2},\ldots,\boldsymbol{A}\_{n})}{\mathbf{1}}=\\\ i,j=1\\\ i\neq j\end{array},\begin{array}{c} \frac{\text{NWTIFBH}\mathbf{H}\mathbf{P}^{\boldsymbol{\upbeta}}(\boldsymbol{A}\_{1},\boldsymbol{A}\_{2},\ldots,\boldsymbol{A}\_{n})}{\mathbf{1}},\frac{\text{N}\mathbf{P}^{\boldsymbol{\upbeta}}(\boldsymbol{A}\_{1},\ldots,\boldsymbol{A}\_{n})}{\mathbf{1}}\\\ i,j=1\\\ i\neq j\end{array},\begin{array}{c} \frac{\text{N}\mathbf{P}^{\boldsymbol{\upbeta}}(\boldsymbol{A}\_{1},\ldots,\boldsymbol{A}\_{n})}{\mathbf{1}},\forall\ \boldsymbol{\upbeta}\_{i}\in\operatorname{\mathit{\upbeta}}\_{i}\forall\ \boldsymbol{\upbeta}\_{i}\forall\ \boldsymbol{\upbeta}\_{i}=\mathbf{1}\\\ i,j=1\\\ i\neq j\end{array}\end{array}\tag{A11}$$

As long as *ai* ≤ *bi* ≤ *ci*, for all *i* = 1, 2, ··· , *n*. By the virtue of the property associated with the NWBHM, one can show that

$$\frac{1}{\begin{array}{c} \frac{\imath}{\imath} \\ i, j = 1 \end{array}} \frac{1}{(1 - w\_i) \overline{a\_i'} \overline{a\_j'}} \le \frac{1}{\left( \sum\_{\begin{subarray}{c} \text{\${u}l \text{s} \text{s}} \\ i, j = 1 \end{subarray}} \frac{1}{\left( \sum\_{\begin{subarray}{c} \text{\${u}l \text{s} \text{s}} \\ i \text{-}l \text{s} \text{s} \end{subarray}} \right) \overline{\frac{1}{\overline{r} + q}}} \le \frac{1}{\left( \sum\_{\begin{subarray}{c} \text{\${u}l \text{s} \text{s}} \\ i \text{-}l \text{s} \text{s} \text{s} \text{s} \end{subarray}} \right) \overline{\frac{1}{\overline{r} + q}}}\tag{A12}$$

Also,

$$0 \le \wedge\_{i=1}^{\mathfrak{n}} \omega\_{A\_i} + \vee\_{i=1}^{\mathfrak{n}} \mathfrak{u}\_{A\_i} \le 1 \tag{A13}$$

From Equations (A12) and (A13), *NWTIFBHMp*,*<sup>q</sup>* is a TIFN.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Type-2 Multi-Fuzzy Sets and Their Applications in Decision Making**

#### **Mohuya B. Kar 1, Bikashkoli Roy 2, Samarjit Kar 2, Saibal Majumder <sup>3</sup> and Dragan Pamucar 4,\***


Received: 30 December 2018; Accepted: 28 January 2019; Published: 1 February 2019

**Abstract:** In a real-life scenario, it is undoable and unmanageable to solve a decision-making problem with the single stand-alone decision-aid method, expert assessment methodology or deterministic approaches. Such problems are often based on the suggestions or feedback of several experts. Usually, the feedback of these experts are heterogeneous imperfect information collected from various more or less reliable sources. In this paper, we introduce the concept of multi-sets over type-2 fuzzy sets. We have tried to propose an extension of type-1 multi-fuzzy sets into a type-2 multi-fuzzy set (T2MFS). After defining T2MFS, we discuss the algebraic properties of these sets including set-theoretic operations such as complement, union, intersection, and others with examples. Subsequently, we define two distance measures over these sets and illustrate a decision-making problem which uses the idea of type-2 multi-fuzzy sets. Furthermore, an application of a medical diagnosis system based on multi-criteria decision making of T2MFS is illustrated with a real-life case study.

**Keywords:** Multi-fuzzy sets; type-2 fuzzy sets; type-2 multi-fuzzy sets; distance measures; set operations

#### **1. Introduction**

According to Cantor, a set is a well-defined collection of distinct objects, but let us see what happens if we consider a collection of objects where one or some or all objects occur more than once. The answer to this question is a different type of set known as multi-set or *m*-set which was first studied by Bruijn [1]. Afterwards, Yager [2] proposed a new and more generalized concept of multi-sets named as multi-fuzzy sets (MFS) which can deal with many real life problems with some degree of ease. Sebastian and Ramakrishnan [3] also studied multi-fuzzy sets and concluded that multi-fuzzy set theory is an extension of Zadeh's fuzzy set theory, Atanassov's intuitionistic fuzzy set theory and *L*-fuzzy set theory. Yang et al. [4] discussed applications of multi-fuzzy soft sets in decision making. Furthermore, Das et al. [5] proposed an approach of group multi-criteria decision-making using intuitionistic multi-fuzzy sets.

A type-2 fuzzy set (T2FS) is an extension of ordinary fuzzy sets, i.e., type-1 fuzzy set (T1FS). The membership value of a type-1 fuzzy set is a real number in the closed interval [0, 1]. On the other hand, the membership value of a T2FS is a type-1 fuzzy set. The concept of T2FS was introduced by Zadeh [6–8]. Mizumoto and Tanaka [9,10], and Dubois and Prade [11] investigated the logical operations of T2FS. Later, many researchers did a lot of theoretical work on the properties of T2FS [12–14] and figured out many applications [15–21].

Although there are various mathematical tools such as fuzzy sets, rough sets, multi-sets, multi-fuzzy sets, intuitionistic fuzzy sets, and type-2 fuzzy sets to deal with uncertainties [22–24], there might be some physical problems where the primary membership function may have more than one secondary membership grade with the same or different values. In those particular cases, the existing mathematical tools might not be adequate. That is why we introduce a new concept of type-2 multi-fuzzy sets. Type-2 multi-fuzzy sets are supposedly a new approach which will be an extension of the existing concepts and shall be helpful to deal with problems related to uncertainties. T2MFS is a type-2 fuzzy set whose primary membership function has a sequence of secondary membership values lying in the closed interval [0, 1]. In this paper, we first give definitions of classical multi-sets, multi-fuzzy sets and type-2 fuzzy sets and also provide examples of each.

The main contributions of this article are highlighted as follows:


The rest of the paper is organized as follows. Some relevant studies of the literature are surveyed in Section 2. The preliminary concepts of our study are discussed in Section 3. In Section 4, we define type-2 multi-fuzzy set (T2MFS) and give examples. Subsequently, in Section 5, the algebraic operations on T2MFS like complement, inclusion, union, and intersection are discussed. Consequently, in Section 6, set-theoretic properties like idempotency, commutativity, associativity, and distributivity are verified for T2MFSs. In Section 7, we define the two distance measures of T2MFS. We provide a real-life application based on a medical diagnosis system, which applies the concept of type-2 multi-fuzzy sets in Section 8. In Section 9, we conduct a case study based on the application presented in Section 8. Finally, the study is concluded in Section 10.

#### **2. Literature Review**

In this section, we present a brief overview of different variants of the multi-fuzzy set (MFS) which have been proposed in previous studies. The survey, by no means, encompasses all the related researches in the literature. However, some related studies having significant contributions are reviewed.

The concept of MFS has originated as an extension of the fuzzy set [25], *L*-fuzzy set [26] and intuitionistic fuzzy set [27]. In a MFS, the membership function is an ordered sequence of ordinary fuzzy membership functions. Here, an element of the universe can repeat itself with possibly the same or different membership values. Motivated by the study of Yager [2] on MFS, several contributions can be observed in the field of multi-fuzzy sets and its variants. Muthuraj and Balamurugan [28] proposed some algebraic structures of multi-fuzzy subgroup and investigated their properties. Sebastian and Ramakrishnan [29] proposed the multi-fuzzy subgroup and normal multi-fuzzy subgroups. Furthermore, various bridge function like lattice homomorphisms, order homomorphisms, *L*-fuzzy lattices, and strong *L*-fuzzy lattices have been developed by Sebastian and Ramakrishnan [30]. Subsequently, multi-fuzzy topology was proposed by Sebastian and Ramakrishnan [31]. In addition, a progressive development of MFS can be observed from the contributions of several researchers [32–34].

Enlightened by the development of MFS, Dey and Pal [35] proposed multi-fuzzy complex numbers and multi-fuzzy complex sets. Using the concepts of their studies, the authors introduced multi-fuzzy complex nilpotent matrices over a distributive lattice [36]. The authors also developed multi-fuzzy vector space and multi-fuzzy linear transformation over a finite-dimensional multi-fuzzy set [37].

Shinoj and John [38] introduced the intuitionistic fuzzy multi-sets (IFMS). After that, several investigations were conducted to develop various features of IFMS. Ejegwa and Awolola [39] determined the binomial probability of IFMS, where for each trial, it was assumed that the probability of the membership degree was constant and the intuitionistic fuzzy multi-set index was negligible. Rajarajeswari and Uma [40] proposed three distance measures and their corresponding similarity measures of IFMS. These measures are based on the Hausdorff distance measure, the geometric distance measure and the normalized distance measure. Subsequently, different studies of IFMS [41–44] can be observed in the literature. Besides, Das et al. [5] proposed an efficient approach for group multi-criteria decision-making (MCDM) based on IMFS.

#### **3. Preliminaries**

Before introducing the concept of type-2 multi-fuzzy sets, we first present some essential concepts of a crisp multi-set or *m*-set, multi-fuzzy sets, type-2 fuzzy sets with examples, and set theoretical operations of multi-fuzzy sets.

#### *3.1. Classical Multi-Sets*

A classical multi-set or *m*-set (in short) is a set, where any element of the set may occur more than once. The definition can be found in the work of (Girish and John [45]). An *m*-set *M* drawn from set X is represented by a function Count-*M* or *CM* defined as *CM* : *X* → *N* where *N* represents the set of non-negative integers.

Here, *CM*(*x*) is the number of occurrences of the element *x* ∈ *X* in the *m*-set *M*. We present the *m*-set *M* drawn from the set *X* = {*x*1, *x*2, ··· , *xn*} as *M* = {*m*1/*x*1, *m*2/*x*2, ··· , *mn*/*xn*}, where *mi* is the number of occurrences of the element *xi*, *i* = 1, 2, ··· , *n* in the *m*-set *M*. However, those elements which are not included in the *m*-set *M* have zero count.

**Example 1:** *Let us consider the universal set of some object as X* = {*a*, *b*, *c*, *d*,*e*} *and let object a appear three times, c appear five times, d appear one time and e appear two times in set M. Then, this appearance of the objects can be represented in set form M* = {*a*, *a*, *a*, *c*, *c*, *c*, *c*, *c*, *d*,*e*,*e*}*. It can also be represented in the form as*

$$M = \{ \mathbf{3}/a, \mathbf{0}/b, \mathbf{5}/c, \mathbf{1}/d, \mathbf{2}/c \}$$

*which is a m*-set.

#### *3.2. Multi-Fuzzy Set*

A multi-fuzzy set (Yager [2]) is a fuzzy set, where for each element of the universal set there may be more than one membership value. It can be defined mathematically as follows. Let *X* be a nonempty set. A multi-fuzzy set (MFS) *A* on *X* is characterized by a function, count membership of *A* denoted by *CMA* such that *CMA* : *X* → *Q*, where *Q* is the set of all crisp multi-sets drawn from the unit interval [0, 1]. Then, for any *x* ∈ *X*, the value *CMA* (*x*) is a crisp multi-set drawn from [0, 1]. For each *x* ∈ *X*, the membership sequence is defined as the decreasingly ordered sequence of elements in *CMA* (*x*). It is denoted by *μ*1 *<sup>A</sup>*(*x*), *<sup>μ</sup>*<sup>2</sup> *<sup>A</sup>*(*x*), ··· , *<sup>μ</sup><sup>p</sup> <sup>A</sup>*(*x*) , where *μ*<sup>1</sup> *<sup>A</sup>*(*x*) ≥ *<sup>μ</sup>*<sup>2</sup> *<sup>A</sup>*(*x*) ≥··· ≥ *<sup>μ</sup><sup>p</sup> <sup>A</sup>*(*x*).

**Example 2:** *Let us consider fuzzy set A as follows*

$$R = \{0.7/\ge, 0.5/\ge, 0.2/\ge, 1.0/y, 1.0/y, 0.4/y, 0.6/y, 0.3/z\}$$

*of the universal set X* = {*x*, *y*, *z*}*. From this fuzzy set, we see that the element x occurs three times with membership values* 0.7, 0.5 *and* 0.2 *respectively; the element y occurs four times with membership values* 1.0, 1.0, 0.4, *and* 0.6 *respectively and the element z occurs once with a membership value* 0.3*. Thus, the set can be rewritten in the form as*

$$R = \{ (0.7, 0.5, 0.2) / \ge (1.0, 1.0, 0.4, 0.6) / y, 0.3/z \}.$$

*which essentially is a multi-fuzzy set.*

The graphical representation of the multi-fuzzy set *R* is shown in Figure 1, where we consider *x* = 3, *y* = 7 and *z* = 9.

**Figure 1.** The membership function of the multi-fuzzy set *R* is represented by four different combinations of membership functions of 3, 7 and 9, where (**a**) the graphical representation of 0.7/3 + 1.0/7 + 0.3/9, (**b**) the graphical representation of 0.5/3 + 1.0/7, (**c**) the graphical representation of 0.2/3 + 0.4/7 and (**d**) the graphical representation of 0.6/7 are presented respectively.

We now discuss some basic operations such as inclusion, equality, union, and intersection of MFSs. Let *A* and *B* be two MFSs defined on *X*.

3.2.1. Inclusion

$$A \subseteq B \iff \mu\_A^j(\mathbf{x}) \le \mu\_B^j(\mathbf{x}), \ j = 1, 2, \dots, \ l\_\star(\mathbf{x}), \ \forall \mathbf{x} \in X,$$
 
$$\text{where } L(\mathbf{x}) = L(\mathbf{x}; A, B) = \max\{L(\mathbf{x}; A), L(\mathbf{x}; B)\} \text{ and } L(\mathbf{x}; A) = \max\{j : \mu\_A^j(\mathbf{x}) \ne 0\}.$$

**Example 3:** *Let us consider two multi-fuzzy sets A and B over a nonempty universe X as*

$$A = \{ (0.8, 0.6, 0.5) / x, (0.6, 0.4, 0.2) / y, (0.7, 0.1) / z \}$$

$$B = \{ (0.8, 0.7, 0.6) / \ge (0.9, 0.8, 0.4) / y, (1.0, 0.8, 0.5) / z \}.$$

*Then, from the definition it is clear that A* ⊆ *B.*

*Symmetry* **2019**, *11*, 170

3.2.2. Equality

$$A = B \iff \mu\_A^j(\mathbf{x}) = \mu\_B^j(\mathbf{x}), \; j = 1, \; 2, \; \cdots, \; \; L(\mathbf{x}), \; \forall \; \mathbf{x} \in X$$

3.2.3. Union

$$\mu\_{A\cup B}^j(\mathbf{x}) = \mu\_A^j(\mathbf{x}) \lor \mu\_B^j(\mathbf{x}),\ j = 1, 2, \cdots, \ \mathbf{L}(\mathbf{x}), \ \forall \ \mathbf{x} \in X$$

3.2.4. Intersection

$$\mu\_{A \cap \ B}^{j}(\mathbf{x}) = \mu\_A^j(\mathbf{x}) \land \mu\_B^j(\mathbf{x}), \ j = 1, \ 2, \ \cdots, \ L(\mathbf{x}), \ \forall \ \mathbf{x} \in X.$$

**Example 4:** *Let us consider two multi-fuzzy sets over a nonempty universe X as*

$$A = \{ (0.8, 0.7, 0.4)/\mathfrak{x} + (1.0, 1.0, 0.8, 0.5)/y + (0.4, 0.3)/z \}$$

*and*

$$B = \{ (0.6, 0.5, 0.2)/x + (1.0, 0.9, 0.7, 0.6)/y + (0.7, 0.6, 0.5)/z \}.$$

*Then*

$$A \cup B = \left\{ \begin{array}{l} (0.8 \lor 0.6, 0.7 \lor 0.5, 0.4 \lor 0.2)/x + \\ (1.0 \lor 1.0, 1.0 \lor 0.9, 0.8 \lor 0.7, 0.5 \lor 0.6)/y + \\ (0.4 \lor 0.7, 0.3 \lor 0.6, 0.0 \lor 0.5)/z \end{array} \right\}$$

$$= \{ (0.8, 0.7, 0.4)/x + (1.0, 1.0, 0.8, 0.6)/y + (0.7, 0.6, 0.5)/z \}$$

*and*

$$A \cap B = \left\{ \begin{array}{c} (0.8 \wedge 0.6, 0.7 \wedge 0.5, \, 0.4 \wedge 0.2)/x + \\ (1.0 \wedge 1.0, \, 1.0 \wedge 0.9, 0.8 \wedge 0.7, \, 0.5 \wedge 0.6)/y + \\ (0.4 \wedge 0.7, \, 0.3 \wedge 0.6, \, 0.0 \wedge 0.5)/z \end{array} \right\}\_{\begin{aligned} \begin{cases} (0.6, 0.7)/x + \\ (0.6, 0.5, \, 0.2)/x + \end{cases} \end{aligned} \right\}\_{\begin{aligned} \begin{cases} (0.6, 0.7)/x + \\ (0.6, 0.3)/z \end{cases} \end{aligned} \right\}\_{\begin{aligned} \begin{cases} (0.6, 0.7)/z \end{cases} \end{aligned} \right\}\_{\begin{aligned} \begin{cases} (0.6, 0.7)/z \end{cases} \end{aligned} \right\}\_{\begin{aligned} \begin{cases} (0.6, 0.7)/z \end{cases} \end{aligned} \right\}\_{\begin{aligned} \begin{cases} (0.6, 0.7)/z \end{cases} \end{aligned} \right\}\_{\begin{aligned} \begin{cases} (0.6, 0.7)/z \end{cases} \end{aligned} \right\}\_{\begin{aligned} \begin{cases} (0.6, 0.7)/z \end{cases} \end{aligned} \right\}\_{\begin{aligned} \begin{cases} (0.6, 0.7)/z \end{cases} \end{aligned}$$

*Here, L*(*x*) = *L*(*x*; *A*, *B*) max{*L*(*x*; *A*), *L*(*x*; *B*)} = max{3, 3} = 3. *Correspondingly, L*(*y*) = max{4, 4} = 4, *L*(*z*) = max{2, 3} = 3.

#### *3.3. Type-2 Fuzzy Set (T2FS)*

A type-2 fuzzy set is a fuzzy set whose membership degree includes uncertainty i.e., membership degree is a type-1 fuzzy set. A T2FS introduces a third dimension to the membership function via the second membership grades. A T2FS *<sup>A</sup>* is mathematically expressed as follows according to (Mendel and John [46])

$$A = \{ ((\mathbf{x}, \
u), \mu\_{\bar{A}}(\mathbf{x}, \
u)) \; : \; \forall \mathbf{x} \in X\_{\prime} \; \forall \; f\_{\mathbf{x}} \subseteq [0, 1] \},$$

where 0 <sup>≤</sup> *<sup>μ</sup>A*(*x*, *<sup>u</sup>*) <sup>≤</sup> 1 is the secondary membership function and *Jx* is the primary membership of *<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>*, which is the domain of *<sup>μ</sup>A*(*x*, *<sup>u</sup>*). *<sup>A</sup>* can be expressed as

$$\tilde{A} = \int\_{x \in \mathcal{X}} \left( \int\_{\{u \in I\_x\}} \mu\_{\overline{A}}(x, u) / u \right) / x, \; J\_x \subseteq [0, 1],$$

where - denotes union over all admissible *x* and *u*. For a discrete universe of discourse, 9 is replaced by ∑ .

For each value of *<sup>x</sup>*, the secondary membership function *<sup>μ</sup>A*(*x*, *<sup>u</sup>*) is defined as

$$
\mu\_{\bar{A}}(\mathbf{x}, \
u) = \int\_{\{\mathbf{u} \in \mathbb{J}\_{\mathbf{x}} : \mathbf{u} \in \mathcal{A}\}} \mu\_{\bar{A}}(\mathbf{x}, \
u) / \mu\_{\bar{A}}
$$

such that for a particular *<sup>u</sup>* <sup>=</sup> *<sup>u</sup>* <sup>∈</sup> *Jx*, the secondary membership grade of (*x*, *<sup>u</sup>*) is called *<sup>μ</sup>A*(*x*, *<sup>u</sup>*).

**Example 5:** *Let the set of infant age be represented by a type-2 fuzzy set <sup>A</sup>. Let youthness be the primary membership function of <sup>A</sup> and the degree of youthness be the secondary membership function. Let E* = {8, 10, 14} *be an age set with the primary membership of the members of E respectively being J*<sup>8</sup> = {0.8, 0.9, 1.0}*, J*<sup>10</sup> = {0.6, 0.7, 0.8}*, J*<sup>14</sup> *=* {0.4, 0.5, 0.6}*. The secondary membership function of* 8 *is <sup>μ</sup>A*(8, *<sup>u</sup>*) <sup>=</sup> (0.9/0.8) <sup>+</sup> (0.7/0.9) <sup>+</sup> (0.6/1.0)*, i.e., <sup>μ</sup>A*(8, 0.8) <sup>=</sup> 0.9 *is the secondary membership grade of* 8 *with primary membership* 0.8*.*

In the same way, *<sup>μ</sup>A*(10, *<sup>u</sup>*) <sup>=</sup> (0.8/0.6) <sup>+</sup> (0.7/0.7) <sup>+</sup> (0.6/0.8) and *<sup>μ</sup>A*(14, *<sup>u</sup>*) <sup>=</sup> (0.9/0.4) <sup>+</sup> (0.8/0.5) + (0.5/0.6).

So the discrete type-2 fuzzy set *<sup>A</sup>* can be represented by

$$\begin{array}{c} \dot{A} = (0.9/0.8) + (0.7/0.9) + (0.6/1.0)/8 + (0.8/0.6) + (0.7/0.7) + (0.6/0.8)/10 \\ \qquad + (0.9/0.4) + (0.8/0.5) + (0.5/0.6)/14. \end{array}$$

#### **4. Type-2 Multi-Fuzzy Sets (T2MFS)**

Let *X* be the universe of discourse. Let *A* be a type-2 fuzzy set defined on *X* and *u* ∈ *Jx* ⊆ [0, 1] be a primary membership value of an element *x* ∈ *X*. Then, *A* is said to be a type-2 multi-fuzzy set if it is characterised by a count function denoted by *CA* and is defined as *CA* : *Jx* → *Q*, where *Q* is the set of all crisp multi-sets taken from the unit interval [0, 1], which are the secondary membership values of *x* ∈ *X*. For each *x* ∈ *X*, the secondary membership sequence is defined in decreasing order as *μ*1 *<sup>A</sup>*(*x*, *<sup>u</sup>*) ≥ *<sup>μ</sup>*<sup>2</sup> *<sup>A</sup>*(*x*, *<sup>u</sup>*) ≥···≥ *<sup>μ</sup><sup>p</sup> <sup>A</sup>*(*x*, *<sup>u</sup>*) and is denoted by *μ*1 *<sup>A</sup>*(*x*, *<sup>u</sup>*), *<sup>μ</sup>*<sup>2</sup> *<sup>A</sup>*(*x*, *<sup>u</sup>*), ··· , *<sup>μ</sup><sup>p</sup> <sup>A</sup>*(*x*, *u*) . Then, set *A* can be represented as

$$A = \sum\_{\mathbf{x} \in X} \left( \sum\_{\mathbf{u} \in f\_{\mathbf{x}} \subseteq [0, 1]} \left( \mu\_A^1(\mathbf{x}, \mathbf{u}), \mu\_A^2(\mathbf{x}, \mathbf{u}), \dots, \mu\_A^p(\mathbf{x}, \mathbf{u}) \right) / \mathbf{u} \right) / \mathbf{x} .$$

if the universe is discrete, whereas if *X* is a continuous universe, then *A* can be written as

$$A = \int\_{\mathbf{x} \in \mathcal{X}} \left( \int\_{\boldsymbol{\mu} \in I\_{\mathcal{X}} \subseteq [0, 1]} \left( \left( \mu\_A^1(\mathbf{x}, \boldsymbol{\mu}), \mu\_A^2(\mathbf{x}, \boldsymbol{\mu}), \dots, \mu\_A^p(\mathbf{x}, \boldsymbol{\mu}) \right) \right) / \boldsymbol{\mu} \right) / \mathbf{x}.$$

Let us illustrate this idea with an example.

**Example 6:** *Let us consider a type-2 fuzzy set T defined in the universal set X* = {*x*, *y*, *z*}

$$T = \begin{pmatrix} (0.8/0.6, 0.5/0.6, 0.2/0.6, 0.3/0.9, 0.7/0.9)/\text{x} + \\ (0.7/0.3, 0.6/0.3, 0.8/0.5, 0.5/0.7, 0.3/0.7 + 0.1/0.7)/y \end{pmatrix}$$

*From the structure of T*, *we see that three x's with primary membership values of* 0.6 *have secondary membership values* 0.8, 0.5 *and* 0.2 *respectively; two x's with primary membership values of* 0.9 *have the corresponding secondary membership values of* 0.3 *and* 0.7*; two y's with a primary membership value of* 0.3 *have secondary membership values of* 0.7 *and* 0.6 *respectively; one y with a primary membership value of* 0.5 *has a secondary* *membership value of* 0.8*; and three y's with primary membership values of* 0.7 *with secondary membership values* 0.5, 0.3 *and* 0.1 *respectively. Therefore, T can be represented as*

> *<sup>T</sup>* <sup>=</sup> ((0.8, 0.5, 0.2)/0.6 <sup>+</sup> (0.3, 0.7)/0.9)/*x*<sup>+</sup> ((0.7, 0.6)/0.3 + 0.8/0.5 + (0.5, 0.3, 0.1)/0.7)/*y*.

**Figure 2.** The membership function of T2MFS *T* is represented by three different combinations of primary membership and secondary membership functions of 3 and 8, where (**a**) the graphical representation of (0.8/0.6 + 0.3/0.9)/3 + (0.7/0.3 + 0.8/0.5 + 0.5/0.7)/8, (**b**) the graphical representation of (0.5/0.6 + 0.7/0.9)/3 + (0.6/0.3 + 0.3/0.7)/8 and (**c**) the graphical representation of (0.2/0.6)/3 + (0.1/0.7)/8 are presented respectively.

*Symmetry* **2019**, *11*, 170

Thus, *T* is a type-2 multi-fuzzy set. The graphical representation of T2MFS *T* is shown in Figure 2, where we consider *x* = 3 and *y* = 8.

Again, let us denote the collection of all T2MFSs over the universe *X* by T2MF(*X*). When we define operations between two T2MFSs, say *A* and *B*, the lengths of their secondary membership sequences *μ*1 *<sup>A</sup>*(*x*, *<sup>u</sup>*), *<sup>μ</sup>*<sup>2</sup> *<sup>A</sup>*(*x*, *<sup>u</sup>*), ··· , *<sup>μ</sup><sup>p</sup> <sup>A</sup>*(*x*, *u*) and *μ*1 *<sup>B</sup>*(*x*, *<sup>u</sup>*), *<sup>μ</sup>*<sup>2</sup> *<sup>B</sup>*(*x*, *<sup>u</sup>*), ··· , *<sup>μ</sup><sup>p</sup> <sup>B</sup>*(*x*, *u*) for a particular primary membership value of *A* (say *u*) and for a particular primary membership value of *B* (say *u* ) should be set to be equal. We hence define the length *L*(*x*, *u*; *A*) as

$$L(\mathfrak{x}, \mathfrak{u}; A) = \max \{ j : \mu\_A^j(\mathfrak{x}, \mathfrak{u}) \not\equiv 0 \}.$$

and

$$L(\mathfrak{x}, \mathfrak{u}, \mathfrak{u}'; A, B) = \max \{ L(\mathfrak{x}, \mathfrak{u}; A), \ L(\mathfrak{x}, \mathfrak{u}'; B) \}.$$

For the sake of simplicity, we write *L*(*x*, *u*, *u* ; *A*, *B*) as *L*(*x*, *u*, *u* ). Let us describe this idea by an example.

**Example 7:** *Let us consider two T2MFSs, say A and B*, *as*

$$A = ((0.8, 0.6, 0.2)/0.7 + (0.9, 0.5)/0.4)/\text{x} + ((0.7, 0.5, 0.3)/0.8 + (0.5, 0.5, 0.4, 0.1)/0.5)/y$$

*and*

$$B = ((0.9, 0.6) / 0.8 + (1.0, 1.0, 0.6, 0.6) / 0.7) / \text{x} + ((0.6, 0.2) / 0.9 + 0.1 / 0.4) / y.$$

*Then*

$$L(\mathbf{x}, 0.7; A) = 3,\\ L(\mathbf{x}, 0.4; A) = 2, \; L(y, 0.8; A) = 3,\\ L(y, 0.5; A) = 4,$$

$$L(\mathbf{x}, 0.8; B) = 2,\\ L(\mathbf{x}, 0.7; B) = 4, \; L(y, 0.9; B) = 2,\\ L(y, 0.4; B) = 1.$$

*Moreover,*

$$L(\mathbf{x}, 0.7; 0.8) = 3, L(\mathbf{x}, 0.7; 0.7) = 4, \; L(\mathbf{x}, 0.4; 0.8) = 2, L(\mathbf{x}, 0.4; 0.7) = 4,$$

$$L(y, 0.8; 0.9) = 3, L(y, 0.8; 0.4) = 3, \; L(y, 0.5; 0.9) = 4, L(y, 0.5; 0.4) = 4.$$

#### **5. Some Operations on T2MFS**

In this section, we discuss four fundamental arithmetical operations: (i) complement, (ii) inclusion, (iii) union, and (iv) intersection of T2MFS.

#### *5.1. Complement*

Let *A* be a T2MFS over some universe *X*. Then, the complement of *A* denoted by *A<sup>c</sup>* is defined as

$$A^\varepsilon = \sum\_{\mathbf{x} \in \mathcal{X}} \left( \sum\_{\mathbf{v} \in f\_\mathbf{x} \subseteq [0,1]} \left( \mu\_A^1(\mathbf{x}, \mathbf{v}), \mu\_A^2(\mathbf{x}, \mathbf{v}), \dots, \mu\_A^p(\mathbf{x}, \mathbf{v}) \right) / v \right) / \mathbf{x}^\varepsilon$$

where *v* = 1 − *u* and *u* is the primary membership function of *A*. Let us give an example to illustrate this idea.

**Example 8:** *Let us consider the T2MFS A used in Example 6.*

$$A^{\varepsilon} = \begin{array}{c} ((0.8, 0.5, 0.2)/0.4 + (0.3, 0.3)/0.1)/x + \\ ((0.7, 0.6)/0.7 + 0.8/0.5 + (0.5, 0.3, 0.1)/0.3)/y. \end{array}$$

*Symmetry* **2019**, *11*, 170

#### *5.2. Inclusion*

Let *A* and *B* be two T2MFSs over some universe *X*. Then we say that *A* ⊆ *B* if and only if

$$\mu \le \mu', \ \mu\_A^j(\mathbf{x}, \boldsymbol{\mu}) \le \mu\_B^j(\mathbf{x}, \boldsymbol{\mu}'), j = 1, 2, \dots, L \\
\text{( $\mathbf{x}$ ,  $\boldsymbol{\mu}$ ,  $\boldsymbol{\mu}'$ ) \,\forall \mathbf{x} \in X\_{\boldsymbol{\mu}}.$$

where *u* and *u* are primary membership functions of *A* and *B*, respectively, and *μ<sup>j</sup> <sup>A</sup>* and *<sup>μ</sup><sup>j</sup> <sup>B</sup>* are the secondary membership functions of *A* and *B* respectively. These two sets *A* and *B* are said to be equal if and only if

$$\mu = \mu', \mu\_A^j(\mathbf{x}, \boldsymbol{\mu}) = \mu\_B^j(\mathbf{x}, \boldsymbol{\mu}'), j = 1, 2, \cdots, L(\mathbf{x}, \boldsymbol{\mu}, \boldsymbol{\mu}') \,\forall \mathbf{x} \in X.$$

Let us illustrate this idea by an example.

**Example 9:** *Let us consider two T2MFSs, say A and B, where*

$$A = \left( \left( (0.8, 0.6, 0.5, 0.1) / 0.8 + (0.9, 0.8, 0.6, 0.3) / 0.7 \right) / \mathbf{x} + \left( (0.7, 0.5, 0.3) / 0.9 + (0.9, 0.4) / 0.5 \right) / \mathbf{y} \right)$$

*and*

$$B = (((0.7, 0.6)/0.6 + (0.8, 0.5, 0.4, 0.1)/0.7)/\mathbf{x} + ((0.6, 0.2)/0.5 + 0.1/0.4)/y)$$

*Then applying the above-mentioned definition, we can see that B* ⊆ *A.*

#### *5.3. Union*

Let *A* and *B* be two T2MFSs over the universe *X*. Then the set *C* = *A* ∪ *B* is defined as

$$\mathcal{C} = \sum\_{\mathbf{x} \in X} \left( \sum\_{\substack{\mathbf{v} \in I\_{\mathbf{x}} \subseteq [0,1]}} \left( \mu\_A^1(\mathbf{x}, \mathbf{v}), \mu\_A^2(\mathbf{x}, \mathbf{v}), \dots, \mu\_A^{L(\mathbf{x}, \mathbf{u}, \mathbf{u}')}(\mathbf{x}, \mathbf{v}) \right) / \mathbf{v} \right) / \mathbf{x} \,, \mathbf{v}$$

where *μ<sup>j</sup> <sup>C</sup>*(*x*, *<sup>v</sup>*) <sup>=</sup> *<sup>μ</sup><sup>j</sup> <sup>A</sup>*<sup>∪</sup> *<sup>B</sup>*(*x*, *<sup>v</sup>*) <sup>=</sup> *<sup>μ</sup><sup>j</sup> <sup>A</sup>*(*x*, *<sup>u</sup>*) <sup>∧</sup> *<sup>μ</sup><sup>j</sup> <sup>B</sup>*(*x*, *u* ) and *j* = 1, 2, ··· , *L*(*x*, *u*, *u* ), ∀*x* ∈ *X*, such that *v* = *u* ∨ *u* , where *u* and *u* are the primary membership functions of *A* and *B* respectively. *μ<sup>j</sup> <sup>A</sup>*(*x*, *u*) and *μ<sup>j</sup> <sup>B</sup>*(*x*, *u* ) are the secondary membership functions of *A* and *B* respectively. Let us illustrate this idea by an example.

**Example 10:** *Let us consider two T2MFSs, say A and B, which are considered in Example 5. Then*

$$A = \left( \left( (0.8, 0.6, 0.2) / 0.7 + (0.9, 0.5) / 0.4 \right) / \mathbf{x} + \left( (0.7, 0.5, 0.3) / 0.8 + (0.5, 0.5, 0.4, 0.1) / 0.5 \right) / \mathbf{y} \right)$$

*and*

$$B = (((0.9, 0.6)/0.8 + (1.0, 0.8, 0.6, 0.6)/0.7)/x + ((0.6, 0.2)/0.9 + 0.1/0.4)/y).$$

*Then*

$$\begin{array}{l} A \cup B \\ = (((0.8, 0.6, 0.0)) / 0.8 + (0.8, 0.6, 0.2, 0.0) / 0.7 + (0.9, 0.5) / 0.8 + (0.9, 0.5, 0.0, 0.0) / 0.7) / x \\ + ((0.6, 0.2, 0.0) / 0.9 + (0.5, 0.2, 0.0, 0.0) / 0.9 + (0.1, 0.0, 0.0) / 0.8 + (0.1, 0.0, 0.0, 0.0) / 0.5) / x \\ = (((\max(0.8, 0.9), \max(0.6, 0.5)) / 0.8 + (\max(0.8, 0.9), \max(0.6, 0.5), \max(0.2, 0.0)) / 0.7) / x \\ + ((\max(0.6, 0.5), \max(0.2, 0.2)) / 0.9 + 0.1 / 0.8 + 0.1 / 0.5) / y \\ = (((0.9, 0.6) / 0.8 + (0.9, 0.6, 0.2) / 0.7) / x + ((0.6, 0.2) / 0.9 + 0.1 / 0.5) / y) . \end{array}$$

#### *5.4. Intersection*

Let *A* and *B* be two T2MFSs over the universe *X*. Then set *C* is the intersection of *A* and *B* where it is denoted as *C* = *A* ∩ *B* and is defined as

$$\mathcal{C} = \sum\_{x \in X} \left( \sum\_{v \in f\_x \subseteq [0, \, 1]} \left( \mu\_A^1(\mathbf{x}, v), \mu\_A^2(\mathbf{x}, v), \dots, \, \_\prime \mu\_A^{L(\mathbf{x}, \mu, \nu')}(\mathbf{x}, v) \right) / v \right) / x^{\nu}$$

Here, *μ<sup>j</sup> <sup>C</sup>*(*x*, *<sup>v</sup>*) <sup>=</sup> *<sup>μ</sup><sup>j</sup> <sup>A</sup>*<sup>∪</sup> *<sup>B</sup>*(*x*, *<sup>v</sup>*) <sup>=</sup> *<sup>μ</sup><sup>j</sup> <sup>A</sup>*(*x*, *<sup>u</sup>*) <sup>∧</sup> *<sup>μ</sup><sup>j</sup> <sup>B</sup>*(*x*, *u* ), *j* = 1, 2, ··· , *L*(*x*, *u*, *u* ), ∀*x* ∈ *X*, where *v* = *u* ∧ *u* , and *u* and *u* are the primary membership functions of *A* and *B* respectively. *μ<sup>j</sup> <sup>A</sup>*(*x*, *u*) and *μj <sup>B</sup>*(*x*, *u* ) are the secondary membership functions of *A* and *B* respectively. Subsequently, we illustrate this idea by an example as shown below.

**Example 11:** *Let us consider two T2MFSs, say A and B, where*

$$A = \left( \left( (0.8, 0.6, 0.2) / 0.7 + (0.9, 0.5) / 0.4 \right) / \mathbf{x} + \left( (0.7, 0.5, 0.3) / 0.8 + (0.5, 0.5, 0.4, 0.1) / 0.5 \right) / y \right)$$

$$B = \left( \left( (0.9, 0.6) / 0.8 + (1.0, 0.8, 0.6, 0.6) / 0.7 \right) / \mathbf{x} + \left( (0.6, 0.2) / 0.9 + 0.1 / 0.4 \right) / y \right)$$

*Then*

*A* ∩ *B* = (((0.8, 0.6, 0.0)/0.7 + (0.8, 0.6, 0.2, 0.0)/0.7 + (0.9, 0.5)/0.4 + (0.9, 0.5, 0.0, 0.0)/0.4)/*x* +((0.6, 0.2, 0.0)/0.8 + (0.5, 0.2, 0.0, 0.0)/0.5 + (0.1, 0.0, 0.0)/0.4 + (0.1, 0.0, 0.0, 0.0)/0.4)/*y*) = (((max(0.8, 0.8), max(0.6, 0.6), max(0.0, 0.2))/0.7 + (max(0.9, 0.9), max(0.5, 0.5))/0.4)/*x* +((0.6, 0.2)/0.8 + (0.5, 0.2)/0.5 + (max(0.1, 0.1))/0.4 + 0.1/0.5)/*y*) = (((0.8, 0.6, 0.2)/0.7 + (0.9, 0.5)/0.4)/*x* + ((0.6, 0.2)/0.8 + (0.5, 0.2)/0.5 + 0.1/0.4)/*y*).

#### **6. Properties of T2MFS**

In this section, we discuss four fundamental properties of T2MFS. Let *A*, *B* and *C* be three T2MFSs over a universe *X*. Then the following relations hold:


The proofs of (i) to (iii) are obvious. We illustrate these results later by example. We now prove the results (iv) and (v).

*Symmetry* **2019**, *11*, 170

**Proof of (iv).**

*A* ∪ (*B* ∪ *C*) = ∑ *x*∈*X* ∑ *u*∈*Jx*⊆[0, 1] *μ*1 *<sup>A</sup>*(*x*, *<sup>u</sup>*), *<sup>μ</sup>*<sup>2</sup> *<sup>A</sup>*(*x*, *<sup>u</sup>*), ..., *<sup>μ</sup>L*(*x*,*u*; *<sup>A</sup>*) *<sup>A</sup>* (*x*, *u*) /*u* /*x* ∪ ∑ *x*∈*X* ∑ *v*,*w*∈*Jx*⊆[0, 1] - *μ*1 *<sup>B</sup>*(*x*, *<sup>v</sup>*) ∧ *<sup>μ</sup>*<sup>1</sup> *<sup>C</sup>*(*x*, *w*) , - *μ*2 *<sup>B</sup>*(*x*, *<sup>v</sup>*) ∧ *<sup>μ</sup>*<sup>2</sup> *<sup>C</sup>*(*x*, *w*) , ..., *μL*(*x*,*v*,*w*) *<sup>B</sup>* (*x*, *<sup>v</sup>*) <sup>∧</sup> *<sup>μ</sup>L*(*x*,*v*,*w*) *<sup>C</sup>* (*x*, *w*) /(*v* ∨ *w*) /*x* = ∑ *x*∈*X* ⎛ ⎜⎝ <sup>∑</sup> *<sup>u</sup>*,*v*,*w*∈*Jx*⊆[0, 1] ⎛ ⎜⎝ - *μ*1 *<sup>A</sup>*(*x*, *<sup>u</sup>*) ∧ *<sup>μ</sup>*<sup>1</sup> *<sup>B</sup>*(*x*, *<sup>v</sup>*) ∧ *<sup>μ</sup>*<sup>1</sup> *<sup>C</sup>*(*x*, *w*) , - *μ*1 *<sup>A</sup>*(*x*, *<sup>u</sup>*) ∧ *<sup>μ</sup>*<sup>2</sup> *<sup>B</sup>*(*x*, *<sup>v</sup>*) ∧ *<sup>μ</sup>*<sup>2</sup> *<sup>C</sup>*(*x*, *w*) , ..., *μL*(*x*,*u*,*v*,*w*) *<sup>A</sup>* (*x*, *<sup>u</sup>*) <sup>∧</sup> *<sup>μ</sup>L*(*x*,*u*,*v*,*w*) *<sup>B</sup>* (*x*, *<sup>v</sup>*) <sup>∧</sup> *<sup>μ</sup>L*(*x*,*u*,*v*,*w*) *<sup>C</sup>* (*x*, *w*) ⎞ ⎟⎠/(*<sup>u</sup>* <sup>∨</sup> *<sup>v</sup>* <sup>∨</sup> *<sup>w</sup>*) ⎞ ⎟⎠/*<sup>x</sup>* = ∑ *x*∈*X* ∑ *u*,*v*∈*Jx*⊆[0, 1] - *μ*1 *<sup>A</sup>*(*x*, *<sup>u</sup>*) ∧ *<sup>μ</sup>*<sup>1</sup> *<sup>B</sup>*(*x*, *v*) , - *μ*2 *<sup>A</sup>*(*x*, *<sup>u</sup>*) ∧ *<sup>μ</sup>*<sup>2</sup> *<sup>B</sup>*(*x*, *v*) , ..., *μL*(*x*,*u*,*v*) *<sup>A</sup>* (*x*, *<sup>u</sup>*) <sup>∧</sup> *<sup>μ</sup>L*(*x*,*u*,*v*) *<sup>B</sup>* (*x*, *v*) /(*u* ∨ *v*) /*x* ∪ ∑ *x*∈*X* ∑ *w*∈*Jx*⊆[0, 1] *μ*1 *<sup>C</sup>*(*x*, *<sup>w</sup>*), *<sup>μ</sup>*<sup>2</sup> *<sup>C</sup>*(*x*, *<sup>w</sup>*), ..., *<sup>μ</sup>L*(*x*,*w*; *<sup>A</sup>*) *<sup>C</sup>* (*x*, *w*) /*w* /*x* = (*A* ∪ *B*) ∪ *C*.

Similarly, we can prove that *A* ∩ (*B* ∩ *C*)=(*A* ∩ *B*) ∩ *C*.

**Proof of (v).**

(*A* ∪ *B*) ∩ *C* = ⎛ <sup>⎝</sup>∑*x*∈*<sup>X</sup>* ⎛ <sup>⎝</sup>∑*u*,*v*∈*Jx*⊆[0, 1] ⎛ ⎝ - *μ*1 *<sup>A</sup>*(*x*, *<sup>u</sup>*) ∧ *<sup>μ</sup>*<sup>1</sup> *<sup>B</sup>*(*x*, *v*) , - *μ*2 *<sup>A</sup>*(*x*, *<sup>u</sup>*) ∧ *<sup>μ</sup>*<sup>2</sup> *<sup>B</sup>*(*x*, *v*) , ··· , *μL*(*x*,*u*,*v*) *<sup>A</sup>* (*x*, *<sup>u</sup>*) <sup>∧</sup> *<sup>μ</sup>L*(*x*,*u*,*v*) *<sup>B</sup>* (*x*, *v*) ⎞ <sup>⎠</sup>/(*<sup>u</sup>* <sup>∨</sup> *<sup>v</sup>*) ⎞ <sup>⎠</sup>/*<sup>x</sup>* ⎞ ⎠∩ ∑*x*∈*<sup>X</sup>* ∑*w*∈*Jx*⊆[0, 1] *μ*1 *<sup>C</sup>*(*x*, *<sup>w</sup>*), *<sup>μ</sup>*<sup>2</sup> *<sup>C</sup>*(*x*, *<sup>w</sup>*), ··· , *<sup>μ</sup>L*(*x*,*w*; *<sup>A</sup>*) *<sup>C</sup>* (*x*, *w*) /*w* /*x* = ∑*x*∈*<sup>X</sup>* ⎛ ⎜⎜⎝ ∑*u*,*v*,*w*∈*Jx*⊆[0, 1] ⎛ ⎜⎜⎝ - *μ*1 *<sup>A</sup>*(*x*, *<sup>u</sup>*) ∧ *<sup>μ</sup>*<sup>1</sup> *<sup>B</sup>*(*x*, *<sup>v</sup>*) ∧ *<sup>μ</sup>*<sup>1</sup> *<sup>C</sup>*(*x*, *w*) , - *μ*1 *<sup>A</sup>*(*x*, *<sup>u</sup>*) ∧ *<sup>μ</sup>*<sup>2</sup> *<sup>B</sup>*(*x*, *<sup>v</sup>*) ∧ *<sup>μ</sup>*<sup>2</sup> *<sup>C</sup>*(*x*, *w*) , ··· , *μL*(*x*,*u*,*v*,*w*) *<sup>A</sup>* (*x*, *<sup>u</sup>*) <sup>∧</sup> *<sup>μ</sup>L*(*x*,*u*,*v*,*w*) *<sup>B</sup>* (*x*, *<sup>v</sup>*) <sup>∧</sup> *<sup>μ</sup>L*(*x*,*u*,*v*,*w*) *<sup>C</sup>* (*x*, *w*) ⎞ ⎟⎟⎠ /(*u* ∨ *v*) ∧ *w* ⎞ ⎟⎟⎠ /*x* = ⎛ <sup>⎝</sup>∑*x*∈*<sup>X</sup>* ⎛ <sup>⎝</sup>∑*u*,*w*∈*Jx*⊆[0, 1] ⎛ ⎝ - *μ*1 *<sup>A</sup>*(*x*, *<sup>u</sup>*) ∧ *<sup>μ</sup>*<sup>1</sup> *<sup>C</sup>*(*x*, *w*) , - *μ*2 *<sup>A</sup>*(*x*, *<sup>u</sup>*) ∧ *<sup>μ</sup>*<sup>2</sup> *<sup>C</sup>*(*x*, *w*) , ··· , *μL*(*x*,*u*,*w*) *<sup>A</sup>* (*x*, *<sup>u</sup>*) <sup>∧</sup> *<sup>μ</sup>L*(*x*,*u*,*w*) *<sup>C</sup>* (*x*, *w*) ⎞ <sup>⎠</sup>/(*<sup>u</sup>* <sup>∧</sup> *<sup>w</sup>*) ⎞ <sup>⎠</sup>/*<sup>x</sup>* ⎞ ⎠∪ ⎛ <sup>⎝</sup>∑*x*∈*<sup>X</sup>* ⎛ <sup>⎝</sup>∑*v*,*w*∈*Jx*⊆[0, 1] ⎛ ⎝ - *μ*1 *<sup>B</sup>*(*x*, *<sup>v</sup>*) ∧ *<sup>μ</sup>*<sup>1</sup> *<sup>C</sup>*(*x*, *w*) , - *μ*2 *<sup>B</sup>*(*x*, *<sup>v</sup>*) ∧ *<sup>μ</sup>*<sup>2</sup> *<sup>C</sup>*(*x*, *w*) , ··· , *μL*(*x*,*v*,*w*) *<sup>B</sup>* (*x*, *<sup>v</sup>*) <sup>∧</sup> *<sup>μ</sup>L*(*x*,*v*,*w*) *<sup>C</sup>* (*x*, *w*) ⎞ <sup>⎠</sup>/(*<sup>v</sup>* <sup>∧</sup> *<sup>w</sup>*) ⎞ <sup>⎠</sup>/*<sup>x</sup>* ⎞ <sup>⎠</sup> = (*<sup>A</sup>* <sup>∩</sup> *<sup>C</sup>*) <sup>∪</sup> (*<sup>B</sup>* <sup>∩</sup> *<sup>C</sup>*)

Similarly, we can prove that (*A* ∩ *B*) ∪ *C* = (*A* ∪ *C*) ∩ (*B* ∪ *C*).

Let us illustrate these results by an example.

**Example 12:** *Let us consider three T2MFSs over a non-empty universe X.*

$$A = (0.8, 0.7, 0.3) / 0.7 / \text{x} + (0.9, 0.5, 0.1) / 0.4 / y + (0.6, 0.5) / 0.3 / z$$

$$B = (1.0, \, 0.9) / 0.9 / \text{x} + (0.7, \, 0.6, \, 0.3) / 0.6 / y + (0.8, \, 0.7, \, 0.1) / 0.2 / z$$

$$C = (0.9, 0.5, 0.2, 0.0) / 0.6 / \text{x} + (0.8, 0.4) / 0.9 / y + (0.8, 0.2, 0.0) / 0.8 / z$$

*For simplicity, we have taken Jx as a singleton set for each x* ∈ *X. Then*

$$A \cup \ A = A \cap \ A = (0.8, 0.7, 0.3) / 0.7 / \mathbf{x} + (0.9, 0.5, 0.1) / 0.4 / y + (0.6, 0.5) / 0.3 / z = A.$$

*Therefore, the idempotent property holds.*

$$(A \cup B)^{\varepsilon} = (0.8, 0.7) / 0.1 / \mathbf{x} + (0.7, 0.5, 0.1) / 0.4 / y + (0.6, 0.5) / 0.7 / z = A^{\varepsilon} \cap B^{\varepsilon}$$

*and*

$$\left(A \cap B\right)^{\varepsilon} = \left(0.8, \, 0.7\right) / 0.3 / \ge + \left(0.7, \, 0.5, \, 0.1\right) / 0.6 / y + \left(0.6, \, 0.5\right) / 0.8 / z = A^{\varepsilon} \cup B^{\varepsilon}.$$

*Hence, De Morgan's laws hold.*

$$A \cup B = (0.8, 0.7) / 0.9 / \text{x} + (0.7, 0.5, 0.1) / 0.6 / y + (0.6, 0.5) / 0.3 / z = B \cup A$$

*and*

$$A \cap B = (0.8, 0.7) / 0.7 / \text{x} + (0.7, 0.5, 0.1) / 0.4 / y + (0.6, 0.5) / 0.2 / z = B \cap A.$$

*Consequently, the commutative property holds.*

$$A \cup (B \cup \ C) = (0.8, 0.5) / 0.9 / \mathbf{x} + (0.7, 0.4) / 0.9 / y + (0.6, 0.2) / 0.8 / z = (A \cup B) \cup \ C$$

*and*

$$A \cap (B \cap \mathbb{C}) = (0.8, 0.5) / 0.6 / \text{x} + (0.7, 0.4) / 0.4 / y + (0.6, 0.2) / 0.2 / z = (A \cap B) \cap \mathbb{C} \dots$$

*Hence, the associative property holds.*

$$(A \cup B) \cap \mathcal{C} = (0.8, 0.5) / 0.6 / \mathbf{x} + (0.7, 0.4) / 0.6 / y + (0.6, 0.2) / 0.3 / z = (A \cap \mathcal{C}) \cup (B \cap \mathcal{C})$$

*and*

$$(A \cap B) \cup \mathcal{C} = (0.8, 0.5) / 0.7 / \mathbf{x} + (0.7, 0.4) / 0.9 / y + (0.6, 0.2) / 0.8 / z = (A \cup \mathcal{C}) \cap (B \cup \mathcal{C}).$$

*As a result, the distributive property holds.*

#### **7. Distance Measures of T2MFS**

Let *A* and *B* be two T2MFSs. Then we can define the following distances as follows:

(1) Hamming Distance

$$d\_H(A,B) = \sum\_{\mathbf{x} \in \mathcal{X}} \sum\_{\boldsymbol{\mu}, \boldsymbol{\mu}' \in \mathcal{J}\_{\mathbf{x}}} \sum\_{j=1}^{L(\mathbf{x}, \boldsymbol{\mu}, \boldsymbol{\mu}'; A, B)} \left| \mu\_A^j(\mathbf{x}, \boldsymbol{\mu}) - \mu\_B^j(\mathbf{x}, \boldsymbol{\mu}') \right|.$$

#### (2) Euclidean Distance

$$d\_E(A,B) = \left[ \sum\_{\mathbf{x} \in \mathcal{X}} \sum\_{\boldsymbol{\mu}, \boldsymbol{\mu}' \in \mathcal{J}\_{\mathcal{X}}} \sum\_{j=1}^{L(\mathbf{x}, \boldsymbol{\mu}, \boldsymbol{\mu}'; A, B)} \left| \mu\_A^j(\mathbf{x}, \boldsymbol{\mu}) - \mu\_B^j(\mathbf{x}, \boldsymbol{\mu}') \right|^2 \right]^{\frac{1}{2}}.$$

where *u* and *u* are the primary membership functions of *A* and *B* respectively, and *μ<sup>A</sup>* and *μ<sup>B</sup>* are the corresponding secondary membership functions of *A* and *B*. Let us now explain this idea with an example.

**Example 13:** *Let X be a non-empty universe. Let A and B be two T2MFSs over X which are given as*

$$A = (0.9, 0.6, 0.4) / 0.8 / \mathbf{x} + (0.8, 0.5, 0.1) / 0.6 / y + (1.0, 0.7) / 0.9 / z$$

$$B = (0.7, 0.7, 0.2) / 0.6 / \mathbf{x} + (0.9, 0.2, 0.1) / 0.3 / y + (0.8, 0.8) / 0.4 / z$$

$$\begin{split} d\_H(A, B) &= |0.9 - 0.7| + |0.6 - 0.7| + |0.4 - 0.2| + |0.8 - 0.9| + |0.5 - 0.2| \\ &+ |0.1 - 0.1| + |1.0 - 0.8| + |0.7 - 0.8| = 1.2 \end{split}$$

*and*

$$d\_{E}(A,B) = \left[\left|0.9-0.7\right|^2 + \left|0.6-0.7\right|^2 + \left|0.4-0.2\right|^2 + \left|0.8-0.9\right|^2 + \left|0.5-0.2\right|^2\right]$$

$$+ \left|0.1-0.1\right|^2 + \left|1.0-0.8\right|^2 + \left|0.7-0.8\right|^2\right|^{\frac{1}{2}} = 0.49.$$

#### **8. Numerical Illustration**

Although fuzzy logic is relatively a newer subject than classical mathematical logic, we use the former extensively in our everyday life. We primarily use fuzzy logic in decision-making problems. However, there may be some cases in decision-making where the primary membership function occurs with a sequence of same or different degrees. In that case, we need to use T2MFS to solve the problem. Let us present an example of medical diagnosis after acquiring the necessary information from Reference [38]. Let *P* = {*P*1, *P*2, *P*3, *P*4} be a set of patients, *D* = {*Viral Fever*, *Tuberculosis*, *Typhoid*, *Throat Disease*} be a set of diseases and *S* = {*Temperature*, *Cough*, *Throat Pain*, *Headache*, *Body Pain*} be a set of symptoms. Let us consider the intensity of the disease symptoms to be the secondary membership functions.

Now, if we look at one set of data, there is a chance that some errors may occur since we know that at different times during a day the disease symptoms have different intensities. To minimize these errors, we study three different samples at three different times of the day. The details of one such example are provided below.

In Table 1, each symptom is described by their primary and secondary membership function.


**Table 1.** Symptoms vs. Diseases.

For proper diagnosis of each patient, we take samples at three different times in a day say at 8 AM, 2 PM and 9 PM. We use the Euclidean distance measure (cf. Section 6) to calculate the distance between the patients and the diseases. Here, the Euclidean distance is determined between each patient *Pi* from every symptoms *Sj* for each diagnosis *dk*, *i*, *k* = 1, 2, 3, 4, *j* = 1, 2, 3, 4, 5.

The data reported in Table 2 is of a T2MFS, where the intensity of the symptoms form a secondary membership sequence. Moreover, in Tables 3 and 4 the Hamming distance and the Euclidean distance are determined respectively for each patient from the set of diseases.

**Table 2.** Patients vs. Symptoms.


**Table 3.** Hamming distance between Patients and Diseases.



**Table 4.** Euclidean distance between Patients and Diseases.

According to the principle of minimum distance point, a lower distance point indicates a proper diagnosis of a particular disease. While comparing the data reported in Tables 3 and 4, a similar interpretation is observed. Here, considering both these Tables 3 and 4, the lowest distance point gives the proper diagnosis, and therefore it can be inferred that patient *P*<sup>1</sup> suffers from *Typhoid* and patients *P*<sup>2</sup> and *P*<sup>3</sup> suffer from *Throat Disease*, whereas, patient *P*<sup>4</sup> suffers from *Viral Fever*. Hence, out of four patients, we observe that two patients are affected with *Throat Disease*. In addition, one patient is affected with each of the two diseases, *Viral Fever* and *Typhoid*, whereas, none of the patients are diagnosed with *Tuberculosis*.

#### **9. Case Study**

For the purpose of simulation, we consider a real case study for 500 patients of the Healthcare Hospital situated in Kolkata, India. Subsequently, we consulted with the specialist doctors of the Healthcare Hospital and received their feedback on the 500 patients at three different times in a day.

A patient suffering from a disease, when visiting a hospital, expresses his/her symptoms, e.g., *Temperature*, *Cough*, *Throat Pain*, *Headache* and *Body Pain* to a doctor. Based on these feedbacks, the doctor analyzes the disease (e.g., *Viral Fever*, *Tuberculosis*, *Typhoid* and *Throat Disease*) the patient is suffering from. These symptoms and the disease of the patients are represented in linguistic terms which involves uncertainty. As an example, in a day, the *Temperature* of a patient can vary in the morning, afternoon and night. Based on these recorded values of the *Temperature*, the evaluation of different doctors might change as well. For example, if the *Temperature* of a patient is recorded 100.5 Fahrenheit (F), 100.2 F and 100.7 F respectively in the morning, afternoon and night, then one doctor might analyze that the patient is having *Viral Fever*, while the other doctor might not agree that the patient is suffering from *Viral Fever* and often agree to observe the *Temperature* for some subsequent days. It can be noted here, that a particular symptom of a patient can fluctuate in a day, and based on these fluctuations, the opinion (analysis) of the experts (doctors) also varies for a particular patient. Therefore, in order to incorporate the uncertainties of the symptoms and the diseases rationally, the parameters of the symptom and the disease of a patient are represented as T2MFS. In our study, the necessary information are received from the doctors of the Healthcare Hospital. It should be mentioned here that while receiving specialist doctors feedback, we consider the set of five symptoms and the four diseases as the same, compared to the one considered in the application mentioned in the numerical illustration section. These data are presented in the tables of the supplementary file. Here, Tables 5 and 6 report the data of the Healthcare Hospital which are similar to the corresponding data presented in Tables 1 and 2. Similar to Table 4, in this case study, we determine the Euclidean distances between patients and diseases. However, instead of the corresponding data of four patients as shown in Tables 2 and 4, Table 6 and Table S1 represent the data of 500 patients. From Table 7, it is observed that 66 patients are affected with *Viral Fever*. Two-hundred-and-forty-two patients are suspected to have *Tuberculosis*; 62 patients are suffering from *Typhoid* and 130 patients are diagnosed with *Throat Disease*.


**Table 5.** Symptoms vs. Diseases in Healthcare Hospital.



**Table 6.** *Cont.*


**Table 6.** *Cont.*


**Table 6.** *Cont.*


**Table 6.** *Cont.*


**Table 6.** *Cont.*


**Table 6.** *Cont.*



**Table 6.** *Cont.*

**Table 7.** Patients diagnosed with a particular disease in Healthcare Hospital.


#### **10. Conclusions**

In this paper, we have tried to extend the concept of multi-fuzzy set theory to type-2 multi-fuzzy sets. The T2MFS may be applied to various applications in daily life. The algebraic properties of these sets have been verified and two types of distance metrics including Hamming distance and Euclidean distance have been discussed. Moreover, a few illustrative examples and a real-life case study of the medical diagnosis system are presented in this article. In the numerical illustration, we measure the Hamming distance and Euclidean distance of each patient for the set of diseases by considering the symptoms of the disease where both types of distance measurements yield a similar diagnostic result. The lowest distance shows proper diagnosis for both the distance measurements. In addition, as an application of T2MFS, the case-study is also conducted on 500 patients undergoing treatment in a hospital.

As far as the limitation of T2MFS is concerned, it is conceptually difficult to define the T2MFS and its necessary algebraic operations in the continuous domain, since the membership functions of such continuous T2MFS will be difficult to represent.

In the future, researchers may attempt to generalize this concept further by studying higher order multi-fuzzy sets in an abstract setting. Also, this research work might be enhancing the study of T2MFS for uncertain group decision-making (GDM) problems by introducing some aggregated operators where GDM is vital due to the lack of information, the expertise of the experts, risk amendment, etc. Besides, the possible extension of T2MFS in various other domains of research including image processing and data mining can be considered as the possible future area of research.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/2073-8994/11/2/170/s1, Table S1: Euclidean distance between Patients and Diseases in Healthcare Hospital.

**Author Contributions:** The individual contribution and responsibilities of the authors were as follows: M.B.K., B.R. and S.M. performed the research study, collected, pre-processed, and analyzed the data and the obtained results, and worked on the development of the paper. S.K. and D.P. provided good advice throughout the research by giving suggestions on, methodology, modelling uncertainty of the patient data, and refinement of the manuscript. All the authors have read and approved the final manuscript.

**Funding:** This research was funded by Department of Science & Technology (DST), Government of India (No. DST/INSPIRE Fellowship/2015/IF150410).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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