T-Spherical Fuzzy Bonferroni Mean Operators and Their Application in Multiple Attribute Decision Making

Abstract

To deal with complicated decision problems with T-Spherical fuzzy values in the aggregation process, T-Spherical fuzzy Bonferroni mean operators are developed by extending the Bonferroni mean and Dombi mean to a T-Spherical fuzzy environment. The T-spherical fuzzy interaction Bonferroni mean operator and the T-spherical fuzzy interaction geometric Bonferroni mean operator are first defined. Then, the T-spherical fuzzy interaction weighted Bonferroni mean operator and the T-spherical fuzzy weighted interaction geometric Bonferroni mean operator are defined. Based on the Dombi mean and the Bonferroni mean operator, some T-Spherical fuzzy Dombi Bonferroni mean operators are proposed, including the T-spherical fuzzy Dombi Bonferroni mean operator, T-spherical fuzzy geometric Dombi Bonferroni mean operator, T-spherical fuzzy weighted Dombi Bonferroni mean operator and the T-spherical fuzzy weighted geometric Dombi Bonferroni mean operator. The properties of these proposed operators are studied. An attribute weight determining method based on the T-spherical fuzzy entropy and symmetric T-spherical fuzzy cross-entropy is developed. A new decision making method based on the proposed T-Spherical fuzzy Bonferroni mean operators is proposed for partly known or completely unknown attribute weight situations. The furniture procurement problem is presented to illustrate the new algorithm, and some comparisons are made.

1. Introduction

Decision problems with fuzzy and uncertain information exist extensively in the real decision making process since decision problems become increasingly complicated. Many useful tools have been developed to model these information, among which Spherical fuzzy sets is an important one developed by Ashraf et al. [1] by extending picture fuzzy sets and intuitionistic fuzzy sets [2,3]. Spherical fuzzy sets have been studied and extended extensively [4,5].

Ashraf and Abdullah [6] developed generalized spherical aggregation operators based on the Archimedean t-norm and t-conorm. Donyatalab et al. [7] defined a spherical fuzzy weighted mean operator and Spherical Fuzzy Einstein weighted Harmonic mean operator. T-spherical fuzzy sets were proposed by Mahmood et al. [8] to generalize Spherical fuzzy sets. Zeng et al. [9] proposed some Einstein geometric averaging interactive aggregation operators.

Zeng et al. [10] introduced T-spherical fuzzy interactive aggregation operators with associate probability. Al-Quran [11] proposed T-spherical hesitant fuzzy sets by combining the hesitant fuzzy and the T-spherical hesitant fuzzy set. Some T-spherical hesitant fuzzy weighted averaging operators have been defined, including the T-spherical hesitant fuzzy weighted averaging operator and the T-spherical hesitant fuzzy weighted geometric operator in [11].

Jan et al. [12] developed the T-spherical fuzzy graph concept and dominant theory of T-spherical fuzzy graphs. Munir et al. [13] defined some Einstein operations based on the Einstein t-norms and t-conorms for T-spherical fuzzy set and developed some T-spherical fuzzy Einstein geometric averaging aggregation operators. Ju et al. [14] extended the TODIM method to the T-spherical fuzzy environment. Xian et al. [15] applied the T-spherical fuzzy c-means method to image segmentation.

Zedam et al. [16] defined the T-spherical fuzzy graph and some related concepts, including subgraphs, the shortest path etc. Guleria and Bajaj [17] applied the T-spherical fuzzy graph notion for supply chain management problems. Mahmood et al. [18] developed T-spherical fuzzy MULTIMOORA method and T-spherical fuzzy Dombi prioritized aggregation operators. Munir et al. [19] introduced some T-spherical fuzzy interactive geometric operators with immediate probability information.

Liu et al. [20] proposed a decision making method based on the T-spherical fuzzy generalized Maclaurin symmetric mean operator and applied it into the problem of selecting the Yunnan Baiyao’s R&D project. Wang and Chen [21] presented the T-spherical fuzzy ELECTRE approach by incorporating two forms of Minkowski distance measures. T-spherical fuzzy correlation coefficients [22] and T-spherical fuzzy similarity measures [23] have been studied, and their applications in clustering were presented.

Khan et al. [24] studied T-spherical fuzzy Schweizer–Sklar weighted geometric Heronian mean operator. Garg et al. [25] proposed the T-spherical fuzzy interactive geometric operators. Mahmood et al. [26] defined interval-valued T-spherical fuzzy soft sets and developed some interval-valued T-spherical fuzzy soft aggregation operators. Garg et al. [27] developed T-spherical fuzzy power aggregation operators.

Ullah et al. [28] defined some T-spherical fuzzy Hamacher aggregation operators. Based on the T-spherical fuzzy values, some new fuzzy sets have been defined and studied [29,30,31,32,33,34,35,36]. Chen et al. [32] presented generalized T-spherical fuzzy geometric aggregation operators; however, the interaction of the operation laws has not been considered.

Though many T-spherical fuzzy multiple attribute decision making methods have been proposed, there are still many decision making problems that cannot be solved using existing methods. Aggregation operators are important in decision making process [36], we develop some new T-spherical fuzzy aggregation operators based on the Bonferroni mean operator and Dombi operator in this paper.

The Bonferroni mean operator is the product of each input value with the average one of the other input values [37,38]. The Bonferroni mean operator has been extended extensively. The Bonferroni geometric mean operator has been studied by Xia et al. [39] and Li et al. [40]. Zhu and Xu [41] developed the hesitant fuzzy Bonferroni mean operator. Zhu et al. [42] studied the hesitant fuzzy geometric Bonferroni mean operator.

He et al. [43] developed the intuitionistic fuzzy geometric power Bonferroni means operators by combing the geometric Bonferroni mean operator with the power mean operator. Park et al. [44] studied optimized weighted geometric Bonferroni means for intuitionistic fuzzy information. Wei et al. [45] proposed the uncertain linguistic Bonferroni mean operators. Liu and Liu [46] defined the intuitionistic uncertain linguistic partitioned Bonferroni mean operators. Chen et al. [47] developed the linguistic 2-tuple geometric Bonferroni mean operator.

Yang et al. [48] studied hesitant Pythagorean fuzzy geometric weight Bonferroni mean operator considering interactions between arguments. Yang et al. [49] studied Pythagorean fuzzy partitioned Bonferroni mean considering interactions. Yang and Pang [50] studied fuzzy Bonferroni mean Dombi aggregation operators in q-rung orthopair fuzzy environments.

Liu and Liu [51] defined normal intuitionistic fuzzy Bonferroni mean operators. Mesiarova-Zemankova et al. [52] introduced the weighted Bonferroni mean considering interactions between inputs. Liang et al. [53] presented interval-valued Pythagorean fuzzy Bonferroni mean operators. Ate and Akay [54] developed picture fuzzy Bonferroni mean operators.

Mahmood and Ahsen [55] presented some picture-hesitant fuzzy Bonferroni mean operators. Although the Bonferroni mean operator has been extended into various environments, the Bonferroni mean operator in T-spherical fuzzy environments has not been studied. Hence, we extend the Bonferroni mean operator to the T-spherical fuzzy environment considering the interaction operations between T-spherical fuzzy values.

The Dombi mean [56] is a flexible aggregation method by a parameter that is based on the Dombi t-norm and Dombi t-conorm. The Dombi aggregation has also received extensive attention. Liu et al. [57] studied Pythagorean fuzzy Dombi power average operators. Jana and Pal [58] presented single-valued neutrosophic aggregation operators.

Wu et al. [59] proposed Dombi Hamy mean operators in interval-valued intuitionistic fuzzy environments. Jana et al. [60] developed some Dombi aggregation operators to aggregate Pythagorean fuzzy information. Shit and Ghorai [61] presented some Fermatean fuzzy Dombi aggregation operators.

Kurama [62] studied the similarity classifier with Dombi aggregation operators. Jana et al. [63] proposed picture fuzzy Dombi aggregation operators. Gulfam et al. [64] defined some bipolar neutrosophic Dombi aggregation operators. Ayub et al. [65] presented cubic fuzzy Dombi aggregation operators using the Heronian mean. Akram et al. [66] proposed complex Pythagorean fuzzy Dombi aggregation operators.

Ali and Mahmood [67] presented some complex q-rung orthopair fuzzy Dombi aggregation operators. Khan et al. [68] studied spherical fuzzy improved Dombi power averaging operators. Saha et al. [69] studied hesitant fuzzy Archimedean Dombi aggregation operators. Kamaci et al. [70] proposed bipolar trapezoidal neutrosophic Dombi operators.

Dombi aggregation operators have been used in typhoon disaster assessment problems [71], personnel evaluation problems [72], green supplier selection problems [73] etc. Since the Bonferroni mean can consider the interaction between input arguments, and the Dombi mean is flexible in aggregation, we further extend the Bonferroni mean to combine the Dombi mean in T-spherical fuzzy environment and develop T-spherical fuzzy Dombi Bonferroni mean operators.

The main contributions of this paper are summarized as follows. (1) The T-spherical fuzzy values are used in the decision-making process to deal with complicated decision problems. (2) The T-spherical fuzzy interaction operation laws are used to reduce the influence of an extremely small membership degree, abstinence degrees or non-membership degree. (3) T-spherical fuzzy interaction Bonferroni mean operators are defined by extending the Bonferroni mean to accommodate the T-spherical fuzzy values by considering interactions. (4) T-spherical fuzzy Dombi Bonferroni mean operators are developed by combining the Bonferroni mean with Dombi mean operator in T-spherical fuzzy environments. (5) A new T-spherical fuzzy entropy measure and a new T-spherical fuzzy cross-entropy measure are proposed. The attribute weights are calculated using the proposed entropy and cross-entropy measure for partly known and completely unknown situations. (6) The decision making method based on the new T-spherical fuzzy Bonferroni mean operators are developed. Some comparisons are conducted to illustrate the practical advantages of the proposed method.

The rest of the paper is organized as follows. In Section 2, some concepts about T-spherical fuzzy sets are reviewed, including the interaction operation laws of T-spherical fuzzy numbers. In Section 3, some T-spherical fuzzy interaction Bonferroni mean operators are defined, including the T-spherical fuzzy interaction Bonferroni mean operator and T-spherical fuzzy interaction weighted Bonferroni mean operator. In Section 4, the T-spherical fuzzy Dombi Bonferroni mean operator and the T-spherical fuzzy weighted Dombi Bonferroni mean operator are introduced.

In Section 5, the T-spherical fuzzy entropy and cross-entropy measures are proposed, and a method to determine attribute weights using the cross-entropy measure is developed. In Section 6, a new T-spherical fuzzy multiple attribute decision making method is presented based on the new proposed operators. In Section 7, a numerical example is proposed to illustrate the new method. In the last section, our conclusions are given.

2. Preliminaries

Definition 1

([8]). For a universal set X, a T-spherical fuzzy set (T-SFS) $\tilde{A}$ on X can be defined as

$$\tilde{A}=\{<x,{\mu }_{\tilde{A}}(x),{\eta }_{\tilde{A}}(x),{\nu }_{\tilde{A}}(x)>\mid x\in X\},$$

(1)

where ${\mu }_{\tilde{A}}(x):X\to [0,1]$, ${\eta }_{\tilde{A}}(x):X\to [0,1]$ and ${\nu }_{\tilde{A}}(x):X\to [0,1]$ denote the membership degree, the abstinence degree and the non-membership degree, respectively, which satisfy the following condition $0\le {\mu }_{\tilde{A}}^t(x)+{\eta }_{\tilde{A}}^t(x)+{\nu }_{\tilde{A}}^t(x)\le 1,t\ge 1.$ The refusal degree is given as ${\pi }_{\tilde{A}}(x)={({\mu }_{\tilde{A}}^t(x)+{\eta }_{\tilde{A}}^t(x)+{\nu }_{\tilde{A}}^t(x))}^{1/t}.$ For simplicity, the triple $p=<{\mu }_{\tilde{A}},{\eta }_{\tilde{A}},{\nu }_{\tilde{A}}>$ is the T-spherical fuzzy number (T-SFN). $p^c=<{\nu }_{\tilde{A}},1-{\eta }_{\tilde{A}},{\mu }_{\tilde{A}}>$. Let H be the set of all the T-spherical fuzzy values.

Definition 2

([14]). Let $\alpha =<\mu ,\eta ,\nu >$ be a T-SFN, then the score function $S(\alpha )$ of α is defined as

$$S(\alpha )=\frac{1}{2}(1+{\mu }^t-{\eta }^t-{\nu }^t),$$

(2)

$S(\alpha )\in [0,1]$.

Definition 3

([14]). Let $\alpha =<\mu ,\eta ,\nu >$ be a T-SFN, then the accuracy function $H(\alpha )$ of α is defined as

$$H(\alpha )={\mu }^t+{\eta }^t+{\nu }^t,$$

(3)

$H(\alpha )\in [0,1]$.

Definition 4

([14]). Let ${\alpha }_1=<{\mu }_1,{\eta }_1,{\nu }_1>$ and ${\alpha }_2=<{\mu }_2,{\eta }_2,{\nu }_2>$ be two T-SFNs. Then,

(1) If$S({\alpha }_1)>S({\alpha }_2)$, then${\alpha }_1>{\alpha }_2$;

(2) If$S({\alpha }_1)=S({\alpha }_2)$, and

            $H({\alpha }_1)>H({\alpha }_2)$, then ${\alpha }_1>{\alpha }_2$;

            $H({\alpha }_1)=H({\alpha }_2)$, then ${\alpha }_1\sim {\alpha }_2$.

Let $\alpha =<{\mu }_{\alpha },{\eta }_{\alpha },{\nu }_{\alpha }>$ and $\beta =<{\mu }_{\beta },{\eta }_{\beta },{\nu }_{\beta }>$ be two T-SFNs, $\lambda >0$. Then, the operational laws of T-spherical fuzzy values can be defined as follows [20]

$(1)\alpha \oplus \beta =<{(1-(1-{\mu }_{\alpha }^t)(1-{\mu }_{\beta }^t))}^{1/t},{\eta }_{\alpha }{\eta }_{\beta },{\nu }_{\alpha }{\nu }_{\beta }>$,

$(2)\alpha \otimes \beta =<{\mu }_{\alpha }{\mu }_{\beta },{(1-(1-{\eta }_{\alpha }^t)(1-{\eta }_{\beta }^t))}^{1/t},{(1-(1-{\nu }_{\alpha }^t)(1-{\nu }_{\beta }^t))}^{1/t}>$,

$(3){\alpha }^{\lambda }=<{\mu }_{\alpha }^{\lambda },{(1-{(1-{\eta }_{\alpha }^t)}^{\lambda })}^{1/t},{(1-{(1-{\nu }_{\alpha }^t)}^{\lambda })}^{1/t}>$,

$(4)\lambda \alpha =<{(1-{(1-{\mu }_{\alpha }^t)}^{\lambda })}^{1/t},{\eta }_{\alpha }^{\lambda },{\nu }_{\alpha }^{\lambda }>$.

Then, by using the T-spherical fuzzy operational laws [20], the T-spherical fuzzy weighted averaging (TSFWA) operator and the the T-spherical fuzzy weighted geometric averaging (TSFWGA) operator can be obtained, respectively, as

$$\begin{array}{ccc}\mathrm{TSFWA}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)\hfill & =\hfill & <{(1-\prod _{i=1}^n{(1-{\mu }_{{\alpha }_i}^t)}^{w_i})}^{1/t},\prod _{i=1}^n{\eta }_{{\alpha }_i}^{w_i},\prod _{i=1}^n{\nu }_{{\alpha }_i}^{w_i}>.\hfill \end{array}$$

$$\begin{array}{ccc}\mathrm{TSFWGA}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)\hfill & =\hfill & <\prod _{i=1}^n{\mu }_{{\alpha }_i}^{w_i},{(1-\prod _{i=1}^n{(1-{\eta }_{{\alpha }_i}^t)}^{w_i})}^{1/t},{(1-\prod _{i=1}^n{(1-{\nu }_{{\alpha }_i}^t)}^{w_i})}^{1/t}>.\hfill \end{array}$$

Example 1.

Let${\alpha }_1=<0.5,0.4,0.0>$,${\alpha }_2=<0.6,0.0,0.4>$,${\alpha }_3=<0.0,0.7,0.3>$,${\alpha }_4=<0.5,0.6,0.4>$,$t=3$. Then, the aggregated results are obtained as$\mathrm{TSFWA}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_4)=<0.4832,0,$$0>$and$\mathrm{TSFWGA}({\alpha }_1,$${\alpha }_2,\cdots ,{\alpha }_4)=<0,0.5632,0.3379>$.

From the above results, we can see that, if 0 is in the membership degree, the abstinence degree and the non-membership degree of the T-SFNs, then there may be 0 in the aggregated the membership degree, the abstinence degree and the non-membership degree even if other values are not zero. In order to avoid information loss, the following interaction operational laws of T-SFNs can be defined.

Let $\alpha =<\mu ,\eta ,\nu >$, ${\alpha }_1=<{\mu }_1,{\eta }_1,{\nu }_1>$ and ${\alpha }_2=<{\mu }_2,{\eta }_2,{\nu }_2>$ be T-SFNs, $\lambda >0$. The operational laws of T-SFNs considering interaction relationships among ${\mu }_i,{\eta }_i,{\nu }_i$ are defined as follows [14]

$$\begin{array}{ccc}\hfill (1){\alpha }_1\oplus {\alpha }_2& =& <{(1-(1-{\mu }_1^t)(1-{\mu }_2^t))}^{1/t},((1-{\mu }_1^t)(1-{\mu }_2^t)-(1-{\mu }_1^t-{\eta }_1^t)(1-{\mu }_2^t-\hfill \\ & & {\eta }_2^t{))}^{1/t},((1-{\mu }_1^t-{\eta }_1^t)(1-{\mu }_2^t-{\eta }_2^t)-(1-{\mu }_1^t-{\eta }_1^t-{\nu }_1^t)(1-{\mu }_2^t-\hfill \\ & & {\eta }_2^t-{\nu }_2^t{))}^{1/t}>;\hfill \end{array}$$

$$\begin{array}{ccc}\hfill (2){\alpha }_1\otimes {\alpha }_2& =& <{((1-{\nu }_1^t-{\eta }_1^t)(1-{\nu }_2^t-{\eta }_2^t)-(1-{\mu }_1^t-{\eta }_1^t-{\nu }_1^t)(1-{\mu }_2^t-{\eta }_2^t-{\nu }_2^t))}^{1/t},\hfill \\ & & {((1-{\nu }_1^t)(1-{\nu }_2^t)-(1-{\nu }_1^t-{\eta }_1^t)(1-{\nu }_2^t-{\eta }_2^t))}^{1/t},(1-(1-{\nu }_1^t)\hfill \\ & & (1-{\nu }_2^t){)}^{1/t}>;\hfill \end{array}$$

$$\begin{array}{ccc}\hfill (3)\lambda \alpha & =& <{(1-{(1-{\mu }^t)}^{\lambda })}^{1/t},{({(1-{\mu }^t)}^{\lambda }-{(1-{\mu }^t-{\eta }^t)}^{\lambda })}^{1/t},({(1-{\mu }^t-{\eta }^t)}^{\lambda }-(1-\hfill \\ & & {\mu }^t-{\eta }^t-{\nu }^t{{)}^{\lambda })}^{1/t}>;\hfill \end{array}$$

$$\begin{array}{ccc}\hfill (4){\alpha }^{\lambda }& =& <{({(1-{\nu }^t-{\eta }^t)}^{\lambda }-{(1-{\mu }^t-{\eta }^t-{\nu }^t)}^{\lambda })}^{1/t},{({(1-{\nu }^t)}^{\lambda }-{(1-{\nu }^t-{\eta }^t)}^{\lambda })}^{1/t},\hfill \\ & & {(1-{(1-{\nu }^t)}^{\lambda })}^{1/t}>.\hfill \end{array}$$

By using the interaction operation laws of T-spherical fuzzy numbers, the TSFWA operator and the TSFWGA operator become the T-spherical fuzzy interaction weighted averaging (TSFIWA) operator and the T-spherical fuzzy interaction weighted geometric averaging (TSFIWGA) operator as

$$\begin{array}{cc}& \mathrm{TSFIWA}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)=<{(1-\prod _{i=1}^n{(1-{\mu }_{{\alpha }_i}^t)}^{w_i})}^{1/t},(\prod _{i=1}^n{(1-{\mu }_{{\alpha }_i}^t)}^{w_i}-\prod _{i=1}^n(1-{\mu }_{{\alpha }_i}^t\hfill \\ & -{\eta }_{{\alpha }_i}^t{{)}^{w_i})}^{1/t},{(\prod _{i=1}^n{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^{w_i}-\prod _{i=1}^n{(1-{\mu }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^{w_i})}^{1/t}>.\hfill \end{array}$$

$$\begin{array}{cc}& \mathrm{TSFIWGA}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)=<{(\prod _{i=1}^n{(1-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^{w_i}-\prod _{i=1}^n{(1-{\mu }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^{w_i})}^{1/t},\hfill \\ & {(\prod _{i=1}^n{(1-{\nu }_{{\alpha }_i}^t)}^{w_i}-\prod _{i=1}^n{(1-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^{w_i})}^{1/t},{(1-\prod _{i=1}^n{(1-{\nu }_{{\alpha }_i}^t)}^{w_i})}^{1/t}>.\hfill \end{array}$$

Example 2.

${\alpha }_i,w_i(i=1,2,\cdots ,4)$and t are the same as that in Example 1. By using the TSFIWA operator and the TSFIWGA operator, the aggregated results can be obtained as$\mathrm{TSFIWA}({\alpha }_1,$${\alpha }_2,\cdots ,{\alpha }_4)=<0.4832,0.5476,0.3393>$,$\mathrm{TSFIWGA}({\alpha }_1,$${\alpha }_2,\cdots ,{\alpha }_4)=<0.4636,$$0.5625,0.3379>$.

Definition 5

([14]). Let ${\alpha }_1=<{\mu }_1,{\eta }_1,{\nu }_1>$, ${\alpha }_2=<{\mu }_2,{\eta }_2,{\nu }_2>$ be two T-SFNs. The T-spherical fuzzy generalized distance measure can be defined as

$$\begin{array}{ccc}& & d{({\alpha }_1,{\alpha }_2)}_{\lambda }=(|{\mu }_1^t-{\mu }_2^t{|}^{\lambda }+|{\eta }_1^t-{\eta }_2^t{|}^{\lambda }+|{\nu }_1^t-{\nu }_2^t{{|}^{\lambda })}^{1/\lambda }.\hfill \end{array}$$

If $\lambda =1$, $d{({\alpha }_1,{\alpha }_2)}_{\lambda }$ becomes the T-spherical fuzzy Hamming distance. If $\lambda =2$, $d{({\alpha }_1,{\alpha }_2)}_{\lambda }$ becomes the T-spherical fuzzy Euclidean distance.

3. T-Spherical Fuzzy Interaction Bonferroni Mean Operator

Definition 6

([37]). The Bonferroni mean aggregation operator of dimension n is a mapping ${(R^{+})}^n\to R^{+}$,

$$\mathrm{BM}(a_1,a_2,\cdots ,a_n)={\left(\frac{1}{n(n-1)}{\oplus }_{i,j=1,i\ne j}^n(a_i^k\otimes a_j^l)\right)}^{\frac{1}{k+l}},$$

where $k,l\ge 0$, $(a_1,a_2,\cdots ,a_n)$ is a collection of nonnegative real numbers.

Definition 7.

Let${\alpha }_i=<{\mu }_{{\alpha }_i},{\eta }_{{\alpha }_i},{\nu }_{{\alpha }_i}>(i=1,2,\cdots ,n)$be T-SFNs. The T-spherical fuzzy interaction Bonferroni mean (TSFIBM) operator can be defined as

$$\begin{array}{ccc}\hfill \mathrm{TSFIBM}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)& =& {\left(\frac{1}{n(n-1)}{\oplus }_{i,j=1,i\ne j}^n({\alpha }_i^k\otimes {\alpha }_j^l)\right)}^{\frac{1}{k+l}},\hfill \end{array}$$

(4)

where$k,t\ge 0$.

Theorem 1.

Let${\alpha }_i=<{\mu }_{{\alpha }_i},{\eta }_{{\alpha }_i},{\nu }_{{\alpha }_i}>(i=1,2,\cdots ,n)$be T-SFNs,$k,t\ge 0$. The aggregated value of TSFIBM operator is still a T-SFN, and

$$\begin{array}{ccc}& \hfill & {\mathrm{TSFIBM}}_{k,l}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)={\left(\frac{1}{n(n-1)}{\oplus }_{i,j=1,i\ne j}^n({\alpha }_i^k\otimes {\alpha }_j^l)\right)}^{\frac{1}{k+l}}\hfill \\ & =\hfill & <((1-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-\hfill \\ & \hfill & {\nu }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}+(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-\hfill \\ & \hfill & {\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t{)}^l{))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}-((\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-\hfill \\ & \hfill & {\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t{)}^l{))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}{)}^{1/t},((1-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t)}^l+\hfill \\ & \hfill & {(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}+\hfill \\ & \hfill & {(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}-\hfill \\ & \hfill & (1-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k\ast \hfill \\ & \hfill & {(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}+(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-{\mu }_{{\alpha }_j}^t-\hfill \\ & \hfill & {\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t{)}^l{))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}{)}^{1/t},(1-(1-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t)}^l+\hfill \\ & \hfill & {(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}+(\prod _{i,j=1,i\ne j}^n((1-{\mu }_{{\alpha }_i}^t-\hfill \\ & \hfill & {\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}{{)}^{\frac{1}{k+l}})}^{1/t}>.\hfill \end{array}$$

(5)

Proof. 

$$\begin{array}{ccc}{\alpha }_i^k\hfill & =\hfill & <{({(1-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k-{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k)}^{1/t},({(1-{\nu }_{{\alpha }_i}^t)}^k-(1-{\nu }_{{\alpha }_i}^t-\hfill \\ & & {\eta }_{{\alpha }_i}^t{{)}^k)}^{1/t},{(1-{(1-{\nu }_{{\alpha }_i}^t)}^k)}^{1/t}>,\hfill \end{array}$$

$$\begin{array}{ccc}{\alpha }_j^l\hfill & =\hfill & <{({(1-{\nu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l-{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l)}^{1/t},({(1-{\nu }_{{\alpha }_j}^t)}^l-(1-{\nu }_{{\alpha }_j}^t-\hfill \\ & & {\eta }_{{\alpha }_j}^t{{)}^l)}^{1/t},{(1-{(1-{\nu }_{{\alpha }_j}^t)}^l)}^{1/t}>,\hfill \end{array}$$

$$\begin{array}{ccc}{\alpha }_i^k\otimes {\alpha }_j^l\hfill & =\hfill & <({(1-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l-{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-{\mu }_{{\alpha }_j}^t-\hfill \\ & & {\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t{{)}^l)}^{1/t},{({(1-{\nu }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t)}^l-{(1-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l)}^{1/t},\hfill \\ & & {(1-{(1-{\nu }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t)}^l)}^{1/t}>,\hfill \end{array}$$

$$\begin{array}{ccc}& & {\oplus }_{i,j=1,i\ne j}^n({\alpha }_i^k\otimes {\alpha }_j^l)\hfill \\ & =& <(1-\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k\ast \hfill \\ & & {(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{1/t},(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l\hfill \\ & & +{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l)-\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t)}^k(1-\hfill \\ & & {\nu }_{{\alpha }_j}^t{)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{1/t},(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t)}^k\ast \hfill \\ & & {(1-{\nu }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l)-\prod _{i,j=1,i\ne j}^n((1-{\mu }_{{\alpha }_i}^t-\hfill \\ & & {\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{1/t}>,\hfill \end{array}$$

$$\begin{array}{ccc}& & \frac{1}{n(n-1)}{\oplus }_{i,j=1,i\ne j}^n({\alpha }_i^k\otimes {\alpha }_j^l)\hfill \\ & =& <(1-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-\hfill \\ & & {\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t{)}^l{))}^{\frac{1}{n(n-1)}}{)}^{1/t},((\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+\hfill \\ & & {(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t)}^k(1-\hfill \\ & & {\nu }_{{\alpha }_j}^t{)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}{)}^{1/t},((\prod _{i,j=1,i\ne j}^n(1-(1-\hfill \\ & & {\nu }_{{\alpha }_i}^t{)}^k{(1-{\nu }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}-\hfill \\ & & {(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l))}^{\frac{1}{n(n-1)}}{)}^{1/t}>,\hfill \end{array}$$

$$\begin{array}{ccc}& & {\left(\frac{1}{n(n-1)}{\oplus }_{i,j=1,i\ne j}^n({\alpha }_i^k\otimes {\alpha }_j^l)\right)}^{\frac{1}{k+l}}\hfill \\ & =\hfill & <((1-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k\ast \hfill \\ & & {(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}+(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-\hfill \\ & & {\nu }_{{\alpha }_j}^t{)}^l{))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}-((\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-\hfill \\ & & {\nu }_{{\alpha }_j}^t{)}^l{))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}{)}^{1/t},((1-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t)}^l+(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-\hfill \\ & & {\nu }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}+(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-{\mu }_{{\alpha }_j}^t-\hfill \\ & & {\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t{)}^l{))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}-(1-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+\hfill \\ & & {(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}+(\prod _{i,j=1,i\ne j}^n((1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-\hfill \\ & & {\nu }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}{)}^{1/t},(1-(1-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t)}^k(1-\hfill \\ & & {\nu }_{{\alpha }_j}^t{)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}+(\prod _{i,j=1,i\ne j}^n((1-\hfill \\ & & {\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}{{)}^{\frac{1}{k+l}})}^{1/t}>.\hfill \end{array}$$

Moreover,

$$\begin{array}{ccc}& & {\mu }^t+{\nu }^t+{\eta }^t\hfill \\ & =\hfill & (1-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k\ast \hfill \\ & & {(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}+(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-{\mu }_{{\alpha }_j}^t-\hfill \\ & & {\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t{)}^l{))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}-((\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-\hfill \\ & & {\nu }_{{\alpha }_j}^t{)}^l{))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}+(1-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t)}^l+(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-\hfill \\ & & {\nu }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}+(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-\hfill \\ & & {\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t{)}^l{))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}-(1-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k(1-{\nu }_{{\alpha }_j}^t-\hfill \\ & & {\eta }_{{\alpha }_j}^t{)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}+(\prod _{i,j=1,i\ne j}^n((1-\hfill \\ & & {\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}})+1-(1-(\prod _{i,j=1,i\ne j}^n(1-\hfill \\ & & {(1-{\nu }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}+\hfill \\ & & {(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}\hfill \\ & =\hfill & 1-{({(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l))}^{\frac{1}{n(n-1)}})}^{\frac{1}{k+l}}.\hfill \end{array}$$

Since $1\le {\mu }_{{\alpha }_i}^t+{\eta }_{{\alpha }_i}^t+{\nu }_{{\alpha }_i}^t\le 1$, $0\le {(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k\le 1$. Similarly, $0\le {(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l\le 1$. Then, $0\le {({({\prod }_{i,j=1,i\ne j}^{n}0\le ({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l))}^{\frac{1}{n(n-1)}})}^{\frac{1}{k+l}}\le 1$ and $0\le 1-{({({\prod }_{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l))}^{\frac{1}{n(n-1)}})}^{\frac{1}{k+l}}\le 1$. Hence, the aggregated value of TSFIBM operator is still a T-SFN. □

In the following, we prove some important properties of the TSFIBM operator, including idempotency and boundedness.

Theorem 2

(Idempotency). If ${\alpha }_i=\alpha ,<{\mu }_{{\alpha }_i},{\eta }_{{\alpha }_i},{\nu }_{{\alpha }_i}>=<{\mu }_{\alpha },{\eta }_{\alpha },{\nu }_{\alpha }>(i=1,2,\cdots ,n)$. Then

$$\begin{array}{ccc}& & {\mathrm{TSFIBM}}_{k,l}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)=\alpha .\hfill \end{array}$$

Proof. 

Let ${\mathrm{TSFIBM}}_{k,l}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)=<\mu ,\eta ,\nu >$.

$$\begin{array}{ccc}\mu \hfill & =\hfill & ((1-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k\ast \hfill \\ & & {(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}+(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-{\mu }_{{\alpha }_j}^t-\hfill \\ & & {\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t{)}^l{))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}-((\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-\hfill \\ & & {\nu }_{{\alpha }_j}^t{)}^l{))}^{\frac{1}{n(n-1)}}{{)}^{\frac{1}{k+l}})}^{1/t}\hfill \\ & =\hfill & ((1-{(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{\alpha }^t-{\eta }_{\alpha }^t)}^{k+l}+{(1-{\mu }_{\alpha }^t-{\eta }_{\alpha }^t-{\nu }_{\alpha }^t)}^{k+l}))}^{\frac{1}{n(n-1)}}+\hfill \\ & & {(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{\alpha }^t-{\eta }_{\alpha }^t-{\nu }_{\alpha }^t)}^{k+l}))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}-((\prod _{i,j=1,i\ne j}^n((1-{\mu }_{\alpha }^t-{\eta }_{\alpha }^t-\hfill \\ & & {\nu }_{\alpha }^t{)}^{k+l}{))}^{\frac{1}{n(n-1)}}{{)}^{\frac{1}{k+l}})}^{1/t}\hfill \\ & =\hfill & {((1-{\nu }_{\alpha }^t-{\eta }_{\alpha }^t)-(1-{\mu }_{\alpha }^t-{\eta }_{\alpha }^t-{\nu }_{\alpha }^t))}^{1/t}={\mu }_{\alpha },\hfill \end{array}$$

$$\begin{array}{ccc}\eta \hfill & =\hfill & ((1-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-\hfill \\ & & {\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t{)}^l{))}^{\frac{1}{n(n-1)}}+(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-{\mu }_{{\alpha }_j}^t-\hfill \\ & & {\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t{)}^l{))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}-(1-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k(1-{\nu }_{{\alpha }_j}^t-\hfill \\ & & {\eta }_{{\alpha }_j}^t{)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}+(\prod _{i,j=1,i\ne j}^n((1-\hfill \\ & & {\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^l{))}^{\frac{1}{n(n-1)}}{{)}^{\frac{1}{k+l}})}^{1/t}\hfill \\ & =\hfill & ({(1-(1-{(1-{\nu }_{\alpha }^t)}^{k+l}+{(1-{\mu }_{\alpha }^t-{\eta }_{\alpha }^t-{\nu }_{\alpha }^t)}^{k+l})+({(1-{\mu }_{\alpha }^t-{\eta }_{\alpha }^t-{\nu }_{\alpha }^t)}^{k+l}))}^{\frac{1}{k+l}}-\hfill \\ & & (1-((1-{(1-{\nu }_{\alpha }^t-{\eta }_{\alpha }^t)}^{k+l}+{(1-{\mu }_{\alpha }^t-{\eta }_{\alpha }^t-{\nu }_{\alpha }^t)}^{k+l})\hfill \\ & & +({(1-{\mu }_{\alpha }^t-{\eta }_{\alpha }^t-{\nu }_{\alpha }^t)}^{k+l}){{)}^{\frac{1}{k+l}})}^{1/t}\hfill \\ & =\hfill & {(1-{\nu }_{\alpha }^t-(1-{\nu }_{\alpha }^t-{\eta }_{\alpha }^t))}^{1/t}={\eta }_{\alpha },\hfill \end{array}$$

$$\begin{array}{ccc}\nu \hfill & =\hfill & (1-(1-(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{{\alpha }_i}^t)}^k{(1-{\nu }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-\hfill \\ & & {\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t{)}^l{))}^{\frac{1}{n(n-1)}}+(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t\hfill \\ & & -{\nu }_{{\alpha }_j}^t{)}^l{))}^{\frac{1}{n(n-1)}}{{)}^{\frac{1}{k+l}})}^{1/t}\hfill \\ & =\hfill & (1-(1-{(\prod _{i,j=1,i\ne j}^n(1-{(1-{\nu }_{\alpha }^t)}^{k+l}+{(1-{\mu }_{\alpha }^t-{\eta }_{\alpha }^t-{\nu }_{\alpha }^t)}^{k+l}))}^{\frac{1}{n(n-1)}}+\hfill \\ & & {(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{\alpha }^t-{\eta }_{\alpha }^t-{\nu }_{\alpha }^t)}^{k+l}))}^{\frac{1}{n(n-1)}}{{)}^{\frac{1}{k+l}})}^{1/t}\hfill \\ & =\hfill & {(1-{({(1-{\nu }_{\alpha }^t)}^{k+l})}^{\frac{1}{k+l}})}^{1/t}={\nu }_{\alpha }.\hfill \end{array}$$

Hence, ${\mathrm{TSFIBM}}_{k,l}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)=<{\mu }_{\alpha },{\eta }_{\alpha },{\nu }_{\alpha }>=\alpha$. □

Theorem 3

(Boundedness). Let $({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)$ be a collection of T-SFNs. If ${\alpha }^{+}=<1,0,0>$, ${\alpha }^{-}=<0,0,1>$, then

$$\begin{array}{ccc}& & {\alpha }^{-}\le {\mathrm{TSFIBM}}_{k,l}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)\le {\alpha }^{+}.\hfill \end{array}$$

Proof. 

Since $0\le {\mu }_{{\alpha }_i}\le 1,0\le {\eta }_{{\alpha }_i}\le 1,0\le {\nu }_{{\alpha }_i}\le 1$, $0\le {\mu }_{{\alpha }_i}^t+{\eta }_{{\alpha }_i}^t+{\nu }_{{\alpha }_i}^t\le 1$, $S({\alpha }_i)=\frac{1}{2}(1+{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)$, then ${\alpha }^{-}\le {\mathrm{TSFIBM}}_{k,l}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)\le {\alpha }^{+}$. □

Definition 8.

Let${\alpha }_i=<{\mu }_{{\alpha }_i},{\eta }_{{\alpha }_i},{\nu }_{{\alpha }_i}>(i=1,2,\cdots ,n)$be T-SFNs. The T-spherical fuzzy interaction geometric Bonferroni mean (TSFIGBM) operator can be defined as

$$\begin{array}{ccc}\hfill \mathrm{TSFIGBM}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)& =& \frac{1}{k+l}{\otimes }_{i,j=1,i\ne j}^n{\left(k{\alpha }_i\oplus l{\alpha }_j\right)}^{\frac{1}{n(n-1)}},\hfill \end{array}$$

(6)

where$k,l\ge 0$.

Theorem 4.

Let${\alpha }_i=<{\mu }_{{\alpha }_i},{\eta }_{{\alpha }_i},{\nu }_{{\alpha }_i}>(i=1,2,\cdots ,n)$be T-SFNs,$k,t\ge 0$. The aggregated value of TSFIGBM operator is still a T-SFN and

$$\begin{array}{ccc}& & \mathrm{TSFIGBM}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)=\frac{1}{k+l}{\otimes }_{i,j=1,i\ne j}^n{\left(k{\alpha }_i\oplus l{\alpha }_j\right)}^{\frac{1}{n(n-1)}}\hfill \\ & =& <(1-(1-(\prod _{i,j=1,i\ne j}^n((1-{(1-{\mu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k\ast \hfill \\ & & {(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{)}^{\frac{1}{n(n-1)}})-\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-{\mu }_{{\alpha }_j}^t-\hfill \\ & & {\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t{)}^l{)}^{\frac{1}{n(n-1)}}{))}^{\frac{1}{k+l}}{)}^{1/t},((1-(\prod _{i,j=1,i\ne j}^n((1-{(1-{\mu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t)}^l+\hfill \\ & & {(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{)}^{\frac{1}{n(n-1)}})-\prod _{i,j=1,i\ne j}^n((1-{\mu }_{{\alpha }_i}^t-\hfill \\ & & {\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{)}^{\frac{1}{n(n-1)}}{))}^{\frac{1}{k+l}}-(1-\prod _{i,j=1,i\ne j}^n(1-(1-{\mu }_{{\alpha }_i}^t-\hfill \\ & & {\eta }_{{\alpha }_i}^t{{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l)}^{\frac{1}{n(n-1)}}+\hfill \\ & & \prod _{i,j=1,i\ne j}^n{({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l)}^{\frac{1}{n(n-1)}}{{)}^{\frac{1}{k+l}})}^{1/t},\hfill \\ & & ((1-\prod _{i,j=1,i\ne j}^n(1-{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-\hfill \\ & & {\nu }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{)}^{\frac{1}{n(n-1)}}+\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-\hfill \\ & & {\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t{)}^l{)}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}-(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-\hfill \\ & & {\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t{)}^l{)}^{\frac{1}{n(n-1)}}{{)}^{\frac{1}{k+l}})}^{1/t}>.\hfill \end{array}$$

Proof. 

$$\begin{array}{ccc}\hfill k{\alpha }_i& =& <{(1-{(1-{\mu }_{{\alpha }_i}^t)}^k)}^{1/t},{({(1-{\mu }_{{\alpha }_i}^t)}^k-{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k)}^{1/t},({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k-\hfill \\ & & {(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{)}^{1/t}>,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill l{\alpha }_i& =& <{(1-{(1-{\mu }_{{\alpha }_j}^t)}^l)}^{1/t},{({(1-{\mu }_{{\alpha }_j}^t)}^l-{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l)}^{1/t},({(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l-\hfill \\ & & {(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{)}^{1/t}>,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill k{\alpha }_i\oplus l{\alpha }_j& =& <{(1-{(1-{\mu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t)}^l)}^{1/t},({(1-{\mu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t)}^l-(1-{\mu }_{{\alpha }_i}^t-\hfill \\ & & {\eta }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l{)}^{1/t},({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l-(1-\hfill \\ & & {\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t{{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l)}^{1/t}>,\hfill \end{array}$$

$$\begin{array}{ccc}& & {\left(k{\alpha }_i\oplus l{\alpha }_j\right)}^{\frac{1}{n(n-1)}}\hfill \\ & =& <({(1-{(1-{\mu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l)}^{\frac{1}{n(n-1)}}-\hfill \\ & & {({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l)}^{\frac{1}{n(n-1)}}{)}^{1/t},((1-{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k\ast \hfill \\ & & {(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{)}^{\frac{1}{n(n-1)}}-(1-\hfill \\ & & {(1-{\mu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{{)}^{\frac{1}{n(n-1)}})}^{1/t},\hfill \\ & & (1-(1-{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-{\mu }_{{\alpha }_j}^t-\hfill \\ & & {\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t{)}^l{{)}^{\frac{1}{n(n-1)}})}^{1/t}>.\hfill \end{array}$$

$$\begin{array}{ccc}& & {\otimes }_{i,j=1,i\ne j}^n{\left(k{\alpha }_i\oplus l{\alpha }_j\right)}^{\frac{1}{n(n-1)}}\hfill \\ & =& <(\prod _{i,j=1,i\ne j}^n((1-{(1-{\mu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-\hfill \\ & & {\nu }_{{\alpha }_i}^t{)}^l{)}^{\frac{1}{n(n-1)}}{)-\prod _{i,j=1,i\ne j}^n{({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l)}^{\frac{1}{n(n-1)}})}^{1/t},\hfill \\ & & (\prod _{i,j=1,i\ne j}^n(1-{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-\hfill \\ & & {\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t{)}^l{)}^{\frac{1}{n(n-1)}}-\prod _{i,j=1,i\ne j}^n(1-{(1-{\mu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t)}^l+(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-\hfill \\ & & {\nu }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{)}^{\frac{1}{n(n-1)}}{)}^{1/t},(1-\prod _{i,j=1,i\ne j}^n(1-{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k\ast \hfill \\ & & {(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{{)}^{\frac{1}{n(n-1)}})}^{1/t}>.\hfill \end{array}$$

$$\begin{array}{ccc}& & \frac{1}{k+l}{\otimes }_{i,j=1,i\ne j}^n{\left(k{\alpha }_i\oplus l{\alpha }_j\right)}^{\frac{1}{n(n-1)}}\hfill \\ & =& <(1-(1-(\prod _{i,j=1,i\ne j}^n((1-{(1-{\mu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k\ast \hfill \\ & & {(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{)}^{\frac{1}{n(n-1)}})-\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-{\mu }_{{\alpha }_j}^t-\hfill \\ & & {\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t{)}^l{)}^{\frac{1}{n(n-1)}}{))}^{\frac{1}{k+l}}{)}^{1/t},((1-(\prod _{i,j=1,i\ne j}^n((1-{(1-{\mu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t)}^l+\hfill \\ & & {(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{)}^{\frac{1}{n(n-1)}})-\prod _{i,j=1,i\ne j}^n((1-{\mu }_{{\alpha }_i}^t-\hfill \\ & & {\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{)}^{\frac{1}{n(n-1)}}{))}^{\frac{1}{k+l}}-(1-\prod _{i,j=1,i\ne j}^n(1-(1-{\mu }_{{\alpha }_i}^t-\hfill \\ & & {\eta }_{{\alpha }_i}^t{{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l)}^{\frac{1}{n(n-1)}}+\hfill \\ & & \prod _{i,j=1,i\ne j}^n{({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l)}^{\frac{1}{n(n-1)}}{{)}^{\frac{1}{k+l}})}^{1/t},\hfill \\ & & ((1-\prod _{i,j=1,i\ne j}^n(1-{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-\hfill \\ & & {\nu }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{)}^{\frac{1}{n(n-1)}}+\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-\hfill \\ & & {\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t{)}^l{)}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}-(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-\hfill \\ & & {\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t{)}^l{)}^{\frac{1}{n(n-1)}}{{)}^{\frac{1}{k+l}})}^{1/t}>.\hfill \end{array}$$

Let $\frac{1}{k+l}{\otimes }_{i,j=1,i\ne j}^n{\left(k{\alpha }_i\oplus l{\alpha }_j\right)}^{\frac{1}{n(n-1)}}=<\mu ,\eta ,\nu >$. Since $0\le {\mu }_{{\alpha }_i}\le 1,0\le {\eta }_{{\alpha }_i}\le 1,0\le {\nu }_{{\alpha }_i}\le 1$, $0\le \mu \le 1,0\le \eta \le 1,0\le \nu \le 1$ can be proved easily. Moreover,

$$\begin{array}{ccc}& & {\mu }^t+{\eta }^t+{\nu }^t=1-{(\prod _{i,j=1,i\ne j}^n{({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l)}^{\frac{1}{n(n-1)}})}^{\frac{1}{k+l}}).\hfill \end{array}$$

$0\le {\mu }_{{\alpha }_i}^t+{\eta }_{{\alpha }_i}^t+{\nu }_{{\alpha }_i}^t\le 1$, $0\le {\mu }_{{\alpha }_j}^t+{\eta }_{{\alpha }_j}^t+{\nu }_{{\alpha }_j}^t\le 1$, $0\le {\prod }_{i,j=1,i\ne j}^n{({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l)}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}\le 1.$ Hence, $0\le {\mu }^t+{\eta }^t+{\nu }^t\le 1.$ Then, the aggregated value of TSFIGBM operator is still a T-SFN. □

Theorem 5

(Idempotency). If ${\alpha }_i=\alpha ,<{\mu }_{{\alpha }_i},{\eta }_{{\alpha }_i},{\nu }_{{\alpha }_i}>=<{\mu }_{\alpha },{\eta }_{\alpha },{\nu }_{\alpha }>(i=1,2,\cdots ,n)$. Then,

$$\begin{array}{ccc}& & {\mathrm{TSFIGBM}}_{k,l}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)=\alpha .\hfill \end{array}$$

Proof. 

$$\begin{array}{ccc}\hfill \widehat{\mu }& =& (1-(1-(\prod _{i,j=1,i\ne j}^n((1-{(1-{\mu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k\ast \hfill \\ & & {(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{)}^{\frac{1}{n(n-1)}})-\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-{\mu }_{{\alpha }_j}^t-\hfill \\ & & {\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t{)}^l{)}^{\frac{1}{n(n-1)}}{))}^{\frac{1}{k+l}}{)}^{1/t}\hfill \\ & =& {(1-{({(1-{\mu }_{\alpha }^t)}^{k+l})}^{\frac{1}{k+l}})}^{1/t}={(1-(1-{\mu }_{\alpha }^t))}^{1/t}={\mu }_{\alpha },\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \widehat{\eta }& =& ((1-(\prod _{i,j=1,i\ne j}^n((1-{(1-{\mu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t)}^l+\hfill \\ & & {(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{)}^{\frac{1}{n(n-1)}})-\prod _{i,j=1,i\ne j}^n((1-{\mu }_{{\alpha }_i}^t-\hfill \\ & & {\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{)}^{\frac{1}{n(n-1)}}{))}^{\frac{1}{k+l}}-(1-\prod _{i,j=1,i\ne j}^n(1-(1-{\mu }_{{\alpha }_i}^t-\hfill \\ & & {\eta }_{{\alpha }_i}^t{{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l)}^{\frac{1}{n(n-1)}}+\hfill \\ & & \prod _{i,j=1,i\ne j}^n{({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l)}^{\frac{1}{n(n-1)}}{{)}^{\frac{1}{k+l}})}^{1/t}\hfill \\ & =& {(1-{\mu }_{\alpha }^t-(1-{\mu }_{\alpha }^t-{\eta }_{\alpha }^t))}^{1/t}={\eta }_{\alpha },\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \widehat{\nu }& =& ((1-\prod _{i,j=1,i\ne j}^n(1-{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^l+(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-\hfill \\ & & {\nu }_{{\alpha }_i}^t{)}^k{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t)}^l{)}^{\frac{1}{n(n-1)}}+\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-\hfill \\ & & {\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t{)}^l{)}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}-(\prod _{i,j=1,i\ne j}^n({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^k(1-\hfill \\ & & {\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_i}^t{)}^l{)}^{\frac{1}{n(n-1)}}{{)}^{\frac{1}{k+l}})}^{1/t}\hfill \\ & =& {((1-{\mu }_{\alpha }^t-{\eta }_{\alpha }^t)-(1-{\mu }_{\alpha }^t-{\eta }_{\alpha }^t-{\nu }_{\alpha }^t))}^{1/t}={\nu }_{\alpha }.\hfill \end{array}$$

□

Theorem 6

(Boundedness). Let $({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)$ be a collection of T-SFNs. If ${\alpha }^{+}=<1,0,0>$, ${\alpha }^{-}=<0,0,1>$, then

$$\begin{array}{ccc}& & {\alpha }^{-}\le {\mathrm{TSFIGBM}}_{k,l}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)\le {\alpha }^{+}.\hfill \end{array}$$

Definition 9.

Let${\alpha }_i=<{\mu }_{{\alpha }_i},{\eta }_{{\alpha }_i},{\nu }_{{\alpha }_i}>,(i=1,2,\cdots ,n)$be T-SFNs. The T-spherical fuzzy interaction weighted Bonferroni mean (TSFIWBM) operator can be defined as

$$\begin{array}{ccc}\hfill \mathrm{TSFIWBM}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)& =& {\left(\frac{1}{n(n-1)}{\oplus }_{i,j=1,i\ne j}^n({(w_i{\alpha }_i)}^k\otimes {(w_j{\alpha }_j)}^l)\right)}^{\frac{1}{k+l}},\hfill \end{array}$$

(7)

where$k,t\ge 0$,$(w_1,w_2,\cdots ,w_n)$is the weight vector of T-SFNs${\alpha }_k(k=1,2,\cdots ,n)$with$w_k\ge 0$and${\sum }_{k=1}^n{w}_k=1$.

Theorem 7.

Let${\alpha }_i=<{\mu }_{{\alpha }_i},{\eta }_{{\alpha }_i},{\nu }_{{\alpha }_i}>(i=1,2,\cdots ,n)$be T-SFNs,$(w_1,w_2,\cdots ,w_n)$be the weight vector of T-SFNs with$w_k\ge 0$and${\sum }_{k=1}^n{w}_k=1$. The aggregated value of TSFIWBM operator is still a T-SFN and

$$\begin{array}{ccc}& \hfill & {\mathrm{TSFWIBM}}_{k,l}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)\hfill \\ & =\hfill & {\left(\frac{1}{n(n-1)}{\oplus }_{i,j=1,i\ne j}^n({(w_i{\alpha }_i)}^k\otimes {(w_j{\alpha }_j)}^l)\right)}^{\frac{1}{k+l}}=<\tilde{\mu },\tilde{\eta },\tilde{\nu }>,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\tilde{\mu }\hfill & =\hfill & ((1+{(\sum _{i,j=1,i\ne j}^n{({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^{w_i})}^k{({(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^{w_j})}^l)}^{\frac{1}{n(n-1)}}-\hfill \\ & & (\sum _{i,j=1,i\ne j}^n(1-{(1-{(1-{\mu }_{{\alpha }_i}^t)}^{w_i}+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^{w_i})}^k(1-{(1-{\mu }_{{\alpha }_j}^t)}^{w_j}+\hfill \\ & & {(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^{w_j}{)}^l+{({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^{w_i})}^k\hfill \\ & & {({(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^{w_j})}^l{))}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}\hfill \\ & & -{({(\sum _{i,j=1,i\ne j}^n{({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^{w_i})}^k{({(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^{w_j})}^l)}^{\frac{1}{n(n-1)}})}^{\frac{1}{k+l}}{)}^{1/t},\hfill \end{array}$$

$$\begin{array}{ccc}\tilde{\eta }\hfill & =\hfill & ((1-(\sum _{i,j=1,i\ne j}^n(1+{({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^{w_i})}^k{({(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^{w_j})}^l-\hfill \\ & & {(1-{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^{w_i}+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^{w_i})}^k(1-{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^{w_i}+\hfill \\ & & {(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^{w_j}{)}^l{))}^{\frac{1}{n(n-1)}}+(\sum _{i,j=1,i\ne j}^n{({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^{w_i})}^k((1-{\mu }_{{\alpha }_j}^t-\hfill \\ & & {\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t{)}^{w_j}{)}^l{)}^{\frac{1}{n(n-1)}}{)}^{\frac{1}{k+l}}-(1+(\sum _{i,j=1,i\ne j}^n{({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^{w_i})}^k((1-{\mu }_{{\alpha }_j}^t-\hfill \\ & & {\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t{)}^{w_j}{)}^l{)}^{\frac{1}{n(n-1)}}-(\sum _{i,j=1,i\ne j}^n(1-(1-{(1-{\mu }_{{\alpha }_i}^t)}^{w_i}+(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-\hfill \\ & & {\nu }_{{\alpha }_i}^t{{)}^{w_i})}^k{(1-{(1-{\mu }_{{\alpha }_j}^t)}^{w_j}+{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^{w_j})}^l+{({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^{w_i})}^k\ast \hfill \\ & & {({(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^{w_j})}^l{))}^{\frac{1}{n(n-1)}}{{)}^{\frac{1}{k+l}})}^{1/t},\hfill \end{array}$$

$$\begin{array}{ccc}\tilde{\nu }\hfill & =\hfill & (1-(1-(\sum _{i,j=1,i\ne j}^n(1+{({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^{w_i})}^k{({(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^{w_j})}^l-\hfill \\ & & {(1-{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t)}^{w_i}+{(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^{w_i})}^k(1-{(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t)}^{w_i}+\hfill \\ & & {(1-{\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t)}^{w_j}{)}^l{))}^{\frac{1}{n(n-1)}}+(\sum _{i,j=1,i\ne j}^n{({(1-{\mu }_{{\alpha }_i}^t-{\eta }_{{\alpha }_i}^t-{\nu }_{{\alpha }_i}^t)}^{w_i})}^k((1-\hfill \\ & & {\mu }_{{\alpha }_j}^t-{\eta }_{{\alpha }_j}^t-{\nu }_{{\alpha }_j}^t{)}^{w_j}{)}^l{)}^{\frac{1}{n(n-1)}}{{)}^{\frac{1}{k+l}})}^{1/t}.\hfill \end{array}$$

Definition 10.

Let${\alpha }_i=<{\mu }_{{\alpha }_i},{\eta }_{{\alpha }_i},{\nu }_{{\alpha }_i}>(i=1,2,\cdots ,n)$be T-SFNs,$k,l\ge 0$. The T-spherical fuzzy weighted interaction geometric Bonferroni mean (TSFIWGBM) operator can be defined as

$$\begin{array}{ccc}\hfill \mathrm{TSFIWGBM}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)& =& \frac{1}{k+l}\left({\otimes }_{i,j=1,i\ne j}^n{((k{\alpha }_i^{w_i})\oplus (l{\alpha }_j^{w_j}))}^{\frac{1}{n(n-1)}}\right).\hfill \end{array}$$

(9)

Theorem 8.

Let${\alpha }_i=<{\mu }_{{\alpha }_i},{\eta }_{{\alpha }_i},{\nu }_{{\alpha }_i}>(i=1,2,\cdots ,n)$be T-SFNs,$(w_1,w_2,\cdots ,w_n)$be the weight vector of T-SFNs with$w_k\ge 0$and${\sum }_{k=1}^n{w}_k=1$. The aggregated value of TSFIWGBM operator is still a T-SFN and

$$\begin{array}{ccc}& \hfill & {\mathrm{TSFWIGBM}}_{k,l}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)\hfill \\ & =\hfill & \frac{1}{k+l}\left({\otimes }_{i,j=1,i\ne j}^n{((k{\alpha }_{}^{}}^{}\right)\hfill \end{array}$$