Aczel–Alsina Hamy Mean Aggregation Operators in T-Spherical Fuzzy Multi-Criteria Decision-Making

Abstract

A T-spherical fuzzy set is a more powerful mathematical tool to handle uncertain and vague information than several fuzzy sets, such as fuzzy set, intuitionistic fuzzy set, Pythagorean fuzzy set, q-rung orthopair fuzzy set, and picture fuzzy set. The Aczel–Alsina t-norm and s-norm are significant mathematical operations with a high premium on affectability with parameter activity, which are extremely conducive to handling imprecise and undetermined data. On the other hand, the Hamy mean operator is able to catch the interconnection among multiple input data and achieve great results in the fusion process of evaluation information. Based on the above advantages, the purpose of this study is to propose some novel aggregation operators (AOs) integrated by the Hamy mean and Aczel–Alsina operations to settle T-spherical fuzzy multi-criteria decision-making (MCDM) issues. First, a series of T-spherical fuzzy Aczel–Alsina Hamy mean AOs are advanced, including the T-spherical fuzzy Aczel–Alsina Hamy mean (TSFAAHM) operator, T-spherical fuzzy Aczel–Alsina dual Hamy mean (TSFAADHM) operator, and their weighted forms, i.e., the T-spherical fuzzy Aczel–Alsina-weighted Hamy mean (TSFAAWHM) and T-spherical fuzzy Aczel–Alsina-weighted dual Hamy mean (TSFAAWDHM) operators. Moreover, some related properties are discussed. Then, a MCDM model based on the proposed AOs is built. Lastly, a numerical example is provided to show the applicability and feasibility of the developed AOs, and the effectiveness of this study is verified by the implementation of a parameters influence test and comparison with available methods.

1. Introduction

As one of the principal branches of decision science, the topics related to MCDM have been extensively focused on in different disciplines, such as engineering, management, and economics. In general, MCDM is utilized to ascertain the optimum solution based on criteria and weights, and the evaluation information is often represented as real numbers and used as decision inputs. However, the complexity of the practical decision system makes it difficult for decision makers to make optimal and effective choices due to the fact that the evaluation values are often ambiguous and uncertain during the decision. While considering the vagueness, uncertainty, and incompleteness of evaluation information, Atanassov [1] extended an intuitionistic fuzzy set (IFS) with the degree of membership (MD) (τ) and the degree of non-membership (ND) (ϑ), based on the classical fuzzy set (FS) [2]. The Pythagorean fuzzy set (PyFS) was proposed by Yager and Abbasov [3] to make up for the deficiency when τ + ϑ > 1. Subsequently, Yager [4] introduced a more flexible q-rung orthopair fuzzy set (q-ROFS) concept, namely, flexibly adjusting the decision range expressed by MD and ND through the parameter q, and meeting the condition: τ^q +ϑ^q ≤ 1, τ, ϑ ∈ [0,1]. However, the evaluation object cannot be fully described by relying only on MD and ND in the above various fuzzy sets. Thus, a picture fuzzy set (PFS) was developed by Cuong [5], containing MD, abstinence degree (AD) (η ∈ [0,1]), and ND, as another form of generalized FS that can describe more information. Although PFS has the ability to describe AD information that IFS, PyFS, and q-ROFS cannot, it still has the limitation that it fails if the total value of 3 degrees is more than 1. In this regard, the concept of the spherical fuzzy set (SFS) was extended by Mahmood et al. [6], then they promoted it to the generalized form, i.e., T-spherical fuzzy set (TSFS), to remove the restriction of decision-makers (DMs) in the allocation of MD, AD, and ND with a larger decision space, enabling them to express DMs’ preferences and opinions more freely. Obviously, the above extended fuzzy sets are all special cases of TSFS, which has been widely studied by numerous scholars because of the generalized form of TSFS with no limitations.

As one of the significant processing tools for fusing evaluation data, the study of AOs in the TSFS environment has also been actively followed by many scholars. The algorithms of TSFNs are important bases for AOs, and currently, t-norm and s-norm operations have emerged, such as Algebraic [7], Hamacher [8], Einstein [9], interaction [10,11,12], Frank [13], Dombi [14], Schweizer–Sklar [15], Aczel–Alsina (AA) [16], etc. Among the above operations, there are no decision-adjustable parameters contained in Algebraic, Hamacher, and Einstein t-norm operations, and interactive operations emphasize the interaction relationship between MD, AD, and ND in any two TSFNs to avoid counterintuitive phenomena caused when the value of the membership function is zero. In addition, Frank, Dombi, Schweizer–Sklar, AA, etc. are the t-norms operations containing decision-tunable parameters, which increase the decision flexibility of the aggregation operator, as well as a limited extent of generalization. In contrast, the AA t-norm and s-norm are more flexible to make decisions [16]. Apart from the arithmetic average and geometric operators, there are various novel AOs developed by integrating them with the Power, Bonferroni mean (BM), Heronian mean (HeM), Maclaurin symmetric mean (MSM), Muirhead mean (MM), etc. For example, some weighted algebraic AOs integrated with Power are advanced by Garg et al. [17] in the TSFS environment, and Wang and Zhang [12] proposed novel T-spherical fuzzy interaction power Heronian mean (TSFIPHeM)aggregation operators that integrated Power and HeM operators and considered the interaction of TSFNs. Liu et al. [18] extended the generalized Maclaurin symmetric mean (GMSM) operator to TSFS and proposed the T-spherical fuzzy GMSM operator (TSFGMSM) and the T-spherical fuzzy weighted GMSM operator (TSFWGMSM). Liu et al. [19] developed some T-spherical fuzzy power Muirhead mean (TSFPMM) and T-spherical fuzzy power dual Muirhead mean (TSFPDMM)aggregation operators based on the advantages of Power integrated with the Muirhead mean. Yang and Pang [20] developed a series of interaction BM (TSFIBM), interaction geometric BM (TSFIGBM), Dombi BM (TSFDBM), and geometric Dombi BM (TSFGDBM) aggregation operators in the TSFS context.

Aczel and Alsina [21] proposed two operations, i.e., AA t-norm and s-norm, emphasizing the significance of adjustable parameters. Currently, some scholars have extended AA operations to different decision environments, such as hesitant fuzzy set [22], IFS [23,24,25,26,27], PyFS [28,29], q-ROFS [30], PFS [31,32], SFS [33,34], TSFS [16], Neutrosophic set [35,36,37], complex q-ROFS [38], bipolar complex fuzzy set [39], and cubic Fermatean fuzzy set [40]. Therefore, a variety of aggregation operators have been developed to solve decision challenges indifferent environments. From the above works, we find that some AA operation laws were extended in different decision-making environments, and in many, the weighted averaging and geometric AOs were developed according to AA operation rules. For instance, Senapati et al. [23,25] advanced some weighted average AOs in IFS and interval-valued IFS successively. Meanwhile, Senapati et al. [24,26] developed some weighted geometric AOs in IFS and interval-valued IFS. Further, Senapati et al. [29,30,31] proposed some weighted average AOs in PyFS, q-ROFS, and PFS. Naeem et al. [32] raised weighted geometric AOs in PFS. Several weighted average and geometric AOs were developed by Hussain et al. [16] in the TSFS environment. Senapati et al. [22] designed a hesitant fuzzy AA weighted BM (HFAAWBM) operator, in addition to developing weighted average and geometric AOs.

From the above review, we grasp that various AOs can be utilized to handle practical MCDM issues. To obtain the optimal option effectively by using the MCDM method, the evaluation information should be handled more expediently and availably. The TSFS, as a generalized shape of IFS, PyFS, q-ROFS, PFS, and SFS, is one of the most effective means to cope with the ambiguity and uncertainty in assessing data. The Hamy mean (HM) is an aggregation function. Like MSM and MM, HM can reflect the interrelationship among multiple attributes, but its calculation process is not as complex as MSM and MM. However, the HM operator has not been extended in the TSFS context. In addition, AA operation is more generalized and flexible than the existing operations, such as Hamcher, Einstein, Frank, and Dombi [16]. At present, it can be combined with various aggregation functions, but unfortunately, the research on the new AOs, based on AA operations with the capacity to capture the interrelationship among multiple variables, has not yet appeared. In order to fill the above two research gaps, it is necessary to develop a series of TSFAAHM AOs, based on the advantages of HM and AA, and to design some aggregation function-based MCDM methods in the T-spherical fuzzy environment.

The contributions of this paper are outlined below:

(1)

We proposed some new AOs for TSFS, which include the TSFAAHM, TSFAADHM, TSFAAWHM, and TSFAAWDHM operators, and some related properties are discussed.

(2)

We designed a novel T-spherical fuzzy MCDM method based on the TSFAAWHM or TSFAAWDHM operator.

(3)

We tested the applicability of our proposed aggregation function-based MCDM method by solving investment decision issues.

(4)

The proposed method is performed by parameter analysis and comparison analysis, with existing methods to show its reliability and effectiveness.

Therefore, the article is structured as follows. Some ideas of TSFS and AA operations of T-spherical fuzzy numbers are briefly reviewed in Section 2. We define a series of TSFAAHM AOs and analyze their relative properties in Section 3. In Section 4, we construct a MCDM model of applying TSFAAWHM and TSFAAWDHM operators. In Section 5, the validity of the method is verified by solving the investment firm decision problem in a numerical example, and the superiorities of the developed model are depicted by sensitivity and comparative study. Finally, we present the final conclusion in Section 6.

2. Preliminaries

Some basic ideas for TSFSs to make the manuscript self-contained are given in this section.

Definition 1 ([6]). 

Suppose X is a universe set, then the TSFS can be defined as

$$\Im =\left\{〈x,\left({\tau }_{\Im }(x),{\eta }_{\Im }(x),{\vartheta }_{\Im }(x)\right)〉|x\in X\right\}$$

(1)

where ${\tau }_{\Im }(x),{\eta }_{\Im }(x),{\vartheta }_{\Im }(x)$ are respectively the MD, AD, and ND of element x ∈ ℑ in X, i.e.,${\tau }_{\Im }(x),{\eta }_{\Im }(x),{\vartheta }_{\Im }(x)\in [0,1]$, and satisfying $0\le {\tau }_{\Im }^q(x)+{\eta }_{\Im }^q(x)+{\vartheta }_{\Im }^q(x)\le 1$, q ≥ 1 for ∀x ∈ X. ${\pi }_{\Im }(x)=\sqrt[q]{1-{\tau }_{\Im }^q(x)-{\eta }_{\Im }^q(x)-{\vartheta }_{\Im }^q(x)}$ is called the degree of refusal. In order to facilitate, δ = (τ, η, ϑ) denoted the T-spherical fuzzy number (TSFN).

Definition 2 ([11]). 

Suppose δ = (τ, η, ϑ) is a TSFN, then the score function sc(δ) is defined as

$$sc(\delta )=\frac{1+{\tau }^q-{\eta }^q-{\vartheta }^q}{2}$$

(2)

and the accuracy function ac(δ) is described as

$$ac(\delta )={\tau }^q+{\eta }^q+{\vartheta }^q$$

(3)

Let δ₁ = (τ₁, η₁, ϑ₁) and δ₂ = (τ₂, η₂, ϑ₂) be two TSFNs, the laws of comparing the two TSFNs are as below:

(1)

If sc(δ₁) is greater than sc(δ₂), then δ₁ is superior to δ₂, i.e., δ₁ > δ₂;

(2)

If sc(δ₁) is equal to sc(δ₂), then (i) if ac(δ₁) is larger than ac(δ₂), then δ₁ is superior to δ₂, i.e., δ₁ > δ₂; (ii) if ac(δ₁) is the same as ac(δ₂), then δ₁ is equal to δ₂, namely, δ₁ = δ₂.

Definition 3 ([6]). 

Suppose δ = (τ, η,ϑ), δ₁ = (τ₁, η₁, ϑ₁), and δ₂ = (τ₂, η₂, ϑ₂) are three arbitrary TSFNs, their basic operations are depicted as follows (λ > 0):

(1)

${\delta }_1\oplus {\delta }_2=\left(\sqrt[q]{{\tau }_1^q+{\tau }_2^q-{\tau }_1^q{\tau }_2^q},{\eta }_1{\eta }_2,{\vartheta }_1{\vartheta }_2\right)$;

(2)

${\delta }_1\otimes {\delta }_2=\left({\tau }_1{\tau }_2,\sqrt[q]{{\eta }_1^q+{\eta }_2^q-{\eta }_1^q{\eta }_2^q},\sqrt[q]{{\vartheta }_1^q+{\vartheta }_2^q-{\vartheta }_1^q{\vartheta }_2^q}\right)$;

(3)

$\lambda \cdot \delta =\left(\sqrt[q]{1-{(1-{\tau }^q)}^{\lambda }},{\eta }^{\lambda },{\vartheta }^{\lambda }\right)$;

(4)

${\delta }^{\lambda }=\left({\tau }^{\lambda },\sqrt[q]{1-{(1-{\eta }^q)}^{\lambda }},\sqrt[q]{1-{(1-{\vartheta }^q)}^{\lambda }}\right)$.

Definition 4 ([21]). 

Suppose x and y are two arbitrary, non-negative real numbers (x, y > 0), then the t-norm and s-norm of Aczel–Alsina can be described as follows:

$$T_A^{\phi }(x,y)=\exp \left\{-{\left({(-\ln x)}^{\phi }+{(-\ln y)}^{\phi }\right)}^{1/\phi }\right\}, \phi >0$$

(4)

$$S_A^{\phi }(x,y)=1-\exp \left\{-{\left({(-\ln (1-x))}^{\phi }+{(-\ln (1-y))}^{\phi }\right)}^{1/\phi }\right\}, \phi >0$$

(5)

Hussain et al. [16] proposed the Aczel–Alsina operation laws for TSFNs based on the TSFS concept and Definition 4.

Definition 5 ([16]). 

Let δ₁ = (τ₁, η₁, ϑ₁) and δ₂ = (τ₂, η₂, ϑ₂) be any two TSFNs, q ≥ 1, λ, φ ≥ 0, and then their AA operation laws are defined as:

(1)

${\delta }_1{\oplus }_{AA}{\delta }_2=\left(\begin{array}{l}\sqrt[q]{1-\exp \left\{-{\left({\left(-\ln (1-{\tau }_1^q)\right)}^{\phi }+{\left(-\ln (1-{\tau }_2^q)\right)}^{\phi }\right)}^{1/\phi }\right\}},\sqrt[q]{\exp \left\{-{\left({\left(-\ln ({\eta }_1^q)\right)}^{\phi }+{\left(-\ln ({\eta }_2^q)\right)}^{\phi }\right)}^{1/\phi }\right\}},\\ \sqrt[q]{\exp \left\{-{\left({\left(-\ln ({\vartheta }_1^q)\right)}^{\phi }+{\left(-\ln ({\vartheta }_2^q)\right)}^{\phi }\right)}^{1/\phi }\right\}}\end{array}\right);$

(2)

${\delta }_1{\otimes }_{AA}{\delta }_2=\left(\begin{array}{l}\sqrt[q]{\exp \left\{-{\left({\left(-\ln ({\tau }_1^q)\right)}^{\phi }+{\left(-\ln ({\tau }_2^q)\right)}^{\phi }\right)}^{1/\phi }\right\}},\sqrt[q]{1-\exp \left\{-{\left({\left(-\ln (1-{\eta }_1^q)\right)}^{\phi }+{\left(-\ln (1-{\eta }_2^q)\right)}^{\phi }\right)}^{1/\phi }\right\}},\\ \sqrt[q]{1-\exp \left\{-{\left({\left(-\ln (1-{\vartheta }_1^q)\right)}^{\phi }+{\left(-\ln (1-{\vartheta }_2^q)\right)}^{\phi }\right)}^{1/\phi }\right\}}\end{array}\right);$

(3)

$\lambda {\cdot }_{AA}{\delta }_1=\left(\sqrt[q]{1-\exp \left\{-{\left(\lambda {\left(-\ln (1-{\tau }_1^q)\right)}^{\phi }\right)}^{1/\phi }\right\}},\sqrt[q]{\exp \left\{-{\left(\lambda {\left(-\ln ({\eta }_1^q)\right)}^{\phi }\right)}^{1/\phi }\right\}},\sqrt[q]{\exp \left\{-{\left(\lambda {\left(-\ln ({\vartheta }_1^q)\right)}^{\phi }\right)}^{1/\phi }\right\}}\right);$

(4)

${\delta }_1^{{\wedge }_{AA}\lambda }=\left(\sqrt[q]{\exp \left\{-{\left(\lambda {\left(-\ln ({\tau }_1^q)\right)}^{\phi }\right)}^{1/\phi }\right\}},\sqrt[q]{1-\exp \left\{-{\left(\lambda {\left(-\ln (1-{\eta }_1^q)\right)}^{\phi }\right)}^{1/\phi }\right\}},\sqrt[q]{1-\exp \left\{-{\left(\lambda {\left(-\ln (1-{\vartheta }_1^q)\right)}^{\phi }\right)}^{1/\phi }\right\}}\right).$

Theorem 1 ([16]). 

Let δ₁ = (τ₁, η₁,ϑ₁) and δ₂ = (τ₂, η₂, ϑ₂) be any two TSFNs, λ is a real number, λ₁, λ₂ ≥ 0, and then the following operation properties are satisfied:

(1)

${\delta }_1{\oplus }_{AA}{\delta }_2={\delta }_2{\oplus }_{AA}{\delta }_1$;

(2)

${\delta }_1{\otimes }_{AA}{\delta }_2={\delta }_2{\otimes }_{AA}{\delta }_1$;

(3)

$\lambda ({\delta }_1{\oplus }_{AA}{\delta }_2)=\lambda {\delta }_1{\oplus }_{AA}\lambda {\delta }_2$;

(4)

${\lambda }_1{\delta }_1{\oplus }_{AA}{\lambda }_2{\delta }_1=({\lambda }_1+{\lambda }_2){\delta }_1$;

(5)

${({\delta }_1{\otimes }_{AA}{\delta }_2)}^{{\wedge }_{AA}}{}^{\lambda }={\delta }_1^{{\wedge }_{AA}\lambda }{\otimes }_{AA}{\delta }_2^{{\wedge }_{AA}\lambda }$;

(6)

${\delta }_1^{{\lambda }_1}{\otimes }_{AA}{\delta }_1^{{\lambda }_2}={\delta }_1^{{\wedge }_{AA}({\lambda }_1+{\lambda }_2)}$.

3. Some TSFAAHM Operators

We propose a series of TSFAAHM AOs involving some AA operations of TSFNs, including the TSFAAHM, TSFAADHM, TSFAAWHM, and TSFAAWDHM operators. Then, we analyze their properties and discuss their special cases.

3.1. HM and DHM Operators

Hara [41] introduced an AO for non-negative real numbers in 1998, i.e., HM, which has the capacity to concern the interconnection among multiple input arguments. The specific definition of HM is described as below.

Definition 6 ([41]). 

Suppose a_i(i = 1, 2, 3, …, n) is a set of non-negative real numbers, γ = 1, 2, …, n. If

$$H{M}^{(\gamma )}(a_1,a_2,\dots ,a_n)=\frac{{\displaystyle \sum _{1\le i_1<\cdots <i_{\gamma }\le n}{\left({\displaystyle \prod _{j=1}^{\gamma }{a}_{i_j}}\right)}^{1/\gamma }}}{C_n^{\gamma }}$$

(6)

Then, HM^(γ) is named as the Hamy mean, where (i₁, i₂, …, i_γ) traverses all the γ-tuple combinations of (1, 2, …, n), and C_n^γ is the binomial coefficient.

Furthermore, a dual form of HM, i.e., dual Hamy mean (DHM), was developed by We et al. [42].

Definition 7 ([42]). 

Suppose a_i(i = 1, 2, 3, …, n) is a set of non-negative real numbers, γ = 1, 2, …, n. If

$$DH{M}^{(\gamma )}(a_1,a_2,\dots ,a_n)={\left({\displaystyle \prod _{1\le i_1<\cdots <i_{\gamma }\le n}\left(\frac{{\displaystyle \sum _{j=1}^{\gamma }{a}_{i_j}}}{\gamma }\right)}\right)}^{1/C_n^{\gamma }}$$

(7)

Then, DHM^(γ) is named as the dual Hamy mean.

3.2. TSFAAHM and TSFAAWHM Operators

Definition 8. 

Let δ_i = (τ_i, η_i, ϑ_i) (i = 1, 2, 3, …, n) be a family of TSFNs, the form of the TSFAAHM operator can be portrayed as:

$$TSFAAH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)=\frac{\underset{1\le i_1<\dots <i_{\gamma }\le n}{\oplus }{\left({\otimes }_{j=1}^{\gamma }{\delta }_{i_j}\right)}^{\frac{1}{\gamma }}}{C_n^{\gamma }}$$

(8)

where γ (γ = 1, 2, …, n) is the parameter of this operator, and$C_n^{\gamma }=\frac{n!}{\gamma !(n-\gamma )!}$is the binomial coefficient with the constraint of 1 ≤ i₁ < i₂ <…< i_γ ≤ n.

According to the AA operation laws of TSFNs, the aggregation results of Equation (8) are as follows.

Theorem 2. 

Suppose δ_i = (τ_i, η_i, ϑ_i) (i = 1, 2, …, n) is a family of TSFNs, the outcome of the TSFAAHM operator is still TSFN based on the Definition 5, i.e.,

$$TSFAAH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)=\left(\begin{array}{l}\sqrt[q]{1-\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\tau }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln (1-{\eta }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln (1-{\vartheta }_{i_j}^q))\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\end{array}\right)$$

(9)

Proof of Theorem 2. 

On the basis of the AA operation laws of TSFNs, there are:

$${\otimes }_{j=1}^{\gamma }{\delta }_{i_j}=\left(\sqrt[q]{\exp \left\{-{\left[{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left({\tau }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\sqrt[q]{1-\exp \left\{-{\left[{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left(1-{\eta }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\sqrt[q]{1-\exp \left\{-{\left[{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left(1-{\vartheta }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\right)$$

and

$${\left({\otimes }_{j=1}^{\gamma }{\delta }_{i_j}\right)}^{\frac{1}{\gamma }}=\left(\begin{array}{l}\sqrt[q]{\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left({\tau }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\sqrt[q]{1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left(1-{\eta }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left(1-{\vartheta }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\end{array}\right)$$

Then,

$$\underset{1\le i_1<\dots <i_{\gamma }\le n}{\oplus }{\left({\otimes }_{j=1}^{\gamma }{\delta }_{i_j}\right)}^{\frac{1}{\gamma }}=\left(\begin{array}{l}\sqrt[q]{1-\exp \left\{-{\left[{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left({\tau }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{\exp \left\{-{\left[{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left(1-{\eta }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{\exp \left\{-{\left[{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left(1-{\vartheta }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\end{array}\right)$$

Therefore,

$$TSFAAH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)=\left(\begin{array}{l}\sqrt[q]{1-\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\tau }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln (1-{\eta }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln (1-{\vartheta }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\end{array}\right)$$

Therefore, Theorem 2 is proved completely. □

Example 1. 

Suppose δ₁= (0.6, 0.7, 0.5), δ₂= (0.9, 0.1, 0.4), δ₃= (0.7, 0.6, 0.6), and δ₄= (0.2, 0.8, 0.3) are four TSFNs, then the values of the parameters are shown as: q = 3, γ = 2, φ = 3. Then, we can obtain the following calculation results by utilizing Equation (9):

The MDs of δ₁, δ₂, δ₃, and δ₄ are aggregated as:

$$\begin{array}{l}\sqrt[3]{1-\exp \left\{-{\left[\frac{1}{6}\left(\begin{array}{l}{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.6}^3\right)\right)}^3+{\left(-\ln \left({0.9}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.6}^3\right)\right)}^3+{\left(-\ln \left({0.7}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\\ +{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.6}^3\right)\right)}^3+{\left(-\ln \left({0.2}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.9}^3\right)\right)}^3+{\left(-\ln \left({0.7}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\\ +{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.9}^3\right)\right)}^3+{\left(-\ln \left({0.2}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.7}^3\right)\right)}^3+{\left(-\ln \left({0.2}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\end{array}\right)\right]}^{\frac{1}{3}}\right\}}\\ =0.736\end{array}$$

The ADs of δ₁, δ₂, δ₃, and δ₄are aggregated as:

$$\begin{array}{l}\sqrt[3]{\exp \left\{-{\left[\frac{1}{6}\left(\begin{array}{l}{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.7}^3\right)\right)}^3+{\left(-\ln \left(1-{0.1}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.7}^3\right)\right)}^3+{\left(-\ln \left(1-{0.6}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\\ +{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.7}^3\right)\right)}^3+{\left(-\ln \left(1-{0.8}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.1}^3\right)\right)}^3+{\left(-\ln \left(1-{0.6}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\\ +{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.1}^3\right)\right)}^3+{\left(-\ln \left(1-{0.8}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.6}^3\right)\right)}^3+{\left(-\ln \left(1-{0.8}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\end{array}\right)\right]}^{\frac{1}{3}}\right\}}\\ =0.518\end{array}$$

Similarly, the NDs of δ₁, δ₂, δ₃, and δ₄ are aggregated as:

$$\begin{array}{l}\sqrt[3]{\exp \left\{-{\left[\frac{1}{6}\left(\begin{array}{l}{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.5}^3\right)\right)}^3+{\left(-\ln \left(1-{0.4}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.5}^3\right)\right)}^3+{\left(-\ln \left(1-{0.6}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\\ +{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.5}^3\right)\right)}^3+{\left(-\ln \left(1-{0.3}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.4}^3\right)\right)}^3+{\left(-\ln \left(1-{0.6}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\\ +{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.4}^3\right)\right)}^3+{\left(-\ln \left(1-{0.3}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.6}^3\right)\right)}^3+{\left(-\ln \left(1-{0.3}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\end{array}\right)\right]}^{\frac{1}{3}}\right\}}\\ =0.304\end{array}$$

Therefore,$TSFAAH{M}^{(2)}({\delta }_1,{\delta }_2,{\delta }_3,{\delta }_4)=(0.736,0.518,0.304)$.

Further, we should analyze the relative properties of the TSFAAHM operator.

Theorem 3. 

Suppose δ_i = (τ_i, η_i, ϑ_i) (i = 1, 2, …, n) are a group of TSFNs, the TSFAAHM operator can meet the below properties:

(1)

(Idempotency) If the values of all TSFNs are equal, i.e., δ_i = δ = (τ, η, ϑ), then

$$TSFAAH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)=\delta$$

(10)

(2)

(Boundness) Let ${\delta }^{-}=\min {\delta }_i$ and ${\delta }^{+}=\max {\delta }_i$, then

$${\delta }^{-}\le TSFAAH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)\le {\delta }^{+}$$

(11)

(3)

(Monotonicity) Let ${{\delta }^{\prime }}_i=({{\tau }^{\prime }}_i,{{\eta }^{\prime }}_i,{{\vartheta }^{\prime }}_i)$ (i = 1, 2, …, n) be another group of TSFNs, if all isatisfy ${\delta }_i\le {{\delta }^{\prime }}_i$, i.e., ${\tau }_i\le {{\tau }^{\prime }}_i$, ${\eta }_i\le {{\eta }^{\prime }}_i$ and ${\vartheta }_i\ge {{\vartheta }^{\prime }}_i$, then

$$TSFAAH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)\le TSFAAH{M}^{(\gamma )}({{\delta }^{\prime }}_1,{{\delta }^{\prime }}_2,\dots ,{{\delta }^{\prime }}_n)$$

(12)

The above properties can be proved as follows.

Proof of Theorem 3. 

(1) (Idempotency) According to Theorem 2, the following can be obtained from the TSFAAHM operator, i.e.,:

$$\begin{array}{l}\sqrt[q]{1-\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\tau }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\\ =\sqrt[q]{1-\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\tau }^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\\ =\sqrt[q]{1-\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[{\displaystyle {\left(-\ln ({\tau }^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\\ =\sqrt[q]{1-\exp \left\{-{\left[{\left(-\ln \left(1-\exp \left\{-{\left[{\left(-\ln ({\tau }^q)\right)}^{\phi }\right]}^{1/\phi }\right\}\right)\right)}^{\phi }\right]}^{1/\phi }\right\}}\\ =\tau \end{array}$$

Similarly,

$$\sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln (1-{\eta }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}=\eta ;$$

$$\sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln (1-{\vartheta }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}=\vartheta .$$

Therefore, $TSFAAH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)=\left(\tau ,\eta ,\vartheta \right)=\delta$.

So, the idempotency of the operator is proved.

(2) (Boundness) Due to ${\delta }^{-}=\min {\delta }_i$, on the basis of the above idempotency of the TSFAAHM operator, then:

$$TSFAAH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)=\frac{\underset{1\le i_1<\dots <i_{\gamma }\le n}{\oplus }{\left({\otimes }_{j=1}^{\gamma }{\delta }_{i_j}\right)}^{\frac{1}{\gamma }}}{C_n^{\gamma }}\ge \frac{\underset{1\le i_1<\dots <i_{\gamma }\le n}{\oplus }{\left({\otimes }_{j=1}^{\gamma }{\delta }^{-}\right)}^{\frac{1}{\gamma }}}{C_n^{\gamma }}=\frac{C_n^{\gamma }{\left({({\delta }^{-})}^{\gamma }\right)}^{\frac{1}{\gamma }}}{C_n^{\gamma }}={\delta }^{-}$$

Similarly, $TSFAAH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)\le {\delta }^{+}$.

Therefore, ${\delta }^{-}\le TSFAAH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)\le {\delta }^{+}$, the boundness of this operator is validated.

(3) (Monotonicity) Due to ${\tau }_i\le {\displaystyle {\tau }_i^{\prime }}$, ${\eta }_i\ge {\displaystyle {\eta }_i^{\prime }}$, ${\vartheta }_i\ge {\displaystyle {\vartheta }_i^{\prime }}$, ${\tau }_i,{\displaystyle {\tau }_i^{\prime }},{\eta }_i,{\displaystyle {\eta }_i^{\prime }},{\vartheta }_i,{\displaystyle {\vartheta }_i^{\prime }}\in [0,1]$, then

$$\begin{array}{l}{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\tau }_{i_j}^q)\right)}^{\phi }}\ge {\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\displaystyle {\tau }_{i_j}^{{\prime }_q}})\right)}^{\phi }}\\ \Rightarrow -{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\tau }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\le -{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\displaystyle {\tau }_{i_j}^{{\prime }_q}})\right)}^{\phi }}\right]}^{1/\phi }\\ \Rightarrow 1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\tau }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\ge 1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\displaystyle {\tau }_{i_j}^{{\prime }_q}})\right)}^{\phi }}\right]}^{1/\phi }\right\}\\ \Rightarrow {\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\tau }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\le {\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\displaystyle {\tau }_{i_j}^{{\prime }_q}})\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\\ \Rightarrow -{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\tau }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\\ \ge -{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\displaystyle {\tau }_{i_j}^{{\prime }_q}})\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\\ \Rightarrow \sqrt[q]{1-\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\tau }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\\ \le \Rightarrow \sqrt[q]{1-\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\displaystyle {\tau }_{i_j}^{{\prime }_q}})\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\end{array}$$

Similarly,

$$\begin{array}{l}\sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln (1-{\eta }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\\ \ge \sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln (1-{\displaystyle {\eta }_{i_j}^{{\prime }_q}})\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\end{array}$$

$$\begin{array}{l}\sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln (1-{\vartheta }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\\ \ge \sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln (1-{\displaystyle {\vartheta }_{i_j}^{{\prime }_q}})\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\end{array}$$

On the basis of Theorem 2, $TSFAAH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)\le TSFAAH{M}^{(\gamma )}({{\delta }^{\prime }}_1,{{\delta }^{\prime }}_2,\dots ,{{\delta }^{\prime }}_n)$ is obtained. Thus, the monotonicity of the operator is certified.

Therefore, the proof of the properties of the TSFAAHM operator is completed. □

Although the TSFAAHM operator is able to consider the interrelationship between various criteria, it neglects the degree of importance of each criterion, for which we develop the TSFAAWHM operator, as below.

Definition 9. 

Suppose δ_i = (τ_i, η_i, ϑ_i) (i = 1, 2, …, n) is a group of TSFNs, w = (w₁, w₂, …, w_n)^T is denoted the weight vector of δ_i(i = 1, 2, …, n), meeting w_i ∈ [0,1] and ${\sum }_{i=1}^n{w}_i=1$. If

$$TSFAAWH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)=\frac{\underset{1\le i_1<\dots <i_{\gamma }\le n}{\oplus }{\left({\otimes }_{j=1}^{\gamma }{w}_{i_j}{\delta }_{i_j}\right)}^{\frac{1}{\gamma }}}{C_n^{\gamma }}$$

(13)

Then, the operator is named as the TSFAAWHM operator.

On the basis of the AA operation laws of TSFNs, the aggregation result of Equation (13) is as follows.

Theorem 4. 

Suppose δ_i = (τ_i, η_i, ϑ_i) (i = 1, 2, …, n) is a family of TSFNs, the aggregation result of the TSFAAWHM operator is still TSFN, i.e.,

$$TSFAAWH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)=\left(\begin{array}{l}\sqrt[q]{1-\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{w}_{i_j}{\left(-\ln ({\tau }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{1-\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{w}_{i_j}{\left(-\ln (1-{\eta }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{1-\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{w}_{i_j}{\left(-\ln (1-{\vartheta }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\end{array}\right)$$

(14)

The aggregation result of the TSFAAWHM operator is still TSFN, and its proof is identical to Theorem 2. Hence, we ellipsis it here.

The TSFAAWHM operator still satisfies monotonicity and boundness, but not idempotency. So, we omit the proof here.

3.3. TSFAADHM and TSFAAWDHM Operators

Definition 10. 

Let δ_i = (τ_i, η_i, ϑ_i) (i = 1, 2, …, n) be a family of TSFNs, the TSFAADHM operator is described as

$$TSFAADH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)={\left(\underset{1\le i_1<\dots <i_{\gamma }\le n}{\otimes }\left(\frac{{\oplus }_{j=1}^{\gamma }{\delta }_{i_j}}{\gamma }\right)\right)}^{\frac{1}{C_n^{\gamma }}}$$

(15)

According to the AA operation laws of TSFNs, the aggregation result of Equation (15) is as below.

Theorem 5. 

Suppose δ_i (i = 1, 2, 3, …, n) is a group of TSFNs. Based on the AA operation laws for TSFNs of Definition 5, then the result of the TSFAADHM operator is still TSFN, i.e.,

$$TSFAADH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)=\left(\begin{array}{l}\sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln (1-{\tau }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\eta }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\vartheta }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\end{array}\right)$$

(16)

Proof of Theorem 5. 

According to the AA operation rules of TSFNs, we have

$${\oplus }_{j=1}^{\gamma }{\delta }_{i_j}=\left(\sqrt[q]{1-\exp \left\{-{\left[{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left(1-{\tau }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\sqrt[q]{\exp \left\{-{\left[{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left({\eta }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\sqrt[q]{\exp \left\{-{\left[{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left({\vartheta }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\right)$$

and

$$\frac{{\oplus }_{j=1}^{\gamma }{\delta }_{i_j}}{\gamma }=\left(\sqrt[q]{1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left(1-{\tau }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\sqrt[q]{\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left({\eta }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\sqrt[q]{\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left({\vartheta }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\right)$$

Then,

$$\underset{1\le i_1<\dots <i_{\gamma }\le n}{\otimes }\left(\frac{{\oplus }_{j=1}^{\gamma }{\delta }_{i_j}}{\gamma }\right)=\left(\begin{array}{l}\sqrt[q]{\exp \left\{-{\left[{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left(1-{\tau }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{\exp \left\{-{\left[{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left({\eta }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}.\\ \sqrt[q]{\exp \left\{-{\left[{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln \left({\vartheta }_{i_j}^q\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\end{array}\right)$$

Thus,

$$TSFAADH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)=\left(\begin{array}{l}\sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln (1-{\tau }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\eta }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{\left(-\ln ({\vartheta }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\end{array}\right)$$

Therefore, the proof of theorem 5 is completed. □

Example 2. 

Suppose δ₁= (0.6, 0.7, 0.5), δ₂= (0.9, 0.1, 0.4), δ₃= (0.7, 0.6, 0.6), and δ₄= (0.2, 0.8, 0.3) are four TSFNs, and the values of the parameters are shown as q = 3, γ = 2, φ = 3. Then, we can obtain the following calculation results by utilizing Equation (16).

The MDs of δ₁, δ₂, δ₃, and δ₄ are aggregated as:

$$\begin{array}{l}\sqrt[3]{\exp \left\{-{\left[\frac{1}{6}\left(\begin{array}{l}{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.6}^3\right)\right)}^3+{\left(-\ln \left(1-{0.9}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.6}^3\right)\right)}^3+{\left(-\ln \left(1-{0.7}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\\ +{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.6}^3\right)\right)}^3+{\left(-\ln \left(1-{0.2}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.9}^3\right)\right)}^3+{\left(-\ln \left(1-{0.7}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\\ +{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.9}^3\right)\right)}^3+{\left(-\ln \left(1-{0.2}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left(1-{0.7}^3\right)\right)}^3+{\left(-\ln \left(1-{0.2}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\end{array}\right)\right]}^{\frac{1}{3}}\right\}}\\ =0.533\end{array}$$

The ADs of δ₁, δ₂, δ₃, and δ₄ are aggregated as:

$$\begin{array}{l}\sqrt[3]{1-\exp \left\{-{\left[\frac{1}{6}\left(\begin{array}{l}{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.7}^3\right)\right)}^3+{\left(-\ln \left({0.1}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.7}^3\right)\right)}^3+{\left(-\ln \left({0.6}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\\ +{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.7}^3\right)\right)}^3+{\left(-\ln \left({0.8}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.1}^3\right)\right)}^3+{\left(-\ln \left({0.6}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\\ +{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.1}^3\right)\right)}^3+{\left(-\ln \left({0.8}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.6}^3\right)\right)}^3+{\left(-\ln \left({0.8}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\end{array}\right)\right]}^{\frac{1}{3}}\right\}}\\ =0.722\end{array}$$

Similarly, The NDs of δ₁, δ₂, δ₃, and δ₄ are aggregated as:

$$\begin{array}{l}\sqrt[3]{1-\exp \left\{-{\left[\frac{1}{6}\left(\begin{array}{l}{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.5}^3\right)\right)}^3+{\left(-\ln \left({0.4}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.5}^3\right)\right)}^3+{\left(-\ln \left({0.6}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\\ +{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.5}^3\right)\right)}^3+{\left(-\ln \left({0.3}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.4}^3\right)\right)}^3+{\left(-\ln \left({0.6}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\\ +{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.4}^3\right)\right)}^3+{\left(-\ln \left({0.3}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left({\left(-\ln \left({0.6}^3\right)\right)}^3+{\left(-\ln \left({0.3}^3\right)\right)}^3\right)}^{\frac{1}{3}}\right]\right\}\right)\right)}^3\end{array}\right)\right]}^{\frac{1}{3}}\right\}}\\ =0.467\end{array}$$

Therefore, $TSFAADH{M}^{(2)}({\delta }_1,{\delta }_2,{\delta }_3,{\delta }_4)=(0.533,0.722,0.467)$.

The TSFAADHM operator satisfies monotonicity, boundness, and idempotency. The proof of these properties resembles Theorem 3.

In order to address the condition that the TSFAADHM operator neglects the importance degree of each criterion, we can define the TSFAAWDHM operator.

Definition 11. 

Suppose δ_i (i = 1, 2, …, n) is a group of TSFNs, w = (w₁, w₂, …, w_n)^T is denoted as the weight vector of δ_i (i = 1, 2, …, n), meeting w_i ∈ [0,1] and ${\sum }_{i=1}^n{w}_i=1$. If

$$TSFAAWDH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)={\left(\underset{1\le i_1<\dots <i_{\gamma }\le n}{\otimes }\left(\frac{{\oplus }_{j=1}^{\gamma }{w}_{i_j}{\delta }_{i_j}}{\gamma }\right)\right)}^{\frac{1}{C_n^{\gamma }}}$$

(17)

Then, the operator is named as the TSFAAWDHM operator.

According to the AA operation laws for TSFNs, the aggregation result of Equation (17) is as below.

Theorem 6. 

Suppose δ_i (i = 1, 2, …, n) is a group of TSFNs. The integration result of the TSFAAWDHM operator is still TSFN, i.e.,

$$TSFAAWDH{M}^{(\gamma )}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)=\left(\begin{array}{l}\sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{w}_{i_j}{\left(-\ln (1-{\tau }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{w}_{i_j}{\left(-\ln ({\eta }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le i_1<\dots <i_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{j=1}^{\gamma }{w}_{i_j}{\left(-\ln ({\vartheta }_{i_j}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\end{array}\right)$$

(18)

The TSFAAWDHM operator still satisfies monotonicity and boundness, but not idempotency. The proof process is omitted here.

4. MCDM Based on the TSFAAHM Aggregation Operators

The T-spherical fuzzy (TSF) MCDM problems can be described as: H = {h₁, h₂, …, h_m} is denoted as a family of alternatives, C = {c₁, c₂, …, c_n} is denoted as a family of criteria, and its weight vector is named as w = (w₁, w₂, …, w_n)^T, meeting 0 ≤ w_i ≤ 1, ${\sum }_{i=1}^n{w}_i=1$. The criteria weight vector is determined by the comprehensive evaluation of the expert group for assignment. D = [d_ij]_m×n is denoted as the initial TSF evaluation matrix given by experts, where $d_{ij}=({\tau }_{ij},{\eta }_{ij},{\vartheta }_{ij})$ is denoted as the expert’s evaluation value of alternative h_i with respect to criterion c_j and is expressed in terms of TSFN and satisfies 0 ≤ τ_ij, η_ij, ϑ_ij ≤ 1 and 0 ≤ (τ_ij)^q + (η_ij)^q + (ϑ_ij)^q ≤ 1 (i = 1, 2, 3, …, m; j = 1, 2, 3, …, n; q ≥ 1).

$$D=\begin{array}{ll}& \begin{array}{llll}\hspace{1em}{c}_1& \hspace{1em}{c}_2& \hspace{1em}\cdots & \hspace{1em}{c}_n\end{array}\\ \begin{array}{l}{h}_1\\ h_2\\ ⋮\\ h_m\end{array}& \left[\begin{array}{cccc}\begin{array}{c}{d}_{11}\\ d_{21}\\ ⋮\\ d_{m1}\end{array}& \begin{array}{c}{d}_{12}\\ d_{22}\\ ⋮\\ d_{m2}\end{array}& \begin{array}{c}\cdots \\ \cdots \\ \ddots \\ \cdots \end{array}& \begin{array}{c}{d}_{1n}\\ d_{2n}\\ ⋮\\ d_{mn}\end{array}\end{array}\right]\end{array}$$

Based on the above depiction of the TSFMCDM problem, we should apply the developed TSFAAWHM and TSFAAWDHM operators to handle the MCDM problem and obtain the best alternative. The specific steps are depicted as below.

Step 1. The initial T-spherical fuzzy evaluation matrix should be normalized. Generally, the criteria in the MCDM problem can be classified into cost- and benefit-type criteria. The former needs to be transformed into a benefit-type criterion, and the transformation process can be implemented by Equation (19). Therefore, the standardized T-spherical fuzzy decision-making matrix R = [r_ij]_m×n is obtained.

$$r_{ij}=\left\{\begin{array}{c}{d}_{ij}=({\tau }_{ij},{\eta }_{ij},{\vartheta }_{ij}),\hspace{1em}\hspace{1em}\hspace{1em}j\in J_1\\ {(d_{ij})}^C=({\vartheta }_{ij},{\eta }_{ij},{\tau }_{ij}),\hspace{1em}\hspace{1em}j\in J_2\end{array}\right.$$

(19)

where (d_ij)^C is the complement of d_ij, J₁ and J₂ express, respectively, the criteria of the benefit- and cost-type.

Step 2. The comprehensive value for each alternative with regard to all criteria evaluation data can be computed by the TSFAAWHM (Equation (20)) or the TSFAAWDHM operator (Equation (21)).

$$f_i=TSFAAWH{M}^{(\gamma )}(r_{i1},r_{i2},\dots ,r_{in})=\left(\begin{array}{l}\sqrt[q]{1-\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le j_1<\dots <j_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{k=1}^{\gamma }{w}_{j_k}{\left(-\ln ({\tau }_{i{j}_k}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{1-\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le j_1<\dots <j_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{k=1}^{\gamma }{w}_{j_k}{\left(-\ln (1-{\eta }_{i{j}_k}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{1-\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le j_1<\dots <j_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{k=1}^{\gamma }{w}_{j_k}{\left(-\ln (1-{\vartheta }_{i{j}_k}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\end{array}\right)$$

(20)

or

$$f_i=TSFAAWDH{M}^{(\gamma )}(r_{i1},r_{i2},\dots ,r_{in})=\left(\begin{array}{l}\sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le j_1<\dots <j_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{k=1}^{\gamma }{w}_{j_k}{\left(-\ln (1-{\tau }_{i{j}_k}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le j_1<\dots <j_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{k=1}^{\gamma }{w}_{j_k}{\left(-\ln ({\eta }_{i{j}_k}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}},\\ \sqrt[q]{\exp \left\{-{\left[\frac{1}{C_n^{\gamma }}{\displaystyle \sum _{1\le j_1<\dots <j_{\gamma }\le n}{\left(-\ln \left(1-\exp \left\{-{\left[\frac{1}{\gamma }{\displaystyle \sum _{k=1}^{\gamma }{w}_{j_k}{\left(-\ln ({\vartheta }_{i{j}_k}^q)\right)}^{\phi }}\right]}^{1/\phi }\right\}\right)\right)}^{\phi }}\right]}^{1/\phi }\right\}}\end{array}\right)$$

(21)

Step 3. The results of the score function sc(f_i) (Equation (2)) and accuracy function ac(f_i) (Equation (3)) are calculated for each alternative. Then, the best option is determined after the descending ranking of alternatives, the bigger the better.

Step 4. End.

5. Numerical Example

The applicability and effectiveness of the developed aggregation operator in a TSF context can be proved by discussing the application of the developed model to evaluate and select the optimal investment strategy (revised from Ullah et al. [43]) in this section.

5.1. Application for Investment

Suppose that an investment company plans to choose the most desirable one to select and invest among four companies (alternatives): (1) a car company (h₁); (2) a food company (h₂); (3) a computer company (h₃); and (4) an arms company (h₄). Through preliminary research and study, the experts selected five factors as evaluation criteria: c₁ (risk of loss of investment capital); c₂ (probability of return on capital investment); c₃ (prospects for market development); c₄ (total capital subject to inflation); and c₅ (industry growth potential). All of the five criteria had a great impact on the success of the investment strategy, and the experts provided their evaluations with all five criteria in mind. Let w = (0.15, 0.2, 0.3, 0.2, 0.15)^T be the weight vector of the criteria. The expert panel expressed the evaluation information for each alternative with each criterion by using TSFN. Therefore, the initial TSF evaluation matrix D was given by the expert panel and is shown in Table 1.

5.2. Decision Analysis

5.2.1. The Method by the Proposed TSFAAWHM Operator

Step 1. Since c₁ is a cost-type criterion and the rest of the criteria are all benefit-type, then the normalized TSF decision matrix R can be obtained from Equation (19), and the results can be listed in Table 2.

Step 2. The comprehensive TSF values f_i (i = 1, 2, 3, 4) for each solution with regard to all criteria are calculated by the TSFAAWHM operator (Equation (20)). Suppose the parameters q = 3, γ = 2, φ = 2.

The DM of aggregation result in the comprehensive value of alternative f₁ is τ₁, that is

$$\begin{array}{l}{\tau }_1=\sqrt[3]{1-\exp \left\{-{\left[\frac{1}{10}\left(\begin{array}{l}{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left(0.15\times {\left(-\ln \left({0.5}^3\right)\right)}^2+0.2\times {\left(-\ln \left({0.9}^3\right)\right)}^2\right)}^{\frac{1}{2}}\right]\right\}\right)\right)}^2+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left(0.15\times {\left(-\ln \left({0.5}^3\right)\right)}^2+0.3\times {\left(-\ln \left({0.8}^3\right)\right)}^2\right)}^{\frac{1}{2}}\right]\right\}\right)\right)}^2\\ +{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left(0.15\times {\left(-\ln \left({0.5}^3\right)\right)}^2+0.2\times {\left(-\ln \left({0.6}^3\right)\right)}^2\right)}^{\frac{1}{2}}\right]\right\}\right)\right)}^2+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left(0.15\times {\left(-\ln \left({0.5}^3\right)\right)}^2+0.15\times {\left(-\ln \left({0.9}^3\right)\right)}^2\right)}^{\frac{1}{2}}\right]\right\}\right)\right)}^2\\ +{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left(0.2\times {\left(-\ln \left({0.9}^3\right)\right)}^2+0.3\times {\left(-\ln \left({0.8}^3\right)\right)}^2\right)}^{\frac{1}{2}}\right]\right\}\right)\right)}^2+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left(0.2\times {\left(-\ln \left({0.9}^3\right)\right)}^2+0.2\times {\left(-\ln \left({0.6}^3\right)\right)}^2\right)}^{\frac{1}{2}}\right]\right\}\right)\right)}^2\\ +{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left(0.2\times {\left(-\ln \left({0.9}^3\right)\right)}^2+0.15\times {\left(-\ln \left({0.9}^3\right)\right)}^2\right)}^{\frac{1}{2}}\right]\right\}\right)\right)}^2+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left(0.3\times {\left(-\ln \left({0.8}^3\right)\right)}^2+0.2\times {\left(-\ln \left({0.6}^3\right)\right)}^2\right)}^{\frac{1}{2}}\right]\right\}\right)\right)}^2\\ +{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left(0.3\times {\left(-\ln \left({0.8}^3\right)\right)}^2+0.15\times {\left(-\ln \left({0.9}^3\right)\right)}^2\right)}^{\frac{1}{2}}\right]\right\}\right)\right)}^2+{\left(-\ln \left(1-\exp \left\{-\left[\frac{1}{2}{\left(0.2\times {\left(-\ln \left({0.6}^3\right)\right)}^2+0.15\times {\left(-\ln \left({0.9}^3\right)\right)}^2\right)}^{\frac{1}{2}}\right]\right\}\right)\right)}^2\end{array}\right)\right]}^{\frac{1}{2}}\right\}}\\ =0.913\end{array}$$

The AD of aggregation result in the comprehensive value of alternative f₁ is η₁, that is

$$\begin{array}{l}{\eta }_1=\text{exception 25:}\end{array}$$