**2. Basic Concepts**

In this section, we overview some basic definition of T2FSs, IT2FS and T2IFSs defined over the universal set *X*.

**Definition 1** ([42])**.** *A type-2 fuzzy set (T2FS) A* ⊆ *X, defined as*

$$A = \{ ((\mathbf{x}, \boldsymbol{\mu}\_A), \boldsymbol{\mu}\_A (\mathbf{x}, \boldsymbol{\mu}\_A)) \mid \mathbf{x} \in X, \boldsymbol{\mu}\_A \in j\_x \subseteq [0, 1] \}\tag{1}$$

*where uA denotes the primary membership function (PMF) of A, μA* ∈ [0, 1] *is called as secondary membership function (SMF) jx* ⊆ [0, 1] *is PMF of x.*

*Another equivalent expression for T2FS A is given as*

$$A = \int\_{\mathbf{x} \in X} \frac{\mu\_A(\mathbf{x})}{\mathbf{x}} = \int\_{\mathbf{x} \in X} \left[ \int\_{\mathbf{u}\_A \in j\_\mathbf{x}} \frac{\left( f\_\mathbf{x}(\mathbf{u}\_A) \right)}{\mathbf{u}\_A} \right] / \mathbf{x} \tag{2}$$

**Definition 2** ([20])**.** *The collection of all PMFs of T2FS is named as "footprint of uncertainty" (FOU), i.e., FOU*(*A*) = A *x*∈*Xjx.*

However, because of high computational burden of T2FSs, researchers prefer using interval type-2 (IT2) fuzzy set (IT2FS) for real-world problems.

**Definition 3** ([44])**.** *A T2FS transform into interval type-2 FS when the grades of all SMFs is equal to 1. Mathematically, an IT2FS A, with a membership function μA*(*<sup>x</sup>*, *uA*)*, may be expressed either as Equation* (3) *or as Equation* (4) *:*

$$A\_{\;\;\;\;\phi} = \{ (\mathbf{x}, \boldsymbol{\mu}\_A), \boldsymbol{\mu}\_A(\mathbf{x}, \boldsymbol{\mu}\_A) = 1 \; | \; \forall \mathbf{x} \in X, \forall \boldsymbol{\mu}\_A \in j\_\mathbf{x} \subseteq [0, 1] \}\tag{3}$$

$$A\_{\;\;\;x\in X} = \int\_{x\in X} \int\_{u\_A \in j\_x} 1/(x, u\_A)\_{\;\;\;x} \, j\_x \subseteq [0, 1] \tag{4}$$

**Definition 4** ([44])**.** *An IT2 FS is normally described by a zone called as FOU, which is limited by two membership functions (MFs), known as lower MF (LMF) <sup>μ</sup>A*(*<sup>x</sup>*, *uA*) *and the upper MF (UMF) <sup>μ</sup>A*(*<sup>x</sup>*, *uA*)*. That is FOU=[μA*(*<sup>x</sup>*, *uA*), *<sup>μ</sup>A*(*<sup>x</sup>*, *uA*)*]. Figure 1 shows the graphical representation of IT2 fuzzy number (IT2 FN) with triangular MF shape.*

**Figure 1.** LMF (dashed), UMF (solid), FOU (shaded) for IT2FS *A*.

**Definition 5** ([38,39])**.** *A T2IFS is a set of ordered pairs consisting of PMFs and SMFs of the element defined as*

$$A = \left\{ \langle (\mathbf{x}, u\_A, v\_A), \mu\_A(\mathbf{x}, u\_A), \nu\_A(\mathbf{x}, v\_A) \rangle \: \mid \: \mathbf{x} \in X, \mu\_A \in j\_{\mathbf{x}\prime}^1 v\_A \in j\_{\mathbf{x}}^2 \right\} \tag{5}$$

*where uA*(*vA*) *represents the primary membership (non-membership) of A denoted by PMF(PNMF), μA*(*<sup>ν</sup>A*) *is secondary membership (non-membership) function of A, denoted by SMF (SNMF) and j* 1 *x*, *j* 2 *x* ⊆ [0, 1] *are PMF and PNMF of x, respectively. When the SMFs μA*(*<sup>x</sup>*, *uA*) = 1*, and SNMF <sup>ν</sup>A*(*<sup>x</sup>*, *vA*) = 0*, a T2IFS translates to an IT2 IFS.*

**Definition 6** ([55])**.** *An IT2 IFS, A, is described by a bounding functions of lower and upper membership and non-membership functions denoted by LMF, UMF, LNMF and UNMF defined as μA, μA and νA, <sup>ν</sup>A with conditions:* 0 ≤ *μA* + *<sup>ν</sup>A* ≤ 1 *and* 0 ≤ *μA* + *νA* ≤ 1*. The FOUs of an IT2IFS is illustrated in Figure 2 with triangular shape and defined mathematically as*

$$FOU(A) = \bigcup\_{\mathbf{x} \in \mathcal{X}} \left[ \underline{\mu}\_A(\mathbf{x}), \overline{\mu}\_A(\mathbf{x}), \underline{\nu}\_A(\mathbf{x}), \overline{\nu}\_A(\mathbf{x}) \right]$$

**Figure 2.** LMF (dashed), UMF (solid), LNMF (doted), UNMF (solid), FOU (shaded) for IT2IFS *A*.

**Definition 7** ([51])**.** *For non-negative real numbers xi*(*<sup>i</sup>* = 1, 2, . . . , *<sup>n</sup>*)*, the Hamy mean (HM) is given as*

$$\Pr\_{\text{HM}^{(k)}(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n)}^{\sum\_{1 \le i\_1 < \dots < i\_k \le n} \left( \prod\_{j=1}^k x\_{i\_j} \right)^{\frac{1}{k}} \tag{6}$$

*where k is the parameter,* (*nk*) = *n*! *k*!(*<sup>n</sup>*−*k*)! *and* (*<sup>i</sup>*1, *i*2,..., *ik*) *crosses all the k*−*tuple mix of* (1, 2, . . . , *<sup>n</sup>*)*.*

#### **3. Proposed Symmetric Triangular Interval T2IFS**

In this section, we present a symmetric triangular IT2IFS and characterize their fundamental operational laws.

**Definition 8.** *Let X be the universal set. A symmetric triangular interval T2 IFS (TIT2IFS) can be represented as follows:*

$$\mathcal{U} = \left\{ \left( \mathbb{Z}\_{\mathbb{A}}(\mathbf{x}), \varrho\_{\mathbb{A}}(\mathbf{x}), \varrho\_{\mathbb{A}}(\mathbf{x}), \varrho\_{\mathbb{A}}^{\*}(\mathbf{x}), \vartheta\_{\mathbb{A}}(\mathbf{x}), \vartheta\_{\mathbb{A}}^{\*}(\mathbf{x}) \right) \mid \mathbf{x} \in X \right\} \tag{7}$$

*where ζα*(*x*), *α*(*x*), *ϕα*(*x*), *ϕ*<sup>∗</sup>*α*(*x*)*, ϑα*(*x*), *<sup>ϑ</sup>*<sup>∗</sup>*α*(*x*) *are the real numbers satisfying the inequalities, ζα*(*x*) ≥ *α*(*x*)*,* 0 ≤ *ϕα*(*x*) ≤ *ϕ*<sup>∗</sup>*α*(*x*) ≤ 1*,* 0 ≤ *<sup>ϑ</sup>*<sup>∗</sup>*α*(*x*) ≤ *ϑα*(*x*) ≤ 1 *such that ϕα*(*x*) + *ϑα*(*x*) ≤ 1 *and ϕ*<sup>∗</sup>*α*(*x*) + *<sup>ϑ</sup>*<sup>∗</sup>*α*(*x*) ≤ 1*.*

For convenience, we represent this pair as *α* = (*ζα*, *α*, *ϕα*, *ϕ*<sup>∗</sup>*α*, *ϑα*, *<sup>ϑ</sup>*<sup>∗</sup>*α*) and called as symmetric triangular IT2 intuitionistic fuzzy (IT2IF) number (STIT2IFN) where *ζα* ≥ *α*, *ϕα* + *ϑα* ≤ 1, *ϕ*∗*α* + *ϑ*∗*α* ≤ 1 and *ϕα* ≤ *ϕ*<sup>∗</sup>*α*, *ϑα* ≥ *ϑ*<sup>∗</sup>*α*. The graphical representation of STIT2IFN is given in Figure 3.

**Definition 9.** *For a STIT2IFN α* = (*ζα*, *α*, *ϕα*, *ϕ*<sup>∗</sup>*α*, *ϑα*, *<sup>ϑ</sup>*<sup>∗</sup>*α*)*, the lower and upper membership and non-membership functions denoted by LMF, UMF, LNMF and UNMF are defined as*

$$\text{LIMF}\_{\mathbf{x}}\left(\mathbf{x}\right) = \begin{cases} \frac{\varrho\_{\mathbf{x}}^{\*}}{\varrho\_{\mathbf{x}}}(\mathbf{x} - \boldsymbol{\zeta}\_{\mathbf{a}} + \varrho\_{\mathbf{a}}), & \boldsymbol{\zeta}\_{\mathbf{a}} - \varrho\_{\mathbf{a}} \leq \mathbf{x} < \boldsymbol{\zeta}\_{\mathbf{a}} \\\\ \varrho\_{\mathbf{x}}^{\*}, & \mathbf{x} - \boldsymbol{\zeta}\_{\mathbf{a}} \\\\ \frac{\varrho\_{\mathbf{x}}^{\*}}{\varrho\_{\mathbf{x}}}(\boldsymbol{\zeta}\_{\mathbf{a}} + \varrho\_{\mathbf{a}} - \mathbf{x}), & \boldsymbol{\zeta}\_{\mathbf{a}} < \mathbf{x} \leq \varrho\_{\mathbf{a}} + \boldsymbol{\zeta}\_{\mathbf{a}} \end{cases}; \quad \text{LINMF}\_{\mathbf{x}}\left(\mathbf{x}\right) = \begin{cases} \frac{(\varrho\_{\mathbf{x}}^{\*} - 1)(\mathbf{x} - \boldsymbol{\zeta}\_{\mathbf{a}} + \varrho\_{\mathbf{a}}) + \varrho\_{\mathbf{a}}}{\varrho\_{\mathbf{x}}}, & \boldsymbol{\zeta}\_{\mathbf{a}} - \varrho\_{\mathbf{a}} \leq \mathbf{x} < \boldsymbol{\zeta}\_{\mathbf{a}} \\\\ \varrho\_{\mathbf{a}}^{\*}, & \mathbf{x} - \boldsymbol{\zeta}\_{\mathbf{a}} \\\\ \frac{(1 - \theta\_{\mathbf{x}}^{\*})(\mathbf{x} - \boldsymbol{\zeta}\_{\mathbf{a}}) + \theta\_{\mathbf{a}}^{\*}\varrho\_{\mathbf{a}}}{\varrho\_{\mathbf{x}}}, & \boldsymbol{\zeta}\_{\mathbf{a}} < \mathbf{x} \leq \varrho\_{\mathbf{a}} + \boldsymbol{\zeta}\_{\mathbf{a}} \end{cases} \tag{8}$$

$$\text{LMF}\_{a}(\mathbf{x}) = \begin{cases} \frac{\varrho\_{a}}{\varrho\_{a}}(\mathbf{x} - \boldsymbol{\zeta}\_{a} + \varrho\_{a}); & \boldsymbol{\zeta}\_{a} - \varrho\_{a} \leq \mathbf{x} < \boldsymbol{\zeta}\_{a} \\\\ \varrho\_{a}; & \mathbf{x} - \boldsymbol{\zeta}\_{a} \end{cases}; \qquad \text{LMF}\_{a}(\mathbf{x}) = \begin{cases} \frac{(\boldsymbol{\theta}\_{a} - 1)(\mathbf{x} - \boldsymbol{\zeta}\_{a} + \varrho\_{a}) + \varrho\_{a}}{\varrho\_{a}}; & \boldsymbol{\zeta}\_{a} - \varrho\_{a} \leq \mathbf{x} < \boldsymbol{\zeta}\_{a} \\\\ \frac{\varrho\_{a}}{\varrho\_{a}}; & \mathbf{x} - \boldsymbol{\zeta}\_{a} \end{cases} \tag{9}$$

**Definition 10.** *The score function of STIT2IFN α* = (*ζα*, *α*, *ϕα*, *ϕ*<sup>∗</sup>*α*, *ϑα*, *<sup>ϑ</sup>*<sup>∗</sup>*α*) *is defined as*

$$\begin{array}{rcl} \mathbf{s}(\boldsymbol{\alpha}) &=& (\mathbf{s}\_{\mathbf{x}}(\boldsymbol{\alpha}), \mathbf{s}\_{\mathbf{y}}(\boldsymbol{\alpha})) \\ &=& \left( \mathbb{J}\_{\boldsymbol{\alpha}} \frac{2\boldsymbol{\varrho}\_{\boldsymbol{\alpha}}\boldsymbol{\varrho}^{\*}\_{\boldsymbol{\alpha}}}{\boldsymbol{\varrho}\_{\boldsymbol{\alpha}} + \boldsymbol{\varrho}^{\*}\_{\boldsymbol{\alpha}}} - \mathbb{J}\_{\boldsymbol{\alpha}} \frac{2\boldsymbol{\theta}\_{\boldsymbol{\alpha}}\boldsymbol{\theta}^{\*}\_{\boldsymbol{\alpha}}}{\boldsymbol{\theta}\_{\boldsymbol{\alpha}} + \boldsymbol{\theta}^{\*}\_{\boldsymbol{\alpha}}}, \frac{\boldsymbol{\theta}\_{\boldsymbol{\alpha}} + \boldsymbol{\varrho}^{\*}\_{\boldsymbol{\alpha}}}{2} - \frac{\boldsymbol{\varrho}\_{\boldsymbol{\alpha}} + \boldsymbol{\theta}^{\*}\_{\boldsymbol{\alpha}}}{2} \right) \end{array} \tag{10}$$

**Definition 11.** *For two STIT2IFNs α and β, an order relation "*(>)*" to compare them is defined as*

$$\begin{aligned} \text{1.} \qquad &\text{If } s\_{\mathfrak{X}}(\alpha) > s\_{\mathfrak{X}}(\beta), \text{ then } \alpha > \beta;\\ \text{2.} \qquad &\text{If } s\_{\mathfrak{X}}(\alpha) = s\_{\mathfrak{x}}(\beta), \text{ then } \begin{cases} s\_{\mathfrak{Y}}(\alpha) > s\_{\mathfrak{Y}}(\beta) \implies \alpha > \beta;\\ s\_{\mathfrak{Y}}(\alpha) = s\_{\mathfrak{Y}}(\beta) \implies \alpha = \beta; \end{cases} \end{aligned}$$

**Definition 12.** *For two STIT2IFNs α* = (*ζα*, *α*, *ϕα*, *ϕ*<sup>∗</sup>*α*, *ϑα*, *<sup>ϑ</sup>*<sup>∗</sup>*α*) *and β* = *ζβ*, *β*, *ϕβ*, *<sup>ϕ</sup>*<sup>∗</sup>*β*, *ϑβ*, *<sup>ϑ</sup>*<sup>∗</sup>*β, λ* > 0*, then the operational laws of it are shown as follows:*

*1. α* ⊕ *β* = *ζα* + *ζβ*, *α* + *β*, *ϕαϕβ*, *ϕ*∗*α* + *ϕ*∗*β* − *<sup>ϕ</sup>*<sup>∗</sup>*αϕ*<sup>∗</sup>*β*, *ϑα* + *ϑβ* − *ϑαϑβ*, *<sup>ϑ</sup>*<sup>∗</sup>*αϑ*<sup>∗</sup>*β;*

$$\text{1.2. } \quad \mathfrak{a} \otimes \mathfrak{b} = \left( \mathbb{J}\_{\mathfrak{a}} \mathbb{J}\_{\mathfrak{b}\prime} \varrho\_{\mathfrak{a}} \varrho\_{\mathfrak{b}\prime}, \varrho\_{\mathfrak{a}} + \varrho\_{\mathfrak{b}} - \varrho\_{\mathfrak{a}} \varrho\_{\mathfrak{b}\prime}, \varrho\_{\mathfrak{a}}^{\*} \varrho\_{\mathfrak{b}\prime}^{\*}, \mathfrak{b}\_{\mathfrak{a}} \mathfrak{b}\_{\mathfrak{b}\prime}, \mathfrak{b}\_{\mathfrak{a}}^{\*} + \mathfrak{b}\_{\mathfrak{b}}^{\*} - \mathfrak{b}\_{\mathfrak{a}}^{\*} \ \mathfrak{b}\_{\mathfrak{b}}^{\*} \right); 2.5. $$

$$\lambda . \qquad \lambda \mathfrak{a} = \left( \lambda \mathbb{Z}\_{a\prime} \lambda \, \varrho\_{a\prime} \left( \varrho\_a \right)^{\lambda}, 1 - (1 - \varrho\_a^\*)^{\lambda}, 1 - (1 - \vartheta\_a)^{\lambda}, (\vartheta\_a^\*)^{\lambda} \right);$$

$$4. \qquad a^{\lambda} = \left(\zeta\_a^{\lambda}, \varrho\_a^{\lambda}, 1 - (1 - \varrho\_a)^{\lambda}, (\varrho\_a^\*)^{\lambda}, (\vartheta\_a)^{\lambda}, 1 - (1 - \vartheta\_a^\*)^{\lambda}\right)$$

**Theorem 1.** *For STIT2IFNs α and β, the operations defined in Definition 12 are again STIT2IFNs.*

**Proof.** Consider two STIT2IFNs *α* = (*ζα*, *α*, *ϕα*, *ϕ*<sup>∗</sup>*α*, *ϑα*, *<sup>ϑ</sup>*<sup>∗</sup>*α*) and *β* = *ζβ*, *β*, *ϕβ*, *<sup>ϕ</sup>*<sup>∗</sup>*β*, *ϑβ*, *<sup>ϑ</sup>*<sup>∗</sup>*β*. So by Definition 8, we have *ζα* ≥ *α*, *ϕα* ≤ *ϕ*<sup>∗</sup>*α*, *ϑα* ≥ *ϑ*<sup>∗</sup>*α*, *ϕα* + *ϑα* ≤ 1, *ϕ*∗*α* + *ϑ*∗*α* ≤ 1, *ζβ* ≥ *β*, *ϕβ* ≤ *<sup>ϕ</sup>*<sup>∗</sup>*β*, *ϑβ* ≥ *ϑ*∗*β ϕβ* + *ϑβ* ≤ 1, *ϕ*∗*β* + *ϑ*∗*β* ≤ 1.

Let *α* ⊕ *β* = *γ* = *ζγ*, *γ*, *ϕγ*, *<sup>ϕ</sup>*<sup>∗</sup>*γ*, *ϑγ*, *ϑ*∗*γ*and thus by Definition 12, we ge<sup>t</sup> *ζγ* = *ζα* + *ζβ*, *γ* = *α* + *β*, *ϕγ* = *ϕαϕβ*, *ϕ*∗*γ* = *ϕ*∗*α* +∗*β* <sup>−</sup>*ϕ*<sup>∗</sup>*αϕ*<sup>∗</sup>*β*, *ϑγ* = *ϑα* + *ϑβ* − *ϑαϑβ*, *ϑ*∗*γ* = *<sup>ϑ</sup>*<sup>∗</sup>*αϑ*<sup>∗</sup>*β*. Now, to show *α* ⊕ *β* is again an STIT2IFN, we need to prove that *ζγ* ≥ *γ*, *ϕγ* ≤ *<sup>ϕ</sup>*<sup>∗</sup>*γ*, *ϑγ* ≥ *<sup>ϑ</sup>*<sup>∗</sup>*γ*, *ϕγ* + *ϑγ* ≤ 1, *ϕ*∗*γ*+ *ϑ*∗*γ*≤ 1.

As *ζα* ≥ *α* and *ζβ* ≥ *β* which implies that *ζγ* ≥ *γ*. Further *ϕα* ≤ *ϕ*<sup>∗</sup>*α*, *ϕβ* ≤ *<sup>ϕ</sup>*<sup>∗</sup>*β*, *ϑα* ≥ *ϑ*<sup>∗</sup>*α*, *ϑβ* ≥ *ϑ*∗*β*, *ϕα* + *ϑα* ≤ 1, *ϕ*∗*α* + *ϑ*∗*α* ≤ 1 which gives that

$$\begin{aligned} \left(\boldsymbol{\varrho}\_{\boldsymbol{\varbeta}} + \boldsymbol{\theta}\_{\boldsymbol{\varbeta}}\right) &=& \boldsymbol{\varrho}\_{\boldsymbol{\varbeta}} \boldsymbol{\uprho}\_{\boldsymbol{\varbeta}} + \left(\boldsymbol{\theta}\_{\boldsymbol{\varbeta}} + \boldsymbol{\theta}\_{\boldsymbol{\varbeta}} - \boldsymbol{\uptheta}\_{\boldsymbol{\varbeta}} \boldsymbol{\uptheta}\_{\boldsymbol{\varbeta}}\right) \\ &=& \boldsymbol{\uprho}\_{\boldsymbol{\varbeta}} \boldsymbol{\uprho}\_{\boldsymbol{\varbeta}} + 1 - \left(1 - \boldsymbol{\uptheta}\_{\boldsymbol{a}}\right) \left(1 - \boldsymbol{\uptheta}\_{\boldsymbol{\varbeta}}\right) \\ &\leq& \boldsymbol{\uprho}\_{\boldsymbol{\varbeta}} \boldsymbol{\uprho}\_{\boldsymbol{\varbeta}} + 1 - \boldsymbol{\uprho}\_{\boldsymbol{a}} \boldsymbol{\uprho}\_{\boldsymbol{\varbeta}} \\ &\leq& 1 \end{aligned}$$

and

$$\begin{aligned} \label{eq:SDAR} \boldsymbol{\theta}\_{\gamma}^{\*} + \boldsymbol{\theta}\_{\gamma}^{\*} &=& \boldsymbol{\varphi}\_{\boldsymbol{\alpha}}^{\*} \boldsymbol{\varphi}\_{\beta}^{\*} - \boldsymbol{\varphi}\_{\boldsymbol{\alpha}}^{\*} \boldsymbol{\varphi}\_{\beta}^{\*} + \boldsymbol{\theta}\_{\boldsymbol{\alpha}}^{\*} \boldsymbol{\theta}\_{\beta}^{\*} \\ &=& 1 - (1 - \boldsymbol{\varphi}\_{\boldsymbol{\alpha}}^{\*}) \left(1 - \boldsymbol{\varphi}\_{\beta}^{\*}\right) + \boldsymbol{\theta}\_{\boldsymbol{\alpha}}^{\*} \boldsymbol{\theta}\_{\beta}^{\*} \\ &\leq& 1 - \boldsymbol{\theta}\_{\boldsymbol{\alpha}}^{\*} \boldsymbol{\theta}\_{\beta}^{\*} + \boldsymbol{\theta}\_{\boldsymbol{\alpha}}^{\*} \boldsymbol{\theta}\_{\beta}^{\*} \\ &\leq& 1 \end{aligned}$$

Finally, *ϕγ* = *ϕαϕβ* ≤ *<sup>ϕ</sup>*<sup>∗</sup>*αϕ*<sup>∗</sup>*β* = *ϕ*∗*γ* and *ϑγ* = *ϑα* + *ϑβ* − *ϑαϑβ* = 1 − (1 − *ϑα*)(1 − *ϑβ*) ≥ 1 − (1 − *<sup>ϑ</sup>*<sup>∗</sup>*α*)(<sup>1</sup> − *ϑ*∗*β*) = *<sup>ϑ</sup>*<sup>∗</sup>*γ*.

Therefore, we conclude that *α* ⊕ *β* becomes STIT2IFN. Similarly, we can prove that *α* ⊗ *β*, *α<sup>λ</sup>* and *λα* are also STIT2IFNs.

#### **4. TIT2IF Hamy Mean Aggregation Operators**

Let Ω be the gathering of all non-empty STIT2IFNs *αi* = -*ζ<sup>i</sup>*, *i*, *ϕi*, *ϕ*∗*i* , *ϑi*, *ϑ*∗*i* . ,(*<sup>i</sup>* = <sup>1</sup>(1)*n*). Here, we present HM-based AOs for STIT2IFNs.

## *4.1. STIT2IFHM Operator*

**Definition 13.** *A STIT2IFHM is a mapping STIT2IFHM* : Ω*n* → Ω *defined as*

$$\bigoplus\_{\substack{1 \le i\_1 < \cdots < i\_k \le n\\1 \le i\_1 \le n\\i\_k \ne i\_1}} \left( \bigotimes\_{j=1}^k a\_{i\_j} \right)^{\frac{1}{k}}$$
 $\text{STIT2}$  $\text{IFHM}(^{k})$  $(a\_1, a\_2, \dots, a\_n) = \frac{\dots \circ i\_k \le n}{\binom{n}{k}} \tag{11}$ 

*then STIT2IHM*(*k*) *is called the symmetric triangular IT2IF Hamy mean operator, where k* = 1, 2, ... , *n is the parameter and* (*nk*) = *n*! *k*!(*<sup>n</sup>*−*k*)! *represent the binomial coefficient.*

**Theorem 2.** *The aggregated value for n STIT2IFNs αi* = -*ζ<sup>i</sup>*, *i*, *ϕi*, *ϕ*∗*i* , *ϑi*, *ϑ*∗*i* . *by using Definition 13 is again STIT2IFN which is given as*

*STIT2IFHM*(*k*)(*<sup>α</sup>*1, *α*2,..., *<sup>α</sup>n*) (12) = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∑<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup> k*∏*j*=1 *ζαij* 1*k* (*nk*) , ∑<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup> k*∏*j*=1 *αij* 1*k* (*nk*) , ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*<sup>j</sup>*=<sup>1</sup>(<sup>1</sup> − *ϕαij*)1*k* ⎞⎠⎞⎟⎠ 1(*nk*) , 1 − ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*j*=1 *ϕ*∗*αij* 1*k* ⎞⎠⎞⎟⎠ 1(*nk*) , 1 − ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*j*=1 *ϑαij* 1*k* ⎞⎠⎞⎟⎠ 1(*nk*) , ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*<sup>j</sup>*=<sup>1</sup>(<sup>1</sup> − *ϑ*∗*αij*)1*k* ⎞⎠⎞⎟⎠ 1(*nk*) ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

**Proof.** The first part of the result can be easily obtained from Theorem 1. So, there is a need to prove only that Equation (12) is kept.

According to the operational laws of STIT2IFNs, we ge<sup>t</sup>

$$\bigotimes\_{j=1}^{k} a\_{i\_j} = \quad \left( \prod\_{j=1}^{k} \zeta\_{a\_{i\_j}}, \prod\_{j=1}^{k} \varrho\_{a\_{i\_j}}, 1 - \prod\_{j=1}^{k} (1 - \varrho\_{a\_{i\_j}}), \prod\_{j=1}^{k} \varrho\_{a\_{i\_j}}^{\*}, \prod\_{j=1}^{k} \vartheta\_{a\_{i\_j}}, 1 - \prod\_{j=1}^{k} (1 - \vartheta\_{a\_{i\_j}}^{\*}) \right)$$

and

$$\begin{pmatrix} \prod\_{j=1}^{n} a\_{i\_{j}} \end{pmatrix}^{\frac{1}{k}} = \begin{pmatrix} \left(\prod\_{j=1}^{k} \zeta\_{a\_{i\_{j}}}\right)^{\frac{1}{k}} \cdot \left(\prod\_{j=1}^{k} \varrho\_{a\_{i\_{j}}}\right)^{\frac{1}{k}} , 1 - \left(\prod\_{j=1}^{k} (1 - \varrho\_{a\_{i\_{j}}})\right)^{\frac{1}{k}} \\\ \vdots \\\ \left(\prod\_{j=1}^{k} \varrho\_{a\_{i\_{j}}}^{\*}\right)^{\overline{k}} \cdot \left(\prod\_{j=1}^{k} \vartheta\_{a\_{i\_{j}}}\right)^{\overline{k}} , 1 - \left(\prod\_{j=1}^{k} (1 - \vartheta\_{a\_{i\_{j}}}^{\*})\right)^{\overline{k}} \end{pmatrix}$$

Therefore,

$$\bigoplus\_{\substack{1\le i\le\cdots\le n\\1\le j\le n}}\left(\bigotimes\_{j=1}^{n}a\_{i\_{j}}\right)^{\frac{1}{k}}=\left(\begin{array}{c}\sum\_{1\le i\le\cdots\le n\\1\le j\le n\end{array}}\left(\prod\_{j=1}^{k}\mathbbm{1}\_{a\_{i\_{j}}}\right)^{\frac{1}{k}},\ \sum\_{1\le i\_{j}\le\cdots\le n\\1\le j\le n\end{array}}\left(\prod\_{j=1}^{k}a\_{a\_{i\_{j}}}\right)^{\frac{1}{k}},\ \prod\_{1\le i\_{1}\le\cdots\le i\_{k}\le n\\1-\prod\_{1\le i\_{1}\le\cdots\le i\_{k}\le n\\\dots-\epsilon\_{k}\le n\end{array}}\left(1-\left(\prod\_{j=1}^{k}\mathbbm{1}\_{a\_{a\_{j}}}\right)^{\frac{1}{k}}\right),\ \begin{aligned} &1-\left(\prod\_{j=1}^{k}\left(1-\left(\prod\_{j=1}^{k}\mathbbm{1}\_{a\_{a\_{j}}}\right)^{\frac{1}{k}}\right),\\&1-\left(\prod\_{j=1}^{k}\left(1-\left(\prod\_{j=1}^{k}\mathbbm{1}\_{a\_{a\_{j}}}\right)^{\frac{1}{k}}\right)\\ &\prod\_{1\le i\_{1}\le\cdots\le i\_{k}\le n\end{aligned}}\left(1-\left(\prod\_{j=1}^{k}\left(1-\theta\_{a\_{i\_{j}}}^{\*}\right)\right)^{\frac{1}{k}}\right),\end{aligned}$$

Subsequently, we have

STIT2IFHM(*k*)(*<sup>α</sup>*1, *α*2,..., *<sup>α</sup>n*) = <sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup> k j*=1 *αij* 1 *k* (*nk*) = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∑<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup> k*∏*j*=1 *ζαij* 1*k* (*nk*) , ∑<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup> k*∏*j*=1 *αij* 1*k* (*nk*) , ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*<sup>j</sup>*=<sup>1</sup>(<sup>1</sup> − *ϕαij*)1*k* ⎞⎠⎞⎟⎠ 1(*nk*) , 1 − ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*j*=1 *ϕ*∗*αij* 1*k* ⎞⎠⎞⎟⎠ 1(*nk*) , 1 − ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*j*=1 *ϑαij* 1*k* ⎞⎠⎞⎟⎠ 1(*nk*) , ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*<sup>j</sup>*=<sup>1</sup>(<sup>1</sup> − *ϑ*∗*αij*)1*k* ⎞⎠⎞⎟⎠ 1(*nk*) ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

In what follows, we investigate the certain property of STIT2IFHM operator.

**Theorem 3.** *(Idempotency) If αi* = *α* = (*ζα*, *α*, *ϕα*, *ϕ*<sup>∗</sup>*α*, *ϑα*, *<sup>ϑ</sup>*<sup>∗</sup>*α*) *for all i, then STIT2IFHM*(*k*)(*<sup>α</sup>*1, *α*2,..., *<sup>α</sup>n*) = *α*.

**Proof.** Since *αi* = *α* = (*ζα*, *α*, *ϕα*, *ϕ*<sup>∗</sup>*α*, *ϑα*, *<sup>ϑ</sup>*<sup>∗</sup>*α*) for all *i* then based on Theorem 2, we have

STIT2IFHM(*k*)(*<sup>α</sup>*, *α*,..., *α*) = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∑<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup> k*∏*j*=1 *ζα*1*k* (*nk*) , ∑<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup> k*∏*j*=1 *α*1*k* (*nk*) , ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*<sup>j</sup>*=<sup>1</sup>(<sup>1</sup> − *ϕα*)1*k* ⎞⎠⎞⎟⎠ 1(*nk*) , 1 − ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*j*=1 *ϕ*∗*α*1*k* ⎞⎠⎞⎟⎠ 1(*nk*) , 1 − ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*j*=1 *ϑα*1*k* ⎞⎠⎞⎟⎠ 1(*nk*) , ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*<sup>j</sup>*=<sup>1</sup>(<sup>1</sup> − *ϑ*∗*α*)1*k* ⎞⎠⎞⎟⎠ 1(*nk*) ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∑<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup> ζ<sup>k</sup>α*<sup>1</sup>*<sup>k</sup>* (*nk*) , ∑<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup> kα* (*nk*) , ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* (1 − (1 − *ϕα*))⎞⎟⎠ 1(*nk*) , 1 − ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* (1 − *ϕ*∗*α*)⎞⎟⎠ 1(*nk*) , 1 − ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* (1 − *ϑα*)⎞⎟⎠ 1(*nk*) , ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* (1 − (1 − *ϑ*∗*α*))⎞⎟⎠ 1(*nk*) ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = ⎛⎝(*nk*)(*ζ<sup>k</sup>α*) 1*k* (*nk*) , (*nk*)(*kα*) 1*k* (*nk*) , 1 − (1 − *ϕα*)(*nk*) (*nk*), 1 − (1 − *ϕ*∗*α*)(*nk*) (*nk*) , 1 − (1 − *ϑα*)(*nk*) (*nk*) ,(<sup>1</sup> − (1 − *<sup>ϑ</sup>*<sup>∗</sup>*α*))(*nk*) (*nk*) ⎞⎠ = (*ζα*, *α*, *ϕα*, *ϕ*<sup>∗</sup>*α*, *ϑα*, *<sup>ϑ</sup>*<sup>∗</sup>*α*) = *α*

**Theorem 4.** *(Commutativity) Let <sup>α</sup>i*(*<sup>i</sup>* = 1, 2, ... , *n*) *be a collection of STIT2IFNs, and αi be any permutation of αi. Then*

$$\text{STIT2IFHM}^{(k)}(\overline{\mathfrak{a}}\_1, \overline{\mathfrak{a}}\_2, \dots, \overline{\mathfrak{a}}\_n) = \text{STIT2IFHM}^{(k)}(\mathfrak{a}\_1, \mathfrak{a}\_2, \dots, \mathfrak{a}\_n)$$

**Proof.** Based on the Definition 13, we have

$$\begin{array}{rcl} \bigoplus\_{\begin{subarray}{c}1\leq i\_{1}\\ \cdot$$

**Theorem 5.** *(Monotonicity) For two different STIT2IFNs αi* = *ζαi* , *αi* , *ϕαi* , *ϕ*∗ *<sup>α</sup>i* , *ϑαi* , *ϑ*∗ *<sup>α</sup>i , and βi* = *ζβi* , *βi* , *ϕβi* , *ϕ*∗ *βi* , *ϑβi* , *ϑ*∗ *βi ,* (*i* = 1, 2, ... , *<sup>n</sup>*)*. If ζαi* ≤ *ζβi , αi* ≥ *βi , ϕαi* ≥ *ϕβi , ϕ*∗ *<sup>α</sup>i* ≤ *ϕ*∗ *βi , ϑαi* ≤ *ϑβi and ϑ*∗ *<sup>α</sup>i* ≥ *ϑ*∗ *βi for all i, then*

$$\text{STIT2IFHM}^{(k)}(\mathfrak{a}\_1, \mathfrak{a}\_2, \dots, \mathfrak{a}\_n) \le \text{STIT2IFHM}^{(k)}(\mathfrak{\beta}\_1, \mathfrak{\beta}\_2, \dots, \mathfrak{\beta}\_n). \tag{13}$$

**Proof.** Let *A* = STIT2IFHM(*k*) (*<sup>α</sup>*1, *α*2, ... , *<sup>α</sup>n*) and *B* = STIT2IFHM(*k*) (*β*1, *β*2, ... , *βn*). Then according to Theorem 2, we ge<sup>t</sup>

$$\begin{split} A &= \begin{array}{c} \text{STIT2}\text{IFHM}^{(k)}(a\_{1},a\_{2},\ldots,a\_{n}) \\ \left(\sum\_{\begin{subarray}{c}1\leq i\_{1}<\cdots\leq i\_{k}\\ \cdots \neq i\_{k}\leq n\end{subarray}} \left(\prod\_{j=1}^{k}\zeta\_{a\_{i\_{j}}}\right)^{\frac{1}{k}}\sum\_{\begin{subarray}{c}1\leq i\_{1}<\cdots\leq i\_{k}\\ \cdots \leq i\_{k}\leq n\end{subarray}} \left(\prod\_{j=1}^{k}\varrho a\_{i\_{j}}\right)^{\frac{1}{k}} \\ &= \left(\prod\_{\begin{subarray}{c}1\leq i\_{1}<\cdots\leq i\_{k}\\ \cdots \leq i\_{k}\leq n\end{subarray}} \left(1-\left(\prod\_{j=1}^{k}\varrho^{\*}\_{a\_{i\_{j}}}\right)^{\frac{1}{k}}\right)\right)^{\frac{1}{k}}, 1-\left(\prod\_{\begin{subarray}{c}1\leq i\_{1}<\cdots\leq i\_{k}\\ \cdots \leq i\_{k}\leq n\end{subarray}} \left(1-\left(\prod\_{j=1}^{k}\varrho a\_{i\_{j}}\right)^{\frac{1}{k}}\right)\right)^{\frac{1}{k}}, \\ & \left(\prod\_{\begin{subarray}{c}1\leq i\_{1}<\cdots\leq i\_{k}\\ \cdots \leq i\_{k}\leq n\end{subarray}} \left(1-\left(\prod\_{j=1}^{k}(1-\theta\_{\boldsymbol{a}\_{i\_{j}}}^{k})\right)^{\frac{1}{k}}\right)^{\frac{1}{k}} \end{array}\right)^{\frac{1}{k}}, \end{split}$$

and

*B*= STIT2IFHM(*k*) (*β*1, *β*2,..., *βn*) = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∑ 1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup> k* ∏ *j*=1 *ζβi j* 1 *k* ( *n k*) , ∑ 1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup> k* ∏ *j*=1 *βi j* 1 *k* ( *n k*) , ⎛ ⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup>* ⎛ ⎝1 − *k* ∏*j*=1 (1 − *ϕβi j* ) 1 *k* ⎞ ⎠ ⎞ ⎟⎠ 1 ( *n k*) , 1 − ⎛ ⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup>* ⎛ ⎝1 − *k* ∏*j*=1 *ϕ*∗ *βi j* 1 *k* ⎞ ⎠ ⎞ ⎟⎠ 1 ( *n k*) , 1 − ⎛ ⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup>* ⎛ ⎝1 − *k* ∏*j*=1 *ϑβi j* 1 *k* ⎞ ⎠ ⎞ ⎟⎠ 1 ( *n k*) , ⎛ ⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup>* ⎛ ⎝1 − *k* ∏*j*=1 (1 − *ϑ*∗ *βi j* ) 1 *k* ⎞ ⎠ ⎞ ⎟⎠ 1 ( *n k*) ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Since *ζαi* ≤ *ζβi* which implies that

$$\frac{\sum\_{\substack{1 \le i\_1 < \dots < i\_k \\ \dots < i\_k \le n}} \left( \prod\_{j=1}^k \mathbb{J}\_{\mathcal{A}\_{i\_j}} \right)^{\frac{1}{k}}}{\binom{n}{k}} \le \frac{\sum\_{\substack{1 \le i\_1 < \dots < i\_k \\ \dots < i\_k \le n}} \left( \prod\_{j=1}^k \mathbb{J}\_{\mathcal{B}\_{i\_j}} \right)^{\frac{1}{k}}}{\binom{n}{k}}$$

Also, *ϕαi* ≥ *ϕβi* implies that

*k* ∏ *j*=1 (1 − *ϕαij*) ≤ *k* ∏ *j*=1 (1 − *ϕβij*) ⇒ *k*∏*<sup>j</sup>*=<sup>1</sup>(<sup>1</sup> − *ϕαij*) 1 *k* ≤ *k*∏*<sup>j</sup>*=<sup>1</sup>(<sup>1</sup> − *ϕβij*) 1 *k* ⇒ ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*<sup>j</sup>*=<sup>1</sup>(<sup>1</sup> − *ϕαij*)1*k* ⎞⎠⎞⎟⎠ 1 (*nk*) ≥ ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*<sup>j</sup>*=<sup>1</sup>(<sup>1</sup> − *ϕβij*)1*k* ⎞⎠⎞⎟⎠ 1 (*nk*)

Similarly for *<sup>ϕ</sup>*<sup>∗</sup>*αi* ≤ *ϕ*∗*βi* , *ϑαi* ≤ *ϑαi* and *ϑ*∗*αi* ≥ *ϑ*∗*βi* for all *i*, we have

$$\begin{split} 1 - \left(\prod\_{\begin{subarray}{c}1\leq i\_{1}<\cdot\\ \dots$$

and

$$1 - \left(\prod\_{\substack{1 \le i\_1 < \\ \dots < i\_k \le n}} \left(1 - \left(\prod\_{j=1}^k \boldsymbol{\varphi}\_{a\_{i\_j}}^\*\right)^{\frac{1}{k}}\right)\right)^{\frac{1}{\binom{n}{k}}} \le 1 - \left(\prod\_{\substack{1 \le i\_1 < \\ \dots < i\_k \le n}} \left(1 - \left(\prod\_{j=1}^k \boldsymbol{\varphi}\_{\beta\_{i\_j}}^\*\right)^{\frac{1}{k}}\right)\right)^{\frac{1}{\binom{n}{k}}}$$

.

Therefore, by using these inequalities and Definition 11, we ge<sup>t</sup>

$$\text{STIT2IFHM}^{(k)}(\mathfrak{a}\_1, \mathfrak{a}\_2, \dots, \mathfrak{a}\_n) \le \text{STIT2IFHM}^{(k)}(\mathfrak{\beta}\_1, \mathfrak{\beta}\_2, \dots, \mathfrak{\beta}\_n)$$

**Theorem 6.** *(Boundedness) For n STIT2IFNs αi, α*<sup>−</sup> = 5min*i* {*ζi*}, max*i* {*i*}, min*i* {*ϕi*}, max*i* {*ϕ*<sup>∗</sup>*i* }, max *i* {*<sup>ϑ</sup>i*}, min*i* {*ϑ*<sup>∗</sup>*i* }6*, and α*<sup>+</sup> = 5max*i* {*ζi*}, min*i* {*i*}, max*i* {*ϕi*}, min*i* {*ϕ*<sup>∗</sup>*i* }, min*i* {*<sup>ϑ</sup>i*}, max*i* {*ϑ*<sup>∗</sup>*i* }6*, we have α* − ≤ *STIT2IFHM*(*k*)(*<sup>α</sup>*1, *α*2,..., *<sup>α</sup>n*) ≤ *α*<sup>+</sup> (14)

**Proof.** Clearly, we ge<sup>t</sup> *α*<sup>−</sup> ≤ *αi* ≤ *<sup>α</sup>*+. Thus, based on Theorems 4 and 5, we have

$$\begin{aligned} \text{STIT2IFHM}^{(k)}(\mathfrak{a}\_1, \mathfrak{a}\_2, \dots, \mathfrak{a}\_n) &\geq \text{STIT2IFHM}^{(k)}(\mathfrak{a}^-, \mathfrak{a}^-, \dots, \mathfrak{a}^-) = \mathfrak{a}^- \\ \text{STIT2IFHM}^{(k)}(\mathfrak{a}\_1, \mathfrak{a}\_2, \dots, \mathfrak{a}\_n) &\leq \text{STIT2IFHM}^{(k)}(\mathfrak{a}^+, \mathfrak{a}^+, \dots, \mathfrak{a}^+) = \mathfrak{a}^+ \end{aligned}$$

**Lemma 1** ([51])**.** *For n non-negative real numbers xi, we have*

> *HM*(1)(*<sup>x</sup>*1, *x*2,..., *xn*) ≥ *HM*(2)(*<sup>x</sup>*1, *x*2,..., *xn*) ≥ ... ≥ *HM*(*n*)(*<sup>x</sup>*1, *x*2,..., *xn*) (15)

*with equality holding iff x*1 = *x*2 = ... = *xn.*

**Lemma 2** ([54])**.** *Let xi*, *yi* > 0 *and n* ∑ *i*=1 *yi* = 1*. Then*

$$\prod\_{i=1}^{n} x\_i^{y\_i} \le \sum\_{i=1}^{n} x\_i y\_i \tag{16}$$

**Theorem 7.** *For given STIT2IFNs αi, the operator STIT2IFHM is monotonically decreasing with parameter k.*

**Proof.** For STIT2IFNs *αi* and *k* = 1, 2, . . . , *n*, we denote

*C*(*k*) = ∑ <sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup> k*∏*j*=1 *ζαij* 1 *k* (*nk*) , Δ(*k*) = ∑ <sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup> k*∏*j*=1 *αij* 1 *k* (*nk*) , *T*(*k*) = ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*<sup>j</sup>*=<sup>1</sup>(<sup>1</sup> − *ϕαij*)1*k* ⎞⎠⎞⎟⎠ 1 (*nk*) , *S*(*k*) = 1 − ⎛⎝∏ <sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*j*=1 *ϕ*∗*αij* 1*k* ⎞⎠⎞⎠ 1 (*nk*) , *T*∗(*k*) = 1 − ⎛⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* 51 − ∏*kj*=<sup>1</sup> *ϑαij* 1*k* 6⎞⎟⎠ 1 (*nk*) , *S*∗(*k*) = ∏ <sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* 1 − 5∏*kj*=<sup>1</sup>(<sup>1</sup> − *<sup>ϑ</sup>*<sup>∗</sup>*<sup>α</sup>ij*)6<sup>1</sup>*<sup>k</sup>* 1 (*nk*)

Based on Theorem 2, we have

STIT2IFHM(*k*)(*<sup>α</sup>*1, *α*2,..., *<sup>α</sup>n*) = (*C*(*k*), <sup>Δ</sup>(*k*), *<sup>T</sup>*(*k*), *<sup>S</sup>*(*k*), *<sup>T</sup>*<sup>∗</sup>(*k*), *S*<sup>∗</sup>(*k*)) and STIT2IFHM(*k*+<sup>1</sup>)(*<sup>α</sup>*1, *α*2,..., *<sup>α</sup>n*) = (*C*(*k* + <sup>1</sup>), Δ(*k* + <sup>1</sup>), *T*(*k* + <sup>1</sup>), *S*(*k* + <sup>1</sup>), *T*<sup>∗</sup>(*k* + <sup>1</sup>), *S*<sup>∗</sup>(*k* + 1))

Following Definition 10 and Lemma 1, we obtained

$$\begin{array}{c} \mathop{\rm s\_{\tiny{x}}}(\text{STIT2IFHM}^{(k)}(\mathfrak{a}\_{1}, \mathfrak{a}\_{2}, \dots, \mathfrak{a}\_{n})) & \mathop{\rm s\_{\tiny{x}}}\limits\_{\begin{subarray}{c} 1 \leq i\_{1} < \cdots \\ i\_{k} \leq n \end{subarray}} \left(\prod\_{j=1}^{k} \mathfrak{f}\_{a} a\_{i\_{j}}\right)^{\frac{1}{k}} \\ & \mathop{\rm s\_{\tiny{x}}}\limits\_{1 \leq i\_{1} < \cdots < i\_{k} \leq n} \left(\prod\_{j=1}^{k+1} \mathfrak{f}\_{a\_{i\_{j}}}\right)^{\frac{1}{k+1}} \\ & \geq \underbrace{\begin{subarray}{c} \sum\_{i\_{1} < \cdots < i\_{k+1} \leq n} \left(\prod\_{j=1}^{k} \mathfrak{f}\_{a\_{i\_{j}}}\right)^{\frac{1}{k+1} \\ (\prod\_{j=1}^{n}) \\ (\prod\_{k+1}^{n}) \end{subarray}} \end{array}$$

Then, two cases are arisen:

Case 1 If *sx* STIT2IFHM(*k*)(*<sup>α</sup>*1, *α*2,..., *<sup>α</sup>n*)> *sx* STIT2IFHM(*k*+<sup>1</sup>)(*<sup>α</sup>*1, *α*2,..., *<sup>α</sup>n*), following the Definition 11 we ge<sup>t</sup>

$$\text{STIT2IFHM}^{(k)}(\mathfrak{a}\_1, \mathfrak{a}\_2, \dots, \mathfrak{a}\_n) \text{ > STIT2IFHM}^{(k+1)}(\mathfrak{a}\_1, \mathfrak{a}\_2, \dots, \mathfrak{a}\_n)$$

Case 2 If *sx* STIT2IFHM(*k*)(*<sup>α</sup>*1, *α*2,..., *<sup>α</sup>n*) = *sx* STIT2IFHM(*k*+<sup>1</sup>)(*<sup>α</sup>*1, *α*2,..., *<sup>α</sup>n*). Then, by Lemmas 1 and 2, we ge<sup>t</sup>

$$S(k) = 1 - \left(\prod\_{1 \le i\_1 < \cdots < i\_k \le n} \left(1 - \left(\prod\_{j=1}^k \boldsymbol{\varphi}\_{a\_{i\_j}}^\*\right)^{\frac{1}{k}}\right)\right)^{\frac{1}{\binom{k}{k}}} \ge 1 - \frac{\sum\_{1 \le i\_1 < \cdots < i\_k \le n} \left(1 - \left(\prod\_{j=1}^k \boldsymbol{\varphi}\_{a\_{i\_j}}^\*\right)^{\frac{1}{k}}\right)}{\binom{n}{k}} = \sum\_{\substack{1 \le i\_1 < \cdots < i\_k \le n\\ \dots < i\_k \le n}} \frac{\left(\prod\_{j=1}^k \boldsymbol{\varphi}\_{a\_{i\_j}}^\*\right)^{\frac{1}{k}}}{\binom{n}{k}}$$

To check the monotonic behavior of *<sup>S</sup>*(*k*), we assume that it is increasing with *k*, i.e.,

$$S(n) > S(n-1) > \dots > S(1) \tag{17}$$

Also since

$$S(1) \ge 1 - \sum\_{1 \le i\_1 \le n} \frac{\prod\_{j=1}^1 \left(1 - \varphi\_{a\_{i\_j}}^\* \right)}{\binom{n}{1}} = 1 - \frac{n - \sum\_{i=1}^n \left(\varphi\_{a\_i}^\* \right)}{n} = \frac{\sum\_{i=1}^n \varphi\_{a\_i}^\*}{n} \tag{18}$$

which implies that

$$S(n) > S(1) = \frac{\sum\_{i=1}^{n} \varphi\_{\alpha\_i}^\*}{n}$$

$$\Rightarrow \quad \left(\prod\_{i=1}^{n} \varphi\_{\alpha\_i}^\*\right)^{\frac{1}{n}} > \frac{\sum\_{i=1}^{n} \varphi\_{\alpha\_i}^\*}{n}$$

which contradict the Lemma 2. Hence with parameter *k*, *S*(*k*) is monotonically decreasing. Similarly, we can ge<sup>t</sup> *T*∗(*k*) is also monotonically decreasing with parameter *k*. Also, the functions *T*(*k*) and *S*∗(*k*) are monotonically increasing with parameter *k*.

Therefore,

$$\begin{aligned} s\_{\mathcal{Y}}\left(\text{STIT2IFHM}^{(k)}(a\_1, a\_2, \dots, a\_n)\right) &=& \frac{S(k) + T^\*(k)}{2} - \frac{T(k) + S^\*(k)}{2} \\ &>& \frac{S(k+1) + T^\*(k+1)}{2} - \frac{T(k+1) + S^\*(k+1)}{2} \\ &=& s\_{\mathcal{Y}}\left(\text{STIT2IFHM}^{(k+1)}(a\_1, a\_2, \dots, a\_n)\right) \end{aligned}$$

Thus, by both the cases, we ge<sup>t</sup> STIT2IFHM(*k*)(*<sup>α</sup>*1, *α*2, ... , *<sup>α</sup>n*) ≥ STIT2IFHM(*k*+<sup>1</sup>)(*<sup>α</sup>*1, *α*2,..., *<sup>α</sup>n*).

Furthermore, we will talk about a few special cases of the STIT2IFHM operator concerning the parameter the *k*.

1. When *k* = 1, Equation (12) reduces to the triangular IT2IF averaging operator.

STIT2IFHM(1) (*<sup>α</sup>*1, *α*2, ··· , *<sup>α</sup>m*) = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∑ 1≤*i*1≤*<sup>n</sup>* 1 ∏ *j*=1 *ζαi j* 1 1 ( *n* 1) , ∑ 1≤*i*1≤*<sup>n</sup>* 1 ∏ *j*=1 *αi j* 1 1 ( *m* 1 ) , ⎛ ⎜⎝ ∏1≤*i*1≤*<sup>n</sup>* ⎛ ⎜⎝1 − ⎛ ⎝ 1 ∏*j*=1 1 − *ϕαi j* ⎞ ⎠ 1 1 ⎞ ⎟⎠ ⎞ ⎟⎠ 1 ( *n* 1) , 1 − ⎛ ⎜⎝ ∏1≤*i*1≤*<sup>n</sup>* ⎛ ⎜⎝1 − ⎛ ⎝ 1 ∏*j*=1 *ϕ*∗ *αi j* ⎞ ⎠ 1 1 ⎞ ⎟⎠ ⎞ ⎟⎠ 1 ( *n* 1) , 1 − ⎛ ⎜⎝ ∏1≤*i*1≤*<sup>n</sup>* ⎛ ⎜⎝1 − ⎛ ⎝ 1 ∏*j*=1 *ϑαi j* ⎞ ⎠ 1 1 ⎞ ⎟⎠ ⎞ ⎟⎠ 1 ( *n* 1) , ⎛ ⎜⎝ ∏1≤*i*1≤*<sup>n</sup>* ⎛ ⎜⎝1 − ⎛ ⎝ 1 ∏*j*=1 1 − *ϑ*<sup>∗</sup> *αi j* ⎞ ⎠ 1 1 ⎞ ⎟⎠ ⎞ ⎟⎠ 1 ( *n* 1) ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ *n* ∑ *i*=1 *ζαi n* , *n* ∑ *i*=1 *αi n* , *n* ∏*i*=1 1 − 1 − *ϕαi j* 1 *n* , 1 − *n* ∏*i*=1 1 − *ϕ*∗ *αi j* 1 *n* , 1 − *n* ∏*i*=1 1 − *ϑαi j* 1 *n* , *n* ∏*i*=1 1 − 1 − *ϑ*<sup>∗</sup> *αi j* 1 *n* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ = ⎛ ⎜⎝ *r* ∑ *i*=1 *ζαi n* , *n* ∑ *i*=1 *αi n* , *n* ∏*i*=1 *ϕαi j* 1 *n* , 1 − *n* ∏*i*=1 1 − *ϕ*∗ *αi j* 1 *n* , 1 − *n* ∏*i*=1 1 − *ϑαi j* 1 *n* , *n* ∏*i*=1 *ϑ*<sup>∗</sup> *αi j* 1 *n* ⎞ ⎟⎠

2. When *k* = *n*, Equation (12) will reduce to triangular IT2IF geometric operator.

STIT2IFHM(*m*) (*<sup>α</sup>*1, *α*2, ··· , *<sup>α</sup>n*) = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∑ 1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup> k* ∏ *j*=1 *ζαi j* 1 *n* ( *n n*) , ∑ 1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup> k* ∏ *j*=1 *γi j* 1 *n* ( *n n*) , ⎛ ⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup>* ⎛ ⎝1 − *k* ∏*j*=1 1 − *ϕγi j* 1 *n* ⎞ ⎠ ⎞ ⎟⎠ 1 ( *n n*) , 1 − ⎛ ⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup>* ⎛ ⎝1 − *k* ∏*j*=1 *ϕ*∗ *<sup>α</sup>i j* 1 *n* ⎞ ⎠ ⎞ ⎟⎠ 1 ( *n n*) , 1 − ⎛ ⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup>* ⎛ ⎝1 − *k* ∏*j*=1 *ϑαi j* 1 *n* ⎞ ⎠ ⎞ ⎟⎠ 1 ( *n n*) , ⎛ ⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup>* 1 − *k* ∏*j*=1 5 1 − *ϑ*∗ *<sup>α</sup>i j* 6 1 *n* 1 ( *n n*) ⎞ ⎟⎠ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = ⎛ ⎜⎜⎜⎜⎝ *k* ∏*j*=1 *ζαi j* 1 *n* , *k* ∏*j*=1 *αi j* 1 *n* , ⎛ ⎝1 − *k* ∏*j*=1 1 − *ϕαi j* 1 *n* ⎞ ⎠ , 1 − ⎛ ⎝1 − *k* ∏*j*=1 *ϕ*∗ *γi j* 1 *n* ⎞ ⎠ , 1 − ⎛ ⎝1 − *k* ∏*j*=1 *ϑαi j* 1 *n* ⎞ ⎠ , ⎛ ⎝1 − *k* ∏*j*=1 5 1 − *ϑ*∗ *<sup>α</sup>i j* 61 *n* ⎞ ⎠ ⎞ ⎟⎟⎟⎟⎠ = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎝ *k* ∏*j*=1 *ζαi j* 1 *n* , *k* ∏*j*=1 *αi j* 1 *n* , ⎛ ⎝1 − *k* ∏*j*=1 1 − *ϕαi j* 1 *n* ⎞ ⎠ , *k* ∏*j*=1 *ϕ*∗ *<sup>α</sup>i j* 1 *n* , *k* ∏*j*=1 *ϑαi j* 1 *n* , 1 − *k* ∏*j*=1 5 1 − *ϑ*∗ *<sup>α</sup>i j* 6 1 *n* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎠
