*4.2. WSTIT2IFHM Operator*

**Definition 14.** *For a collection of n STIT2IFNs, αi, w* = (*<sup>w</sup>*1, *w*2, ··· , *wn*)*<sup>T</sup> is weight vector of αi, where wi* > 0 *and* ∑*ni*=<sup>1</sup> *wi* = 1*, we define WSTIT2IFHM operator as*

$$\text{WSTIT2IFHM}\_{w}^{(k)}(a\_1, a\_2, \cdots, a\_m) = \begin{cases} \bigoplus\_{\substack{1 \le i\_1 < \cdots < i\_k \\ \cdots < i\_k \le n}} \left( 1 - \sum\_{j=1}^k w\_{i\_j} \right) \binom{k}{\bigotimes\_{j=1}^k a\_{i\_j}} \text{,} \\ \frac{\binom{n-1}{k}}{\bigvee\_{j=1}^k \binom{n-1}{k}} & \text{,} 1 \le k < n \\ \bigotimes\_{j=1}^k a\_j^{\frac{1-w\_j}{n-1}} & \text{,} k = n \end{cases} \tag{19}$$

*then WSTIT2IFHM*(*k*) *w is stated as weighted symmetric triangular IT2IF Hamy mean operator.*

**Theorem 8.** *For n STIT2IFNs αi* = -*ζ<sup>i</sup>*, *i*, *ϕi*, *ϕ*∗*i* , *ϑi*, *ϑ*∗*i* . (*i* = 1, 2, ... , *<sup>n</sup>*)*, the value obtained through Equation* (19) *is also STIT2IFN, and is given as*

*WSTIT2IFHM*(*k*) *w* (*<sup>α</sup>*1, *α*2, ··· , *<sup>α</sup>m*) =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∑<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* 1 − *k*∑*j*=1 *wij k*∏*j*=1 *ζαij* 1*k* (*<sup>n</sup>*−<sup>1</sup> *k* ) , ∑<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* 1 − *k*∑*j*=1 *wij k*∏*j*=1 *αij* 1*k* (*<sup>n</sup>*−<sup>1</sup> *k* ) , ⎛⎜⎜⎜⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*j*=1 1 − *ϕαij* 1*k* ⎞⎠1− *k*∑*j*=1 *wij*⎞⎟⎟⎟⎟⎠ 1 (*<sup>n</sup>*−<sup>1</sup> *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− *k*∑*j*=1 *wij*⎞⎟⎟⎟⎟⎠ 1 (*<sup>n</sup>*−<sup>1</sup> *k* ) , 1 − ⎛⎜⎜⎜⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*j*=1 *ϑαij* 1*k* ⎞⎠1− *k*∑*j*=1 *wij*⎞⎟⎟⎟⎟⎠ 1 (*<sup>n</sup>*−<sup>1</sup> *k* ) , ⎛⎜⎜⎜⎜⎝ ∏<sup>1</sup>≤*i*1<sup>&</sup>lt; ...<sup>&</sup>lt;*ik*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*j*=1 51 − *ϑ*∗*αij* 61*k* ⎞⎠1− *k*∑*j*=1 *wij*⎞⎟⎟⎟⎟⎠ 1 (*<sup>n</sup>*−<sup>1</sup> *k* ) ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ; *if* 1 ≤ *k* < *n*

*and*

$$= \begin{pmatrix} \mathsf{WSTT2I}\mathsf{I}\mathsf{F}\mathsf{H}\mathsf{M}\_{w}^{(k)}(\mathsf{a}\_{1}, \mathsf{a}\_{2}, \cdots, \mathsf{a}\_{n})\\ \displaystyle \prod\_{j=1}^{k} \mathsf{T}\_{a\_{j}}^{\frac{1-w\_{j}}{n-1}} \mathsf{I} \prod\_{j=1}^{k} \mathsf{q}\_{a\_{j}^{\frac{1-w\_{j}}{n-1}}}^{\frac{1-w\_{j}}{n-1}} \mathsf{I} - \prod\_{j=1}^{k} \left(1 - \mathsf{q}\_{a\_{j}}\right)^{\frac{1-w\_{j}}{n-1}}\\ \displaystyle \prod\_{j=1}^{k} \left(\mathsf{q}\_{a\_{j}}^{\*}\right)^{\frac{1-w\_{j}}{n-1}} \mathsf{I} \prod\_{j=1}^{k} \left(\mathsf{q}\_{a\_{j}}\right)^{\frac{1-w\_{j}}{n-1}} , 1 - \prod\_{j=1}^{k} \left(1 - \mathsf{q}\_{a\_{j}}^{\*}\right)^{\frac{1-w\_{j}}{n-1}} \end{pmatrix}\_{j=1}^{1-w\_{j}}; \text{if } \begin{array}{l} k=n\\ \end{array}$$

**Proof.** Similar to the proof of Theorem 2.

**Theorem 9.** *The operator STIT2IFHM is a special case of the WSTIT2IFHM operator.*

**Proof.** Assume that *w* = 5 1*n* , 1*n* , ··· , 1*n*6*T*, then by Theorem 5, we have 1.if1≤*k*<*n*,wehave

WSTIT2IFHM(*k*) *w* (*<sup>α</sup>*1, *α*2, ··· , *<sup>α</sup>n*) = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∑ 1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup>* 1 − *k n k* ∏ *j*=1 *ζαi j* 1 *k* ( *<sup>n</sup>*−1 *k* ) , ∑ 1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup>* 1 − *k m k* ∏ *j*=1 *αi j* 1 *k* ( *<sup>n</sup>*−1 *k* ) , ⎛ ⎜⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup>* ⎛ ⎝1 − *k* ∏*j*=1 1 − *ϕαi j* 1 *k* ⎞ ⎠ (<sup>1</sup>− *k m* ) ⎞ ⎟⎟⎠ 1 ( *<sup>n</sup>*−1 *k* ) , 1 − ⎛ ⎜⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup>* ⎛ ⎝1 − *k* ∏*j*=1 *ϕ*∗ *<sup>α</sup>i j* 1 *k* ⎞ ⎠ (<sup>1</sup>− *k m* ) ⎞ ⎟⎟⎠ 1 ( *<sup>n</sup>*−1 *k* ) , 1 − ⎛ ⎜⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ...<*i k*≤*<sup>n</sup>* ⎛ ⎝1 − *k* ∏*j*=1 *ϑαi j* 1 *k* ⎞ ⎠ (<sup>1</sup>− *k m* ) ⎞ ⎟⎟⎠ 1 ( *<sup>n</sup>*−1 *k* ) , ⎛ ⎜⎜⎝ ∏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 *k* ⎞ ⎠ (<sup>1</sup>− *k m* ) ⎞ ⎟⎟⎠ 1 ( *<sup>n</sup>*−1 *k* ) ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∑ 1≤*i* 1<sup>&</sup>lt; ···<*i k*≤*<sup>n</sup>* 1 − *k n k* ∏ *j*=1 *ζαi j* 1 *k* ( *n k*) *<sup>n</sup>*−*k n* , ∑ 1≤*i* 1<sup>&</sup>lt; ···<*i k*≤*<sup>n</sup>* 1 − *k n k* ∏ *j*=1 *αi j* 1 *k* ( *n k*) *<sup>n</sup>*−*k n* , ⎛ ⎜⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ···<*i k*≤*<sup>n</sup>* ⎛ ⎝1 − *k* ∏*j*=1 1 − *ϕαi j* 1 *k* ⎞ ⎠ (<sup>1</sup>− *k n* ) ⎞ ⎟⎟⎠ 1 ( *n k*) *<sup>n</sup>*−*k n* , 1 − ⎛ ⎜⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ···<*i k*≤*<sup>n</sup>* ⎛ ⎝1 − *k* ∏*j*=1 *ϕ*∗ *<sup>α</sup>i j* 1 *k* ⎞ ⎠ (<sup>1</sup>− *k n* ) ⎞ ⎟⎟⎠ 1 ( *n k*) *<sup>n</sup>*−*k n* , 1 − ⎛ ⎜⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ···<*i k*≤*<sup>n</sup>* ⎛ ⎝1 − *k* ∏*j*=1 *ϑαi j* 1 *k* ⎞ ⎠ (<sup>1</sup>− *k n* ) ⎞ ⎟⎟⎠ 1 ( *n k*) *<sup>n</sup>*−*k 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 *k* ⎞ ⎠ 1− *k n* ⎞ ⎟⎟⎠ 1 ( *n k*) *<sup>n</sup>*−*k n* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∑ 1≤*i* 1<sup>&</sup>lt; ···<*i k*≤*<sup>n</sup> k* ∏ *j*=1 *ζαi j* 1 *k* ( *n k*) , ∑ 1≤*i* 1<sup>&</sup>lt; ···<*i k*≤*<sup>n</sup> k* ∏ *j*=1 *αi j* 1 *k* ( *n k*) , ⎛ ⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ···<*i k*≤*<sup>n</sup>* ⎛ ⎝1 − *k* ∏*j*=1 1 − *ϕαi j* 1 *k* ⎞ ⎠ ⎞ ⎟⎠ 1 ( *n k*) , 1 − ⎛ ⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ···<*i k*≤*<sup>n</sup>* ⎛ ⎝1 − *k* ∏*j*=1 *ϕ*∗ *<sup>α</sup>i j* 1 *k* ⎞ ⎠ ⎞ ⎟⎠ 1 ( *n k*) , 1 − ⎛ ⎜⎝ ∏1≤*i* 1<sup>&</sup>lt; ···<*i k*≤*<sup>n</sup>* ⎛ ⎝1 − *k* ∏*j*=1 *ϑαi j* 1 *k* ⎞ ⎠ ⎞ ⎟⎠ 1 ( *n k*) , ⎛ ⎜⎝ ∏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 *k* ⎞ ⎠ ⎞ ⎟⎠ 1 ( *n k*) ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ =STIT2IFHM(*k*) (*<sup>α</sup>*1,*α*2,···*<sup>α</sup>n*)

 2. If *k* = *n*, we have  ,

$$\begin{split} & \quad \text{WSTIT2IFHM}\_{\mathcal{X}}^{(k)}(\mathfrak{a}\_{1},\mathfrak{a}\_{2},\cdots,\mathfrak{a}\_{n}) \\ &= \quad \left( \prod\_{j=1}^{k} \zeta\_{\mathfrak{a}\_{j}^{-1}}^{\frac{1}{k-1}}, \prod\_{j=1}^{k} \varrho\_{\mathfrak{a}\_{j}^{-1}}^{\frac{1}{k-1}}, 1 - \prod\_{j=1}^{k} \left(1 - \varrho\_{\mathfrak{a}\_{j}}\right)^{\frac{1-\frac{1}{k}}{k-1}} \right) \\ & \quad \prod\_{j=1}^{k} \left(\varrho\_{\mathfrak{a}\_{j}}^{\*}\right)^{\frac{1-\frac{1}{k}}{k-1}}, \prod\_{j=1}^{k} \left(\mathfrak{a}\_{j}\right)^{\frac{1-\frac{1}{k}}{k-1}}, 1 - \prod\_{j=1}^{k} \left(1 - \mathfrak{a}\_{\mathfrak{a}\_{j}}^{\*}\right)^{\frac{1-\frac{1}{k}}{k-1}} \right) \\ &= \quad \left( \prod\_{j=1}^{k} \zeta\_{\mathfrak{a}\_{j}^{-1}}^{\frac{1}{k}}, \prod\_{j=1}^{k} \varrho\_{\mathfrak{a}\_{j}}^{\frac{1}{k}}, 1 - \prod\_{j=1}^{k} (1 - \varrho\_{\mathfrak{a}\_{j}})^{\frac{1}{k}}\right) \\ &= \quad \quad \text{STT2IFHM}^{(k)}(\mathfrak{a}\_{1}, \mathfrak{a}\_{2}, \cdots, \mathfrak{a}\_{n}) \\ &= \quad \quad \quad \text{STT2IFHM}^{(k)}(\mathfrak{a}\_{1}, \mathfrak{a}\_{2}, \cdots, \mathfrak{a}\_{n}) \end{split}$$

#### **5. An Approach to MCDM Based on the Proposed WSTIT2IFHM Operator**

In this section, an MCDM approach is developed under the triangular IT2IF (TIT2IF) environment. The description of the problem, as well as the procedure steps, are explained as below.

Assume an MCDM problem which consists of '*n*' different alternatives *A*1, *A*2, ... , *An* and a set of ' *m*' attributes *C*1, *C*2, ... , *Cm* whose weight vector is *w* = ( *w*1, *w*2, ··· , *wm*)*<sup>T</sup>*, satisfying *wj* > 0 and ∑*m j*=1 *wj* = 1. An expert has evaluated these given alternatives and rate them under TIT2IF environment denoted by *lpj*(*p* = 1, 2, ... , *n*; *j* = 1, 2, ... , *m*) where *lpj* represent the linguistic information about the alternatives. Furthermore, the importance of the attributes plays a dominant role during the decision-making process. During handling the MCDM problems, if the sum of the relative coefficient w.r.t. each criterion is small, it relates that such criteria demonstrate a major impact on the overall values of the alternative. Similarly, if the relative coefficient sum is large then it shows such criterion play a less significant role. Hence, the relative coefficient of the alternative under the certain criteria is inversely proportional to the corresponding weights of criteria. Therefore, the weight of the criteria is determined by using the Spearman method [56] which main steps are summarized in Algorithm 1.

**Algorithm 1** Weight determination using Spearman coefficient method.

1: Take two criteria *Ck* and *Cj* and then compute their relative coefficients as

$$\Delta\_{kj} = 1 - \frac{6\sum\_{p=1}^{n} (l\_{pk} - l\_{pj})^2}{m(m-1)}\tag{20}$$

and hence construct the matrix Δ*m*×*m* = ( <sup>Δ</sup>*kj*)*m*×*m* as

$$
\Delta\_{m \times m} = \begin{pmatrix}
\Delta\_{11} & \Delta\_{12} & \cdots & \Delta\_{1m} \\
\Delta\_{21} & \Delta\_{22} & \cdots & \Delta\_{2m} \\
\cdots & \cdots & \ddots & \cdots \\
\Delta\_{m1} & \Delta\_{m2} & \cdots & \Delta\_{mm}
\end{pmatrix} \tag{21}
$$

2: Compute the relative coefficient sum of each criteria by using Equation (22).

$$
\Delta\_{\bar{j}} = \sum\_{k=1 \atop k \neq j}^{m} \Delta\_{jk} \tag{22}
$$

3: Compute the weight of each criteria as

$$w\_j = \frac{\sigma\_j}{\sum\_{j=1}^{m} \sigma\_j} \tag{23}$$

where *σj* = 1 Δ*j* represent the contribution index of the criteria.

By using this weight vector, we summarized the following steps based on the proposed AO to rank the alternatives under TIT2IFS environment.

Step 1: Arrange the information of each alternative in decision matrix *L* as

$$\mathbf{T} = \begin{array}{c} \mathbf{C}\_1 & \mathbf{C}\_2 & \dots & \mathbf{C}\_n \\ A\_1 & \begin{pmatrix} \overline{l}\_{11} & \overline{l}\_{12} & \dots & \overline{l}\_{1n} \\ \overline{l}\_{21} & \overline{l}\_{22} & \dots & \overline{l}\_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ \overline{l}\_{m1} & \overline{l}\_{m2} & \dots & \overline{l}\_{mn} \end{pmatrix} \\ \tag{24}$$

where *lpj* = *ζ pj*, *pj*, *ϕpj*, *<sup>ϕ</sup>*<sup>∗</sup>*pj*, *<sup>ϑ</sup>pj*, *ϑ*∗*pj*be the STIT2IFNs provided by an expert. Step 2: Compute the normalized decision matrix *L* from *L* by using the normalized formula

$$\mathcal{I}\_{pj} = \begin{cases} \left< \overline{\mathbb{Q}}\_{pj}, \overline{\mathbb{Q}}\_{pj}, \overline{\mathbb{P}}\_{pj}, \overline{\mathbb{P}}\_{pj}^\*, \overline{\mathbb{P}}\_{pj}, \overline{\mathbb{P}}\_{pj}^\* \right> & \text{; for the benefit type criteria} \\ \left< \overline{\mathbb{Q}}\_{pj}, \overline{\mathbb{Q}}\_{pj}, \overline{\mathbb{P}}\_{pj}, \overline{\mathbb{Q}}\_{pj}^\*, \overline{\mathbb{Q}}\_{pj}, \overline{\mathbb{P}}\_{pj}^\* \right> & \text{; for the cost type criteria} \end{cases} \tag{25}$$

Step 3: Compute the weight vector to each criteria by using Algorithm 1.

Step 4: Combine the different values of STIT2IFNs *lpj*(*j* = 1, 2, ... , *m*) into the single one *lp* of each alternative *Ap*(*p* = 1, 2, . . . , *n*) by using WSTIT2IFHM operator as follows:

*lp* = WSTIT2IFHM(*k*) *w* (*lp*1, *lp*2, ··· , *lpn*) = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∑<sup>1</sup>≤*p*1<sup>&</sup>lt; ...<sup>&</sup>lt;*pk*≤*<sup>n</sup>* 1 − *k*∑*j*=1 *wpj k*∏*j*=1 *ζ pj* 1*k* (*<sup>n</sup>*−<sup>1</sup> *k* ) , ∑<sup>1</sup>≤*p*1<sup>&</sup>lt; ...<sup>&</sup>lt;*pk*≤*<sup>n</sup>* 1 − *k*∑*j*=1 *wpj k*∏*j*=1 *pj* 1*k* (*<sup>n</sup>*−<sup>1</sup> *k* ) , ⎛⎜⎜⎜⎝ ∏<sup>1</sup>≤*p*1<sup>&</sup>lt; ...<sup>&</sup>lt;*pk*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*j*=1 1 − *ϕpj* 1*k* ⎞⎠1− *k*∑*j*=1 *wpj* ⎞⎟⎟⎟⎠ 1 (*<sup>n</sup>*−<sup>1</sup> *k* ) , 1 − ⎛⎜⎜⎜⎝ ∏<sup>1</sup>≤*p*1<sup>&</sup>lt; ...<sup>&</sup>lt;*pk*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*j*=1 *ϕ*∗*pj* 1*k* ⎞⎠1− *k*∑*j*=1 *wpj* ⎞⎟⎟⎟⎠ 1 (*<sup>n</sup>*−<sup>1</sup> *k* ) , 1 − ⎛⎜⎜⎜⎝ ∏<sup>1</sup>≤*p*1<sup>&</sup>lt; ...<sup>&</sup>lt;*pk*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*j*=1 *ϑpj* 1*k* ⎞⎠1− *k*∑*j*=1 *wpj* ⎞⎟⎟⎟⎠ 1 (*<sup>n</sup>*−<sup>1</sup> *k* ) , ⎛⎜⎜⎜⎝ ∏<sup>1</sup>≤*p*1<sup>&</sup>lt; ...<sup>&</sup>lt;*pk*≤*<sup>n</sup>* ⎛⎝1 − *k*∏*j*=1 1 − *ϑ*∗*pj* 1*k* ⎞⎠1− *k*∑*j*=1 *wpj* ⎞⎟⎟⎟⎠ 1 (*<sup>n</sup>*−<sup>1</sup> *k* ) ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Step 5: Compute the score value of the *lp* by using Equation (10).

Step 6: Rank all the alternatives by using an order relation defined in Definition 11 and hence select the most feasible alternative(s).

## **6. Illustrative Example**

The above mentioned approach has been illustrate with a numerical example which is stated as below.

#### *6.1. A Case Study*

Jharkhand is the eastern state of the India, which has the 40 percent mineral resources of the country and second leading state of the mineral wealth after Chhattisgarh state. It is also known for its vast forest resources. Jamshedpur, Bokaro and Dhanbad cities of the Jharkhand are famous for industries in all over the world. After that, it is the widespread poverty state of the India because it is the primarily a rural state as 76 percent of the population live in the villages which depend on the agriculture and wages. Only 30 percent villages are connected by roads while only 55 percent villages have accessed to electricity and other facilities. But in the today's life, everyone is changing fast to himself for a better life, therefore, everyone moves to the urban cities for a better job. To stop this emigration, Jharkhand governmen<sup>t</sup> wants to set up the industries based on the agriculture in the rural areas. For this, the governmen<sup>t</sup> has been organized "MOMENTUM JHARKHAND" global investor submit 2017 in Ranchi to invite the companies for investment in the rural areas. Government announced the various facilities for setup the five food processing plants in the rural areas and consider the six attributes required for company selection to setup them, namely, project cost (*<sup>G</sup>*1), completion time (*<sup>G</sup>*2), technical capability (*<sup>G</sup>*3), financial status (*<sup>G</sup>*4), company background (*<sup>G</sup>*5), reference from previous project (*<sup>G</sup>*6) and assign the weights of relative importance of each attributes. The six companies taken as in the form of the alternatives, namely, Surya Food and Agro Pvt. Ltd. (*<sup>A</sup>*1), Mother Dairy Fruit and Vegetable Pvt. Ltd. (*<sup>A</sup>*2), Parle Products Ltd. (*<sup>A</sup>*3), Heritage Food Ltd. (*<sup>A</sup>*4), Verka Pvt. Ltd. (*<sup>A</sup>*5) and Reliance Pvt. Ltd. (*<sup>A</sup>*6) interested for these projects. Then the main object of the governmen<sup>t</sup> is to choose the best company among them for the task. In order to find the best feasible alternative(s) for the required task, the authority called an expert to evaluate these alternatives and rate their preferences in terms of linguistic terms (LTs). The standardized LTs such as "Very High" (VH), "High"(H), "Medium"(M), "Medium Low"(ML), "Low"(L), "Very Low"(VL) are defined in terms of STIT2IFNs given in Table 1. Furthermore, the complementary relation corresponding to LTs is presented in Table 2.

**Table 1.** Linguistic grade and coressponding values.


**Table 2.** Linguistic grades and compliments.


The above mentioned steps are executed to locate the best alternative(s).

Step 1: An expert has evaluated each alternative and present their rating values in terms of LTs which are summarized as

$$
\begin{array}{ccccccccc}
 & \mathbf{C\_1} & \mathbf{C\_2} & \mathbf{C\_3} & \mathbf{C\_4} & \mathbf{C\_5} & \mathbf{C\_6} & \mathbf{C\_7} \\
 & A\_1 & \begin{pmatrix} VH & H & M & MH & H & VH & H \\ M & ML & H & VH & H & VH & VH \end{pmatrix} \\
\mathbf{T} = & A\_3 & \begin{pmatrix} H & VH & VH & M & MH & L & VL \\ M & VL & MH & M & MH & L & VL \\ MH & VL & MH & H & VL & MH & L \\ VL & VL & VL & H & M & VL & L \\ M & VL & VL & M & VL & L & H \end{pmatrix} \end{array} \tag{26}
$$

Step 2: As the criteria *C*1 and *C*2 are the cost type, so we normalize their rating values by using Table 2 and Equation (25), we ge<sup>t</sup>

$$L = \begin{array}{c} \mathbb{C}\_1 & \mathbb{C}\_2 & \mathbb{C}\_3 & \mathbb{C}\_4 & \mathbb{C}\_5 & \mathbb{C}\_6 & \mathbb{C}\_7\\ A\_1 & VL & L & M & MH & H & VH\\ M & MH & H & VL & H & VH & VL\\ L & VL & VL & M & MH & L & VL\\ M L & VL & MH & H & VL & MH & H\\ V L & L & VL & H & M & VL & L\\ M H & VH & VH & M & VL & L & H \end{array} \tag{27}$$

	- (a) By using Equation (20), construct the relative coefficient matrix Δ for each criteria as


(b) The relative coefficient sum of each criteria is computed by using Equation (22) and ge<sup>t</sup>

> Δ1 = 5.564, Δ2 = 5.558, Δ3 = 5.554, Δ4 = 5.618, Δ5 = 5.440, Δ6 = 5.612, Δ7 = 5.668.

(c) By using Equation (23), the weight vector of each criteria is obtained as

> *w*1 = 0.1431, *w*2 = 0.1432, *w*3 = 0.1433, *w*4 = 0.1417, *w*5 = 0.1463, *w*6 = 0.1419, *w*7 = 0.1405.

Step 4: Aggregate all the values by using WSTIT2IFHM operator into a collective one *lp*(*p* = 1, 2, . . . , <sup>6</sup>). Here, without loss of generality, we take *k* = 2 and the obtained results are

*l*1= WSTIT2IFHM(2) *w* (*l*11, *l*12, ··· , *l*17) = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∑ 1≤*p*1<*p*2≤7 1 − 2∑*j*=1 *w*1*j* 2∏*j*=1 *ζ*1*j* 12 (62) , ∑ 1≤*p*1<*p*2≤7 1 − 2∑*j*=1 *wpj* 2∏*j*=1 *pj* 12 (62) , ⎛⎜⎜⎜⎝ ∏ 1≤*p*1<*p*2≤7 ⎛⎝1 − 2∏*j*=1 1 − *ϕpj* 12 ⎞⎠1− 2∑*j*=1 *wpj* ⎞⎟⎟⎟⎠ 1(62) , 1 − ⎛⎜⎜⎜⎝ ∏ 1≤*p*1<*p*2≤7 ⎛⎝1 − 2∏*j*=1 *ϕ*∗*pj* 12 ⎞⎠1− 2∑*j*=1 *wpj* ⎞⎟⎟⎟⎠ 1(62) , 1 − ⎛⎜⎜⎜⎝ ∏ 1≤*p*1<*p*2≤7 ⎛⎝1 − 2∏*j*=1 *ϑpj* 12 ⎞⎠1− 2∑*j*=1 *wpj* ⎞⎟⎟⎟⎠ 1(62) , ⎛⎜⎜⎜⎝ ∏ 1≤*p*1<*p*2≤7 ⎛⎝1 − 2∏*j*=1 1 − *ϑ*∗*pj* 12 ⎞⎠1− 2∑*j*=1 *wpj* ⎞⎟⎟⎟⎠ 1(62) ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = (0.5154, 0.2276, 0.7820, 0.8314, 0.1596, 0.1339)

Similarly, we have

> *l*2 = (0.6950, 0.3239, 0.8546, 0.9053, 0.0974, 0.0687); *l*3 = (0.3846, 0.1681, 0.7166, 0.7633, 0.2243, 0.1927); *l*4 = (0.5481, 0.2612, 0.7952, 0.8449, 0.1436, 0.1210); *l*5 = (0.3201, 0.1342, 0.6769, 0.7244, 0.2642, 0.2292); *l*6 = (0.5272, 0.2390, 0.7914, 0.8414, 0.1536, 0.1232)

Step 5: The score values of *lp*(*p* = 1, 2, . . . , 6) are computed by Equation (10) and ge<sup>t</sup>

$$\begin{array}{rcl} \mathrm{s(l\_1)} &=& (0.3404, 0.0375); \mathrm{s(l\_2)} = (0.5550, 0.0396); \mathrm{s(l\_3)} = (0.2046, 0.0392) \\ \mathrm{s(l\_4)} &=& (0.3770, 0.0362); \mathrm{s(l\_5)} = (0.1455, 0.0412); \mathrm{s(l\_6)} = (0.3579, 0.0402) \end{array}$$

Step 6: Since *sx*(*l*2) > *sx*(*l*4) > *sx*(*l*6) > *sx*(*l*1) > *sx*(*l*3) > *sx*(*l*5) and thus by Definition 11, we ge<sup>t</sup> the ranking order of the alternatives as *A*2 *A*4 *A*6 *A*1 *A*3 *A*5. Here "" means "preferred to". Therefore, *A*2 is the best alternative.

#### *6.2. Influence of k on Alternatives*

Keeping in mind the end goal to investigate the impact of the parameter *k* on to the final positioning order of the alternatives, we use an alternate estimation of *k* in our test. Here *n* is 7 in our case, so we shift *k* from 1 to 7 and their outcomes relating to the proposed technique have been outlined in Table 3. From this table, it is seen that with the expansion of the interaction of the multi-input options, the general score estimations of it diminishes which recommend that the proposed operator reflect the risk preferences to the decision makers. This examination will propose the distinctive decisions to the analyst as indicated by his/her decision. For example, in the event that he will cover the risk parameters during the aggregation then they will allocate a little incentive to the parameter *k* with the goal that score esteems increments while, if the analyst is pessimistic in nature towards the choice then the bigger estimation of *k* can be allocated during the procedure.


**Table 3.** Effect of *k* on to ranking of alternatives.

Furthermore, in some other existing Bonferroni mean (BM) and generalized Bonferroni mean (GBM) operators, the information takes only two or three arguments during an aggregation. Also, in BM operator there is need of two additional parameters (*p*, *q*) while the three parameters (*p*, *q*,*<sup>r</sup>*) for GBM from an infinite rational set. Thus, the computational complexity is too high in such cases. On the other hand, in the proposed operator, there is only one parameter *k* from a finite integer set and hence the computational complexity is low and easier to understand. Finally, the several operators such as averaging, BM and geometric for the T2IFNs can be deduced from the proposed ones by setting *k* = 1, *k* = 2 and *k* = *n* respectively. Subsequently, our proposed operator and the strategy are more summed up and adaptable to tackle the decision-making problems.
