**Algorithm 1** *Cont.*

or

W*ij* = q-ROFEPWG(P (1) *ij* ,P (2) *ij* , . . . P (*p*) *ij* ) = √*q* 2 ∏ *p <sup>z</sup>*=<sup>1</sup> <sup>P</sup><sup>ˇ</sup> i˘ *z j* ∑ *n j*=1 i˘ *z j j q* vuut ∏ *p z*=1 (2−((P<sup>ˇ</sup> *q ij*) *z* ) i˘ *z j* ∑ *n j*=1 i˘ *z <sup>j</sup>* +∏ *p z*=1 (((Pˇ *q ij*) *z* ) i˘ *z j* ∑ *n j*=1 i˘ *z j* , *q* vuuuuut ∏ *p z*=1 (1+((q *q ij*) *z* ) i˘ *z j* ∑ *n j*=1 i˘ *z <sup>j</sup>* −∏ *p z*=1 (1−((q *q ij*) *z* ) i˘ *z j* ∑ *n j*=1 i˘ *z j* ∏ *p z*=1 (1+((q *q ij*) *z* ) i˘ *z j* ∑ *n j*=1 i˘ *z <sup>j</sup>* +∏ *p z*=1 (1−((q *q ij*) *z* ) i˘ *z j* ∑ *n j*=1 i˘ *z j* (47)

To aggregate all individual q-ROF decision matrices *Y* (*p*) = (P (*p*) *ij* )*m*×*<sup>n</sup>* into one cumulative assessments matrix of the alternatives *<sup>W</sup>*(*p*) = (W*ij*)*m*×*<sup>n</sup>* **Step 5:**

Calculate the values of i˘ *ij* by the following formula.

$$
\breve{\Delta}\_{ij} = \prod\_{k=1}^{j-1} \breve{\Xi}(\mathcal{W}\_{ik}) \quad (j = 2 \dots \dots n), \tag{48}
$$

$$
\breve{\Delta}\_{i1} = 1
$$

**Step 6:**

Aggregate the q-ROF values W*ij* for each alternative X¨ *<sup>i</sup>* by the q-ROFEPWA (or q-ROFEPWG) operator:

$$\begin{array}{rcl}\mathbb{W}\_{i}^{\boldsymbol{\pi}} = \operatorname{q-ROFEPWA}(\mathcal{P}\_{\mathcal{I}1}, \mathcal{P}\_{\mathcal{I}2}, \dots, \mathcal{P}\_{\mathcal{I}n}) &=& \begin{pmatrix} \frac{\mathsf{\top}\_{i}}{\mathsf{q}} & \frac{\mathsf{\top}\_{i}}{\mathsf{q}\mathsf{q}}\\ \vdots\\ \frac{\mathsf{\top}\_{i-1}}{\mathsf{q}} (1 + (\mathsf{\mathsf{\tilde{}}}{\mathsf{\tilde{}}})^{\mathsf{q}})^{\frac{\mathsf{\tilde{}}\_{i}}{\mathsf{p}} - \mathsf{\tilde{}}\_{i}\mathsf{\tilde{}}} - \Pi^{\mathsf{e}}\_{\mathcal{I}i}(1 - (\mathsf{\tilde{}}\mathsf{\tilde{}})^{\mathsf{q}})^{\frac{\mathsf{\tilde{}}\_{i}}{\mathsf{p}} - \mathsf{\tilde{}}\_{i}\mathsf{\tilde{}}} \end{pmatrix} \\ & & \begin{array}{rcl} \frac{\mathsf{\top}\_{i}}{\mathsf{q}} & \frac{\mathsf{\top}\_{i}}{\mathsf{q}}\\ \frac{\mathsf{\top}\_{i}}{\mathsf{p}} & \frac{\mathsf{\top}\_{i}}{\mathsf{p}} & \frac{\mathsf{\top}\_{i}}{\mathsf{p}}\\ \frac{\mathsf{\top}\_{i}}{\mathsf{p}} & \frac{\mathsf{\top}\_{i}}{\mathsf{p}} & \frac{\mathsf{\top}\_{i}}{\mathsf{p}} \end{pmatrix} \\ \end{array} \end{array} \tag{49}$$

or

W*<sup>i</sup>* = q-ROFPWG(P*i*<sup>1</sup> ,P*i*<sup>2</sup> , . . . P*in*) = √*q* 2 ∏ *n <sup>j</sup>*=<sup>1</sup> <sup>P</sup><sup>ˇ</sup> i˘ *j* ∑ *n j*=1 i˘ *j ij q* vuut ∏ *n j*=1 (2−(P<sup>ˇ</sup> *ij*) *q* ) i˘ *j* ∑ *n j*=1 i˘ *<sup>j</sup>* +∏ *n j*=1 ((Pˇ *ij*) *q* ) i˘ *j* ∑ *n j*=1 i˘ *j* , *q* vuuuuut ∏ *n j*=1 (1+(q*ij*) *q* ) i˘ *j* ∑ *n j*=1 i˘ *<sup>j</sup>* <sup>−</sup><sup>∏</sup> *n j*=1 (1−(q*ij*) *q* ) i˘ *j* ∑ *n j*=1 i˘ *j* ∏ *n j*=1 (1+(q*ij*) *q* ) i˘ *j* ∑ *n j*=1 i˘ *<sup>j</sup>* +∏ *n j*=1 (1−(q*ij*) *q* ) i˘ *j* ∑ *n j*=1 i˘ *j* (50)

**Step 7:**

Evaluate the score of the all cumulative alternative assessments.

### **Step 8:**

Ranked the alternatives by the score function and ultimately choose the most appropriate alternative.

### **5. Illustrative Example**

We provide a numerical illustration to explain the approach suggested below.

Let us assume an inviting bid process whereby the investor is trying to find out the optimal biding scheme. In order to catch up with the advancement of modern manufacturing industries and to enhance the city's ecosystem equality, steel and iron works are planning to build a palletizing plant in its primary iron ore production area with a production capacity of more than 1.45 million tons per year. The builder will request bidding for the construction project, taking into account the project regulations, and will choose from five bidders as per six attributes as follows:

**Example 3.** *Consider a set of alternatives* <sup>X</sup>¨ <sup>=</sup> {X¨ <sup>1</sup>, X¨ <sup>2</sup>, X¨ <sup>3</sup>, X¨ <sup>4</sup>, X¨ <sup>5</sup>} *and* |¯ = {|¯ <sup>1</sup> , |¯ <sup>2</sup> , |¯ <sup>3</sup> , |¯ <sup>4</sup> , |¯ <sup>5</sup> , |¯ <sup>6</sup>} *is the finite set of criterions given in Table 2. Prioritization is given between the criteria presented by the linear order* |¯ <sup>1</sup> |¯ <sup>2</sup> |¯ <sup>3</sup> . . . ¯|<sup>6</sup> *indicates criteria* |¯ *<sup>J</sup> has a higher priority than* |¯*<sup>i</sup> if j* > *i.* K = {K1,K2,K3} *is the group of decision makers and decision makers (DMs) do not have the equal importance. Prioritization given between the DMs presented by the linear order* K<sup>1</sup> K<sup>2</sup> K<sup>3</sup> *indicates DM* K*<sup>ζ</sup> has a higher priority than* K*\$ if ζ* > *\$. Decision makers provide a matrix of their own opinion D*(*p*) = (B (*p*) *ij* )*m*×*n, where* B (*p*) *ij is given for the alternatives* X¨ *<sup>i</sup>* <sup>∈</sup> <sup>X</sup>¨ *with respect to the criteria* <sup>|</sup>¯*<sup>j</sup>* <sup>∈</sup> <sup>|</sup>¯ *by* <sup>K</sup>*<sup>p</sup> decision maker in the form of q-ROFNs. We take q* <sup>=</sup> <sup>3</sup>*.*

**Table 2.** Criterions for evaluating the best alternative.


*Step 1:*

*Acquire a decision/assessment matrix D*(*p*) = (B (*p*) *ij* )*m*×*<sup>n</sup> in the form of q-ROFNs from the decision makers. Assessment matrix acquired from* K<sup>1</sup> *is given in Table 3.*



*Assessment matrix acquired from* K<sup>2</sup> *is given in Table 4.*

**Table 4.** Assessment matrix acquired from K2.


*Assessment matrix acquired from* K<sup>3</sup> *is given in Table 5.*


**Table 5.** Assessment matrix acquired from K3.

### *Step 2:*

*Normalize the decision matrixes acquired by DMs using Equation (44). In Table 2, there are two types of criterions.* |¯ <sup>2</sup> *is cost type criteria and others are benefit type criterions. Normalized assessment matrix acquired from* K<sup>1</sup> *is given in Table 6.*

**Table 6.** Normalized assessment matrix acquired from K<sup>1</sup> .


*Normalized assessment matrix acquired from* K<sup>2</sup> *is given in Table 7.*

**Table 7.** Normalized assessment matrix acquired from K2.


*Normalized assessment matrix acquired from* K<sup>3</sup> *is given in Table 8.*

**Table 8.** Normalized assessment matrix acquired from K3.



$$
\mathfrak{D}\_{ij}^{(1)} = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{pmatrix}
$$

$$
\begin{aligned}
\cline{^2}
\spadesuit{^2} &= \begin{pmatrix}
0.8645 & 0.3841 & 0.7093 & 0.9270 & 0.7109 & 0.5378 \\
0.9209 & 0.2575 & 0.5754 & 0.7093 & 0.5000 & 0.5198 \\
0.8054 & 0.5618 & 0.7031 & 0.5832 & 0.6159 & 0.5456 \\
0.6895 & 0.2421 & 0.6356 & 0.5136 & 0.7031 & 0.3105 \\
0.7482 & 0.3920 & 0.5061 & 0.3644 & 0.6356 & 0.3705
\end{pmatrix} \\
\spadesuit{^2}
\dot{\spadesuit{^3}} &= \begin{pmatrix}
0.6078 & 0.1653 & 0.5629 & 0.8593 & 0.5319 & 0.4640 \\
0.5355 & 0.1065 & 0.3062 & 0.4891 & 0.3119 & 0.3695 \\
0.6092 & 0.2060 & 0.2985 & 0.3232 & 0.3042 & 0.3590 \\
0.3878 & 0.1087 & 0.3186 & 0.2334 & 0.3507 & 0.1821 \\
0.5878 & 0.1341 & 0.2804 & 0.1623 & 0.3489 & 0.3522
\end{pmatrix}
\end{aligned}
$$

*Step 4:*

*Use q-ROFEPWA to aggregate all individual q-ROF decision matrices Y* (*p*) = (P (*p*) *ij* )*m*×*<sup>n</sup> into one cumulative assessments matrix of the alternatives W*(*p*) = (W*ij*)*m*×*<sup>n</sup> using Equation (46) given in Table 9.*

**Table 9.** Collective q-ROF assessment matrix.


### *Step 5:*

*Evaluate the values of* i˘ *ij by using Equation (48).*

$$
\mathfrak{A}\_{ij} = \begin{pmatrix}
1 & 0.8205 & 0.3152 & 0.2378 & 0.2073 & 0.1527 \\
1 & 0.8125 & 0.2529 & 0.1444 & 0.0978 & 0.0544 \\
1 & 0.8011 & 0.4007 & 0.2469 & 0.1473 & 0.0874 \\
1 & 0.6301 & 0.1988 & 0.1149 & 0.0597 & 0.0361 \\
1 & 0.7365 & 0.2788 & 0.1550 & 0.0669 & 0.0407
\end{pmatrix}
$$

### *Step 6:*

*Aggregate the q-ROF values* W*ij for each alternative* X¨ *<sup>i</sup> by the q-ROFPWA operator using Equation (49) given in Table 10.*

**Table 10.** q-ROF Aggregated values W*<sup>i</sup>*

.


*Step 7:*

*Calculate the score of all q-ROF aggregated values* W*<sup>i</sup> .*

$$\tilde{\Xi}(\mathcal{W}\_1) = 0.7312$$

$$
\Xi(\mathbb{X}\_3) = 0.6761
$$

$$
\tilde{\Xi}(\mathbb{X}\_4) = 0.5830
$$

$$
\tilde{\Xi}(\mathbb{X}\_5) = 0.6039
$$

*Step 8:*

*So,*

*Ranks by score function values.*

W<sup>1</sup> W<sup>2</sup> W<sup>3</sup> W<sup>5</sup> W<sup>4</sup> X¨ <sup>X</sup>¨ <sup>X</sup>¨ <sup>X</sup>¨ <sup>X</sup>¨ 

### *Comparison Analysis*

The proposed q-ROFEPWA operator is compared as shown in the Table 11 below, which lists the comparative results in the completed ranking of top five alternatives. The best selection made by the proposed operator and current operators supports the efficiency and validity of the suggested methods, can be found in the comparison Table 11. Comparison analysis represented that our top alternative is not changed when we use our proposed AOs. This show the feasibility and consistency of results.

**Table 11.** Comparison analysis of the proposed operators and existing operators in the given numerical example.


### **6. Conclusions**

We introduced q-rung orthopair fuzzy Einstein prioritized weighted averaging (q-ROFEPWA) operator and q-rung orthopair fuzzy Einstein prioritized weighted geometric (q-ROFEPWG) operator. The proposed operators are more efficient and flexible for information fusion and superior than existing aggregation operators (AOs) for decision-making process under q-ROF information. Einstein sums and Einstein products are good alternatives to algebraic sums and algebraic products because they provide a very smooth approximation. So the suggested operators are suitable for prioritized relationship in the criterion and a smooth approximation of q-ROF information. The significant contribution of the defined q-ROF prioritized AOs is that they take into account prioritization between attributes and DMs. We addressed many of the basic characteristics of the defined operators, namely idempotency, non-compensatory, boundary and monotonicity. A novel approach for MCGDM issues with q-ROFNs is also provided on the basis of the proposed operators. After this, an illustrative example is presented

to demonstrate the effectiveness of the suggested approach. Additionally, the Einstein prioritized aggregation operators are used to discuss the symmetry of attributes and their symmetrical roles under q-ROF information. The MCGDM process has been designed to study the prioritization relationship between parameters and DMs, which have become necessary to obtain symmetrical aspects in decision analysis. For further studies, taking into account the advanced simulation capabilities of q-ROFSs, in the q-ROF context we may further examine different kinds of AOs and apply them to realistic decision-making situations. Moreover, the methodological advances for many fields like machine learning, robotics, green supply chain management (GSCM), medical diagnosis, weather forecasting, intelligence, informatics and sustainable energy planning decision making are promising areas for future studies. We believe that there are substantial growth and opportunities to understand our world in the convergence of these key climate-centric organizational research fields.

**Author Contributions:** M.R., D.P. and Y.-M.C. conceived and worked together to achieve this manuscript, M.R. and D.P. construct the ideas and algorithms for data analysis and design the model of the manuscript, H.M.A.F. and H.K. processed the data collection and wrote the paper. Finally all authors have read and agreed to the published version of the manuscript.

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

**Acknowledgments:** The authors are highly thankful to editor-in-chief and referees for their valuable comments and suggestions for the improvement of our manuscript.

### **References**


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