WASPAS method and Aczel-Alsina aggregation operators for managing complex interval-valued intuitionistic fuzzy information and their applications in decision-making

Introduction

MADM strategies aims to identify the best of a few relative other options or positioning choices as per their importance as far as the assessed objective. The techniques are utilized for choosing the most acceptable other option/arrangement, because there is no such option for which all rules’ esteems are awesome. MADM strategy is the sub-part of the decision-making technique that has been used in the region of discrete fields. However, it is massively difficult to apply the MADM technique to the phenomena of fuzzy sets rather than crisp sets. To achieve this idea in the real scenario, Zadeh (1965) explored the fuzzy set (FS), which only depends on the supporting grade (SG) $\overline{\overline{{\mathcal{M}}_{\overline{\overline{{\mathfrak{C}}_C}}}}}\in \left[0,1\right]$.

In facilitating that sort of situation, FS suffers from an obvious deficiency for not describing the data in the shape of yes or no, not addressing expert opinion, namely, non-SG (NSG). To conquer this imperfection, Atanassov (1986) proposed the methodology of intuitionistic FS (IFS) with an SG and NSG. The well-known prominent of IFS is as followed: $0\le \overline{\overline{{\mathcal{M}}_{\overline{\overline{R}}}}}+\overline{\overline{{\mathcal{N}}_{\overline{R}}}}\le 1$. Interval-valued (IV) data is mostly utilized to depict the ambiguity and problematic occurrences, like the difference in temperature, the vacillation of stock cost, and the scope of circulatory strain. Besides, the IV information might be gotten from various areas or sources. For this, the IV intuitionistic FS (IVIFS), was stated by Atanassov & Gargov (1989) with SG $\left[{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R}}}}}}^U\left(\widetilde{x_E}\right)\right]$ and NSG $\left[{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R}}}}}}^U\left(\widetilde{x_E}\right)\right]$ such that $0\le {\overline{\overline{{\mathcal{M}}_{\overline{\overline{R}}}}}}^U\left(\widetilde{x_E}\right)+{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R}}}}}}^U\left(\widetilde{x_E}\right)\le 1$.

Based on above discussions, we have obtained the result that the prevailing theories neglect to manage two-domination data in the shape of SG and NSG, and simultaneously neglect to survive with inconsistent and fluctuational at a provided phase of time. However, the data got from “medical research” such that the biometric and facial acknowledgment data set consistently changes with the entry of the time. Along these lines, Ramot et al. (2002) extended the scope of SG from a genuine subset to the unit circle of the complicated plane and henceforth established the principle of complex FS (CFS). The mathematical structure of SG in the circumstances of CFS is of the form $\overline{\overline{{\mathcal{M}}_{\overline{\overline{{\mathfrak{C}}_C}}}}}\left(\widetilde{x_E}\right)=\overline{\overline{{\mathcal{M}}_{\overline{\overline{R}}}}}\left(\widetilde{x_E}\right)e^{i2\pi \left(\overline{\overline{{\mathcal{M}}_I}}\left(\widetilde{x_E}\right)\right)}$ with $\overline{\overline{{\mathcal{M}}_{\overline{\overline{R}}}}}\left(\widetilde{x_E}\right),\overline{\overline{{\mathcal{M}}_{\overline{\overline{I}}}}}\left(\widetilde{x_E}\right)\in \left[0,1\right]$. Since CFS restricts only up to SG and does not take into account NSG, Alkouri & Salleh (2012) produced the principle of complex IFS (CIFS) in the shape of SG $\overline{\overline{{\mathcal{M}}_{\overline{\overline{{\mathfrak{C}}_C}}}}}\left(\widetilde{x_E}\right)=\overline{\overline{{\mathcal{M}}_{\overline{\overline{R}}}}}\left(\widetilde{x_E}\right)e^{i2\pi \left(\overline{\overline{{\mathcal{M}}_{\overline{\overline{I}}}}}\left(\widetilde{x_E}\right)\right)}$ and NSG $\overline{\overline{{\mathcal{N}}_{\overline{\overline{{\mathfrak{C}}_C}}}}}\left(\widetilde{x_E}\right)=\overline{\overline{{\mathcal{N}}_{\overline{\overline{R}}}}}\left(\widetilde{x_E}\right)e^{i2\pi \left(\overline{\overline{{\mathcal{N}}_{\overline{\overline{I}}}}}\left(\widetilde{x_E}\right)\right)}$, with $0\le \overline{\overline{{\mathcal{M}}_{\overline{\overline{R}}}}}\left(\widetilde{x_E}\right)+\overline{\overline{{\mathcal{N}}_{\overline{\overline{R}}}}}\left(\widetilde{x_E}\right)\le 1$ and $0\le \overline{\overline{{\mathcal{M}}_{\overline{\overline{I}}}}}\left(\widetilde{x_E}\right)+\overline{\overline{{\mathcal{N}}_{\overline{\overline{I}}}}}\left(\widetilde{x_E}\right)\le 1$. Recently, Garg & Rani (2019a) studied the form of CIFS in the interval environment and proposed the mathematical structure of CIVIFS. Because of its strong ability in dealing with uncertain information, CIVIFS has been promoted in many ways, but the results in Aczel-Alsina operational laws still need to be enriched.

Literature Review

Under the powerful characteristic of FS, many scholars have conducted a lot of extended research. For illustration, ordered weighted averaging aggregation operators (Yager, 1988), immediate probabilities (Yager, Engemann & Filev, 1995), modeling decision-making under immediate probabilities (Engemann, Filev & Yager, 1996), mixed uncertain satisfaction (Yager, 2017), aggregation function (Durante & Ricci, 2018), deviation-based aggregation (Decký, Mesiar & Stupňanová, 2018), generalized averaging aggregation operators (Beliakov et al., 2011; Liu et al., 2016; Yang & Yao, 2021), and analysis of fuzzy research under bibliometric indicators (Merigó, Gil-Lafuente & Yager, 2015).

Due to strong data-inclusive features, IFS and IVIFS have been extended in distinct regions, including bipolar soft sets (Mahmood, 2020), analysis of image quality under measures (Hassaballah & Ghareeb, 2017), decision-making framework (Gao et al., 2021; Zeng, Hu & Llopis-Albert, 2023), distance and similarity measures (Garg & Rani, 2021; Peng, Xiaohe & Jianbo, 2021), hybrid variable approach (Liu et al., 2021; Xue, Deng & Garg, 2021; Zhang et al., 2022), construction of shadowed sets (Yang & Yao, 2021), time-series mapping (Bas, Yolcu & Egrioglu, 2021), transportation problem (Bharati, 2021), and combined compromise solution approach (Alrasheedi et al., 2021; Su et al., 2023).

Due to its dominant structure, several scholars have shown their interest in CFS and applied it to diverse regions. For convenience, cross-entropy measures (Liu, Ali & Mahmood, 2020), complex fuzzy soft sets (CFSS) (Thirunavukarasu, Suresh & Ashokkumar, 2017), IV CFSS (Selvachandran & Singh, 2018; Dai, Bi & Hu, 2019), and complex multi-fuzzy soft sets (Al-Qudah & Hassan, 2019). Further, Garg & Rani (2019a); Garg & Rani (2020a) invented the CIVIFS and the advanced aggregation operators under CIFS. Garg & Rani (2020b) modified the theory of robust and geometric aggregation operators under CIFS. Garg & Rani (2019b) proposed the methodology of aggregation operators under generalized CIFS. Ali et al. (2021) combined the principle of CIFS and soft set and explored some aggregation operators. Especially, the statistical metrics evaluated by Menger (1942), Einstein aggregation operators for IFS invented by Wang & Liu (2012), Archimedean aggregation operators for IFS stated by Xia, Xu & Zhu (2012), Hamacher aggregation operators for interval-valued IFS presented by Liu (2013), and so on.

Motivation and Main Contribution

CIVIFS theory is the modified version of the FS, IFS, IVIFS, CFS, and CIFS because of their valuable and dominant structure. Further, the theory of Aczel-Alsina is also very famous and reliable because it is the generation of the algebraic t-norm and t-conorm. Moreover, discovering the theory of aggregation operators in the presence of Aczel-Alsina information for managing CIVIF values is a very challenging task for new fuzzy scholars, because up to date no one can derive the theory of Aczel-Alsina aggregation operators for CIVIF values. Furthermore, deriving the theory of the WASPAS technique (Zavadskas et al., 2012) is also a very awkward and challenging task for fuzzy researchers.

Keeping the benefits of the above prevailing operators, the major contribution of this analysis is illustrated below:

1. To initiate the Aczel-Alsina operational laws and their related results.

2. To invent the principle of CIVIFAAWA, CIVIFAAOWA, CIVIFAAHA, CIVIFAAWG, CIVIFAAOWG, and CIVIFAAHG operators, and illustrated their well-known properties and results.

3. To derive the theory of the WASPAS method for CIVIFSs.

4. To demonstrate the MADM strategy under the invented works.

5. To express the supremacy and dominancy of the invented works with the help of sensitive analysis and geometrical shown of the explored works.

Presentation of our analysis is implemented in the shape: Section 2 covers all the prevailing methodologies. In Section 3, we initiate the Aczel-Alsina operational laws and their related results. Section 4 produces the principle of CIVIFAAWA, CIVIFAAOWA, CIVIFAAHA, CIVIFAAWG, CIVIFAAOWG, and CIVIFAAHG operators, and illustrates their well-known properties. In Section 5, we derive the WASPAS method for CIVIFSs. In Section 6, we demonstrate the effectiveness of the MADM strategy under the invented works. The conclusion of this study is illustrated in Section 7.

Before starting the proposed work, all variables and indexes used in this study are defined in Table 1.

Preliminaries

Here, we utilized the weighted sum model (WSM) and the weighted product model (WPM) to review the concept of the WASPAS method (Zavadskas et al., 2012). Moreover, the extended WASPAS method was derived from Zavadskas et al. (2013). Some valuable and effective steps of the WASPAS method are listed below:

Step 1: The input data of the technique is represented in the form of a matrix of alternatives and attributes, which is based on the data received from the expert.

Step 2: Normalize the decision matrix in the presence of the information in Eq. (1): (1)${\displaystyle {\overline{\overline{{\mathfrak{C}}_{C_{ij}}}}}^{{}^{\prime }}=\left\{\begin{array}{cc}\hfill \frac{\overline{\overline{{\mathfrak{C}}_{C_{ij}}}}}{{max}_i\overline{\overline{{\mathfrak{C}}_{C_{ij}}}}}\hfill & \hfill ifi\in B\hfill \\ \hfill \frac{{min}_i\overline{\overline{{\mathfrak{C}}_{C_{ij}}}}}{\overline{\overline{{\mathfrak{C}}_{C_{ij}}}}}\hfill & \hfill ifi\in C\hfill \end{array}\right.}$

where B represented the benefit types of data and C stated the cost type of criteria.

Step 3: Compute WSM and WPM of each alternative: (2)${\displaystyle WS{M}_i=\sum _{j=1}^m{W}_j{\overline{\overline{{\mathfrak{C}}_{C_{ij}}}}}^{{}^{\prime }}}$ (3)${\displaystyle WP{M}_i=\prod _{j=1}^m{\left({\overline{\overline{{\mathfrak{C}}_{C_{ij}}}}}^{{}^{\prime }}\right)}^{W_j}}$

Step 4: Calculate the score value by using the theory of WSM and WPM information referring to the following way: (4)${\displaystyle S_i=°F\ast WS{M}_i+\left(1-°F\right)WP{M}_i.}$There exist some special cases: when °F = 1 in Eq. (4), S_i = WSM_i; when °F = 0 , S_i = WPM_i.

Step 5: Deriving the best preference by the score value in Step 4.

Next, the algebraic theories of some prevailing principles like CIVIFSs, the concept of Aczel-Alsina t-norm and t-conorm will be discussed. Of note, the notation $\widetilde{{\mathbb{X}}_U}$, stated for universal sets.

Definition 1:(Garg & Rani, 2019a) The mathematical structure of CIVIFS $\overline{\overline{{\mathfrak{C}}_C}}$ is shown in the shape of: (5)${\displaystyle \overline{\overline{{\mathfrak{C}}_C}}=\left\{\left(\overline{\overline{{\mathcal{M}}_{\overline{\overline{{\mathfrak{C}}_C}}}}}\left(\widetilde{x_E}\right),\overline{\overline{{\mathcal{N}}_{\overline{\overline{{\mathfrak{C}}_C}}}}}\left(\widetilde{x_E}\right)\right):\widetilde{x_E}\in \widetilde{{\mathbb{X}}_U}\right\}}$where the term $\overline{\overline{{\mathcal{M}}_{\overline{\overline{{\mathfrak{C}}_C}}}}}\left(\widetilde{x_E}\right)=\left[{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R}}}}}}^U\left(\widetilde{x_E}\right)\right]e^{i2\pi \left(\left[{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I}}}}}}^U\left(\widetilde{x_E}\right)\right]\right)}$ and $\overline{\overline{{\mathcal{N}}_{\overline{\overline{{\mathfrak{C}}_C}}}}}\left(\widetilde{x_E}\right)=\left[{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R}}}}}}^U\left(\widetilde{x_E}\right)\right]e^{i2\pi \left(\left[{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I}}}}}}^U\left(\widetilde{x_E}\right)\right]\right)}$ indicate the TD and FD with $0\le {\overline{\overline{{\mathcal{M}}_{\overline{\overline{R}}}}}}^U\left(\widetilde{x_E}\right)+{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R}}}}}}^U\left(\widetilde{x_E}\right)\le 1$ and $0\le {\overline{\overline{{\mathcal{M}}_{\overline{\overline{I}}}}}}^U\left(\widetilde{x_E}\right)+{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I}}}}}}^U\left(\widetilde{x_E}\right)\le 1$. Moreover, $\overline{\overline{{\mathfrak{R}}_{\overline{\overline{{\mathfrak{C}}_C}}}}}\left(\widetilde{x_E}\right)=\left[{\overline{\overline{{\mathfrak{R}}_{\overline{\overline{R}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathfrak{R}}_{\overline{\overline{R}}}}}}^U\left(\widetilde{x_E}\right)\right]e^{i2\pi \left(\left[{\overline{\overline{{\mathfrak{R}}_{\overline{\overline{I}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathfrak{R}}_{\overline{\overline{I}}}}}}^U\left(\widetilde{x_E}\right)\right]\right)}=\left[\left(1-\left({\overline{\overline{{\mathcal{M}}_{\overline{\overline{R}}}}}}^L\left(\widetilde{x_E}\right)+{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R}}}}}}^L\left(\widetilde{x_E}\right)\right)\right),\left(1-\left({\overline{\overline{{\mathcal{M}}_{\overline{\overline{R}}}}}}^U\left(\widetilde{x_E}\right)+{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R}}}}}}^U\left(\widetilde{x_E}\right)\right)\right)\right]e^{i2\pi \left[\left(1-\left({\overline{\overline{{\mathcal{M}}_{\overline{\overline{I}}}}}}^L\left(\widetilde{x_E}\right)+{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I}}}}}}^L\left(\widetilde{x_E}\right)\right)\right),\left(1-\left({\overline{\overline{{\mathcal{M}}_{\overline{\overline{I}}}}}}^U\left(\widetilde{x_E}\right)+{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I}}}}}}^U\left(\widetilde{x_E}\right)\right)\right)\right]}$ states the neutral grade, and

$\overline{\overline{{\mathfrak{C}}_{C_j}}}=\left(\left[{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_j}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_j}}}}}}^U\left(\widetilde{x_E}\right)\right]e^{i2\pi \left(\left[{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_j}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_j}}}}}}^U\left(\widetilde{x_E}\right)\right]\right)},\right.$

$\left.\left[{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_j}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_j}}}}}}^U\left(\widetilde{x_E}\right)\right]e^{i2\pi \left(\left[{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_j}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_j}}}}}}^U\left(\widetilde{x_E}\right)\right]\right)}\right)$ denotes the complex interval-valued intuitionistic fuzzy number (CIVIFN).

Definition 2:(Garg & Rani, 2019a) Suppose there are two CIVIFNs

$\overline{\overline{{\mathfrak{C}}_{C_j}}}=\left(\begin{array}{c}\hfill \left[{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_j}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_j}}}}}}^U\left(\widetilde{x_E}\right)\right]e^{i2\pi \left(\left[{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_j}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_j}}}}}}^U\left(\widetilde{x_E}\right)\right]\right)},\hfill \\ \hfill \left[{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_j}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_j}}}}}}^U\left(\widetilde{x_E}\right)\right]e^{i2\pi \left(\left[{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_j}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_j}}}}}}^U\left(\widetilde{x_E}\right)\right]\right)}\hfill \end{array}\right),j=1,2$, then: (6)${\displaystyle \overline{\overline{{\mathfrak{C}}_{C_1}}}\oplus \overline{\overline{{\mathfrak{C}}_{C_2}}}=\left(\begin{array}{c}\hfill \left[{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_1}}}}}}^L+{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_2}}}}}}^L-{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_1}}}}}}^L{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_2}}}}}}^L,{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_1}}}}}}^U+{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_2}}}}}}^U-{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_1}}}}}}^U{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_2}}}}}}^U\right]\hfill \\ \hfill e^{i2\pi \left[{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_1}}}}}}^L+{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_2}}}}}}^L-{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_1}}}}}}^L{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_2}}}}}}^L,{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_1}}}}}}^U+{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_2}}}}}}^U-{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_1}}}}}}^U{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_2}}}}}}^U\right]},\hfill \\ \hfill \left[{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_1}}}}}}^L{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_2}}}}}}^L,{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_1}}}}}}^U{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_2}}}}}}^U\right]e^{i2\pi \left[{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_1}}}}}}^L{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_2}}}}}}^L,{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_1}}}}}}^U{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_2}}}}}}^U\right]}\hfill \end{array}\right)}$ (7)${\displaystyle \overline{\overline{{\mathfrak{C}}_{C_1}}}\otimes \overline{\overline{{\mathfrak{C}}_{C_2}}}=\left(\begin{array}{c}\hfill \left[{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_1}}}}}}^L{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_2}}}}}}^L,{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_1}}}}}}^U{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_2}}}}}}^U\right]e^{i2\pi \left[{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_1}}}}}}^L{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_2}}}}}}^L,{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_1}}}}}}^U{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_2}}}}}}^U\right]},\hfill \\ \hfill \left[{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_1}}}}}}^L+{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_2}}}}}}^L-{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_1}}}}}}^L{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_2}}}}}}^L,{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_1}}}}}}^U+{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_2}}}}}}^U-{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_1}}}}}}^U{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_2}}}}}}^U\right]\hfill \\ \hfill e^{i2\pi \left[{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_1}}}}}}^L+{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_2}}}}}}^L-{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_1}}}}}}^L{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_2}}}}}}^L,{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_1}}}}}}^U+{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_2}}}}}}^U-{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_1}}}}}}^U{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_2}}}}}}^U\right]}\hfill \end{array}\right)}$ (8)${\displaystyle \overline{\overline{{\psi }_S}}\overline{\overline{{\mathfrak{C}}_{C_1}}}=\left(\begin{array}{c}\hfill \left[\left(1-{\left(1-{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_1}}}}}}^L\right)}^{\overline{\overline{{\psi }_S}}}\right),\left(1-{\left(1-{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_1}}}}}}^U\right)}^{\overline{\overline{{\psi }_S}}}\right)\right]e^{i2\pi \left[\left(1-{\left(1-{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_1}}}}}}^L\right)}^{\overline{\overline{{\psi }_S}}}\right),\left(1-{\left(1-{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_1}}}}}}^U\right)}^{\overline{\overline{{\psi }_S}}}\right)\right]},\hfill \\ \hfill \left[{{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_1}}}}}}^L}^{\overline{{\psi }_S}},{{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_1}}}}}}^U}^{\overline{{\psi }_S}}\right]e^{i2\pi \left[{{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_1}}}}}}^L}^{\overline{\overline{{\psi }_S}}},{{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_1}}}}}}^U}^{\overline{\overline{{\psi }_S}}}\right]}\hfill \end{array}\right)}$ (9)${\displaystyle {\overline{\overline{{\mathfrak{C}}_{C_1}}}}^{\overline{\overline{{\psi }_S}}}=\left(\begin{array}{c}\hfill \left[{{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_1}}}}}}^L}^{\overline{\overline{{\psi }_S}}},{{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_1}}}}}}^U}^{\overline{\overline{{\psi }_S}}}\right]e^{i2\pi \left[{{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_1}}}}}}^L}^{\overline{\overline{{\psi }_S}}},{{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_1}}}}}}^U}^{\overline{\overline{{\psi }_S}}}\right]},\hfill \\ \hfill \left[\left(1-{\left(1-{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_1}}}}}}^L\right)}^{\overline{\overline{{\psi }_S}}}\right),\left(1-{\left(1-{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_1}}}}}}^U\right)}^{\overline{\overline{{\psi }_S}}}\right)\right]e^{i2\pi \left[\left(1-{\left(1-{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_1}}}}}}^L\right)}^{\overline{\overline{{\psi }_S}}}\right),\left(1-{\left(1-{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_1}}}}}}^U\right)}^{\overline{\overline{{\psi }_S}}}\right)\right]}\hfill \end{array}\right)}$

Definition 3:(Garg & Rani, 2019b) By taking any two CIVIFNs

$\overline{\overline{{\mathfrak{C}}_{C_j}}}=\left(\begin{array}{c}\hfill \left[{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_j}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_j}}}}}}^U\left(\widetilde{x_E}\right)\right]e^{i2\pi \left(\left[{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_j}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_j}}}}}}^U\left(\widetilde{x_E}\right)\right]\right)},\hfill \\ \hfill \left[{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_j}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_j}}}}}}^U\left(\widetilde{x_E}\right)\right]e^{i2\pi \left(\left[{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_j}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_j}}}}}}^U\left(\widetilde{x_E}\right)\right]\right)}\hfill \end{array}\right)$, then the score value (SV) and accuracy value (AV) are determined by the following formulas: (10)${\displaystyle \overline{\overline{{\mathcal{S}}_{SV}}}\left(\overline{\overline{{\mathfrak{C}}_{C_1}}}\right)=\frac{1}{4}\left({\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_1}}}}}}^L+{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_1}}}}}}^L-{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_1}}}}}}^L-{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_1}}}}}}^L+{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_1}}}}}}^U+{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_1}}}}}}^U-{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_1}}}}}}^U-{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_1}}}}}}^U\right),}$ (11)${\displaystyle \overline{\overline{{\mathcal{H}}_{AV}}}\left(\overline{\overline{{\mathfrak{C}}_{C_1}}}\right)=\frac{1}{4}\left({\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_1}}}}}}^L+{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_1}}}}}}^L+{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_1}}}}}}^L+{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_1}}}}}}^L+{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_1}}}}}}^U+{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_1}}}}}}^U+{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_1}}}}}}^U+{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_1}}}}}}^U\right)}$Definition 4:(Garg & Rani, 2019a) By taking any two CIVIFNs $\overline{\overline{{\mathfrak{C}}_{C_j}}}=\left(\begin{array}{c}\hfill \left[{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_j}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{M}}_{\overline{\overline{R_j}}}}}}^U\left(\widetilde{x_E}\right)\right]e^{i2\pi \left(\left[{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_j}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{M}}_{\overline{\overline{I_j}}}}}}^U\left(\widetilde{x_E}\right)\right]\right)},\hfill \\ \hfill \left[{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_j}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{N}}_{\overline{\overline{R_j}}}}}}^U\left(\widetilde{x_E}\right)\right]e^{i2\pi \left(\left[{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_j}}}}}}^L\left(\widetilde{x_E}\right),{\overline{\overline{{\mathcal{N}}_{\overline{\overline{I_j}}}}}}^U\left(\widetilde{x_E}\right)\right]\right)}\hfill \end{array}\right)$, j = 1, 2, then

1. If $\overline{\overline{{\mathcal{S}}_{SV}}}\left(\overline{\overline{{\mathfrak{C}}_{C_1}}}\right)>\overline{\overline{{\mathcal{S}}_{SV}}}\left(\overline{\overline{{\mathfrak{C}}_{C_2}}}\right),$ then $\overline{\overline{{\mathfrak{C}}_{C_1}}}>\overline{\overline{{\mathfrak{C}}_{C_2}}}$;

2. If $\overline{\overline{{\mathcal{S}}_{SV}}}\left(\overline{\overline{{\mathfrak{C}}_{C_1}}}\right)<\overline{\overline{{\mathcal{S}}_{SV}}}\left(\overline{\overline{{\mathfrak{C}}_{C_2}}}\right),$ then $\overline{\overline{{\mathfrak{C}}_{C_1}}}<\overline{\overline{{\mathfrak{C}}_{C_2}}}$;

3. If $\overline{\overline{{\mathcal{S}}_{SV}}}\left(\overline{\overline{{\mathfrak{C}}_{C_1}}}\right)=\overline{\overline{{\mathcal{S}}_{SV}}}\left(\overline{\overline{{\mathfrak{C}}_{C_2}}}\right),$ then

4. If $\overline{\overline{{\mathcal{H}}_{AV}}}\left(\overline{\overline{{\mathfrak{C}}_{C_1}}}\right)>\overline{\overline{{\mathcal{H}}_{AV}}}\left(\overline{\overline{{\mathfrak{C}}_{C_2}}}\right)$, then $\overline{\overline{{\mathfrak{C}}_{C_1}}}>\overline{\overline{{\mathfrak{C}}_{C_2}}}$;

5. If $\overline{\overline{{\mathcal{H}}_{AV}}}\left(\overline{\overline{{\mathfrak{C}}_{C_1}}}\right)<\overline{\overline{{\mathcal{H}}_{AV}}}\left(\overline{\overline{{\mathfrak{C}}_{C_2}}}\right)$, then $\overline{\overline{{\mathfrak{C}}_{C_1}}}<\overline{\overline{{\mathfrak{C}}_{C_2}}}$;

6. If $\overline{\overline{{\mathcal{H}}_{AV}}}\left(\overline{\overline{{\mathfrak{C}}_{C_1}}}\right)=\overline{\overline{{\mathcal{H}}_{AV}}}\left(\overline{\overline{{\mathfrak{C}}_{C_2}}}\right)$, then $\overline{\overline{{\mathfrak{C}}_{C_1}}}=\overline{\overline{{\mathfrak{C}}_{C_2}}}$.

Definition 5:(Klement & Mesiar, 1997) Suppose $\overline{\overline{{\mathbb{T}}_{TN}}}:\left[0,1\right]\times \left[0,1\right]\to \left[0,1\right]$ states a TN, then

1. $\overline{\overline{{\mathbb{T}}_{TN}}}\left(\widetilde{x_E},{\widetilde{x_E}}^{{}^{\prime }}\right)=\overline{\overline{{\mathbb{T}}_{TN}}}\left({\widetilde{x_E}}^{{}^{\prime }},\widetilde{x_E}\right),\widetilde{x_E},{\widetilde{x_E}}^{{}^{\prime }}\in \left[0,1\right]$;

2. $\overline{\overline{{\mathbb{T}}_{TN}}}\left(\widetilde{x_E},{\widetilde{x_E}}^{{}^{\prime }}\right)\le \overline{\overline{{\mathbb{T}}_{TN}}}\left(\widetilde{x_E},{\widetilde{x_E}}^{{}^{\prime \prime }}\right)$, if ${\widetilde{x_E}}^{{}^{\prime }}\le {\widetilde{x_E}}^{{}^{\prime \prime }}$;

3. $\overline{\overline{{\mathbb{T}}_{TN}}}\left(\widetilde{x_E},\overline{\overline{{\mathbb{T}}_{TN}}}\left({\widetilde{x_E}}^{{}^{\prime }},{\widetilde{x_E}}^{{}^{\prime \prime }}\right)\right)=\overline{\overline{{\mathbb{T}}_{TN}}}\left(\overline{\overline{{\mathbb{T}}_{TN}}}\left(\widetilde{x_E},{\widetilde{x_E}}^{{}^{\prime }}\right),{\widetilde{x_E}}^{{}^{\prime \prime }}\right)$;

4. $\overline{\overline{{\mathbb{T}}_{TN}}}\left(\widetilde{x_E},1\right)=\widetilde{x_E}$.

Definition 6:(Klement & Mesiar, 1997) Suppose $\overline{\overline{{\mathbb{S}}_{TN}}}:\left[0,1\right]\times \left[0,1\right]\to \left[0,1\right]$ states a TCN, then