Aggregation Operator and Its Application in Assessing First-Class Discipline Construction in Industry-Characteristic Universities

Abstract

To effectively deal with the uncertainty of value assessments of industry-characteristic universities, this paper proposes a new fuzzy multi-attribute assessment method. Firstly, we define the complex cubic fractional orthotriple fuzzy set (CCFOFS) for expressing ambiguous information and present some basic operational rules and information measures. Then, we present the complex cubic fractional orthotriple fuzzy Dombi-weighted power-partitioned Muirhead mean (CCFOFDWPPMM) operator, which combines the superiority of the Dombi operations, power average (PA) operator, and partitioned Muirhead mean (PMM) operator. Further, a multi-attribute assessment method is constructed based on the CCFOFDWPPMM operator and the Integrated Determination of Objective Criteria Weights (IDOCRIW) method. Furthermore, we constructed a novel assessment index system for the construction of first-class disciplines. Finally, this paper verifies the validity and applicability of the method by applying the novel multi-attribute assessment method to a practical case of first-class discipline construction in industry-characteristic universities.

1. Introduction

In the strategic context of the development of higher education and the construction of “world-class” universities, industry-characteristic universities are an important part of the higher education system, and the quality of their discipline construction is directly related to the realization of national industrial transformation and upgrading and innovation-driven development strategies. Industry-characteristic universities, with their deep integration with specific industries, have developed unique educational advantages and disciplinary characteristics. How to guide the construction of world-class disciplines through scientific assessment has become an important issue in current higher education research [1,2]. The essential characteristics of industry-characteristic universities are reflected in their positioning of “relying on industry and serving industry” [3]. Compared with comprehensive universities, these types of universities have a significant industry agglomeration effect in specific disciplinary fields [4], and their disciplinary layout, talent cultivation, and scientific research are all centered around industry needs [5]. This uniqueness determines that the construction of first-class disciplines must follow a “characteristic-led and demand-oriented” development path, which must both conform to the general laws of higher education and meet the special needs of industry development. Assessments of first-class discipline construction in industry-characteristic universities can effectively reflect the effectiveness of their construction, guide disciplines to accurately match national strategies and industry needs, and strengthen their distinctive advantages. Through assessment, shortcomings in discipline construction can be diagnosed, resource allocation optimized, and service industry capabilities enhanced. Assessment can promote the deep integration of industry, academia, and research, and accelerate the transformation of scientific and technological achievements. Categorized assessment helps improve the higher education assessment system, avoids a “one size fits all” approach, and encourages industry-characteristic universities to take a differentiated, high-quality development path to better serve economic and social development. However, the existing relevant research mostly focuses on first-class discipline construction in comprehensive universities [1], which has obvious limitations in its applicability to industry-characteristic universities. First-class discipline construction in industry-characteristic universities has distinct practical characteristics: on the one hand, it emphasizes the deep integration of discipline chains and industrial chains and improves service capabilities by constructing a “discipline–industry” collaborative innovation mechanism [6]; on the other hand, it focuses on the refinement of characteristic directions and forms irreplaceable competitive advantages in sub-fields. This construction model requires the assessment system to break through the traditional mindset of discipline assessment and establish an assessment model that can reflect industry characteristics. At present, the assessment of first-class discipline construction in industry-characteristic universities faces multiple complexities: it is difficult to quantify the interdisciplinary nature and uniqueness of industries, and it is difficult to balance universality and professionalism in assessment standards; discipline construction elements are intertwined, and there are interactions between their attributes; and administrative intervention, resource dependence, and industry volatility further amplify the uncertainty of the results. The assessment of first-class discipline construction in industry-characteristic universities is a complex, uncertain, and multidimensional systematic project, which is neither the independent influence of a single attribute nor the simple sum of multiple attributes. Therefore, the assessment of first-class discipline construction in industry-characteristic universities should not only be analyzed from a multidimensional and multi-attribute perspective, but also focus on researching the overall combined effect of the interactive and interrelated influences of multiple attributes on the assessment of first-class discipline construction in industry-characteristic universities. This requires that the assessment of first-class discipline construction in industry-characteristic universities introduces advanced and scientific multi-attribute assessment methods and explores the best way to construct an assessment model for first-class discipline construction in industry-characteristic universities.

The multiple complexities of assessing first-class discipline construction in industry-characteristic universities pose challenges of complexity and uncertainty for experts in the assessment process. Assessment information is often uncertain. Zadeh [7] proposed fuzzy sets that can characterize fuzzy phenomena and solve fuzzy assessment problems to address the uncertainty of assessment information in the assessment process. Traditional fuzzy sets can only express either–or assessment information and are constrained by their inability to appropriately represent fuzzy assessment information. Atanassov [8] proposed intuitive fuzzy sets (IFS) that can represent both–and assessment information, but IFS has certain limitations, requiring $u+v\le 1$, and cannot handle situations where the sum of the membership degree (MD) $u$ and non-membership degree (NMD) $v$ is greater than 1. To address such issues, Yager [9] defined Pythagorean fuzzy sets (PyFS), extending the boundary values of MD and NMD to $u^2+v^2\le 1$. Yager [10] further relaxed the restrictions on MD and NMD and proposed q-rung orthopair fuzzy sets (q-ROFS), which satisfy $0\le u^q+v^q\le 1$, are more versatile and flexible, and are more suitable for complex fuzzy assessment environments. On this basis, Abosuliman et al. [11] proposed fractional orthotriple fuzzy sets (FOFS), which added hesitation degrees (HD) $\pi$ (degree of uncertainty) and $0\le u^q+{\pi }^q+v^q\le 1$ to MD and NMD. This ternary structure is more advantageous for representing uncertain assessment information. Scholars have researched fractional orthotriple fuzzy multi-attribute decision-making methods and have verified their applicability in scenarios such as optimal selection for an EEG of depression patients [12], an assessment of the Government Information Disclosure on Public Health Emergencies [13], and an assessment of project team members [14], among other scenarios. In addition, with the deepening of research and practice, higher requirements have been put forward for data types in actual assessment scenarios. Zhou et al. [15] combined the characteristics of complex fuzzy sets (CFS) [16] and cubic fuzzy sets (CuFS) [17] to propose the concept of complex cubic fuzzy sets (CCFS), which is a new extension of fuzzy set theory. The core of CCFS lies in their ability to simultaneously integrate the interval description capability of CuFS and the phase information expression capability of CFS. Compared with traditional fuzzy sets and their extensions, CCFS have multidimensional information expression capabilities: (1) they can introduce periodic or time information through phase terms; (2) they can simultaneously characterize the fuzzy range (interval value) and accurately assess (point value) of MD, avoiding information loss. Based on existing research results, this paper focuses on the ingenious combination of CCFS and FOFS to define a new CCFOFS, which can contain more comprehensive uncertain information than a single fuzzy set and is more suitable for handling complex and uncertain multi-attribute assessment problems. We will also try to apply it to the complex and uncertain field of first-class discipline construction assessment in industry-characteristic universities. Therefore, this study uses CCFOFS as the underlying information representation framework for first-class discipline construction assessment in industry-characteristic universities. The core motivation for this choice lies in its excellent ability to meet the practical needs of the assessment process of first-class discipline construction in industry-characteristic universities, and to adapt to the core behavioral characteristics of information expression preference assessments when facing high uncertainty in the actual assessment process. By constructing a comprehensive assessment method based on CCFOFS, this study aims to improve the ability to represent preference information in the assessment of first-class discipline construction in industry-characteristic universities and provide theoretical and methodological support for promoting scientific decision-making and long-term stable development in this field.

The aggregation of assessment information is a key component of the multi-attribute assessment process. Scholars attach great importance to information aggregation operators, viewing them as an effective means of aggregating assessment information. Currently, basic operators such as WA [18], WG [19], OWA [20], and OWG [19] are widely used in multi-attribute assessment, but the above aggregation operators are based on the assumption of independence between attributes. In actual multi-attribute assessment problems, there are often correlations between attributes. Considering the correlations between attributes, scholars have applied the Bonferroni mean (BM) operator [21] and the Heronian mean (HM) operator [22] to multi-attribute assessment. However, these two operators can only handle the correlation between two attributes. Muirhead [23] proposed the Muirhead mean (MM) operator, which can handle the interdependencies among multiple attributes. Building on this, Yang et al. [24] considered the adverse effects of extreme values on aggregation results and proposed the Power MM operator by integrating the PA operator with the MM operator. In practical applications, not all attributes are mutually related. Therefore, when certain attributes are unrelated, they should be divided into different partitions. Qin et al. [25] combined the PA operator with the PMM operator to propose the power-partitioned Muirhead mean (PPMM) operator, further advancing the research on aggregation operators. However, the aforementioned aggregation operators are all based on algebraic operations, with single-parameter adjustments, making them difficult to adapt to complex decision-making requirements. The Dombi operation can effectively adjust parameters, demonstrating significant flexibility and robustness in the information aggregation process, and has thus garnered widespread attention from scholars [26]. Jana et al. [27] explored picture fuzzy Dombi aggregation operators. Deveci et al. [28] constructed the Dombi–Bonferroni operator. Liu et al. [29] proposed single-valued neutrosophic credibility numbers Dombi extended power weight-aggregation operators. Zang et al. [30] proposed an advanced linguistic complex T-spherical fuzzy Dombi-weighted power-partitioned Heronian mean operator. The existing research has fully demonstrated the superiority of Dombi operations in information aggregation processes under different fuzzy environments. Given the above analysis, this paper innovatively extends the PA operator, PMM operator, and Dombi operation to the newly proposed complex cubic fractional orthotriple (CCFOF) fuzzy environment, thereby expanding the application scope of the Dombi operation while providing more effective technical support for the efficient aggregation of CCFOF information. We mentioned above that the assessment of first-class discipline construction in industry-characteristic universities is a complex, uncertain, multi-dimensional, and multi-attribute systematic assessment process. The high degree of complexity and multi-source uncertainty faced in the assessment process makes it difficult for experts to express their preferences with precise numerical values. More importantly, the assessment results are not isolated superpositions or linear combinations of the effects of each attribute, but originate from the complex nonlinear interactions between them. This internal correlation and system integration make it impossible for traditional mixed methods to meet the high standards and multi-objective requirements of the assessment of first-class discipline construction in industry-characteristic universities. We attempt to apply the newly proposed operator to solve the problem of attribute information aggregation in the assessment of first-class discipline construction in industry-characteristic universities.

In fuzzy multi-attribute assessment, attribute weight allocation is the core link connecting indicator assessment information and the final result, directly determining the directionality and reliability of the assessment results. Attribute weighting methods are mainly divided into subjective weighting methods and objective weighting methods. The weights in subjective weighting methods are derived from the knowledge, experience, preferences, and subjective judgments of decision-makers or experts, which may not reflect the objective characteristics of the indicator data in fuzzy multi-attribute assessment and may lack transparency due to a lack of data support [31]. The objective weighting method uses mathematical mechanisms to directly extract objective evidence about the importance of fuzzy data attributes, minimizing subjective bias, enhancing the objectivity, transparency, and repeatability of the assessment process, and effectively utilizing the intrinsic characteristics of fuzzy data. It is an important and effective tool for dealing with attribute weighting issues in fuzzy environments, especially in scenarios that require an emphasis on objectivity or data-driven approaches or lack reliable expert judgment [32]. The entropy method is a classic weighting allocation method, an objective weighting approach entirely based on the degree of data dispersion. Its core principle is that if the data variability of a certain indicator is greater (lower entropy value), the information it provides is more significant, and thus its weight should be higher. It has been applied to green low-carbon port assessments [33], water resource management assessments [34], and green supplier performance assessments [35]. However, the entropy method relies solely on the degree of data dispersion (information entropy). If the data variability of an indicator is high (low entropy value), its weight is high; conversely, its weight is low. Its essence is a unidimensional measurement of information content, which is unable to reflect the interactive influences between indicators. To effectively solve the above problems, Tajik et al. [36] proposed the Criterion impact loss (CILOS) method, which is based on the principle of “loss compensation” and describes the impact loss of a certain alternative attribute caused by selecting that indicator when other indicators are selected as the best, implying the interaction and conflict between the measurement indicators. The CILOS method implies conflict perception, has more balanced weights, and is more in line with the needs of indicator trade-offs in actual assessments. Through a competitive compensation mechanism and dynamic loss calculation, CILOS solves the key defects of the entropy method in correlation processing and data adaptability. In fuzzy multi-attribute decision-making, its weighting results are more realistic and interpretable. Scholars have extended it to hydrological geographical regions assessment [37] and water pollution and water quality assessments [38] in fuzzy assessment environments. Zavadskas et al. [39] combined the entropy method and the CILOS method to propose the IDOCRIW method, in which the entropy method provides the basis for the data, CILOS supplements the competitive perspective, and IDOCRIW realizes comprehensive optimization, effectively solving the problem of single-objective methods ignoring correlations between indicators. The IDOCRIW method has been widely used in intuitive fuzzy multi-attribute assessment environments [40], triangular fuzzy multi-attribute assessment environments [41], and probabilistic double-hierarchy linguistic (PDHL) environments [42]. This paper introduces it into the newly proposed CCFOF multi-attribute assessment environment to effectively expand the applicable field of the IDOCRIW method.

In summary, this paper aims to solve the problem of assessing first-class discipline construction in industry-characteristic universities. First, we propose a new CCFOFS to characterize the assessment information provided by experts. We also provide the calculation rules and information measurement formulas of the CCFOFS to provide theoretical support for follow-up research. Then, we integrate the Dombi operations, PA operator, and PMM operator to propose the CCFOFDPMM operator and CCFOFDWPPMM operator, which are applied to assessment-information aggregation. Furthermore, we construct an indicator system suitable for first-class discipline construction in industry-characteristic universities. Finally, we propose a new multi-attribute assessment method by combining the CCFOFDWPPMM operator and the IDOCRIW method, and apply it to solve the problem of assessing first-class discipline construction in industry-characteristic universities.

The contributions of this paper are as follows:

• We combine the advantages of CFS, CuFS, and FOFS in representing uncertain information to propose a new type of fuzzy set called CCFOFS. This new type of fuzzy set can represent uncertain information more comprehensively and is more suitable for dealing with fuzzy multi-attribute assessment problems. We apply it to the assessment of first-class discipline construction in industry-characteristic universities.
• We propose the CCFOFDWPPMM operator, which can consider the variability of parameter operations and is more compatible than traditional operation rules. This operator can effectively avoid the influence of singular values and solve decision-making problems involving interrelationships between attributes in different partitions, which is more in line with the reality of multi-attribute assessment.
• Taking into account the different degrees of importance of each attribute, we combine the IDOCRIW-weighting method with the proposed CCFOFDWPPMM operator to propose a new multi-attribute assessment method, and apply this method to the assessment of first-class discipline construction in industry-characteristic universities.
• We provide an indicator system suitable for first-class discipline construction in industry-characteristic universities, focusing on the four dimensions of “talent cultivation”, “scientific research”, “social services”, and “open education,” highlighting the core characteristics of first-class discipline construction in industry-characteristic universities. This indicator system breaks through the constraints of the comprehensive university assessment system framework and highlights the special mission of industry-characteristic universities in serving national strategies and specific industries.

The structure of this paper is described as follows: Section 2 introduces some basic concepts, including FOFS, CCFS, PA operator, MM operator, and PMM operator. Section 3 presents detailed information on CCFOFS and aggregation operators. Section 4 constructs an assessment indicator system for first-class discipline construction in industry-characteristic universities. Section 5 introduces a multi-attribute assessment method based on the CCFOFDWPPMM operator and IDOCRIW method. Section 6 further verifies the feasibility and applicability of the proposed method through actual cases of assessing first-class discipline construction in industry-characteristic universities. The superiority and effectiveness of the proposed method are illustrated through a sensitivity analysis and comparative analysis. Finally, a discussion is presented. Section 7 is the conclusion.

2. Preliminaries

This section introduces some basic concepts, including FOFS, CCFS, PA operator, MM operator, and PMM operator.

Table 1. Nomenclature.

$R=\left\{r_1,r_2,\dots ,r_n\right\}$	A non-empty finite set	$Z_i=\left\{Z_1,Z_2,\dots ,Z_m\right\}$	A collection of alternatives
$b_i$	A set of non-negative real numbers	$F_j=\left\{F_1,F_2,\dots ,F_n\right\}$	A group of attributes
$d\left(b_i,b_j\right)$	The distance of $b_i$ and $b_j$	$D_e=\left\{e_1,e_2,\dots ,e_c\right\}$	A group of experts
$Sup\left(x_i\right.\left.,x_j\right)$	Support of $x_j$ for $x_i$	${\varpi }_j=\left\{{\varpi }_1,{\varpi }_2,\dots ,{\varpi }_n\right\}$	The weights of attributes
$P=\left({\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_n\right)\in P^n$	A vector of parameters	${\omega }_e=\left\{{\omega }_1,{\omega }_2,\dots ,{\omega }_c\right\}$	Weights of experts
$O_h\left(h=1,2,\dots ,b\right)$	b subregions	$M={\left({\Omega }_{ije}\right)}_{m\times n}$	The initial assessment matrix
$\left|O_h\right|$	The number of arguments in the partition $O_h$	$M^{\prime }={\left({\Omega }_{ij}\right)}_{m\times n}$	The comprehensive assessment matrix
${\upsilon }_i$	Complex cubic fractional orthotriple fuzzy number	$A_{ij}={\left[{\eta }_{ij}\right]}_{m\times n}$	The score matrix
$S\left({\upsilon }_i\right)$	The score function of ${\upsilon }_i$	$w_j^s$	The attribute weights according to the entropy method
$H\left({\upsilon }_i\right)$	The accuracy function of ${\upsilon }_i$	$w_j^t$	The attribute weights according to the CILOS method
$E_j$	The entropy of the j-th attribute	$g_{ij}$	The ratio of the j-th attribute of the i-th alternative

displays the nomenclature used in the current study.

2.1. Fractional Orthotriple Fuzzy Set

Definition 1. 

[11] Let $R=\left\{r_1,r_2,\dots ,r_n\right\}$ be a non-empty finite set. Then, for each $r\in R$, the FOFS F can be expressed by

$$F=\left\{r,{\alpha }_F\left(r\right),{\beta }_F\left(r\right),{\chi }_F\left(r\right)\left|r\in R\right.\right\}$$

(1)

where ${\alpha }_F\left(r\right),{\beta }_F\left(r\right),{\chi }_F\left(r\right):R\to \left[0,1\right]$ represent the MD, HD, and NMD, respectively, and satisfy the condition: $0\le {\left({\alpha }_F\left(r\right)\right)}^{\kappa }+{\left({\beta }_F\left(r\right)\right)}^{\kappa }+{\left({\chi }_F\left(r\right)\right)}^{\kappa }\le 1, \kappa \ge 1.$ ${\tau }_F\left(r\right)=\left(1-{\left({\alpha }_F\left(r\right)\right)}^{\kappa }\right.-{\left({\beta }_F\left(r\right)\right)}^{\kappa }-{\left.{\left({\chi }_F\left(r\right)\right)}^{\kappa }\right)}^{1/\kappa }$ represents the rejection degree of r.

2.2. Complex Cubic Fuzzy Set

Definition 2. 

[15] Let $R=\left\{r_1,r_2,\dots ,r_n\right\}$ be a non-empty finite set. Then, for each $r\in R$ , the CCFS can be expressed by

$$CCFS=\left\{r,A\left(r\right),B\left(r\right)\left|r\in R\right.\right\}$$

(2)

where $A\left(r\right)=\left[{\alpha }_{CCFS}^L\left(r\right),{\alpha }_{CCFS}^U\left(r\right)\right]e^{j\left[{\epsilon }_{CCFS}^L\left(r\right),{\epsilon }_{CCFS}^U\left(r\right)\right]}$ represents an interval-value denoted as MD, $B\left(r\right)=\left[{\beta }_{CCFS}\left(r\right)\right]e^{j{\varphi }_{CCFS}\left(r\right)}$ is a single value denoted as NMD, and $j=\sqrt{-1}$.

2.3. The Dombi Operations

Definition 3. 

[26]  Let $g,t$ be two real numbers, $g,t\in \left[0,1\right]$, $\lambda >0$. Dombi t-norm and Dombi t-conrom are described as follows:

$$D_f\left(g,t\right)={\left\{1+{\left[{\left({\displaystyle \frac{1-g}{g}}\right)}^{\lambda }+{\left({\displaystyle \frac{1-t}{t}}\right)}^{\lambda }\right]}^{{\displaystyle \frac{1}{\lambda }}}\right\}}^{-1}$$

(3)

$${D_f}^c\left(g,t\right)=1-{\left\{1+{\left[{\left({\displaystyle \frac{g}{1-g}}\right)}^{\lambda }+{\left({\displaystyle \frac{t}{1-t}}\right)}^{\lambda }\right]}^{{\displaystyle \frac{1}{\lambda }}}\right\}}^{-1}$$

(4)

2.4. PA Operator

Definition 4. 

[43] Let $b_i\left(i=1,2,\dots ,n\right)$ be a set of non-negative real numbers; then, the definition of PA operator is described as follows:

$$PA\left(b_1,b_2,\dots ,b_n\right)={\displaystyle \frac{{{\displaystyle \sum }}_{i=1}^n\left(\left(1+T\left(b_i\right)\right)b_i\right)}{{{\displaystyle \sum }}_{i=1}^n\left(1+T\left(b_i\right)\right)}}$$

(5)

where

$$T\left(b_i\right)={{\displaystyle \sum }}_{j=1,j\ne i}^{n}Sup\left(b_i\right.,\left.b_j\right)$$

(6)

$$Sup\left(b_i,b_j\right)=1-d\left(b_i,b_j\right)$$

(7)

where $d\left(b_i,b_j\right)$ denotes the distance of $b_i$ and $b_j$, and $Sup\left(x_i\right.\left.,x_j\right)$ represents the support of $x_j$ for $x_i$, which satisfies the following conditions:

(1)

$Sup\left(b_i\right.,\left.b_j\right)\in \left[0,\left.1\right]\right.$;

(2)

$Sup\left(b_i\right.,\left.b_j\right)=Sup\left(b_j\right.,\left.b_i\right)$;

(3)

If $d\left(b_i,b_j\right)\le d\left(x,y\right)$, then $Sup\left(b_i\right.,\left.b_j\right)\ge Sup\left(x\right.,\left.y\right)$.

2.5. MM Operator

Definition 5. 

[23] Let $r_i\left(i=1,2,\dots ,\left.n\right)\right.$, be a series of crisp numbers; $P=\left({\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_n\right)\in P^n$ is a vector of parameters, satisfying the condition ${\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_n\ge 0$, but not concurrently ${\vartheta }_1={\vartheta }_2=\dots ={\vartheta }_n=0$. MM operator is defined as follows:

$$M{M}^P\left(r_1,r_2,\dots ,r_n\right)={({\displaystyle \frac{1}{n!}}{{\displaystyle \sum }}_{a\in A_n}{{\displaystyle \prod }}_{i=1}^n{r}_{a_i}^{{\vartheta }_i})}^{{\displaystyle \frac{1}{{{\displaystyle \sum }}_{i=1}^n{\vartheta }_i}}}$$

(8)

where $a_i\left(i=1,2,\dots ,n\right)$ is any permutation of $\left(1,2,\dots ,n\right)$, and $A_n$ is the combination of all the permutations.

When there are different parameter vectors, the following special forms are provided.

(1)

When the parameter vector $P=\left(\vartheta ,\vartheta ,\dots ,\vartheta \right)\left({\vartheta }_i=\vartheta ,i=1,2,\dots ,n\right)$, the MM operator degrades to the geometric mean (GM) operator:

$$M{M}^{\left(\vartheta ,\vartheta ,\dots ,\vartheta \right)}\left(r_1,r_2,\dots ,r_n\right)={\left({{\displaystyle \prod }}_{i=1}^n{r}_i\right)}^{{\left(n\right)}^{-1}}$$

(9)

(2)

When $P=\left(1,0,\dots ,0\right)\left({\vartheta }_1=1,{\vartheta }_i=0,i=2,3,\dots ,n\right)$, the MM operator degrades to the arithmetic mean (AM) operator:

$$M{M}^{\left(1,0,\dots ,0\right)}\left(r_1,r_2,\dots ,r_n\right)={\left(n\right)}^{-1}{{\displaystyle \sum }}_{i=1}^n{r}_i$$

(10)

(3)

When $P=\left({\vartheta }_1,{\vartheta }_2,0,\dots ,0\right)\left({\vartheta }_1,{\vartheta }_2\ne 0,{\vartheta }_i=0,i=3,4,\dots ,n\right)$, the MM operator degrades to the BM operator:

$$M{M}^{\left({\vartheta }_1,{\vartheta }_2,0,\dots ,0\right)}\left(r_1,r_2,\dots ,r_n\right)={\left({\left(n\times \left(n-1\right)\right)}^{-1}{{\displaystyle \sum }}_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n{r}_i^{{\vartheta }_1}{r}_j^{{\vartheta }_2}\right)}^{{\left({\vartheta }_1+{\vartheta }_2\right)}^{-1}}$$

(11)

(4)

When $P=\left(\stackrel{k}{\overbrace{1,1,\dots ,1}},\stackrel{n-k}{\overbrace{0,0,\dots ,0}}\right){\left(\vartheta \right.}_1={\vartheta }_2=\dots ={\vartheta }_k=1,{\vartheta }_{k+1}=$ ${\vartheta }_{k+2}=\dots ={\vartheta }_n=\left.0\right)$, the MM operator degrades to the Maclaurin symmetric mean (MSM) operator [33]:

$$M{M}^{\left(\stackrel{k}{\overbrace{1,1,\dots ,1}},\stackrel{n-k}{\overbrace{0,0,\dots ,0}}\right)}\left(r_1,r_2,\dots ,r_n\right)={\left(\left({{\displaystyle \sum }}_{1\le i_1<\dots <i_k\le n}{{\displaystyle \prod }}_{j=1}^k{r}_{i_j}\right)/C_n^k\right)}^{{\left(k\right)}^{-1}}$$

(12)

2.6. PMM Operator

Definition 6. 

[25]  Let $\left(r_1,r_2,\dots ,r_n\right)$ be a series of non-negative real numbers and $O=\left\{r_1,r_2,\dots ,r_n\right\}$ means the set of $r_1,r_2,\dots ,r_n$, $O_h=\left\{r_1,r_2,\dots ,r_{\left|O_h\right|}\right\}\left(h=1,2,\dots ,b\right)$ be b subregions of R, based on the situation $O_1\cup O_2\cup \dots \cup O_b=O$, and $O_1\cap O_2\cap \dots \cap O_b=\varnothing$; the PMM operator is shown as follows:

$$PM{M}^H\left(r_1,r_2,\dots ,r_n\right)={\displaystyle \frac{1}{b}}{\displaystyle \sum }_{h=1}^b{\left({\displaystyle \frac{1}{\left|O_h\right|!}}{\displaystyle \sum }_{a\in A_{\left|O_h\right|}}{\displaystyle \prod }_{i_h=1}^{\left|O_h\right|}{r}_{a_{i_h}}^{{\vartheta }_{i_h}}\right)}^{{\displaystyle \frac{1}{{{\displaystyle \sum }}_{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}}$$

(13)

where $\left|O_h\right|$ means the number of arguments in the partition $O_h=\left\{r_{1,}{r}_2,\dots ,r_{\left|O_h\right|}\right\}\left(h=1,2,\dots ,b\right)$, $a_{i_h}$ represents any permutation of $\left(1,2,\dots ,\left|O_h\right|\right)$, and $A_{\left|O_h\right|}$ is the convergence of all permutations of $\left(1,2,\dots ,\left|O_h\right|\right)$. $S=\left({\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_n\right)$ collects n real numbers, satisfying ${\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_n\ge 0$ but not concurrently satisfying ${\vartheta }_1={\vartheta }_2=\dots ={\vartheta }_n=0$.

3. Complex Cubic Fractional Orthotriple Fuzzy Aggregation Operators

3.1. Complex Cubic Fractional Orthotriple Fuzzy Set

Definition 7. 

Let $R=\left\{r_1,r_2,\dots ,r_n\right\}$ be a non-empty finite set. Then, for each $r\in R$, the CCFOFS can be expressed by

$$CCFOFS=\left\{r,\left(\Omega \left(r\right),\Xi \left(r\right),\Psi \left(r\right)\right),\left.\left(\eta \left(r\right),\iota \left(r\right),\mu \left(r\right)\right)\right|r\in R\right\}$$

(14)

$\left(\Omega \left(r\right)=\left[{\alpha }_{CCFOFS}^L\left(r\right),{\alpha }_{CCFOFS}^U\left(r\right)\right]e^{j\left[{\epsilon }_{CCFOFS}^L\left(r\right),{\epsilon }_{CCFOFS}^U\left(r\right)\right]},\Xi \left(r\right)=\left[{\beta }_{CCFOFS}^L\left(r\right),\right.\left.{\beta }_{CCFOFS}^U\left(r\right)\right]\right.$ $e^{j\left[{\varphi }_{CCFOFS}^L\left(r\right),{\varphi }_{CCFOFS}^U\left(r\right)\right]},\left.\Psi \left(r\right)=\left[{\chi }_{CCFOFS}^L\left(r\right)\right.,\left.{\chi }_{CCFOFS}^U\left(r\right)\right]e^{j\left[{\phi }_{CCFOFS}^L\left(r\right),{\phi }_{CCFOFS}^U\left(r\right)\right]}\right)$ represents the complex interval-valued fractional orthotriple MD, and $\left(\eta \left(r\right)={\alpha }_{CCFOFS}\left(r\right)e^{j{\epsilon }_{CCFOFS}\left(r\right)}, \right.\iota \left(r\right)={\beta }_{CCFOFS}\left(r\right)e^{j{\varphi }_{CCFOFS}\left(r\right)},\mu \left(r\right)=\left.{\chi }_{CCFOFS}\left(r\right)e^{j{\phi }_{CCFOFS}\left(r\right)}\right)$ represents complex fractional orthotriple NMD, and $j=\sqrt{-1}$. A complex cubic fractional orthotriple fuzzy number (CCFOFN) can be represented as follows: $\upsilon \left(r\right)=\left(\left(\left[{\alpha }_{\upsilon }^L,{\alpha }_{\upsilon }^U\right]\right.\right.\left.e^{j\left[{\epsilon }_{\upsilon }^L,{\epsilon }_{\upsilon }^U\right]},\left[{\beta }_{\upsilon }^L,{\beta }_{\upsilon }^U\right]e^{j\left[{\varphi }_{\upsilon }^L,{\varphi }_{\upsilon }^U\right]},\left[{\chi }_{\upsilon }^L,{\chi }_{\upsilon }^U\right]e^{j\left[{\phi }_{\upsilon }^L,{\phi }_{\upsilon }^U\right]}\right)\left.,\left({\alpha }_{\upsilon }{e}^{j{\epsilon }_{\upsilon }},{\beta }_{\upsilon }{e}^{j{\varphi }_{\upsilon }},{\chi }_{\upsilon }{e}^{j{\phi }_{\upsilon }}\right)\right)$.

Definition 8. 

Let ${\upsilon }_1$ and ${\upsilon }_2$ be two CCFOFNs; then, ${\upsilon }_1=\left(\left({\left[\alpha \right.}_{{\upsilon }_1}^L\right.\left.,\left.{\alpha }_{{\upsilon }_1}^U\right]\right)\right.e^{j\left[{\epsilon }_{{\upsilon }_1}^L,{\epsilon }_{{\upsilon }_1}^U\right]},\left[{\beta }_{{\upsilon }_1}^L,{\beta }_{{\upsilon }_1}^U\right]e^{j\left[{\varphi }_{{\upsilon }_1}^L,{\varphi }_{{\upsilon }_1}^U\right]},\left[{\chi }_{{\upsilon }_1}^L,{\chi }_{{\upsilon }_1}^U\right]e^{j\left[{\phi }_{{\upsilon }_1}^L,{\phi }_{{\upsilon }_1}^U\right]},\left({\alpha }_{{\upsilon }_1}\right.e^{j{\epsilon }_{{\upsilon }_1}},{\beta }_{{\upsilon }_1}{e}^{j{\varphi }_{{\upsilon }_1}},\left.{\chi }_{{\upsilon }_1}\left.e^{j{\phi }_{{\upsilon }_1}}\right)\right),$ ${\upsilon }_2=\left(\left({\left[\alpha \right.}_{{\upsilon }_2}^L\right.\left.,\left.{\alpha }_{{\upsilon }_2}^U\right]\right)\right.e^{j\left[{\epsilon }_{{\upsilon }_2}^L,{\epsilon }_{{\upsilon }_2}^U\right]},\left[{\beta }_{{\upsilon }_2}^L,{\beta }_{{\upsilon }_2}^U\right]e^{j\left[{\varphi }_{{\upsilon }_2}^L,{\varphi }_{{\upsilon }_2}^U\right]},\left[{\chi }_{{\upsilon }_2}^L,{\chi }_{{\upsilon }_2}^U\right]e^{j\left[{\phi }_{{\upsilon }_2}^L,{\phi }_{{\upsilon }_2}^U\right]},\left({\alpha }_{{\upsilon }_2}\right.e^{j{\epsilon }_{{\upsilon }_2}},{\beta }_{{\upsilon }_2}{e}^{j{\varphi }_{{\upsilon }_2}},\left.{\chi }_{{\upsilon }_2}\left.e^{j{\phi }_{{\upsilon }_2}}\right)\right).$ $\lambda >0$, $j=\sqrt{-1}$, and the relevant operational rules are shown as follows:

$$\begin{array}{ll}\left(1\right){\upsilon }_1\oplus {\upsilon }_2=& \left(\left(\left[{\left({\left({\alpha }_{{\upsilon }_1}^L\right)}^{\kappa }+{\left({\alpha }_{{\upsilon }_2}^L\right)}^{\kappa }-{\left({\alpha }_{{\upsilon }_1}^L\right)}^{\kappa }{\left({\alpha }_{{\upsilon }_2}^L\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }+{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }},\right.\right.\right.\\ & \left.{\left({\left({\alpha }_{{\upsilon }_1}^U\right)}^{\kappa }+{\left({\alpha }_{{\upsilon }_2}^U\right)}^{\kappa }-{\left({\alpha }_{{\upsilon }_1}^U\right)}^{\kappa }{\left({\alpha }_{{\upsilon }_2}^U\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }+{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }}\right],\\ & \left[{\left({\left({\beta }_{{\upsilon }_1}^L\right)}^{\kappa }{\left({\beta }_{{\upsilon }_2}^L\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }},{\left({\left({\beta }_{{\upsilon }_1}^U\right)}^{\kappa }{\left({\beta }_{{\upsilon }_2}^U\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }}\right],\\ & \left.\left[{\left({\left({\chi }_{{\upsilon }_1}^L\right)}^{\kappa }{\left({\chi }_{{\upsilon }_2}^L\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }},{\left({\left({\chi }_{{\upsilon }_1}^U\right)}^{\kappa }{\left({\chi }_{{\upsilon }_2}^U\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }}\right]\right),\\ & \left({\left({\left({\alpha }_{{\upsilon }_1}\right)}^{\kappa }+{\left({\alpha }_{{\upsilon }_2}\right)}^{\kappa }-{\left({\alpha }_{{\upsilon }_1}\right)}^{\kappa }{\left({\alpha }_{{\upsilon }_2}\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }+{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }},\right.\\ & \left.\left.{\left({\left({\beta }_{{\upsilon }_1}\right)}^{\kappa }{\left({\beta }_{{\upsilon }_2}\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }},{\left({\left({\chi }_{{\upsilon }_1}\right)}^{\kappa }{\left({\chi }_{{\upsilon }_2}\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }}\right)\right)\end{array}$$

(15)

$$\begin{array}{ll}\left(2\right){\upsilon }_1\otimes {\upsilon }_2=& \left(\left(\left[{\left({\left({\alpha }_{{\upsilon }_1}^L\right)}^{\kappa }{\left({\alpha }_{{\upsilon }_2}^L\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }},\right.\right.\right.\left.{\left({\left({\alpha }_{{\upsilon }_1}^U\right)}^{\kappa }{\left({\alpha }_{{\upsilon }_2}^U\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }}\right],\\ & \left[{\left({\left({\beta }_{{\upsilon }_1}^L\right)}^{\kappa }+{\left({\beta }_{{\upsilon }_2}^L\right)}^{\kappa }-{\left({\beta }_{{\upsilon }_1}^L\right)}^{\kappa }{\left({\beta }_{{\upsilon }_2}^L\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }+{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }},\right.\\ & \left.{\left({\left({\beta }_{{\upsilon }_1}^U\right)}^{\kappa }+{\left({\beta }_{{\upsilon }_2}^U\right)}^{\kappa }-{\left({\beta }_{{\upsilon }_1}^U\right)}^{\kappa }{\left({\beta }_{{\upsilon }_2}^U\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }+{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }}\right],\\ & \left[{\left({\left({\chi }_{{\upsilon }_1}^L\right)}^{\kappa }+{\left({\chi }_{{\upsilon }_2}^L\right)}^{\kappa }-{\left({\chi }_{{\upsilon }_1}^L\right)}^{\kappa }{\left({\chi }_{{\upsilon }_2}^L\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }+{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }},\right.\\ & \left.\left.{\left({\left({\chi }_{{\upsilon }_1}^U\right)}^{\kappa }+{\left({\chi }_{{\upsilon }_2}^U\right)}^{\kappa }-{\left({\chi }_{{\upsilon }_1}^U\right)}^{\kappa }{\left({\chi }_{{\upsilon }_2}^U\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }+{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }}\right]\right),\\ & \left({\left({\left({\alpha }_{{\upsilon }_1}\right)}^{\kappa }{\left({\alpha }_{{\upsilon }_2}\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }}\right.,\\ & {\left({\left({\beta }_{{\upsilon }_1}\right)}^{\kappa }+{\left({\beta }_{{\upsilon }_2}\right)}^{\kappa }-{\left({\beta }_{{\upsilon }_1}\right)}^{\kappa }{\left({\beta }_{{\upsilon }_2}\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }+{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }},\\ & \left.\left.{\left({\left({\chi }_{{\upsilon }_1}\right)}^{\kappa }+{\left({\chi }_{{\upsilon }_2}\right)}^{\kappa }-{\left({\chi }_{{\upsilon }_1}\right)}^{\kappa }{\left({\chi }_{{\upsilon }_2}\right)}^{\kappa }\right)}^{1/\kappa }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }+{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }}\right)\right)\end{array}$$

(16)

$$\begin{array}{ll}\left(3\right)\lambda {\upsilon }_1=& \left(\left(\left[{\left(1-{\left(1-{\left({\alpha }_{{\upsilon }_1}^L\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }{e}^{j2\pi {\left(1-{\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }}\right.,\left.{\left(1-{\left(1-{\left({\alpha }_{{\upsilon }_1}^U\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }{e}^{j2\pi {\left(1-{\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }}\right]\right.\right.,\\ & \left[{\left({\beta }_{{\upsilon }_1}^L\right)}^{\lambda }{e}^{j2\pi {\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\lambda }},{\left({\beta }_{{\upsilon }_1}^U\right)}^{\lambda }{e}^{j2\pi {\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\lambda }}\right],\left.\left[{\left({\chi }_{{\upsilon }_1}^L\right)}^{\lambda }{e}^{j2\pi {\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\lambda }},{\left({\chi }_{{\upsilon }_1}^U\right)}^{\lambda }{e}^{j2\pi {\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\lambda }}\right]\right),\\ & \left({\left(1-{\left(1-{\left({\alpha }_{{\upsilon }_1}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }{e}^{j2\pi {\left(1-{\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }},{\left({\beta }_{{\upsilon }_1}\right)}^{\lambda }{e}^{j2\pi {\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}}{2\pi }}\right)}^{\lambda }},\right.\left.\left.{\left({\chi }_{{\upsilon }_1}\right)}^{\lambda }{e}^{j2\pi {\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}}{2\pi }}\right)}^{\lambda }}\right)\right)\end{array}$$

(17)

$$\begin{array}{ll}\left(4\right){{\upsilon }_1}^{\lambda }=& \left(\left(\left[{\left({\alpha }_{{\upsilon }_1}^L\right)}^{\lambda }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }},{\left({\alpha }_{{\upsilon }_1}^U\right)}^{\lambda }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }}\right]\right.\right.,\\ & \left[{\left(1-{\left(1-{\left({\beta }_{{\upsilon }_1}^L\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }{e}^{j2\pi {\left(1-{\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }},{\left(1-{\left(1-{\left({\beta }_{{\upsilon }_1}^U\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }{e}^{j2\pi {\left(1-{\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }}\right],\\ & \left.\left[{\left(1-{\left(1-{\left({\chi }_{{\upsilon }_1}^L\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }{e}^{j2\pi {\left(1-{\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }},{\left(1-{\left(1-{\left({\chi }_{{\upsilon }_1}^U\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }{e}^{j2\pi {\left(1-{\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }}\right]\right),\\ & \left({\left({\alpha }_{{\upsilon }_1}\right)}^{\lambda }{e}^{j2\pi {\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)}^{1/\kappa }},{\left(1-{\left(1-{\left({\beta }_{{\upsilon }_1}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }{e}^{j2\pi {\left(1-{\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }},\right.\\ & \left.\left.{\left(1-{\left(1-{\left({\chi }_{{\upsilon }_1}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }{e}^{j2\pi {\left(1-{\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\kappa }}\right)\right)\end{array}$$

(18)

Definition 9. 

Let ${\upsilon }_1=\left(\left(\left[{\alpha }_{{\upsilon }_1}^L,{\alpha }_{{\upsilon }_1}^U\right]\right.e^{j\left[{\epsilon }_{{\upsilon }_1}^L,{\epsilon }_{{\upsilon }_1}^U\right]},\right.\left[{\beta }_{{\upsilon }_1}^L,{\beta }_{{\upsilon }_1}^U\right]e^{j\left[{\varphi }_{{\upsilon }_1}^L,{\varphi }_{{\upsilon }_1}^U\right]},\left[{\chi }_{{\upsilon }_1}^L,{\chi }_{{\upsilon }_1}^U\right]e\left.j\left[{\phi }_{{\upsilon }_1}^L,{\phi }_{{\upsilon }_1}^U\right]\right),$$\left({\alpha }_{{\upsilon }_1}\right.e^{j{\epsilon }_{{\upsilon }_1}},{\beta }_{{\upsilon }_1}{e}^{j{\varphi }_{{\upsilon }_1}},\left.{\chi }_{{\upsilon }_1}\left.e^{j{\phi }_{{\upsilon }_1}}\right)\right)$, ${\upsilon }_2=\left(\left(\left[{\alpha }_{{\upsilon }_2}^L\right.\right.\right.,\left.{\alpha }_{{\upsilon }_2}^U\right]e^{j\left[{\epsilon }_{{\upsilon }_2}^L,{\epsilon }_{{\upsilon }_2}^U\right]},\left[{\beta }_{{\upsilon }_2}^L,{\beta }_{{\upsilon }_2}^U\right]e^{j\left[{\varphi }_{{\upsilon }_2}^L,{\varphi }_{{\upsilon }_2}^U\right]},\left[{\chi }_{{\upsilon }_2}^L,{\chi }_{{\upsilon }_2}^U\right]$$e^{j\left[{\phi }_{{\upsilon }_2}^L,{\phi }_{{\upsilon }_2}^U\right]},\left({\alpha }_{{\upsilon }_2}\right.e^{j{\epsilon }_{{\upsilon }_2}},{\beta }_{{\upsilon }_2}{e}^{j{\varphi }_{{\upsilon }_2}},\left.{\chi }_{{\upsilon }_2}\left.e^{j{\phi }_{{\upsilon }_2}}\right)\right)$ be any two CCFOFNs, $j=\sqrt{-1}$; then, the distance between ${\upsilon }_1$ and ${\upsilon }_2$ can be represented as follows:

$$\begin{array}{ll}d\left({\upsilon }_1,{\upsilon }_2\right)& ={\displaystyle \frac{1}{12}}\left(\left|{\left({\alpha }_{{\upsilon }_1}^L\right)}^{\kappa }-{\left({\alpha }_{{\upsilon }_2}^L\right)}^{\kappa }\right|+\left|{\left({\alpha }_{{\upsilon }_1}^U\right)}^{\kappa }-{\left({\alpha }_{{\upsilon }_2}^U\right)}^{\kappa }\right|+\left|{\left({\beta }_{{\upsilon }_1}^L\right)}^{\kappa }-{\left({\beta }_{{\upsilon }_2}^L\right)}^{\kappa }\right|+\left|{\left({\beta }_{{\upsilon }_1}^U\right)}^{\kappa }-{\left({\beta }_{{\upsilon }_2}^U\right)}^{\kappa }\right|\right.+\left|{\left({\chi }_{{\upsilon }_1}^L\right)}^{\kappa }-{\left({\chi }_{{\upsilon }_2}^L\right)}^{\kappa }\right|\\ & +\left|{\left({\chi }_{{\upsilon }_1}^U\right)}^{\kappa }\right.\left.-{\left({\chi }_{{\upsilon }_2}^U\right)}^{\kappa }\right|+{\displaystyle \frac{1}{2\pi }}\left(\left|{\left({\epsilon }_{{\upsilon }_1}^L\right)}^{\kappa }-{\left({\epsilon }_{{\upsilon }_2}^L\right)}^{\kappa }\right|\right.+\left|{\left({\epsilon }_{{\upsilon }_1}^U\right)}^{\kappa }-{\left({\epsilon }_{{\upsilon }_2}^U\right)}^{\kappa }\right|+\left|{\left({\varphi }_{{\upsilon }_1}^L\right)}^{\kappa }-{\left({\varphi }_{{\upsilon }_2}^L\right)}^{\kappa }\right|+\left|{\left({\varphi }_{{\upsilon }_1}^U\right)}^{\kappa }-{\left({\varphi }_{{\upsilon }_2}^U\right)}^{\kappa }\right|\\ & +\left|{\left({\phi }_{{\upsilon }_1}^L\right)}^{\kappa }-{\left({\phi }_{{\upsilon }_2}^L\right)}^{\kappa }\right|+\left.\left.\left|{\left({\phi }_{{\upsilon }_1}^U\right)}^{\kappa }-{\left({\phi }_{{\upsilon }_2}^U\right)}^{\kappa }\right|\right)\right)+{\displaystyle \frac{1}{6}}\left(\left|{\left({\alpha }_{{\upsilon }_1}\right)}^{\kappa }-{\left({\alpha }_{{\upsilon }_2}\right)}^{\kappa }\right|+\left|{\left({\beta }_{{\upsilon }_1}\right)}^{\kappa }-{\left({\beta }_{{\upsilon }_2}\right)}^{\kappa }\right|+\left|{\left({\chi }_{{\upsilon }_1}\right)}^{\kappa }-{\left({\chi }_{{\upsilon }_2}\right)}^{\kappa }\right|+\right.\\ & \left({\displaystyle \frac{1}{2\pi }}\left(\left|{\left({\epsilon }_{{\upsilon }_1}\right)}^{\kappa }-{\left({\epsilon }_{{\upsilon }_2}\right)}^{\kappa }\right|+\right.\right.\left.\left.\left|{\left({\varphi }_{{\upsilon }_1}\right)}^{\kappa }-{\left({\varphi }_{{\upsilon }_2}\right)}^{\kappa }\right|+\left|{\left({\phi }_{{\upsilon }_1}\right)}^{\kappa }-{\left({\phi }_{{\upsilon }_2}\right)}^{\kappa }\right|\right)\right)\end{array}$$

(19)

Definition 10. 

Let ${\upsilon }_1=\left(\left(\left[{\alpha }_{{\upsilon }_1}^L\right.,\left.{\alpha }_{{\upsilon }_1}^U\right]e^{j\left[{\epsilon }_{{\upsilon }_1}^L,{\epsilon }_{{\upsilon }_1}^U\right]}\right.\right.,\left[{\beta }_{{\upsilon }_1}^L,{\beta }_{{\upsilon }_1}^U\right]e^{j\left[{\varphi }_{{\upsilon }_1}^L,{\varphi }_{{\upsilon }_1}^U\right]},\left[{\chi }_{{\upsilon }_1}^L,{\chi }_{{\upsilon }_1}^U\right]\left.e^{j\left[{\phi }_{{\upsilon }_1}^L,{\phi }_{{\upsilon }_1}^U\right]}\right)$, $\left.\left({\alpha }_{{\upsilon }_1}{e}^{j{\epsilon }_{{\upsilon }_1}},{\beta }_{{\upsilon }_1}{e}^{j{\varphi }_{{\upsilon }_1}},{\chi }_{{\upsilon }_1}{e}^{j{\phi }_{{\upsilon }_1}}\right)\right)$ be a CCFOFN; then, its score function $S\left({\upsilon }_1\right)$ can be expressed as

$$\begin{array}{ll}S\left({\upsilon }_1\right)& ={\displaystyle \frac{1}{12}}\left(\left({\left({\alpha }_{{\upsilon }_1}^L\right)}^{\kappa }+{\left({\alpha }_{{\upsilon }_1}^U\right)}^{\kappa }\right)-\left({\left({\beta }_{{\upsilon }_1}^L\right)}^{\kappa }+{\left({\beta }_{{\upsilon }_1}^U\right)}^{\kappa }\right)\right.\left.-\left({\left({\chi }_{{\upsilon }_1}^L\right)}^{\kappa }+{\left({\chi }_{{\upsilon }_1}^U\right)}^{\kappa }\right)\right)+{\left({\displaystyle \frac{1}{2\pi }}\right)}^{\kappa }\left(\left({\left({\epsilon }_{{\upsilon }_1}^L\right)}^{\kappa }+{\left({\epsilon }_{{\upsilon }_1}^U\right)}^{\kappa }\right)-\right.\left({\left({\varphi }_{{\upsilon }_1}^L\right)}^{\kappa }+{\left({\varphi }_{{\upsilon }_1}^U\right)}^{\kappa }\right)\\ & \left.-\left({\left({\phi }_{{\upsilon }_1}^L\right)}^{\kappa }+{\left({\phi }_{{\upsilon }_1}^U\right)}^{\kappa }\right)\right)+{\displaystyle \frac{1}{6}}\left({\left({\alpha }_{{\upsilon }_1}\right)}^{\kappa }-{\left({\beta }_{{\upsilon }_1}\right)}^{\kappa }\right.-{\left({\chi }_{{\upsilon }_1}\right)}^{\kappa }+{\left({\displaystyle \frac{1}{2\pi }}\right)}^{\kappa }\left({\left({\epsilon }_{{\upsilon }_1}\right)}^{\kappa }-\right.\left.\left.{\left({\varphi }_{{\upsilon }_1}\right)}^{\kappa }-{\left({\phi }_{{\upsilon }_1}\right)}^{\kappa }\right)\right)\end{array}$$

(20)

and its accuracy function $H\left({\upsilon }_1\right)$ can be expressed as

$$\begin{array}{ll}H\left({\upsilon }_1\right)& ={\displaystyle \frac{1}{12}}\left(\left({\left({\alpha }_{{\upsilon }_1}^L\right)}^{\kappa }+{\left({\alpha }_{{\upsilon }_1}^U\right)}^{\kappa }\right)+\left({\left({\beta }_{{\upsilon }_1}^L\right)}^{\kappa }+{\left({\beta }_{{\upsilon }_1}^U\right)}^{\kappa }\right)\right.\left.+\left({\left({\chi }_{{\upsilon }_1}^L\right)}^{\kappa }+{\left({\chi }_{{\upsilon }_1}^U\right)}^{\kappa }\right)\right)+{\left({\displaystyle \frac{1}{2\pi }}\right)}^{\kappa }\left(\left({\left({\epsilon }_{{\upsilon }_1}^L\right)}^{\kappa }+{\left({\epsilon }_{{\upsilon }_1}^U\right)}^{\kappa }\right)+\right.\left({\left({\varphi }_{{\upsilon }_1}^L\right)}^{\kappa }+{\left({\varphi }_{{\upsilon }_1}^U\right)}^{\kappa }\right)\\ & \left.+\left({\left({\phi }_{{\upsilon }_1}^L\right)}^{\kappa }+{\left({\phi }_{{\upsilon }_1}^U\right)}^{\kappa }\right)\right)+{\displaystyle \frac{1}{6}}\left({\left({\alpha }_{{\upsilon }_1}\right)}^{\kappa }+{\left({\beta }_{{\upsilon }_1}\right)}^{\kappa }\right.+{\left({\chi }_{{\upsilon }_1}\right)}^{\kappa }+{\left({\displaystyle \frac{1}{2\pi }}\right)}^{\kappa }\left({\left({\epsilon }_{{\upsilon }_1}\right)}^{\kappa }+\right.\left.\left.{\left({\varphi }_{{\upsilon }_1}\right)}^{\kappa }+{\left({\phi }_{{\upsilon }_1}\right)}^{\kappa }\right)\right)\end{array}$$

(21)

Therefore, we can sort CCFOFNs by comparing the score function and accuracy function. Let ${\upsilon }_1$ and ${\upsilon }_2$ be two CCFOFNs. The summarization of the comparison rules can be presented in the following manner:

(1)

If $S\left({\upsilon }_1\right)<S\left({\upsilon }_2\right)$, then ${\upsilon }_1<{\upsilon }_2$;

(2)

If $S\left({\upsilon }_1\right)>S\left({\upsilon }_2\right)$, then ${\upsilon }_1>{\upsilon }_2$;

(3)

If $S\left({\upsilon }_1\right)=S\left({\upsilon }_2\right)$, then

(a)

If $H\left({\upsilon }_1\right)<H\left({\upsilon }_2\right)$, then ${\upsilon }_1<{\upsilon }_2$;

(b)

If $H\left({\upsilon }_1\right)>H\left({\upsilon }_2\right)$, then ${\upsilon }_1>{\upsilon }_2$;

(c)

If $H\left({\upsilon }_1\right)=H\left({\upsilon }_2\right)$, then ${\upsilon }_1={\upsilon }_2$.

3.2. Dombi Operations for Complex Cubic Fractional Orthotriple Fuzzy Set

Definition 11. 

Suppose there are two CCFOFNs: ${\upsilon }_1=\left(\left(\left[{\alpha }_{{\upsilon }_1}^L,{\alpha }_{{\upsilon }_1}^U\right]e^{j\left[{\epsilon }_{{\upsilon }_1}^L,{\epsilon }_{{\upsilon }_1}^U\right]}\right.,\right.\left[{\beta }_{{\upsilon }_1}^L\right.,\left.{\beta }_{{\upsilon }_1}^U\right]$$e^{j\left[{\varphi }_{{\upsilon }_1}^L,{\varphi }_{{\upsilon }_1}^U\right]},\left[{\chi }_{{\upsilon }_1}^L,{\chi }_{{\upsilon }_1}^U\right]\left.e^{j\left[{\phi }_{{\upsilon }_1}^L,{\phi }_{{\upsilon }_1}^U\right]}\right),\left.\left({\alpha }_{{\upsilon }_1}{e}^{j{\epsilon }_{{\upsilon }_1}},{\beta }_{{\upsilon }_1}{e}^{j{\varphi }_{{\upsilon }_1}},{\chi }_{{\upsilon }_1}{e}^{j{\phi }_{{\upsilon }_1}}\right)\right),$ ${\upsilon }_2=\left(\left(\left[{\alpha }_{{\upsilon }_2}^L,{\alpha }_{{\upsilon }_2}^U\right]e^{j\left[{\epsilon }_{{\upsilon }_2}^L,{\epsilon }_{{\upsilon }_2}^U\right]}\right.,\right.\left[{\beta }_{{\upsilon }_2}^L\right.,\left.{\beta }_{{\upsilon }_2}^U\right]e^{j\left[{\varphi }_{{\upsilon }_2}^L,{\varphi }_{{\upsilon }_2}^U\right]},$$\left[{\chi }_{{\upsilon }_2}^L,{\chi }_{{\upsilon }_2}^U\right]\left.e^{j\left[{\phi }_{{\upsilon }_2}^L,{\phi }_{{\upsilon }_2}^U\right]}\right),\left.\left({\alpha }_{{\upsilon }_2}{e}^{j{\epsilon }_{{\upsilon }_2}},{\beta }_{{\upsilon }_2}{e}^{j{\varphi }_{{\upsilon }_2}},{\chi }_{{\upsilon }_2}{e}^{j{\phi }_{{\upsilon }_2}}\right)\right),$ $\lambda >0$, $j=\sqrt{-1}$; then, the operational rules of the CCFOFS based on Dombi operations are presented as follows:

$$\begin{array}{ll}\left(1\right){\upsilon }_1{\oplus }_{Dom}{\upsilon }_2=& \left(\left(\left[{\left(1-{\left(1+{\left({\left({\left({\alpha }_{{\upsilon }_1}^L\right)}^{\kappa }/\left(1-{\left({\alpha }_{{\upsilon }_1}^L\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\alpha }_{{\upsilon }_2}^L\right)}^{\kappa }/\left(1-{\left({\alpha }_{{\upsilon }_2}^L\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right.\right.\right.,\\ & \left.\left.{\left(1-{\left(1+{\left({\left({\left({\alpha }_{{\upsilon }_1}^U\right)}^{\kappa }/\left(1-{\left({\alpha }_{{\upsilon }_1}^U\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\alpha }_{{\upsilon }_2}^U\right)}^{\kappa }/\left(1-{\left({\alpha }_{{\upsilon }_2}^U\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right)\right]\\ & e^{j2\pi \left[{\left(1-{\left(1+{\left({\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },\left.{\left(1-{\left(1+{\left({\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]\right.},\\ & \left[{\left({\left(1+{\left({\left(\left(1-{\left({\beta }_{{\upsilon }_1}^L\right)}^{\kappa }\right)/{\left({\beta }_{{\upsilon }_1}^L\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\beta }_{{\upsilon }_2}^L\right)}^{\kappa }\right)/{\left({\beta }_{{\upsilon }_2}^L\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right.,\\ & \left.{\left({\left(1+{\left({\left(\left(1-{\left({\beta }_{{\upsilon }_1}^U\right)}^{\kappa }\right)/{\left({\beta }_{{\upsilon }_1}^U\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\beta }_{{\upsilon }_2}^U\right)}^{\kappa }\right)/{\left({\beta }_{{\upsilon }_2}^U\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]\\ & e^{j2\pi \left[{\left({\left(1+{\left({\left(\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },\left.{\left({\left(1+{\left({\left(\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]\right.},\\ & \left[{\left({\left(1+{\left({\left(\left(1-{\left({\chi }_{{\upsilon }_1}^L\right)}^{\kappa }\right)/{\left({\chi }_{{\upsilon }_1}^L\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\chi }_{{\upsilon }_2}^L\right)}^{\kappa }\right)/{\left({\chi }_{{\upsilon }_2}^L\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },\right.\\ & \left.{\left({\left(1+{\left({\left(\left(1-{\left({\chi }_{{\upsilon }_1}^U\right)}^{\kappa }\right)/{\left({\chi }_{{\upsilon }_1}^U\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\chi }_{{\upsilon }_2}^U\right)}^{\kappa }\right)/{\left({\chi }_{{\upsilon }_2}^U\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]\\ & \left.e^{j2\pi \left[{\left({\left(1+{\left({\left(\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },\left.{\left({\left(1+{\left({\left(\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]\right.}\right),\\ & \left({\left(1-{\left(1+{\left({\left({\left({\alpha }_{{\upsilon }_1}\right)}^{\kappa }/\left(1-{\left({\alpha }_{{\upsilon }_1}\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\alpha }_{{\upsilon }_2}\right)}^{\kappa }/\left(1-{\left({\alpha }_{{\upsilon }_2}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right.,\\ & {\left({\left(1+{\left({\left(\left(1-{\left({\beta }_{{\upsilon }_1}\right)}^{\kappa }\right)/{\left({\beta }_{{\upsilon }_1}\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\beta }_{{\upsilon }_2}\right)}^{\kappa }\right)/{\left({\beta }_{{\upsilon }_2}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },\\ & \left.{\left({\left(1+{\left({\left(\left(1-{\left({\chi }_{{\upsilon }_1}\right)}^{\kappa }\right)/{\left({\chi }_{{\upsilon }_1}\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\chi }_{{\upsilon }_2}\right)}^{\kappa }\right)/{\left({\chi }_{{\upsilon }_2}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right)\\ & \left.e^{\begin{array}{l}j2\pi \left({\left(1-{\left(1+{\left({\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left({\left(1+{\left({\left(\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },\right.\\ \left.{\left({\left(1+{\left({\left(\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right)\end{array}}\right)\end{array}$$

(22)

$$\begin{array}{ll}\left(2\right){\upsilon }_1{\otimes }_{Dom}{\upsilon }_2=& \left(\left(\left[{\left({\left(1+{\left({\left(\left(1-{\left({\alpha }_{{\upsilon }_1}^L\right)}^{\kappa }\right)/{\left({\alpha }_{{\upsilon }_1}^L\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\alpha }_{{\upsilon }_2}^L\right)}^{\kappa }\right)/{\left({\alpha }_{{\upsilon }_2}^L\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },\right.\right.\right.\\ & \left.{\left({\left(1+{\left({\left(\left(1-{\left({\alpha }_{{\upsilon }_1}^U\right)}^{\kappa }\right)/{\left({\alpha }_{{\upsilon }_1}^U\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\alpha }_{{\upsilon }_2}^U\right)}^{\kappa }\right)/{\left({\alpha }_{{\upsilon }_2}^U\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]\\ & e^{j2\pi \left[{\left({\left(1+{\left({\left(\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left({\left(1+{\left({\left(\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]},\\ & \left[{\left(1-{\left(1+{\left({\left({\left({\beta }_{{\upsilon }_1}^L\right)}^{\kappa }/\left(1-{\left({\beta }_{{\upsilon }_1}^L\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\beta }_{{\upsilon }_2}^L\right)}^{\kappa }/\left(1-{\left({\beta }_{{\upsilon }_2}^L\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },\right.\\ & \left.{\left({\left(1-{\left(1+\left({\left({\left({\beta }_{{\upsilon }_1}^U\right)}^{\kappa }/\left(1-{\left({\beta }_{{\upsilon }_1}^U\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\beta }_{{\upsilon }_2}^U\right)}^{\kappa }/\left(1-{\left({\beta }_{{\upsilon }_2}^U\right)}^{\kappa }\right)\right)}^{\lambda }\right)\right.}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]\\ & e^{j2\pi \left[{\left(1-{\left(1+{\left({\left({\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left(1-{\left(1+{\left({\left({\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]},\\ & \left[{\left(1-{\left(1+{\left({\left({\left({\chi }_{{\upsilon }_1}^L\right)}^{\kappa }/\left(1-{\left({\chi }_{{\upsilon }_1}^L\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\chi }_{{\upsilon }_2}^L\right)}^{\kappa }/\left(1-{\left({\chi }_{{\upsilon }_2}^L\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },\right.\\ & \left.{\left(1-{\left(1+{\left({\left({\left({\chi }_{{\upsilon }_1}^U\right)}^{\kappa }/\left(1-{\left({\chi }_{{\upsilon }_1}^U\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\chi }_{{\upsilon }_2}^U\right)}^{\kappa }/\left(1-{\left({\chi }_{{\upsilon }_2}^U\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]\\ & \left.e^{j2\pi \left[{\left(1-{\left(1+{\left({\left({\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}^L}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left(1-{\left(1+{\left({\left({\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}^U}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]}\right),\\ & \left({\left({\left(1+{\left({\left(\left(1-{\left({\alpha }_{{\upsilon }_1}\right)}^{\kappa }\right)/{\left({\alpha }_{{\upsilon }_1}\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\alpha }_{{\upsilon }_2}\right)}^{\kappa }\right)/{\left({\alpha }_{{\upsilon }_2}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },\right.\\ & {\left(1-{\left(1+{\left({\left({\left({\beta }_{{\upsilon }_1}\right)}^{\kappa }/\left(1-{\left({\beta }_{{\upsilon }_1}\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\beta }_{{\upsilon }_2}\right)}^{\kappa }/\left(1-{\left({\beta }_{{\upsilon }_2}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },\\ & \left.{\left(1-{\left(1+{\left({\left({\left({\chi }_{{\upsilon }_1}\right)}^{\kappa }/\left(1-{\left({\chi }_{{\upsilon }_1}\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\chi }_{{\upsilon }_2}\right)}^{\kappa }/\left(1-{\left({\chi }_{{\upsilon }_2}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right)\\ & \left.e^{j2\pi \left(\begin{array}{l}{\left({\left(1+{\left({\left(\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }+{\left(\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left(1-{\left(1+{\left({\left({\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },\\ {\left(1-{\left(1+{\left({\left({\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }+{\left({\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_2}}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\end{array}\right)}\right)\end{array}$$

(23)

$$\begin{array}{ll}\left(3\right)\zeta {\upsilon }_1=& \left(\left(\left[{\left(1-{\left(1+{\left(\zeta {\left({\left({\alpha }_{{\upsilon }_1}^L\right)}^{\kappa }/\left(1-{\left({\alpha }_{{\upsilon }_1}^L\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },\left.{\left(1-{\left(1+{\left(\zeta {\left({\left({\alpha }_{{\upsilon }_1}^U\right)}^{\kappa }/\left(1-{\left({\alpha }_{{\upsilon }_1}^U\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]\right.\right.\right.\\ & e^{j2\pi \left[{\left(1-{\left(1+{\left(\zeta {\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left(1-{\left(1+{\left(\zeta {\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]},\\ & \left[{\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\beta }_{{\upsilon }_1}^L\right)}^{\kappa }\right)/{\left({\beta }_{{\upsilon }_1}^L\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\beta }_{{\upsilon }_1}^U\right)}^{\kappa }\right)/{\left({\beta }_{{\upsilon }_1}^U\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]\\ & e^{j2\pi \left[{\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]},\\ & \left[{\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\chi }_{{\upsilon }_1}^L\right)}^{\kappa }\right)/{\left({\chi }_{{\upsilon }_1}^L\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\chi }_{{\upsilon }_1}^U\right)}^{\kappa }\right)/{\left({\chi }_{{\upsilon }_1}^U\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]\\ & \left.e^{j2\pi \left[{\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]}\right),\\ & \left({\left(1-{\left(1+{\left(\zeta {\left({\left({\alpha }_{{\upsilon }_1}\right)}^{\kappa }/\left(1-{\left({\alpha }_{{\upsilon }_1}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\beta }_{{\upsilon }_1}\right)}^{\kappa }\right)/{\left({\beta }_{{\upsilon }_1}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },\right.\\ & \left.{\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\chi }_{{\upsilon }_1}\right)}^{\kappa }\right)/{\left({\chi }_{{\upsilon }_1}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right)\\ & \left.e^{j2\pi \left({\left(1-{\left(1+{\left(\zeta {\left({\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right)}\right)\end{array}$$

(24)

$$\begin{array}{ll}\left(4\right){\upsilon }_1^{\zeta }=& \left(\left(\left[{\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\alpha }_{{\upsilon }_1}^L\right)}^{\kappa }\right)/{\left({\alpha }_{{\upsilon }_1}^L\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\alpha }_{{\upsilon }_1}^U\right)}^{\kappa }\right)/{\left({\alpha }_{{\upsilon }_1}^U\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]\right.\right.\\ & e^{j2\pi \left[{\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]},\\ & \left[{\left(1-{\left(1+{\left(\zeta {\left({\left({\beta }_{{\upsilon }_1}^L\right)}^{\kappa }/\left(1-{\left({\beta }_{{\upsilon }_1}^L\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left(1-{\left(1+{\left(\zeta {\left({\left({\beta }_{{\upsilon }_1}^{\mathrm{U}}\right)}^{\kappa }/\left(1-{\left({\beta }_{{\upsilon }_1}^{\mathrm{U}}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]\\ & e^{j2\pi \left[{\left(1-{\left(1+{\left(\zeta {\left({\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left(1-{\left(1+{\left(\zeta {\left({\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]},\\ & \left[{\left(1-{\left(1+{\left(\zeta {\left({\left({\chi }_{{\upsilon }_1}^L\right)}^{\kappa }/\left(1-{\left({\chi }_{{\upsilon }_1}^L\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left(1-{\left(1+{\left(\zeta {\left({\left({\chi }_{{\upsilon }_1}^U\right)}^{\kappa }/\left(1-{\left({\chi }_{{\upsilon }_1}^U\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]\\ & \left.e^{j2\pi \left[{\left(1-{\left(1+{\left(\zeta {\left({\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^L}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left(1-{\left(1+{\left(\zeta {\left({\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}^U}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right]}\right),\\ & \left({\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\alpha }_{{\upsilon }_1}\right)}^{\kappa }\right)/{\left({\alpha }_{{\upsilon }_1}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },\right.{\left(1-{\left(1+{\left(\zeta {\left({\left({\beta }_{{\upsilon }_1}\right)}^{\kappa }/\left(1-{\left({\beta }_{{\upsilon }_1}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },\\ & \left.{\left(1-{\left(1+{\left(\zeta {\left({\left({\chi }_{{\upsilon }_1}\right)}^{\kappa }/\left(1-{\left({\chi }_{{\upsilon }_1}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right)\\ & \left.e^{j2\pi \left({\left({\left(1+{\left(\zeta {\left(\left(1-{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)/{\left({\displaystyle \frac{{\epsilon }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left(1-{\left(1+{\left(\zeta {\left({\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\varphi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa },{\left(1-{\left(1+{\left(\zeta {\left({\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }/\left(1-{\left({\displaystyle \frac{{\phi }_{{\upsilon }_1}}{2\pi }}\right)}^{\kappa }\right)\right)}^{\lambda }\right)}^{1/\lambda }\right)}^{-1}\right)}^{1/\kappa }\right)}\right)\end{array}$$

(25)

3.3. Complex Cubic Fractional Orthotriple Fuzzy Dombi-Partitioned Muirhead Mean Operator

To integrate the flexible generalization capabilities of the Dombi operator, the local adaptability of the partition concept, and the advantages of the Muirhead mean in capturing complex correlations, this paper proposes the CCFOFDPMM operator.

Definition 12. 

Let $\left\{{\upsilon }_1,{\upsilon }_2,\dots ,{\upsilon }_n\right\}\left(i=1,2,\dots ,n\right)$ be a set of CCFOFNs, ${\upsilon }_i=\left(\left(\left[{\alpha }_{{\upsilon }_i}^L,{\alpha }_{{\upsilon }_i}^U\right]\right.\right.$$e^{j\left[{\epsilon }_{{\upsilon }_i}^L,{\epsilon }_{{\upsilon }_i}^U\right]}\left[{\beta }_{{\upsilon }_i}^L,{\beta }_{{\upsilon }_i}^U\right]e^{j\left[{\varphi }_{{\upsilon }_i}^L,{\varphi }_{{\upsilon }_i}^U\right]},$ $\left.\left[{\chi }_{{\upsilon }_i}^L,{\chi }_{{\upsilon }_i}^U\right]e^{j\left[{\phi }_{{\upsilon }_i}^L,{\phi }_{{\upsilon }_i}^U\right]}\right),\left.\left({\alpha }_{{\upsilon }_i}{e}^{j{\epsilon }_{{\upsilon }_i}},{\beta }_{{\upsilon }_i}{e}^{j{\varphi }_{{\upsilon }_i}},{\chi }_{{\upsilon }_i}{e}^{j{\phi }_{{\upsilon }_i}}\right)\right)$, $j=\sqrt{-1}$, and CCFOFNs are divided into b partitions $O_1,O_2,\dots ,O_b,$ $O_h=\left\{{\theta }_1,{\theta }_2,\dots ,{\theta }_{\left|O_h\right|}\right\}$$\left(h=1,2,\right.\dots ,\left.b\right)$, and satisfy the following conditions: $O_1\cup O_2\cup \dots \cup O_b=O,$ $O_1\cap O_2\cap \dots \cap O_b=\varnothing$, $\left|O_1\right|+\left|O_2\right|+\dots +\left|O_b\right|=n$. Then, the CCFOFDPMM operator can be defined as

$$CCFOFDPMM\left({\upsilon }_1,{\upsilon }_2,\dots ,{\upsilon }_n\right)={\left\{{\displaystyle \frac{1}{b}}{\displaystyle \sum _{h=1}^b{\left(\frac{1}{\left|O_h\right|!}{\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\displaystyle \prod _{i_h=1}^{\left|O_h\right|}{\upsilon }_{a_{i_h}}^{{\vartheta }_{i_h}}}}\right)}^{\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}}}\right\}}_{Dom}$$

(26)

where $\left({\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_n\right)$ is a set of n real numbers, satisfying ${\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_n\ge 0$, but not concurrently satisfying ${\vartheta }_1={\vartheta }_2=\dots ={\vartheta }_n=0$. $a_i\left(i=1,2,\dots ,n\right)$ is any permutation of $1,2,\dots ,n$, and $A_n$ is the combination of all the permutations.

Theorem 1. 

Let $\left\{{\upsilon }_1,{\upsilon }_2,\dots ,{\upsilon }_n\right\}$ be a set of CCFOFNs, $j=\sqrt{-1}$. Applying the CCFOFDPMM operator to aggregate, the result is still a CCFOFN:

$$\begin{array}{l}CCFOFDPMM\left({\upsilon }_1,{\upsilon }_2,\dots ,{\upsilon }_n\right)=\\ \left(\left(\left[{{\rm I}}_{\upsilon }^L,{{\rm I}}_{\upsilon }^U\right]e^{j2\pi \left[{{\rm N}}_{\upsilon }^L,{\mathrm{N}}_{\upsilon }^U\right]},\left[{{\rm K}}_{\upsilon }^L,{{\rm K}}_{\upsilon }^U\right]e^{j2\pi \left[{\Pi }_{\upsilon }^L,{\Pi }_{\upsilon }^U\right]},\left[{\Lambda }_{\upsilon }^L,{\Lambda }_{\upsilon }^U\right]e^{j2\pi \left[{\Theta }_{\upsilon }^L,{\Theta }_{\upsilon }^U\right]}\right),\right.\left.\left({{\rm I}}_{\upsilon }{e}^{j2\pi {{\rm N}}_{\upsilon }},{{\rm K}}_{\upsilon }{e}^{j2\pi {\Pi }_{\upsilon }},{\Lambda }_{\upsilon }{e}^{j2\pi {\Theta }_{\upsilon }}\right)\right)\end{array}$$

(27)

where

$$\begin{array}{ll}\left[{{\rm I}}_{\upsilon }^L,{{\rm I}}_{\upsilon }^U\right]=& \left[{\left(1-{\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{{\left({\alpha }_{{\upsilon }_i}^L\right)}^{\kappa }}{1-{\left({\alpha }_{{\upsilon }_i}^L\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}},\right.\\ & \left.{\left(1-{\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{{\left({\alpha }_{{\upsilon }_i}^U\right)}^{\kappa }}{1-{\left({\alpha }_{{\upsilon }_i}^U\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\right]\end{array}$$

(28)

$$\begin{array}{ll}\left[{{\rm N}}_{\upsilon }^L,N_{\upsilon }^U\right]=& \left[{\left(1-{\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{{\left(\frac{{\epsilon }_{{\upsilon }_i}^L}{2\pi }\right)}^{\kappa }}{1-{\left(\frac{{\epsilon }_{{\upsilon }_i}^L}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}},\right.\\ & \begin{array}{c}\left.{\left(1-{\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{{\left(\frac{{\epsilon }_{{\upsilon }_i}^U}{2\pi }\right)}^{\kappa }}{1-{\left(\frac{{\epsilon }_{{\upsilon }_i}^U}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\right]\end{array}\end{array}$$

(29)

$$\begin{array}{ll}\left[{{\rm K}}_{\upsilon }^L,{{\rm K}}_{\upsilon }^U\right]=& \left[{\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{1-{\left({\beta }_{{\upsilon }_i}^L\right)}^{\kappa }}{{\left({\beta }_{{\upsilon }_i}^L\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}},\right.\\ & \left.{\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{1-{\left({\beta }_{{\upsilon }_i}^U\right)}^{\kappa }}{{\left({\beta }_{{\upsilon }_i}^U\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\right]\end{array}$$

(30)

$$\begin{array}{ll}\left[{\Pi }_{\upsilon }^L,{\Pi }_{\upsilon }^U\right]=& \left[{\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{1-{\left(\frac{{\varphi }_{{\upsilon }_i}^L}{2\pi }\right)}^{\kappa }}{{\left(\frac{{\varphi }_{{\upsilon }_i}^L}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}},\right.\\ & \begin{array}{c}\left.{\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{1-{\left(\frac{{\varphi }_{\upsilon }^U}{2\pi }\right)}^{\kappa }}{{\left(\frac{{\varphi }_{\upsilon }^U}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\right]\end{array}\end{array}$$

(31)

$$\begin{array}{ll}\left[{\Lambda }_{\upsilon }^L,{\Lambda }_{\upsilon }^U\right]=& \left[{\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{1-{\left({\chi }_{{\upsilon }_i}^L\right)}^{\kappa }}{{\left({\chi }_{{\upsilon }_i}^L\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}},\right.\\ & \left.{\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{1-{\left({\chi }_{{\upsilon }_i}^U\right)}^{\kappa }}{{\left({\chi }_{{\upsilon }_i}^U\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\right]\end{array}$$

(32)

$$\begin{array}{ll}\left[{\Theta }_{\upsilon }^L,{\Theta }_{\upsilon }^U\right]=& \left[{\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{1-{\left(\frac{{\phi }_{{\upsilon }_i}^L}{2\pi }\right)}^{\kappa }}{{\left(\frac{{\phi }_{{\upsilon }_i}^L}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}},\right.\\ & \left.{\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{1-{\left(\frac{{\phi }_{{\upsilon }_i}^U}{2\pi }\right)}^{\kappa }}{{\left(\frac{{\phi }_{{\upsilon }_i}^U}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\right]\end{array}$$

(33)

$$\begin{array}{ll}{{\rm I}}_{\upsilon }{e}^{j2\pi {{\rm N}}_{\upsilon }}=& \left(\left(1-{\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{{\left({\alpha }_{{\upsilon }_i}\right)}^{\kappa }}{1-{\left({\alpha }_{{\upsilon }_i}\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)\right.\\ & e^{j2\pi {\left(1-{\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{{\left(\frac{{\alpha }_{{\upsilon }_i}}{2\pi }\right)}^{\kappa }}{1-{\left(\frac{{\alpha }_{{\upsilon }_i}}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}}\end{array}$$

(34)

$$\begin{array}{ll}{{\rm K}}_{\upsilon }{e}^{j2\pi {\Pi }_{\upsilon }}=& {\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{1-{\left({\beta }_{{\upsilon }_i}\right)}^{\kappa }}{{\left({\beta }_{{\upsilon }_i}\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\\ & \begin{array}{c}{e}^{j2\pi {\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{1-{\left(\frac{{\varphi }_{\upsilon }}{2\pi }\right)}^{\kappa }}{{\left(\frac{{\varphi }_{\upsilon }}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}}\end{array}\end{array}$$

(35)

$$\begin{array}{ll}{\Lambda }_{\upsilon }{e}^{j2\pi {\Theta }_{\upsilon }}=& {\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{1-{\left({\chi }_{{\upsilon }_i}\right)}^{\kappa }}{{\left({\chi }_{{\upsilon }_i}\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\\ & e^{j2\pi {\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_{\mathrm{h}}\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}{{\displaystyle \sum _{a\in A_{\left|O_h\right|}}\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_i{\left(\frac{1-{\left(\frac{{\phi }_{{\upsilon }_i}}{2\pi }\right)}^{\kappa }}{{\left(\frac{{\phi }_{{\upsilon }_i}}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}\right)}}^{-1}\right)}^{-1}\right)}^{-1}}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}}\end{array}$$

(36)

When there are different parameter vectors, the subsequent special forms are given.

(1)

When the parameter vector $P=\left(\vartheta ,\vartheta ,\dots ,\vartheta \right)\left({\vartheta }_i=\vartheta ,i=1,2,\dots ,n\right)$, the CCFOFDPMM operator degrades to the CCFOFDPGM operator:

$$CCFOFDPM{M}^{\left(\vartheta ,\vartheta ,\dots ,\vartheta \right)}\left(r_1,r_2,\dots ,r_n\right)={\left\{{\left(b\right)}^{-1}{\displaystyle \sum _{h=1}^b{\left({\displaystyle \prod _{i_h=1}^{\left|O_h\right|}{r}_{i_h}}\right)}^{{\left|O_h\right|}^{-1}}}\right\}}_{Dom}$$

(37)

(2)

When $P=\left(1,0,\dots ,0\right)\left({\vartheta }_1=1,{\vartheta }_i=0,i=2,3,\dots ,n\right)$, the CCFOFDPMM operator degrades to the CCFOFDPAM operator:

$$CCFOFDPM{M}^{\left(1,0,\dots ,0\right)}\left(r_1,r_2,\dots ,r_n\right)={\left\{{\left(b\right)}^{-1}{\displaystyle \sum _{h=1}^b\left({\left|O_h\right|}^{-1}{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{r}_{i_h}}\right)}\right\}}_{Dom}$$

(38)

(3)

When $P=\left({\vartheta }_1,{\vartheta }_2,0,\dots ,0\right)\left({\vartheta }_1,{\vartheta }_2\ne 0,{\vartheta }_i=0,i=3,4,\dots ,n\right)$, the CCFOFDPMM operator degrades to the CCFOFDPBM operator:

$$CCFOFDPM{M}^{\left({\vartheta }_1,{\vartheta }_2,0,\dots ,0\right)}\left(r_1,r_2,\dots ,r_n\right)={\left\{{\left(b\right)}^{-1}{\displaystyle \sum _{h=1}^b{\left({\left(\left|O_h\right|\times \left(\left|O_h\right|-1\right)\right)}^{-1}{\displaystyle \sum _{\begin{array}{l}{i}_h,j_h=1\\ i_h\ne j_h\end{array}}^{\left|O_h\right|}{r}_{i_h}^{{\vartheta }_1}{r}_{j_h}^{{\vartheta }_2}}\right)}^{{\left({\vartheta }_1+{\vartheta }_2\right)}^{-1}}}\right\}}_{Dom}$$

(39)

(4)

When $P=\left(\stackrel{k}{\overbrace{1,1,\dots ,1}},\stackrel{n-k}{\overbrace{0,0,\dots ,0}}\right)\left({\vartheta }_1={\vartheta }_2=\dots ={\vartheta }_k=1\right)\left({\vartheta }_{k+1}=\right.{\vartheta }_{k+2}=\left.\dots ={\vartheta }_n=0\right)$, the CCFOFDMM operator degrades to the CCFOFDPMSM operator:

$$CCFOFDPM{M}^{\left(\stackrel{k}{\overbrace{1,1,\dots ,1}},\stackrel{n-k}{\overbrace{0,0,\dots ,0}}\right)}\left(r_1,r_2,\dots ,r_n\right)={\left\{{\left(b\right)}^{-1}{\displaystyle \sum _{h=1}^b{\left(\left({\displaystyle \sum _{1\le i_1<\dots <i_k\le \left|O_h\right|}{\displaystyle \prod _{j=1}^k{r}_{i_j}}}\right)/C_{\left|O_h\right|}^k\right)}^{{\left(k\right)}^{-1}}}\right\}}_{Dom}$$

(40)

3.4. Complex Cubic Fractional Orthotriple Dombi-Weighted Power-Partitioned Muirhead Mean Operator

Definition 13. 

Let $\left\{{\upsilon }_1,{\upsilon }_2,\dots ,{\upsilon }_n\right\}\left(i=1,2,\dots ,n\right)$ be a collection of CCFOFNs, where ${\upsilon }_i=\left(\left(\left[{\alpha }_{{\upsilon }_i}^L,{\alpha }_{{\upsilon }_i}^U\right]\right.e^{j\left[{\epsilon }_{{\upsilon }_i}^L,{\epsilon }_{{\upsilon }_i}^U\right]}\right.,\left[{\beta }_{{\upsilon }_i}^L,{\beta }_{{\upsilon }_i}^U\right]e^{j\left[{\varphi }_{{\upsilon }_i}^L,{\varphi }_{{\upsilon }_i}^U\right]},$$\left[{\chi }_{{\upsilon }_i}^L,{\chi }_{{\upsilon }_i}^U\right]\left.e^{j\left[{\phi }_{{\upsilon }_i}^L,{\phi }_{{\upsilon }_i}^U\right]}\right),\left.\left({\alpha }_{{\upsilon }_i}{e}^{j{\epsilon }_{{\upsilon }_i}},{\beta }_{{\upsilon }_i}{e}^{j{\varphi }_{{\upsilon }_i}},{\chi }_{{\upsilon }_i}{e}^{j{\phi }_{{\upsilon }_i}}\right)\right)$, $j=\sqrt{-1}$, and CCFOFNs are divided into b partitions: $O_1,O_2,\dots ,O_b,O_h=\left\{{\theta }_1,{\theta }_2,\dots ,{\theta }_{\left|O_h\right|}\right\}\left(h=1,2,\right.\dots ,\left.b\right)$, $\left|O_h\right|$ denotes the number of attributes in the h-th partition, and satisfies the subsequent conditions: $O_1\cup O_2\cup \dots \cup O_b=O,O_1\cap O_2\cap \dots \cap O_b=\varnothing$, $\left|O_1\right|+\left|O_2\right|+\dots +\left|O_h\right|=n$. Let ${\varpi }_i$ be the weight of ${\upsilon }_i$, ${\varpi }_i\in \left[0,1\right]$, ${\varpi }_1+{\varpi }_2+\dots +{\varpi }_n=1$; then, the CCFOFDWPPMM operator can be defined as follows:

$$CCFOFDWPPMM\left({\upsilon }_1,{\upsilon }_2,\dots ,{\upsilon }_n\right)={\left\{{\displaystyle \frac{1}{b}}{\displaystyle \sum _{h=1}^b{\left(\frac{1}{\left|O_h\right|!}{\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\displaystyle \prod _{i_h=1}^{\left|O_h\right|}{\left(\frac{n{\omega }_{a_{i_h}}(1+T({\upsilon }_{a_{i_h}}))}{\underset{k=1}{\overset{n}{\Sigma }}{\omega }_k(1+T({\upsilon }_k))}{\upsilon }_{a_{i_h}}\right)}^{{\vartheta }_{i_h}}}}\right)}^{\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}}}\right\}}_{Dom}$$

(41)

where the parameter vector $\left({\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_n\right)$ is a collection of n real numbers, satisfying ${\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_n\ge 0$, but not concurrently satisfying ${\vartheta }_1={\vartheta }_2=\dots ={\vartheta }_n=0$. $a_i\left(i=1,2,\dots ,n\right)$ is any permutation of $1,2,\dots ,n$, and $A_n$ is a combination of all the permutations. $T({\upsilon }_{i_h})=\underset{\begin{array}{l}{j}_h=1\\ j_h\ne i_h\end{array}}{\overset{\left|O_h\right|}{\Sigma }}Sup({\upsilon }_{i_h},{\upsilon }_{j_h})$, $Sup\left({\upsilon }_{i_h},{\upsilon }_{j_h}\right)=1-d\left({\upsilon }_{i_h},{\upsilon }_{j_h}\right)$, where $d\left(b_i,b_j\right)$ indicates the distance of $b_i$ and $b_j$, and $Sup\left({\upsilon }_{i_h},{\upsilon }_{j_h}\right)$ indicates the support from ${\upsilon }_{j_h}$ to ${\upsilon }_{i_h}$, meeting the following conditions:

(1)

$Sup\left({\upsilon }_{i_h},{\upsilon }_{j_h}\right)\in \left[0,1\right]$;

(2)

$Sup\left({\upsilon }_{i_h},{\upsilon }_{j_h}\right)=Sup\left({\upsilon }_{j_h},{\upsilon }_{i_h}\right)$;

(3)

If $d\left({\upsilon }_{i_h},{\upsilon }_{j_h}\right)\le d\left({\upsilon }_{x_h},{\upsilon }_{y_h}\right)$, then $Sup\left({\upsilon }_{i_h},{\upsilon }_{j_h}\right)\ge Sup\left({\upsilon }_{x_h}\right.,\left.{\upsilon }_{y_h}\right)$.

To simplify, let ${\varpi }_i^{\prime }={\displaystyle \frac{1+T\left({\upsilon }_i\right)}{{{\displaystyle \sum }}_{i=1}^n\left(1+T\left({\upsilon }_i\right)\right)}}$, where ${\varpi }_i^{\prime }\in \left[0\right.,\left.1\right]$ and ${\displaystyle \sum }_{i=1}^n{\varpi }_i^{\prime }=1$. Therefore, the equation can be reduced to the following:

$$CCFOFDWPPMM\left({\upsilon }_1,{\upsilon }_2,\dots ,{\upsilon }_n\right)={\left\{{\displaystyle \frac{1}{b}}{\displaystyle \sum _{h=1}^b{\left(\frac{1}{\left|O_h\right|!}{\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\displaystyle \prod _{i_h=1}^{\left|O_h\right|}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\upsilon }_{a_{i_h}}\right)}^{{\vartheta }_{i_h}}}}\right)}^{\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}}}\right\}}_{Dom}$$

(42)

Theorem 2. 

Let $\left\{{\upsilon }_1,{\upsilon }_2,\dots ,{\upsilon }_n\right\}$ be a collection of CCFOFNs, $j=\sqrt{-1}$. Applying the CCFOFDWPPMM operator to aggregate, the result is still a CCFOFN:

$$\begin{array}{l}CCFOFDWPPMM\left({\upsilon }_1,{\upsilon }_2 ,\dots ,{\upsilon }_n\right)=\\ \left(\left(\left[{{\rm P}}_{\upsilon }^L,{{\rm P}}_{\upsilon }^U\right]e^{j2\pi \left[{\Omega }_{\upsilon }^L,{\Omega }_{\upsilon }^U\right]},\left[{\Sigma }_{\upsilon }^L,{\Sigma }_{\upsilon }^U\right]e^{j2\pi \left[{\Xi }_{\upsilon }^L,{\Xi }_{\upsilon }^U\right]},\left[{{\rm T}}_{\upsilon }^L,{{\rm T}}_{\upsilon }^U\right]e^{j2\pi \left[{\Psi }_{\upsilon }^L,{\Psi }_{\upsilon }^U\right]}\right),\right.\left.\left({{\rm P}}_{\upsilon }{e}^{j2\pi {\Omega }_{\upsilon }},{\Sigma }_{\upsilon }{e}^{j2\pi {\Xi }_{\upsilon }},{{\rm T}}_{\upsilon }{e}^{j2\pi {\Psi }_{\upsilon }}\right)\right)\end{array}$$

(43)

where

$$\begin{array}{ll}\left[{{\rm P}}_{\upsilon }^L,{{\rm P}}_{\upsilon }^U\right]=& \left[{\left(1-{\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{{\left({\alpha }_{{\upsilon }_i}^L\right)}^{\kappa }}{1-{\left({\alpha }_{{\upsilon }_i}^L\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}},\right.\\ & \left.{\left(1-{\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{{\left({\alpha }_{{\upsilon }_i}^U\right)}^{\kappa }}{1-{\left({\alpha }_{{\upsilon }_i}^U\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\right]\end{array}$$

(44)

$$\begin{array}{ll}\left[{\Omega }_{\upsilon }^L,{\Omega }_{\upsilon }^U\right]=& \left[{\left(1-{\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{{\left(\frac{{\epsilon }_{{\upsilon }_i}^L}{2\pi }\right)}^{\kappa }}{1-{\left(\frac{{\epsilon }_{{\upsilon }_i}^L}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}},\right.\\ & \left.{\left(1-{\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{{\left(\frac{{\epsilon }_{{\upsilon }_i}^U}{2\pi }\right)}^{\kappa }}{1-{\left(\frac{{\epsilon }_{{\upsilon }_i}^U}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\right]\end{array}$$

(45)

$$\begin{array}{ll}\left[{\Sigma }_{\upsilon }^L,{\Sigma }_{\upsilon }^U\right]=& \left[{\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{1-{\left({\beta }_{{\upsilon }_i}^L\right)}^{\kappa }}{{\left({\beta }_{{\upsilon }_i}^L\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}},\right.\\ & \left.{\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{1-{\left({\beta }_{{\upsilon }_i}^U\right)}^{\kappa }}{{\left({\beta }_{{\upsilon }_i}^U\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\right]\end{array}$$

(46)

$$\begin{array}{ll}\left[{\Xi }_{\upsilon }^L,{\Xi }_{\upsilon }^U\right]=& \left[{\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{1-{\left(\frac{{\varphi }_{{\upsilon }_i}^L}{2\pi }\right)}^{\kappa }}{{\left(\frac{{\varphi }_{{\upsilon }_i}^L}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}},\right.\\ & \left.{\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{1-{\left(\frac{{\varphi }_{{\upsilon }_i}^U}{2\pi }\right)}^{\kappa }}{{\left(\frac{{\varphi }_{{\upsilon }_i}^U}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\right]\end{array}$$

(47)

$$\begin{array}{ll}\left[{{\rm T}}_{\upsilon }^L,{{\rm T}}_{\upsilon }^U\right]=& \left[{\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{1-{\left({\chi }_{{\upsilon }_i}^L\right)}^{\kappa }}{{\left({\chi }_{{\upsilon }_i}^L\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}},\right.\\ & \left.{\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{1-{\left({\chi }_{{\upsilon }_i}^U\right)}^{\kappa }}{{\left({\chi }_{{\upsilon }_i}^U\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\right]\end{array}$$

(48)

$$\begin{array}{ll}\left[{\Psi }_{\upsilon }^L,{\Psi }_{\upsilon }^U\right]=& \left[{\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{1-{\left(\frac{{\phi }_{{\upsilon }_i}^L}{2\pi }\right)}^{\kappa }}{{\left(\frac{{\phi }_{{\upsilon }_i}^L}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}},\right.\\ & \left.{\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_b\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{1-{\left(\frac{{\phi }_{{\upsilon }_i}^U}{2\pi }\right)}^{\kappa }}{{\left(\frac{{\phi }_{{\upsilon }_i}^U}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\right]\end{array}$$

(49)

$$\begin{array}{ll}{{\rm P}}_{\upsilon }{e}^{j2\pi {\Omega }_{\upsilon }}=& {\left(1-{\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{{\left({\alpha }_{{\upsilon }_i}\right)}^{\kappa }}{1-{\left({\alpha }_{{\upsilon }_i}\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\\ & e^{j2\pi {\left(1-{\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{{\left(\frac{{\epsilon }_{{\upsilon }_i}}{2\pi }\right)}^{\kappa }}{1-{\left(\frac{{\epsilon }_{{\upsilon }_i}}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}}\end{array}$$

(50)

$$\begin{array}{ll}{\Sigma }_{\upsilon }{e}^{j2\pi {\Xi }_{\upsilon }}=& {\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{1-{\left({\beta }_{{\upsilon }_i}\right)}^{\kappa }}{{\left({\beta }_{{\upsilon }_i}\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\\ & e^{j2\pi {\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{1-{\left(\frac{{\varphi }_{{\upsilon }_i}}{2\pi }\right)}^{\kappa }}{{\left(\frac{{\varphi }_{{\upsilon }_i}}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}}\end{array}$$

(51)

$$\begin{array}{ll}{T}_{\upsilon }{e}^{j2\pi {\Psi }_{\upsilon }}=& {\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{1-{\left({\chi }_{{\upsilon }_i}\right)}^{\kappa }}{{\left({\chi }_{{\upsilon }_i}\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}\\ & e^{j2\pi {\left({\left(1+{\left({\displaystyle \frac{1}{b}}\left({\displaystyle \sum _{h=1}^b\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\vartheta }_{i_h}}}{\left(\frac{1}{\left|O_h\right|!}\left({\displaystyle \sum _{a\in A_{\left|O_h\right|}}{\left({\displaystyle \sum _{i_h=1}^{\left|O_h\right|}\left({\vartheta }_{i_h}{\left(\frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}{\left(\frac{1-{\left(\frac{{\phi }_{{\upsilon }_i}}{2\pi }\right)}^{\kappa }}{{\left(\frac{{\phi }_{{\upsilon }_i}}{2\pi }\right)}^{\kappa }}\right)}^{\lambda }\right)}^{-1}\right)}\right)}^{-1}}\right)\right)}^{-1}\right)}\right)\right)}^{{\displaystyle \frac{1}{\lambda }}}\right)}^{-1}\right)}^{{\displaystyle \frac{1}{\kappa }}}}\end{array}$$

(52)

The proof of Theorem 2 is shown in Appendix A.

When there are different parameter vectors, the following special forms are required. For convenience, let

$${\displaystyle \frac{n{\varpi }_{a_{i_h}}{{\varpi }_{a_{i_h}}}^{\prime }}{\underset{k=1}{\overset{n}{\Sigma }}{\varpi }_k{\varpi }_k^{\prime }}}={\wp }_{a_{i_h}}$$

(53)

(1)

When $P=\left(\vartheta ,\vartheta ,\dots ,\vartheta \right)\left({\vartheta }_i=\vartheta ,i=1,2,\dots ,n\right)$, the CCFOFDWPPMM operator degrades to the CCFOFDWPPGM operator:

$$CCFOFDWPPM{M}^{\left(\vartheta ,\vartheta ,\dots ,\vartheta \right)}\left(r_1,r_2,\dots ,r_n\right)={\left\{{\left(b\right)}^{-1}{\displaystyle \sum _{h=1}^b{\left({\displaystyle \prod _{i_h=1}^{\left|O_h\right|}{\wp }_{a_{i_h}}{r}_{i_h}}\right)}^{{\left|O_h\right|}^{-1}}}\right\}}_{Dom}$$

(54)

(2)

When $P=\left(1,0,\dots ,0\right)\left({\vartheta }_1=1,{\vartheta }_i=0,i=2,3,\dots ,n\right)$, the CCFOFDWPPMM operator degrades to the CCFOFDWPPAM operator:

$$CCFOFDWPPM{M}^{\left(1,0,\dots ,0\right)}\left(r_1,r_2,\dots ,r_n\right)={\left\{{\left(b\right)}^{-1}{\displaystyle \sum _{h=1}^b\left({\left|O_h\right|}^{-1}{\displaystyle \sum _{i_h=1}^{\left|O_h\right|}{\wp }_{a_{i_h}}{r}_{i_h}}\right)}\right\}}_{Dom}$$

(55)

(3)

When $P=\left({\vartheta }_1,{\vartheta }_2,0,\dots ,0\right)\left({\vartheta }_1,{\vartheta }_2\ne 0,{\vartheta }_i=0,i=3,4,\dots ,n\right)$, the CCFOFDWPPMM operator degrades to the CCFOFDWPPBM operator:

$$CCFOFDWPPM{M}^{\left({\vartheta }_1,{\vartheta }_2,0,\dots ,0\right)}\left(r_1,r_2,\dots ,r_n\right)={\left\{{\left(b\right)}^{-1}{\displaystyle \sum _{h=1}^b{\left({\left(\left|O_h\right|\times \left(\left|O_h\right|-1\right)\right)}^{-1}{\displaystyle \sum _{\begin{array}{l}{i}_h,j_h=1\\ i_h\ne j_h\end{array}}^{\left|O_h\right|}{\wp }_{a_{i_h}}{r}_{i_h}^{{\vartheta }_1}{r}_{j_h}^{{\vartheta }_2}}\right)}^{{\left({\vartheta }_1+{\vartheta }_2\right)}^{-1}}}\right\}}_{Dom}$$

(56)

(4)

When $P=\left(\stackrel{k}{\overbrace{1,1,\dots ,1}},\stackrel{n-k}{\overbrace{0,0,\dots ,0}}\right)\left({\vartheta }_1\right.={\vartheta }_2=\dots ={\vartheta }_k=1,{\vartheta }_{k+1}={\vartheta }_{k+2}=\dots ={\vartheta }_n=\left.0\right)$, the CCFOFDWPPMM operator degrades to the CCFOFDWPPMSM operator:

$$CCFOFDWPPM{M}^{\left(\stackrel{k}{\overbrace{1,1,\dots ,1}},\stackrel{n-k}{\overbrace{0,0,\dots ,0}}\right)}\left(r_1,r_2,\dots ,r_n\right)={\left\{{\left(b\right)}^{-1}{{\displaystyle \sum }}_{h=1}^b{\left(\left({{\displaystyle \sum }}_{1\le i_1<\dots <i_k\le \left|O_h\right|}{{\displaystyle \prod }}_{j=1}^k{\wp }_{a_{i_h}}{r}_{i_j}\right)/C_{\left|O_h\right|}^k\right)}^{{\left(k\right)}^{-1}}\right\}}_{Dom}$$

(57)

4. Constructing an Assessment Indicator System for the Construction of First-Class Disciplines

To enhance the competitiveness of first-class disciplines in industry-characteristic universities and optimize their disciplinary layout, it is necessary to assess the effectiveness of first-class discipline construction in industry-characteristic universities through a first-class discipline construction assessment. Due to the obvious social nature of first-class disciplines, the effectiveness of their construction can directly reflect the value and characteristics of university disciplines and majors. Therefore, first-class discipline construction assessment is a complex process that requires analysis from a multidimensional and multilevel perspective.

In selecting assessment indicators, we strictly followed the principles of scientificity, objectivity, operability, and universality, and used a three-stage mixed method to construct the assessment indicator system. Firstly, based on the WOS database, the relevant literature was searched with the keywords of ‘World Class Discipline evaluation OR assessment‘ and ‘First Class Discipline evaluation OR assessment’. At the same time, we referred to the policy documents issued by the Ministry of Education (MOE) and other authoritative institutions of China, such as the “Interim Measures for Evaluating the Effectiveness of Double First-Class Initiative”, “Guidelines on building a strong educational country (2024–2035)”, “Reform Plan for Adjusting and Optimizing the Discipline and Major Settings of Ordinary Higher Education”, and other policy documents, and closely combined the basic functions of universities, namely talent cultivation, scientific research, social services, and open education, to establish 22 initial candidate indicators. According to their connotations and characteristics, they were classified into their corresponding dimensions, and a preliminary indicator database was constructed. Secondly, based on three criteria, namely, frequency of occurrence ≥ 3 times, expert-assessed feasibility, and strong correlation with the assessment of first-class discipline construction in industry-characteristic universities, four indicators were screened out. The reasons for exclusion are shown in

Table 2. Reason for exclusion of an indicator.

Exclusion Indicator	Violated Standards	Specific Reasons
Industry fiscal resource investment [44]	Strong correlation with the assessment of first-class discipline construction in industry-characteristic universities	This indicator reflects the scale of resources rather than the effectiveness of their use, but first-class discipline construction in industry-characteristic universities emphasizes “output orientation” rather than input scale, and the correlation between the two is weak.
Industry teacher title structure [45]	Expert-assessed feasibility	The proportion of senior professional titles cannot directly reflect the quality of teaching staff, and experts find it difficult to quantify “structural rationality,” which can easily lead to the misconception of “focusing solely on titles.”
Industry recognition Academic reputation [46]	Expert-assessed feasibility	Relies on subjective impression surveys; experts may be subject to assessment bias due to differences in regional or disciplinary backgrounds, making it impossible to establish universal assessment standards.
Industry Student Satisfaction [47]	Frequency of occurrence ≥ 3 times	Only two documents mention this, and there is a lack of universal assessment value.

. Eighteen indicators were ultimately retained. Subsequently, the Delphi method was used for expert consultation. Ten senior experts in the fields of higher education assessment, discipline construction, and university governance were invited to score the indicators. Based on the opinions of the experts, the 18 indicators were selected and fine-tuned. Finally, this paper constructed an assessment indicator system for first-class discipline construction in industry-characteristic universities, which includes four dimensions and sixteen indicators. The specific assessment indicators are shown in

Table 3. First-class discipline construction assessment indicator system.

Construction Assessment for First-Class Disciplines	Talent Cultivation (O1)	Competitiveness of Graduates in Core Industry Positions (F1)	[48,49]
		Industry Target Alignment (F2)	[50,51]
		Industry Practice Platform Support (F3)	[52,53]
		Industry–Academia Integration (F4)	[54,55]
	Scientific Research (O2)	Industry Standards and Patent Value (F5)	[56,57]
		Industry Think-Tank Influence Strength (F6)	[58,59]
		Contributions to Breakthroughs in Key Technologies in the Industry (F7)	[60,61]
		Industry Innovation Platform Level (F8)	[62,63]
	Social Service (O3)	High-End Training Brand in the Industry (F9)	[64,65]
		Adequacy of Emergency Service Response (F10)	[66,67]
		Effectiveness of Technology Transfer (F11)	[68,69]
		Industry Ecosystem Co-Construction Participation Rate (F12)	[70,71]
	Open Education (O4)	International Rule-Making Authority (F13)	[72,73]
		Depth of International Cooperation (F14)	[74,75]
		International Student Industry Suitability (F15)	[76,77]
		Industry Resource Coordination Density (F16)	[78,79]

.

Talent cultivation in first-class discipline construction in industry-characteristic universities is a comprehensive expression of the discipline in terms of talent cultivation and related achievements, and can fundamentally reflect its effectiveness in educating people. It includes the following four aspects: the competitiveness of graduates in core industry positions, industry target alignment, industry practice platform support, and industry–academia integration. The competitiveness of graduates in core industry positions refers to the proportion of graduates entering leading enterprises in the industry and key positions, as well as their career development potential. It directly reflects the market suitability of the talent output of the discipline and the core needs of the industry. Its competitiveness is the core yardstick for the social recognition of the discipline. Industry target alignment refers to the degree of dynamic matching between the discipline’s training program and the industry’s competency requirements, including the speed of the curriculum system’s response to technological changes. This degree of alignment determines the solidity of the undergraduate’s industry literacy foundation and guarantees the quality of talent cultivation. Industry practice platform support refers to the degree of realism and the utilization rate of industry–academia collaboration laboratories, training bases, and other platforms in simulating real production scenarios, directly influencing students’ ability to solve complex engineering problems. Industry–academia integration refers to the proportion of faculty with industry backgrounds and the intensity of industry experts’ substantive involvement in teaching. The depth of this integration directly shapes students’ industry-oriented thinking and practical skills.

Scientific research for first-class discipline construction in industry-characteristic universities is systematic, creative work carried out by universities to enrich disciplinary knowledge and use this knowledge to invent new technologies. It covers industry standards and patent value, industry think-tank influence strength, contributions to breakthroughs in key technologies in the industry, and the industry innovation platform level. Industry standards and patent value refer to the hierarchical level of the number of industry technical standards and specifications led by the discipline, as well as the economic benefits generated by patent conversion within a fuzzy range. Industry think-tank influence strength refers to the decision-making level at which the policy recommendations submitted to the government or industry associations are adopted, and the breadth of diffusion of industry trend forecast reports cited by leading enterprises. Contributions to breakthroughs in key technologies in the industry refer to the level of industry-critical technological R&D projects that are undertaken and the penetration depth of their outcomes within the industrial chain. Industry innovation platform level refers to the equipment-sharing rate and industrial service activity level of national-level or provincial/ministerial-level key laboratories and engineering centers established by the discipline.

Social services aim to solve industry pain points and promote social progress as their ultimate goal, referring to the actual contribution of industry-characteristic universities’ first-class disciplines to national strategic needs and industrial development. This requires consideration of the high-end training reputation in the industry, adequacy of emergency service responses, effectiveness of technology transfer, and the industry ecosystem co-construction participation rate. High-end training reputation in the industry refers to the authority of certified training programs for senior technical managers in the industry and the distribution of participants’ job levels, reflecting the accuracy of academic knowledge, which feeds back into industry talent renewal and brand premium. The adequacy of emergency service response refers to the response time for providing technical solutions to major industry accidents and the fuzzy valuation of loss recovery, reflecting the academic disciplines’ fulfillment of their social responsibility to protect the industry’s safety bottom line. The effectiveness of technology transfer refers to the coverage of scientific and technological achievements in industrial chain enterprises and the contribution to industrial upgrading, quantifying the efficiency of disciplinary technology diffusion and industrial value-creation capabilities. Industry ecosystem co-construction participation rate refers to the number of industrial innovation consortia and common technology diffusion platforms led by the discipline, as well as the coverage rate of leading enterprises, measuring the organizational capacity of the industry ecosystem.

Open education focuses on the role and position of industry-characteristic universities’ first-class disciplines in international educational exchange networks, referring to the degree to which disciplines are integrated into the global educational ecosystem. It includes four levels: international rule-making authority, the depth of international cooperation, the suitability for international students in industry, and the industry resource coordination density. The international rule-making authority refers to the weight of experts’ positions in international industry standard-setting organizations and the frequency of their playing a leading role in revising standards, reflecting the strategic ability of disciplines to establish technical rules in global industry. The depth of international cooperation refers to the number and level of jointly established laboratories and collaborative research projects with top international industry institutions, as well as the depth of participation in technological breakthroughs. The suitability for international students in industry refers to the proportion of international students from industry-related fields and the degree of localization of the training programs. The industry resource coordination density refers to the annual intensity of leading enterprises’ investment in jointly building industry–university–research platforms and the assessment of project sustainability, which represents the discipline’s ability to attract core industry resources.

5. Multi-Attribute Assessment Method Based on the CCFOFDWPPMM Operator and the IDOCRIW Method

In this section, we propose a new multi-attribute assessment method based on the presented CCFOFDWPPMM operator and the IDOCRIW method. Suppose $Z_i=\left\{Z_1,Z_2,\dots ,Z_m\right\}$ represents a collection of alternatives, $F_j=\left\{F_1,F_2,\dots ,F_n\right\}$ means a group of attributes, and the weights of attributes are expressed by ${\varpi }_j=\left\{{\varpi }_1,{\varpi }_2,\dots ,{\varpi }_n\right\}$, where ${\varpi }_j\in \left[0,1\right]$ and ${\varpi }_1+{\varpi }_2+\dots +{\varpi }_n=1$. Then, the attributes are divided into b partitions, denoted as $O_1,O_2,\dots ,O_b$, where $O_h=\left\{{\theta }_1,{\theta }_2,\dots ,{\theta }_{\left|O_h\right|}\right\}\left(h=1,2,\dots ,b\right)$. $\left|O_h\right|$ represents the number of attributes in the h-th partition, where $O_1\cup O_2\cup \dots \cup O_b=O,O_1\cap O_2\cap \dots \cap O_b=\varnothing$ and $\left|O_1\right|+\left|O_2\right|+\dots +\left|O_b\right| =n$. We invite a group of experts $D_e=\left\{e_1,e_2,\dots ,e_c\right\}$ by applying CCFOFNs to characterize their assessment information. The weights of experts are denoted by ${\omega }_e=\left\{{\omega }_1,{\omega }_2,\dots ,{\omega }_c\right\}$, where ${\omega }_e\in \left[0,1\right]$ and ${\omega }_1+{\omega }_2+\dots +{\omega }_c=1$. The assessment matrix is expressed as $M={\left({\Omega }_{ije}\right)}_{m\times n}$, where $i=1,2,\dots ,m;j=1,2,\dots ,n;e=1,2,\dots ,c$. ${\Omega }_{ije}=\left(\left(\left[{\alpha }_{{\Omega }_{ije}}^L,{\alpha }_{{\Omega }_{ije}}^U\right]e^{j\left[{\epsilon }_{{\Omega }_{ij}e}^L,{\epsilon }_{{\Omega }_{ij}e}^U\right]},\right.\right.\left[{\beta }_{{\Omega }_{ije}}^L,{\beta }_{{\Omega }_{ije}}^U\right]e^{j\left[{\varphi }_{{\Omega }_{ije}}^L,{\varphi }_{{\Omega }_{ij}e}^U\right]},\left[{\chi }_{{\Omega }_{ije}}^L,{\chi }_{{\Omega }_{ije}}^U\right]\left.e^{j\left[{\phi }_{{\Omega }_{ij}e}^L,{\phi }_{{\Omega }_{ije}}^U\right]}\right),$$\left({\alpha }_{{\Omega }_{ije}}{e}^{j{\epsilon }_{{\Omega }_{ije}}}\right.,{\beta }_{{\Omega }_{ije}}{e}^{j{\varphi }_{{\Omega }_{ije}}},\left.\left.{\chi }_{{\Omega }_{ije}}{e}^{j{\phi }_{{\Omega }_{ije}}}\right)\right)$ is a CCFOFN, which shows the assessment value with the attribute $F_j$ of alternative $Z_i$ given by expert $D_e$. To address the complex multi-attribute assessment problem, we apply the proposed multi-attribute assessment method based on the CCFOFDWPPMM operator and the IDOCRIW method, which is shown in Figure 1. Specific procedures are as follows:

Step 1: Normalize the expert assessment matrix. Attributes include cost attributes and benefit attributes, which have positive or negative effects on the aggregation results, respectively. To eliminate the effects of different attribute types, we can convert cost attributes into benefit attributes. The conversion rules for the CCFOFNs ${\Omega }_{ije}$ are as follows:

$${{\Omega }_{ije}}^{\prime }=\left\{\begin{array}{l}\left(\left(\left[{\alpha }_{{\Omega }_{ije}}^L,{\alpha }_{{\Omega }_{ije}}^U\right]\right.\right.\left.e^{j\left[{\epsilon }_{{\Omega }_{ije}}^L,{\epsilon }_{{\Omega }_{ije}}^U\right]},\left[{\beta }_{{\Omega }_{ije}}^L,{\beta }_{{\Omega }_{ije}}^U\right]e^{j\left[{\varphi }_{{\Omega }_{ije}}^L,{\varphi }_{{\Omega }_{ije}}^U\right]},\left[{\chi }_{{\Omega }_{ije}}^L,{\chi }_{{\Omega }_{ije}}^U\right]e^{j\left[{\phi }_{{\Omega }_{ije}}^L,{\phi }_{{\Omega }_{ije}}^U\right]}\right)\left.,\left({\alpha }_{{\Omega }_{ije}}{e}^{j{\epsilon }_{{\Omega }_{ije}}},{\beta }_{{\Omega }_{ije}}{e}^{j{\varphi }_{{\Omega }_{ije}}},{\chi }_{{\Omega }_{ije}}{e}^{j{\phi }_{{\Omega }_{ije}}}\right)\right),\\ \begin{array}{c}\mathrm{if} F_j \mathrm{is} \mathrm{a} \mathrm{benefit} \mathrm{attribute}\end{array}\\ \left(\left(\left[{\chi }_{{\Omega }_{ije}}^L,{\chi }_{{\Omega }_{ije}}^U\right]e^{j\left[{\phi }_{{\Omega }_{ije}}^L,{\phi }_{{\Omega }_{ije}}^U\right]}\right.\right.\left.,\left[{\beta }_{{\Omega }_{ije}}^L,{\beta }_{{\Omega }_{ije}}^U\right]e^{j\left[{\varphi }_{{\Omega }_{ije}}^L,{\varphi }_{{\Omega }_{ije}}^U\right]},\left[{\alpha }_{{\Omega }_{ije}}^L,{\alpha }_{{\Omega }_{ije}}^U\right]e^{j\left[{\epsilon }_{{\Omega }_{ije}}^L,{\epsilon }_{{\Omega }_{ije}}^U\right]}\right)\left.,\left({\chi }_{{\Omega }_{ije}}{e}^{j{\phi }_{{\Omega }_{ije}}},{\beta }_{{\Omega }_{ije}}{e}^{j{\varphi }_{{\Omega }_{ije}}},{\alpha }_{{\Omega }_{ije}}{e}^{j{\epsilon }_{{\Omega }_{ije}}}\right)\right)\\ \begin{array}{c}\mathrm{if} F_j \mathrm{is} \mathrm{a} \cos \mathrm{t} \mathrm{attribute}\end{array}\end{array}\right.$$

(58)

Step 2: According to the normalized expert assessment matrix, we can apply the CCFOFDWPPMM operator based on Equation (41) to obtain the comprehensive assessment matrix $M^{\prime }={\left({\Omega }_{ij}\right)}_{m\times n}$.

Step 3: Calculate the attribute weights $w_j^s$ according to the entropy method.

Step 3.1: According to Equation (20), construct the score matrix: $A_{ij}={\left[{\eta }_{ij}\right]}_{m\times n}$.

Step 3.2: Calculate the ratio of the j-th attribute of the i-th alternative; the result is denoted by $g_{ij}$.

$$g_{ij}={\displaystyle \frac{{\eta }_{ij}}{{{\displaystyle \sum }}_{i=1}^n{\eta }_{ij}}}$$

(59)

Step 3.3: Compute the entropy of the j-th attribute, namely $E_j$.

$$E_j=-{\displaystyle \frac{1}{\ln n}}{{\displaystyle \sum }}_{j=1}^n{\eta }_{ij}\ln \left({\eta }_{ij}\right)$$

(60)

Step 3.4: Calculates the weight of the j-th attribute, which is expressed by $w_j^s$.

$$w_j^s={\displaystyle \frac{1-E_j}{{{\displaystyle \sum }}_{j=1}^n\left(1-E_j\right)}}$$

(61)

Step 4: Calculate the attribute weights $w_j^t$ according to the CILOS method.

Step 4.1: According to Equation (20), construct score matrix $A_{ij}={\left[{\eta }_{ij}\right]}_{m\times n}$.

Step 4.2: Compute the maximum value of each column j based on the score matrix ${\eta }_j={\max }_i{x}_{ij}=x_{c_{j}j}$, where $c_j$ denotes the row of j-th column. Therefore, construct the square matrix $Y={\left[y_{ij}\right]}_{n\times n}\left(y_{ii}={\eta }_i,y_{ij}={\eta }_{c_{j}j}\right)$.

Step 4.3: Construct square matrix P of the relative loss of each attribute.

$$P={\left[p_{ij}\right]}_{n\times n}$$

(62)

$$p_{ij}={\displaystyle \frac{{\eta }_j-y_{ij}}{j}}={\displaystyle \frac{y_{ii}-y_{ij}}{y_{ii}}}$$

(63)

Step 4.4: Build matrix R.

$$R=\left[\begin{array}{cccc}-{\displaystyle \sum _{i=1}^n{p}_{i1}}& p_{12}& \dots & p_{1n}\\ p_{21}& -{\displaystyle \sum _{i=1}^n{p}_{2i}}& \dots & p_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ p_{n1}& p_{n2}& \dots & -{\displaystyle \sum _{i=1}^n{p}_{nn}}\end{array}\right]$$

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Step 4.5: Calculate the weight $w_j^t$, where ${{\displaystyle \sum }}_{j=1}^n{w}_j^t=1$ and $R\times w_j^t=0$.

Step 5: Compute the comprehensive weight value $\varpi$ by aggregating $w_j^s$ and $w_j^t$.

$$\varpi ={\displaystyle \frac{w_j^s{w}_j^t}{{{\displaystyle \sum }}_{j=1}^n{w}_j^s{w}_j^t}}$$

(65)

Step 6: According to Equation (41), we can apply the CCFOFDWPPMM operator to aggregate the attributes of each alternative.

Step 7: Determine the score values of each alternative according to Equation (20). Then, we can obtain the optimal alternative based on the ranked score values.

6. Example Analysis

In order to specifically analyze the practice patterns of first-class disciplines construction in industry-characteristic universities, four typical and representative cases were selected for in-depth assessment in this study. These universities show distinctive features and advantages in their respective industry backgrounds and discipline development. The four universities are Beijing Jiaotong University (Z₁), North China Electric Power University (Z₂), Beijing University of Chemical Technology (Z₃), and Beijing University of Posts and Telecommunications (Z₄). Four experts $\left\{{\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_3,{\mathrm{e}}_4\right\}$ were invited to assess the construction of first-class disciplines in four industry-characteristic universities, according to the attributes shown in Table 3. Assessment information is represented by CCFOFN. The assessment matrices given by experts are shown in Appendix B (see

Table A1. Assessment matrix from ${\mathrm{e}}_1$.

Partition	Attribute	Z1	Z2
O1	F1	(([0.12,0.60]ej[π/2,π], [0.41,0.81]ej[π/2,π/2], [0.39,0.8]ej[π/2,π/2]), (0.35,0.61,0.5)ej(π/2,π,π))	(([0.12,0.61]ej[π/2,π/2], [0.39,0.79]ej[π/2,π/2], [0.4,0.8]ej[π/2,π/2]), (0.37,0.59,0.4)ej(π/2,π/2,π/2))
	F2	(([0.23,0.76]ej[π/2,π/2], [0.24,0.65]ej[π/2,π/2], [0.24,0.7]ej[π/2,π/2]), (0.50,0.45,0.6)ej(π/2,π,π/2))	(([0.20,0.73]ej[π/2,π/2], [0.27,0.68]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.47,0.47,0.4)ej(π/2,π/2,π/2))
	F3	(([0.29,0.88]ej[π/3,π/2], [0.12,0.48]ej[π/2,π/2], [0.1,0.4]ej[π/2,π/2]), (0.58,0.30,0.4)ej(π/2,π/2,π/2))	(([0.20,0.73]ej[π/2,π/2], [0.27,0.68]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.47,0.48.0,4)ej(π/2,π/2,π/2))
	F4	(([0.13,0.80]ej[π/2,π/2], [0.20,0.60]ej[π/2,π/2], [0.20,0.60]ej[π/2,π/2]), (0.46,0.40,0.38)ej(π/2,π/2,π/2))	(([0.25,0.80]ej[π/2,π/2], [0.20,0.60]ej[π/2,π/2], [0.24,0.63]ej[π/2,π/2]), (0.53,0.40,0.4)ej(π/2,π/2,π/2))
O2	F5	(([0.05,0.71]ej[π/2,π/2], [0.29,0.71],ej[π/2,π/2], [0.2,0.7],ej[π/2,π/2]), (0.38,0.50,0.4)ej(π/2,π/2,π))	([0.06,0.70]ej[π/2,π/2], [0.29,0.71]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.38,0.50,0.5)ej(π/2,π/2,π/2))
	F6	(([0.02,0.85],ej[π/3,π/2], [0.15,0.52]ej[π/2,π/2], [0.2,0.5]ej[π/2,π/2]), (0.44,0.33,0.4)ej(π/2,π/2,π/2))	(([0.05,0.89]ej[π/2,π/2], [0.11,0.45]ej[π/2,π/2], [0.2,0.5]ej[π/2,π/2]), (0.47,0.28,0.3)ej(π/2,π/2,π/2))
	F7	(([0.04,0.62]ej[π/2,π/2], [0.39,0.79]ej[π/2,π/2], [0.4,0.8]ej[π/2,π/2]), (0.33,0.58,0.6)ej(π/2,π/2,π/2))	(([0.01,0.69]ej[π/2,π/2], [0.31,0.73]ej[π/2,π/2], [0.3,0.6]ej[π/2,π/2]), (0.35,0.52,0.4)ej(π/2,π/2,π/2))
	F8	(([0.12,0.81]ej[π/2,π], [0.19,0.59]ej[π/2,π/2], [0.20.6]ej[π/2,π/2]), (0.46,0.39,0.4)ej(π/2,π/2,π/2))	(([0.20,0.70]ej[π/2,π/2], [0.30,0.71]ej[π/2,π/2], [0.30,0.6]ej[π/2,π/2]), (0.45,0.50,0.4)ej(π/2,π/2,π/2))
O3	F9	(([0.05,0.79]ej[π/2,π/2], [0.21,0.61]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.42,0.41,0.3)ej(π/2,π/2,π/2))	(([0.10,0.87]ej[π/2,π/2], [0.13,0.50]ej[π/2,π/2], [0.2,0.50]ej[π/2,π/2]), (0.48,0.31,0.42)ej(π/2,π/2,π/2))
	F10	(([0.02,0.84]ej[π/2,π/2], [0.16,0.54]ej[π/3,π/2], [0.2,0.5]ej[π/3,π/2]), (0.43,0.35,0.4)ej(π/2,π/2,π/2))	(([0.07,0.89]ej[π/2,π/2], [0.10,0.45]ej[π/2,π/2], [0.2,0.5]ej[π/2,π/2]), (0.48,0.28,0.3)ej(π/2,π/2,π/2))
	F11	(([0.05,0.83]ej[π/2,π/2], [0.17,0.55]ej[π/2,π/2], [0.2,0.5]ej[π/2,π/2]), (0.44,0.36,0.3)ej(π/2,π,π))	(([0.10,0.87]ej[π/2,π/2], [0.13,0.49]ej[π/3,π/2], [0.2,0.5]ej[π/3,π/2]), (0.49,0.31,0.3)ej(π/2,π/2,π/2))
	F12	(([0.21,0.78]ej[π/2,π], [0.22,0.62]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.49,0.42,0.3)ej(π/2,π/2,π/2))	(([0.25,0.80]ej[π/2,π/2], [0.20,0.60]ej[π/2,π/2], [0.3,0.50]ej[π/2,π/2]), (0.53,0.40,0.3)ej(π/2,π/2,π/2))
O4	F13	(([0.03,0.78]ej[π/2,π/2], [0.22,0.62]ej[π/2,π/2], [0.3,0.6]ej[π/2,π/2]), (0.41,0.42,0.3)ej(π/2,π,π))	(([0.04,0.81]ej[π/2,π/2], [0.19,0.59]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.43,0.39,0.3)ej(π/2,π/2,π/2))
	F14	(([0.04,0.85]ej[π/2,π/2], [0.15,0.52]ej[π/2,π/2], [0.21,0.5]ej[π/2,π/2]), (0.45,0.33,0.34)ej(π/2,π/2,π/2))	(([0.04,0.86]ej[π/2,π/2], [0.14,0.51]ej[π/2,π/2], [0.2,0.5]ej[π/2,π/2]), (0.45,0.33,0.36)ej(π/2,π,π))
	F15	(([0.05,0.81]ej[π/2,π/2], [0.19,0.59]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.43,0.39,0.3)ej(π/2,π/2,π/2))	(([0.12,0.87]ej[π/2,π/2], [0.13,0.49]ej[π/2,π/2], [0.2,0.49]ej[π/2,π/2]), (0.49,0.31,0.35)ej(π/2,π/2,π/2))
	F16	(([0.10,0.85]ej[π/2,π/2], [0.15,0.52]ej[π/2,π/2], [0.2,0.5]ej[π/2,π/2]), (0.48,0.34,0.35)ej(π/2,π,π)	(([0.07,0.59]ej[π/2,π/2], [0.41,0.81]ej[π/2,π/2], [0.4,0.8]ej[π/2,π/2]), (0.33,0.61,0.5)ej(π/2,π/2,π/2))
Partition	Attribute	Z3	Z4
O1	F1	(([0.05,0.52]ej[π/2,π/2], [0.48,0.85]ej[π/2,π/2], [0.4,0.8]ej[π/2,π/2]), (0.29,0.67,0.5)ej(π/3,π/2,π/2))	(([0.34,0.87]ej[π/3,π/2], [0.13,0.50]ej[π/2,π/2], [0.2,0.50]ej[π/2,π/2]), (0.60,0.31,0.4)ej(π/2,π/2,π/2))
	F2	(([0.1,0.71]ej[π/3,π/2], [0.29,0.71]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.45,0.50,0.4)ej(π/3,π/2,π/2))	(([0.3,0.81]ej[π/2,π/2], [0.19,0.59]ej[π/2,π/2], [0.28,0.61]ej[π/2,π/2], (0.50,0.39,0.4)ej(π/2,π/2,π/2))
	F3	(([0.04,0.74]ej[π/2,π], [0.26,0.68]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.39,0.46,0.4)ej(π/2,π/2,π/2))	(([0.32,0.71]ej[π/2,π/2], [0.29,0.71]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.46,0.50,0.5)ej(π/2,π/2,π/2))
	F4	(([0.1,0.74]ej[π/2,π/2], [0.26,0.67]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.50,0.46,0.4)ej(π/2,π/2,π/2))	(([0.3,0.81]ej[π/2,π/2], [0.19,0.59]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.50,0.39,0.4)ej(π/2,π/2,π/2))
O2	F5	(([0.05,0.66]ej[π/2,π], [0.34,0.75]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.35,0.54,0.4)ej(π/2,π/2,π/2))	(([0.22,0.80]ej[π/2,π/2], [0.20,0.59]ej[π/2,π/2], [0.20,0.6]ej[π/2,π/2]), (0.55,0.40,0.3)ej(π/2,π/2,π/2))
	F6	(([0.05,0.85]ej[π/2,π], [0.15,0.53]ej[π/2,π/2], [0.2,0.5]ej[π/2,π/2]), (0.45,0.34,0.3)ej(π/2,π/2,π/2))	(([0.21,0.78]ej[π/2,π/2], [0.22,0.63]ej[π/2,π/2], [0.3,0.6]ej[π/2,π/2]), (0.49,0.43,0.4)ej(π/2,π/2,π/2))
	F7	(([0.01,0.66]ej[π/2,π/2], [0.34,0.75]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.34,0.54,0.4)ej(π/2,π/2,π/2))	(([0.13,0.90]ej[π/2,π/2], [0.10,0.43]ej[π/2,π/2], [0.15,0.45]ej[π/2,π/2]), (0.52,0.27,0.34)ej(π/2,π/2,π/2))
	F8	(([0.04,0.63]ej[π/2,π], [0.37,0.77]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.39,0.57,0.4)ej(π/2,π/2,π/2))	(([0.17,0.82]ej[π/3,π/2], [0.18,0.57]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.49,0.37,0.43)ej(π/2,π/2,π))
O3	F9	(([0.07,0.82]ej[π/2,π/2], [0.18,0.57]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.45,0.37,0.2)ej(π/2,π/2,π/2))	(([0.06,0.78]ej[π/2,π/2], [0.22,0.62]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.42,0.42,0.3)ej(π/2,π/2,π/2))
	F10	(([0.11,0.85]ej[π/2,π], [0.15,0.53]ej[π/2,π/2], [0.2,0.5]ej[π/2,π/2]), (0.48,0.34,0.4)ej(π/2,π/2,π/2))	(([0.19,0.87]ej[π/2,π/2], [0.13,0.49]ej[π/2,π/2], [0.2,0.51]ej[π/2,π/2]), (0.53,0.31,0.4)ej(π/2,π/2,π/2))
	F11	(([0.05,0.74]ej[π/2,π/2], [0.26,0.67]ej[π/2,π/2], [0.31,0.69]ej[π/2,π/2]), (0.39,0.46,0.3)ej(π/2,π/2,π/2))	(([0.25,0.84]ej[π/2,π], [0.16,0.54]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.55,0.35,0.4)ej(π/2,π/2,π/2))
	F12	(([0.18,0.53]ej[π/2,π], [0.47,0.85]ej[π/2,π/2], [0.5,0.8]ej[π/2,π/2]), (0.35,0.66,0.5)ej(π/2,π/2,π/2))	(([0.13,0.74]ej[π/2,π/2], [0.26,0.67]ej[π/2,π/2], [0.3,0.6]ej[π/2,π/2]), (0.44,0.47,0.3)ej(π/2,π,π))
O4	F13	(([0.05,0.75]ej[π/3,π/2], [0.25,0.66]ej[π/2,π/2], [0.3,0.6]ej[π/2,π/2]), (0.40,0.46,0.4)ej(π/2,π/2,π/2))	(([0.06,0.88]ej[π/2,π/2], [0.12,0.47]ej[π/2,π/2], [0.2,0.4]ej[π/2,π/2]), (0.55,0.29,0.4)ej(π/2,π/2,π/2))
	F14	(([0.04,0.80]ej[π/2,π/2], [0.20,0.60]ej[π/2,π/2], [0.20,0.60]ej[π/2,π/2]), (0.42,0.40,0.34)ej(π/2,π/2,π/2))	(([0.06,0.89]ej[π/2,π/2], [0.11,0.46]ej[π/2,π/2], [0.2,0.5]ej[π/2,π/2]), (0.48,0.28,0.3)ej(π/2,π/2,π/2))
	F15	(([0.07,0.83]ej[π/2,π/2], [0.17,0.55]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.45,0.36,0.4)ej(π/2,π/2,π/2))	(([0.09,0.75]ej[π/2,π/2], [0.25,0.66]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.42,0.46,0.35)ej(π/2,π/2,π/2))
	F16	(([0.09,0.47]ej[π/3,π/2], [0.53,0.88]ej[π/2,π/2], [0.52,0.7]ej[π/2,π/2]), (0.28,0.71,0.7)ej(π/2,π/2,π/2))	(([0.07,0.45]ej[π/2,π/2], [0.55,0.89]ej[π/2,π/2], [0.44,0.69]ej[π/2,π/2]), (0.26,0.72,0.45)ej(π/2,π,π))

,

Table A2. Assessment matrix from ${\mathrm{e}}_2$.

Partition	Attribute	Z1	Z2
O1	F1	(([0.1,0.56]ej[π/2,π], [0.4,0.8]ej[π/2,π], [0.36,0.81]ej[π/2,π/2]), (0.37,0.6,0.5)ej(π/2,π/2,π))	(([0.15,0.62]ej[π/2,π/2], [0.4,0.8]ej[π/2,π/2], [0.45,0.81]ej[π/2,π/2]), (0.39,0.6,0.35)ej(π/2,π,π))
	F2	(([0.2,0.6]ej[π/2,π/2], [0.25,0.6]ej[π/2,π/2], [0.2,0.7]ej[π/2,π/2]), (0.55,0.5,0.61)ej(π/2,π,π/2))	(([0.23,0.7]ej[π/2,π/2], [0.24,0.7]ej[π/2,π/2], [0.34,0.75]ej[π/2,π]), (0.47,0.47,0.4)ej(π/2,π/2,π))
	F3	(([0.31,0.82]ej[π/3,π/2], [0.15,0.5]ej[π/2,π/2], [0.15,0.39]ej[π/2,π/2]), (0.6,0.31,0.42)ej(π/2,π,π))	(([0.22,0.7]ej[π/2,π/2], [0.3,0.65]ej[π/2,π/2], [0.35,0.71]ej[π/2,π/2]), (0.5,0.42.0,4)ej(π/2,π/2,π/2))
	F4	(([0.15,0.82]ej[π/2,π/2], [0.22,0.64]ej[π/2,π/2], [0.25,0.63]ej[π/2,π]), (0.45,0.45,0.35)ej(π/2,π/2,π/2))	(([0.26,0.76]ej[π/2,π], [0.21,0.64]ej[π/2,π/2], [0.2,0.62]ej[π/2,π/2]), (0.47,0.39,0.38)ej(π/2,π/2,π/2))
O2	F5	(([0.06,0.74]ej[π/2,π/2], [0.29,0.7],ej[π/2,π/2], [0.21,0.67],ej[π/2,π]), (0.4,0.50,0.45)ej(π/2,π/2,π))	(([0.06,0.70]ej[π/2,π/2], [0.29,0.71]ej[π/3,π/2], [0.32,0.75]ej[π/2,π]), (0.4,0.45,0.35)ej(π/2,π/2,π))
	F6	(([0.05,0.8],ej[π/3,π/2], [0.2,0.5]ej[π/3,π/2], [0.3,0.45]ej[π/2,π/2]), (0.41,0.32,0.44)ej(π/2,π/2,π/2))	(([0.05,0.83]ej[π/2,π/2], [0.15,0.43]ej[π/2,π/2], [0.23,0.58]ej[π/2,π/2]), (0.48,0.32,0.31)ej(π/2,π/2,π))
	F7	(([0.04,0.62]ej[π/2,π/2], [0.4,0.79]ej[π/2,π/2], [0.45,0.78]ej[π/3,π]), (0.35,0.56,0.5)ej(π/2,π/2,π/2))	(([0.02,0.64]ej[π/2,π/2], [0.32,0.71]ej[π/2,π/2], [0.31,0.64]ej[π/2,π/2]), (0.4,0.51,0.34)ej(π/2,π/2,π/2))
	F8	(([0.15,0.83]ej[π/2,π], [0.24,0.56]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.5,0.42,0.41)ej(π/2,π/2,π/2))	(([0.18,0.70]ej[π/2,π/2], [0.39,0.71]ej[π/2,π/2], [0.30,0.7]ej[π/2,π/2]), (0.54,0.45,0.5)ej(π/2,π/2,π/2))
O3	F9	(([0.03,0.73]ej[π/3,π/2], [0.2,0.59]ej[π/2,π/2], [0.21,0.56]ej[π/2,π/2]), (0.4,0.41,0.32)ej(π/2,π/2,π/2))	(([0.08,0.87]ej[π/3,π/2], [0.13,0.50]ej[π/2,π/2], [0.2,0.50]ej[π/2,π/2]), (0.48,0.31,0.42)ej(π/2,π/2,π/2))
	F10	(([0.02,0.72]ej[π/2,π/2], [0.16,0.53]ej[π/2,π/2], [0.13,0.35]ej[π/3,π/2]), (0.38,0.4,0.34)ej(π/2,π,π))	(([0.05,0.78]ej[π/2,π/2], [0.12,0.43]ej[π/2,π/2], [0.22,0.38]ej[π/2,π/2]), (0.44,0.38,0.36)ej(π/2,π,π))
	F11	(([0.03,0.77]ej[π/2,π/2], [0.18,0.56]ej[π/2,π/2], [0.22,0.54]ej[π/2,π/2]), (0.45,0.39,0.39)ej(π/2,π,π))	(([0.11,0.82]ej[π/3,π/2], [0.13,0.42]ej[π/3,π/2], [0.21,0.5]ej[π/3,π/2]), (0.43,0.25,0.23)ej(π/2,π/2,π/2))
	F12	(([0.05,0.71]ej[π/2,π], [0.25,0.6]ej[π/2,π/2], [0.25,0.62]ej[π/2,π/2]), (0.44,0.49,0.36)ej(π/2,π/2,π/2))	(([0.13,0.75]ej[π/2,π/2], [0.24,0.65]ej[π/2,π/2], [0.23,0.55]ej[π/2,π/2]), (0.51,0.31,0.33)ej(π/2,π/2,π))
O4	F13	(([0.04,0.8]ej[π/2,π/2], [0.21,0.52]ej[π/2,π/2], [0.38,0.46]ej[π/2,π/2]), (0.42,0.43,0.36)ej(π/2,π,π))	(([0.03,0.75]ej[π/2,π/2], [0.13,0.52]ej[π/2,π/2], [0.2,0.56]ej[π/2,π/2]), (0.41,0.4,0.3)ej(π/2,π/2,π/2))
	F14	(([0.05,0.8]ej[π/2,π/2], [0.17,0.53]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.42,0.43,0.49)ej(π/2,π/2,π/2))	(([0.04,0.56]ej[π/2,π/2], [0.12,0.5]ej[π/2,π/2], [0.2,0.51]ej[π/2,π/2]), (0.45,0.44,0.46)ej(π/2,π,π))
	F15	(([0.12,0.72]ej[π/2,π/2], [0.13,0.5]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.41,0.36,0.3)ej(π/2,π/2,π/2))	(([0.15,0.68]ej[π/2,π/2], [0.14,0.41]ej[π/2,π/2], [0.2,0.49]ej[π/2,π/2]), (0.49,0.3,0.3)ej(π/2,π/2,π/2))
	F16	(([0.10,0.81]ej[π/2,π/2], [0.19,0.5]ej[π/2,π/2], [0.21,0.45]ej[π/2,π/2]), (0.5,0.4,0.5)ej(π/2,π,π))	(([0.1,0.69]ej[π/2,π/2], [0.35,0.8]ej[π/2,π/2], [0.4,0.81]ej[π/2,π/2]), (0.36,0.71,0.51)ej(π/2,π/2,π/2))
Partition	Attribute	Z3	Z4
O1	F1	(([0.1,0.5]ej[π/2,π/2], [0.42,0.68]ej[π/2,π/2], [0.34,0.7]ej[π/2,π/2]), (0.3,0.62,0.45)ej(π/2,π/2,π/2))	(([0.24,0.78]ej[π/3,π/2], [0.13,0.4]ej[π/2,π/2], [0.2,0.4]ej[π/2,π/2]), (0.55,0.25,0.35)ej(π/2,π/2,π/2))
	F2	([0.13,0.7]ej[π/3,π/2], [0.29,0.71]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.45,0.50,0.4)ej(π/3,π/2,π))	(([0.2,0.7]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2], [0.2,0.55]ej[π/2,π/2]), (0.50,0.4,0.4)ej(π/2,π/2,π/2))
	F3	(([0.14,0.74]ej[π/2,π], [0.26,0.68]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.4,0.46,0.4)ej(π/2,π/2,π/2))	(([0.22,0.71]ej[π/3,π/2], [0.25,0.72]ej[π/2,π/2], [0.2,0.7]ej[π/2,π/2]), (0.46,0.50,0.5)ej(π/2,π/2,π/2))
	F4	(([0.1,0.8]ej[π/2,π/2], [0.26,0.67]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.50,0.46,0.4)ej(π/2,π/2,π/2))	(([0.20,0.78]ej[π/2,π/2], [0.19,0.59]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.5,0.39,0.4)ej(π/2,π/2,π))
O2	F5	(([0.06,0.66]ej[π/2,π/2], [0.4,0.79]ej[π/2,π/2], [0.35,0.7]ej[π/2,π/2]), (0.4,0.54,0.4)ej(π/2,π/2,π/2))	(([0.36,0.78]ej[π/2,π/2], [0.23,0.59]ej[π/2,π/2], [0.24,0.6]ej[π/2,π/2]), (0.6,0.43,0.36)ej(π/3,π/2,π/2))
	F6	(([0.05,0.85]ej[π/2,π], [0.25,0.58]ej[π/2,π/2], [0.22,0.45]ej[π/2,π]), (0.45,0.34,0.3)ej(π/2,π/2,π/2))	(([0.2,0.8]ej[π/3,π/2], [0.32,0.73]ej[π/2,π/2], [0.31,0.66]ej[π/2,π/2]), (0.45,0.43,0.34)ej(π/2,π/2,π/2))
	F7	(([0.02,0.68]ej[π/2,π/2], [0.4,0.65]ej[π/2,π/2], [0.5,0.6]ej[π/2,π/2]), (0.4,0.5,0.45)ej(π/2,π/2,π))	(([0.21,0.86]ej[π/2,π/2], [0.13,0.45]ej[π/2,π/2], [0.14,0.42]ej[π/2,π/2]), (0.53,0.34,0.3)ej(π/2,π/2,π/2))
	F8	(([0.06,0.6]ej[π/2,π], [0.3,0.7]ej[π/2,π/2], [0.33,0.67]ej[π/2,π/2]), (0.4,0.62,0.56)ej(π/2,π/2,π))	(([0.17,0.78]ej[π/3,π/2], [0.23,0.5]ej[π/2,π/2], [0.22,0.63]ej[π/2,π/2]), (0.5,0.4,0.33)ej(π/2,π/2,π/2))
O3	F9	(([0.08,0.76]ej[π/3,π/2], [0.18,0.57]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.45,0.37,0.2)ej(π/2,π/2,π/2))	(([0.13,0.81]ej[π/3,π/2], [0.27,0.64]ej[π/2,π/2], [0.23,0.64]ej[π/2,π/2]), (0.41,0.42,0.5)ej(π/2,π/2,π/2))
	F10	(([0.1,0.8]ej[π/2,π], [0.12,0.51]ej[π/2,π/2], [0.12,0.5]ej[π/2,π/2]), (0.44,0.54,0.54)ej(π/2,π,π))	(([0.2,0.83]ej[π/2,π/2], [0.23,0.7]ej[π/2,π/2], [0.27,0.74]ej[π/2,π/2]), (0.53,0.31,0.4)ej(π/2,π,π))
	F11	(([0.04,0.74]ej[π/2,π/2], [0.26,0.67]ej[π/2,π/2], [0.31,0.69]ej[π/2,π/2]), (0.39,0.5,0.3)ej(π/2,π/2,π/2))	(([0.3,0.84]ej[π/2,π], [0.16,0.54]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.55,0.35,0.45)ej(π/2,π/2,π/2))
	F12	(([0.11,0.52]ej[π/2,π], [0.43,0.82]ej[π/2,π/2], [0.2,0.7]ej[π/2,π/2]), (0.35,0.69,0.53)ej(π/2,π/2,π/2))	(([0.19,0.64]ej[π/2,π/2], [0.29,0.7]ej[π/2,π/2], [0.32,0.75]ej[π/2,π/2]), (0.44,0.48,0.32)ej(π/2,π,π))
O4	F13	([0.05,0.7]ej[π/3,π/2], [0.22,0.69]ej[π/2,π/2], [0.34,0.6]ej[π/2,π/2], (0.41,0.47,0.45)ej(π/2,π/2,π/2))	([0.22,0.89]ej[π/2,π/2], [0.22,0.47]ej[π/2,π/2], [0.26,0.4]ej[π/2,π/2], (0.5,0.3,0.4)ej(π/2,π/2,π/2))
	F14	(([0.06,0.67]ej[π/2,π/2], [0.21,0.63]ej[π/2,π/2], [0.20,0.65]ej[π/2,π/2]), (0.42,0.40,0.39)ej(π/2,π/2,π/2))	(([0.2,0.7]ej[π/2,π/2], [0.19,0.49]ej[π/2,π/2], [0.26,0.6]ej[π/2,π/2]), (0.49,0.28,0.34)ej(π/2,π/2,π/2))
	F15	(([0.08,0.8]ej[π/2,π/2], [0.15,0.57]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.46,0.35,0.4)ej(π/2,π/2,π/2))	(([0.19,0.76]ej[π/2,π/2], [0.22,0.65]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.45,0.41,0.32)ej(π/2,π/2,π/2))
	F16	(([0.09,0.42]ej[π/3,π/2], [0.51,0.8]ej[π/2,π/2], [0.42,0.71]ej[π/2,π/2]), (0.21,0.71,0.76)ej(π/2,π/2,π/2))	(([0.18,0.5]ej[π/2,π/2], [0.45,0.8]ej[π/2,π/2], [0.4,0.6]ej[π/2,π/2]), (0.21,0.76,0.4)ej(π/2,π,π))

,

Table A3. Assessment matrix from ${\mathrm{e}}_3$.

Partition	Attribute	Z1	Z2
O1	F1	(([0.15,0.62]ej[π/2,π/2], [0.4,0.78]ej[π/2,π/2], [0.35,0.68]ej[π/2,π/2]), (0.3,0.56,0.5)ej(π/2,π/2,π))	(([0.16,0.56]ej[π/2,π/2], [0.38,0.75]ej[π/2,π/2], [0.41,0.78]ej[π/2,π/2]), (0.36,0.51,0.44)ej(π/2,π/2,π/2))
	F2	(([0.25,0.78]ej[π/2,π/2], [0.26,0.67]ej[π/2,π/2], [0.26,0.76]ej[π/2,π/2]), (0.35,0.36,0.56)ej(π/2,π/2,π/2))	(([0.23,0.72]ej[π/2,π/2], [0.21,0.65]ej[π/2,π/2], [0.31,0.67]ej[π/2,π/2]), (0.45,0.44,0.47)ej(π/2,π/2,π/2))
	F3	(([0.2,0.5]ej[π/3,π/2], [0.12,0.4]ej[π/3,π/2], [0.1,0.4]ej[π/3,π/2]), (0.58,0.30,0.4)ej(π/2,π/2,π/2))	(([0.21,0.72]ej[π/2,π/2], [0.23,0.7]ej[π/2,π/2], [0.33,0.7]ej[π/2,π/2]), (0.46,0.44.0,94)ej(π/2,π/2,π/2))
	F4	(([0.2,0.80]ej[π/2,π/2], [0.25,0.68]ej[π/2,π/2], [0.21,0.62]ej[π/2,π/2]), (0.5,0.43,0.4)ej(π/2,π/2,π/2))	(([0.25,0.82]ej[π/2,π/2], [0.22,0.67]ej[π/2,π/2], [0.24,0.63]ej[π/2,π/2]), (0.58,0.49,0.44)ej(π/2,π/2,π/2))
O2	F5	(([0.04,0.68]ej[π/2,π/2], [0.25,0.72]ej[π/2,π/2], [0.21,0.67]ej[π/2,π/2]), (0.39,0.50,0.54)ej(π/2,π/2,π))	(([0.06,0.70]ej[π/2,π/2], [0.26,0.61]ej[π/2,π/2], [0.34,0.72]ej[π/2,π/2]), (0.34,0.50,0.54)ej(π/2,π/2,π/2))
	F6	(([0.03,0.81]ej[π/3,π/2], [0.14,0.51]ej[π/2,π/2], [0.21,0.45]ej[π/2,π/2]), (0.43,0.43,0.49)ej(π/2,π/2,π/2))	(([0.05,0.82]ej[π/2,π/2], [0.1,0.44]ej[π/2,π/2], [0.21,0.5]ej[π/2,π/2]), (0.46,0.29,0.3)ej(π/2,π/2,π/2))
	F7	(([0.05,0.62]ej[π/2,π/2], [0.4,0.79]ej[π/2,π/2], [0.4,0.8]ej[π/2,π/2]), (0.33,0.58,0.55)ej(π/2,π/2,π/2))	(([0.01,0.69]ej[π/2,π/2], [0.31,0.72]ej[π/2,π/2], [0.27,0.63]ej[π/2,π/2]), (0.37,0.54,0.46)ej(π/2,π/2,π/2))
	F8	(([0.1,0.81]ej[π/2,π], [0.19,0.6]ej[π/2,π/2], [0.20.6]ej[π/2,π/2]), (0.46,0.39,0.45)ej(π/2,π/2,π/2))	(([0.21,0.70]ej[π/2,π/2], [0.30,0.71]ej[π/2,π/2], [0.32,0.6]ej[π/2,π/2]), (0.45,0.54,0.4)ej(π/2,π/2,π/2))
O3	F9	(([0.04,0.72]ej[π/2,π/2], [0.23,0.66]ej[π/2,π/2], [0.21,0.62]ej[π/2,π/2]), (0.42,0.41,0.3)ej(π/2,π/2,π/2))	(([0.09,0.84]ej[π/2,π/2], [0.15,0.54]ej[π/2,π/2], [0.21,0.55]ej[π/2,π/2]), (0.49,0.3,0.4)ej(π/2,π/2,π/2))
	F10	(([0.03,0.78]ej[π/2,π/2], [0.2,0.5]ej[π/3,π/2], [0.24,0.54]ej[π/3,π/2]), (0.42,0.34,0.42)ej(π/2,π/2,π/2))	(([0.05,0.81]ej[π/2,π/2], [0.13,0.41]ej[π/2,π/2], [0.21,0.56]ej[π/2,π/2]), (0.49,0.38,0.31)ej(π/2,π/2,π/2))
	F11	(([0.05,0.76]ej[π/2,π/2], [0.18,0.58]ej[π/2,π/2], [0.21,0.45]ej[π/2,π/2]), (0.45,0.39,0.35)ej(π/2,π,π))	(([0.09,0.82]ej[π/2,π/2], [0.12,0.45]ej[π/3,π/2], [0.21,0.56]ej[π/3,π/2]), (0.41,0.32,0.3)ej(π/2,π/2,π/2))
	F12	(([0.23,0.72]ej[π/2,π], [0.21,0.69]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.5,0.47,0.38)ej(π/2,π/2,π/2))	(([0.15,0.7]ej[π/2,π/2], [0.21,0.61]ej[π/2,π/2], [0.34,0.50]ej[π/2,π/2]), (0.53,0.43,0.3)ej(π/2,π/2,π/2))
O4	F13	(([0.03,0.7]ej[π/2,π/2], [0.21,0.6]ej[π/2,π/2], [0.33,0.61]ej[π/2,π/2]), (0.41,0.45,0.38)ej(π/2,π,π))	(([0.05,0.73]ej[π/2,π/2], [0.29,0.5]ej[π/2,π/2], [0.21,0.56]ej[π/2,π/2)], (0.41,0.39,0.3)ej(π/2,π/2,π/2))
	F14	(([0.04,0.86]ej[π/2,π/2], [0.13,0.51]ej[π/2,π/2], [0.22,0.56]ej[π/2,π/2]), (0.41,0.39,0.35)ej(π/2,π/2,π/2))	(([0.07,0.8]ej[π/2,π/2], [0.12,0.45]ej[π/2,π/2], [0.21,0.51]ej[π/2,π/2]), (0.49,0.23,0.26)ej(π/2,π,π))
	F15	(([0.05,0.8]ej[π/2,π/2], [0.14,0.5]ej[π/2,π/2], [0.21,0.6]ej[π/2,π/2]), (0.4,0.39,0.31)ej(π/2,π/2,π/2))	(([0.1,0.83]ej[π/2,π/2], [0.12,0.5]ej[π/2,π/2], [0.25,0.5]ej[π/2,π/2]), (0.41,0.41,0.4)ej(π/2,π/2,π/2))
	F16	(([0.10,0.59]ej[π/2,π/2], [0.19,0.51]ej[π/2,π/2], [0.21,0.58]ej[π/2,π/2]), (0.5,0.4,0.5)ej(π/2,π,π))	(([0.07,0.51]ej[π/2,π/2], [0.42,0.71]ej[π/2,π/2], [0.43,0.6]ej[π/2,π/2]), (0.34,0.51,0.6)ej(π/2,π/2,π/2))
Partition	Attribute	Z3	Z4
O1	F1	(([0.11,0.58]ej[π/3,π/2], [0.34,0.39]ej[π/3,π/2], [0.34,0.4]ej[π/3,π/2]), (0.29,0.67,0.5)ej(π/3,π/2,π/2))	(([0.21,0.83]ej[π/2,π/2], [0.2,0.59]ej[π/2,π/2], [0.23,0.6]ej[π/2,π/2]), (0.56,0.39,0.44)ej(π/2,π/2,π/2))
	F2	(([0.09,0.54]ej[π/3,π/2], [0.21,0.61]ej[π/2,π/2], [0.23,0.67]ej[π/2,π/2]), (0.4,0.45,0.49)ej(π/3,π/2,π/2))	(([0.23,0.6]ej[π/2,π/2], [0.19,0.59]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2),] (0.50,0.39,0.4)ej(π/2,π/2,π/2))
	F3	(([0.04,0.74]ej[π/2,π], [0.23,0.61]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.39,0.46,0.41)ej(π/2,π/2,π/2))	(([0.26,0.71]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2], [0.23,0.77]ej[π/2,π/2]), (0.5,0.4,0.45)ej(π/2,π/2,π/2))
	F4	(([0.12,0.67]ej[π/2,π/2], [0.24,0.67]ej[π/2,π/2], [0.23,0.7]ej[π/2,π/2]), (0.45,0.46,0.4)ej(π/2,π/2,π/2))	(([0.23,0.81]ej[π/2,π/2], [0.2,0.59]ej[π/2,π/2], [0.22,0.6]ej[π/2,π/2]), (0.51,0.39,0.4)ej(π/2,π/2,π/2))
O2	F5	(([0.04,0.66]ej[π/2,π], [0.38,0.75]ej[π/2,π/2], [0.34,0.67]ej[π/2,π/2]), (0.33,0.51,0.44)ej(π/2,π/2,π/2))	(([0.26,0.78]ej[π/2,π/2], [0.21,0.6]ej[π/2,π/2], [0.22,0.65]ej[π/2,π/2]), (0.54,0.42,0.39)ej(π/2,π/2,π/2))
	F6	(([0.05,0.8]ej[π/2,π], [0.15,0.53]ej[π/2,π/2], [0.22,0.5]ej[π/2,π/2]), (0.45,0.34,0.3)ej(π/2,π/2,π/2))	(([0.21,0.78]ej[π/2,π/2], [0.22,0.63]ej[π/2,π/2], [0.3,0.6]ej[π/2,π/2]), (0.49,0.43,0.4)ej(π/2,π/2,π/2))
	F7	(([0.03,0.65]ej[π/2,π/2], [0.1,0.57]ej[π/2,π/2], [0.19,0.6]ej[π/2,π/2]), (0.31,0.52,0.43)ej(π/2,π/2,π/2))	(([0.39,0.87]ej[π/2,π/2], [0.33,0.47]ej[π/2,π/2], [0.19,0.49]ej[π/2,π/2]), (0.52,0.27,0.34)ej(π/2,π/2,π/2))
	F8	(([0.14,0.63]ej[π/2,π], [0.37,0.77]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.39,0.57,0.4)ej(π/2,π/2,π/2))	(([0.2,0.8]ej[π/3,π/2], [0.15,0.56]ej[π/2,π/2], [0.21,0.56]ej[π/2,π/2]), (0.51,0.38,0.4)ej(π/2,π/2,π))
O3	F9	(([0.06,0.8]ej[π/2,π/2], [0.2,0.54]ej[π/2,π/2], [0.23,0.56]ej[π/2,π/2]), (0.4,0.39,0.27)ej(π/2,π/2,π/2))	(([0.18,0.8]ej[π/2,π/2], [0.23,0.6]ej[π/2,π/2], [0.21,0.62]ej[π/2,π/2]), (0.4,0.41,0.43)ej(π/2,π/2,π/2))
	F10	(([0.12,0.83]ej[π/2,π], [0.16,0.54]ej[π/2,π/2], [0.21,0.53]ej[π/2,π/2]), (0.43,0.36,0.43)ej(π/2,π/2,π/2))	(([0.2,0.8]ej[π/2,π/2], [0.19,0.56]ej[π/2,π/2], [0.25,0.58]ej[π/2,π/2]), (0.51,0.41,0.44)ej(π/2,π/2,π/2))
	F11	(([0.04,0.71]ej[π/2,π/2], [0.22,0.65]ej[π/2,π/2], [0.32,0.61]ej[π/2,π/2]), (0.32,0.48,0.34)ej(π/2,π/2,π/2))	(([0.28,0.87]ej[π/2,π], [0.2,0.54]ej[π/2,π/2], [0.27,0.6]ej[π/2,π/2]), (0.59,0.36,0.42)ej(π/2,π/2,π/2))
	F12	(([0.08,0.54]ej[π/2,π], [0.3,0.7]ej[π/2,π/2], [0.4,0.76]ej[π/2,π/2]), (0.33,0.45,0.5)ej(π/2,π/2,π/2))	(([0.18,0.72]ej[π/2,π/2], [0.21,0.69]ej[π/2,π/2], [0.29,0.7]ej[π/2,π/2]), (0.41,0.5,0.37)ej(π/2,π,π))
O4	F13	(([0.06,0.71]ej[π/3,π/2], [0.28,0.61]ej[π/2,π/2], [0.3,0.6]ej[π/2,π/2]), (0.42,0.5,0.4)ej(π/2,π/2,π/2))	(([0.22,0.77]ej[π/2,π/2], [0.13,0.48]ej[π/2,π/2], [0.21,0.43]ej[π/2,π/2]), (0.5,0.3,0.4)ej(π/2,π/2,π/2))
	F14	(([0.04,0.76]ej[π/2,π/2], [0.21,0.61]ej[π/2,π/2], [0.21,0.61]ej[π/2,π/2]), (0.32,0.40,0.34)ej(π/2,π/2,π/2))	(([0.16,0.8]ej[π/2,π/2], [0.14,0.48]ej[π/2,π/2], [0.21,0.59]ej[π/2,π/2]), (0.47,0.3,0.3)ej(π/2,π/2,π/2))
	F15	(([0.07,0.82]ej[π/2,π/2], [0.16,0.59]ej[π/2,π/2], [0.26,0.63]ej[π/2,π/2]), (0.47,0.39,0.45)ej(π/2,π/2,π/2))	(([0.19,0.7]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2], [0.31,0.65]ej[π/2,π/2]), (0.45,0.46,0.36)ej(π/2,π/2,π/2))
	F16	(([0.08,0.46]ej[π/3,π/2], [0.51,0.82]ej[π/2,π/2], [0.51,0.72]ej[π/2,π/2]), (0.3,0.72,0.7)ej(π/2,π/2,π/2))	(([0.27,0.41]ej[π/2,π/2], [0.43,0.67]ej[π/2,π/2], [0.44,0.56]ej[π/2,π/2]), (0.2,0.51,0.45)ej(π/2,π,π))

and

Table A4. Assessment matrix from ${\mathrm{e}}_4$.

Partition	Attribute	Z1	Z2
O1	F1	(([0.1,0.56]ej[π/2,π], [0.13,0.5]ej[π/2,π/2], [0.3,0.6]ej[π/2,π/2]), (0.36,0.56,0.51)ej(π/2,π,π))	(([0.1,0.62]ej[π/2,π/2], [0.36,0.72]ej[π/2,π/2], [0.4,0.8]ej[π/2,π/2]), (0.37,0.59,0.4)ej(π/2,π/2,π/2))
	F2	(([0.19,0.77]ej[π/2,π/2], [0.27,0.69]ej[π/2,π/2], [0.23,0.7]ej[π/2,π/2]), (0.48,0.46,0.56)ej(π/2,π/2,π/2))	(([0.3,0.72]ej[π/2,π/2], [0.25,0.66]ej[π/2,π/2], [0.32,0.71]ej[π/2,π/2]), (0.48,0.49,0.5)ej(π/2,π/2,π/2))
	F3	(([0.29,0.88]ej[π/3,π/2], [0.4,0.7]ej[π/2,π/2], [0.2,0.5]ej[π/2,π/2]), (0.58,0.30,0.4)ej(π/2,π/2,π/2))	(([0.15,0.6]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2], [0.1,0.67]ej[π/2,π/2]), (0.49,0.44.0,49)ej(π/2,π/2,π/2))
	F4	(([0.21,0.80]ej[π/2,π/2], [0.20,0.60]ej[π/2,π/2], [0.20,0.60]ej[π/2,π/2]), (0.46,0.40,0.38)ej(π/2,π/2,π/2))	(([0.25,0.80]ej[π/2,π/2], [0.21,0.60]ej[π/2,π/2], [0.24,0.63]ej[π/2,π/2]), (0.54,0.41,0.47)ej(π/2,π/2,π/2))
O2	F5	(([0.07,0.75]ej[π/2,π/2], [0.28,0.73],ej[π/2,π/2], [0.21,0.72],ej[π/2,π/2]), (0.39,0.51,0.41)ej(π/2,π/2,π))	(([0.06,0.70]ej[π/2,π/2], [0.25,0.71]ej[π/2,π/2], [0.23,0.67]ej[π/2,π/2]), (0.35,0.51,0.51)ej(π/2,π/2,π/2))
	F6	(([0.04,0.82],ej[π/3,π/2], [0.15,0.52]ej[π/2,π/2], [0.2,0.5]ej[π/2,π/2]), (0.44,0.3,0.4)ej(π/2,π/2,π/2))	(([0.04,0.89]ej[π/2,π/2], [0.11,0.45]ej[π/2,π/2], [0.2,0.5]ej[π/2,π/2]), (0.47,0.3,0.3)ej(π/2,π/2,π/2))
	F7	(([0.03,0.61]ej[π/2,π/2], [0.39,0.79]ej[π/2,π/2], [0.4,0.8]ej[π/2,π/2]), (0.33,0.58,0.6)ej(π/2,π/2,π/2))	(([0.02,0.63]ej[π/2,π/2], [0.31,0.73]ej[π/2,π/2], [0.3,0.6]ej[π/2,π/2]), (0.35,0.52,0.4)ej(π/2,π/2,π/2))
	F8	(([0.02,0.81]ej[π/2,π], [0.19,0.59]ej[π/2,π/2], [0.21.64]ej[π/2,π/2]), (0.45,0.48,0.4)ej(π/2,π/2,π/2))	(([0.02,0.73]ej[π/2,π/2], [0.23,0.71]ej[π/2,π/2], [0.31,0.58]ej[π/2,π/2]), (0.45,0.50,0.4)ej(π/2,π/2,π/2))
O3	F9	(([0.05,0.79]ej[π/2,π/2], [0.22,0.56]ej[π/2,π/2], [0.21,0.6]ej[π/2,π/2]), (0.43,0.41,0.34)ej(π/2,π/2,π/2))	(([0.11,0.83]ej[π/2,π/2], [0.15,0.52]ej[π/2,π/2], [0.21,0.59]ej[π/2,π/2]), (0.44,0.32,0.41)ej(π/2,π/2,π/2))
	F10	(([0.03,0.8]ej[π/2,π/2], [0.2,0.5]ej[π/3,π/2], [0.2,0.54]ej[π/3,π/2]), (0.4,0.35,0.45)ej(π/2,π/2,π/2))	(([0.07,0.78]ej[π/2,π/2], [0.13,0.5]ej[π/2,π/2], [0.21,0.45]ej[π/2,π/2]), (0.42,0.38,0.33)ej(π/2,π/2,π/2))
	F11	(([0.05,0.81]ej[π/2,π/2], [0.16,0.55]ej[π/2,π/2], [0.2,0.5]ej[π/2,π/2]), (0.48,0.4,0.43)ej(π/2,π,π))	(([0.09,0.78]ej[π/2,π/2], [0.15,0.45]ej[π/3,π/2], [0.28,0.51]ej[π/3,π/2]), (0.5,0.41,0.3)ej(π/2,π/2,π/2))
	F12	(([0.11,0.78]ej[π/2,π], [0.29,0.62]ej[π/2,π/2], [0.2,0.61]ej[π/2,π/2], (0.5,0.4,0.25)ej(π/2,π/2,π/2))	(([0.15,0.72]ej[π/2,π/2], [0.11,0.67]ej[π/2,π/2], [0.36,0.58]ej[π/2,π/2]), (0.54,0.40,0.3)ej(π/2,π/2,π/2))
O4	F13	(([0.03,0.7]ej[π/2,π/2], [0.21,0.72]ej[π/2,π/2], [0.31,0.61]ej[π/2,π/2]), (0.42,0.4,0.3)ej(π/2,π,π))	(([0.04,0.67]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2], [0.21,0.67]ej[π/2,π/2]), (0.45,0.4,0.3)ej(π/2,π/2,π/2))
	F14	(([0.05,0.83]ej[π/2,π/2], [0.16,0.62]ej[π/2,π/2], [0.22,0.51]ej[π/2,π/2]), (0.45,0.4,0.4)ej(π/2,π/2,π/2))	(([0.05,0.67]ej[π/2,π/2], [0.15,0.52]ej[π/2,π/2], [0.21,0.55]ej[π/2,π/2]), (0.46,0.33,0.39)ej(π/2,π,π))
	F15	(([0.05,0.69]ej[π/2,π/2], [0.11,0.51]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.45,0.36,0.31)ej(π/2,π/2,π/2))	(([0.11,0.83]ej[π/2,π/2], [0.17,0.42]ej[π/2,π/2], [0.21,0.56]ej[π/2,π/2]), (0.5,0.35,0.35)ej(π/2,π/2,π/2))
	F16	(([0.10,0.61]ej[π/2,π/2], [0.17,0.56]ej[π/2,π/2], [0.23,0.59]ej[π/2,π/2]), (0.5,0.34,0.5)ej(π/2,π,π))	(([0.06,0.6]ej[π/2,π/2], [0.4,0.76]ej[π/2,π/2], [0.41,0.78]ej[π/2,π/2]), (0.34,0.65,0.5)ej(π/2,π/2,π/2))
Partition	Attribute	Z3	Z4
O1	F1	(([0.04,0.54]ej[π/2,π/2], [0.43,0.76]ej[π/2,π/2], [0.34,0.6]ej[π/2,π/2]), (0.24,0.29,0.3)ej(π/3,π/2,π/2))	(([0.27,0.84]ej[π/3,π/2], [0.2,0.54]ej[π/2,π/2], [0.21,0.51]ej[π/2,π/2]), (0.61,0.32,0.34)ej(π/2,π/2,π/2))
	F2	(([0.09,0.7]ej[π/3,π/2], [0.29,0.74]ej[π/2,π/2], [0.3,0.57]ej[π/2,π/2]), (0.5,0.50,0.4)ej(π/3,π/2,π/2))	(([0.2,0.81]ej[π/2,π/2], [0.19,0.59]ej[π/2,π/2], [0.22,0.56]ej[π/2,π/2]), (0.54,0.4,0.39)ej(π/2,π/2,π/2))
	F3	(([0.03,0.73]ej[π/2,π], [0.26,0.66]ej[π/2,π/2], [0.3,0.7]ej[π/2,π/2]), (0.34,0.46,0.48)ej(π/2,π/2,π/2))	(([0.26,0.78]ej[π/2,π/2], [0.23,0.71]ej[π/2,π/2], [0.31,0.67]ej[π/2,π/2]), (0.47,0.51,0.56)ej(π/2,π/2,π/2))
	F4	(([0.05,0.74]ej[π/2,π/2], [0.27,0.64]ej[π/2,π/2], [0.32,0.67]ej[π/2,π/2]), (0.46,0.46,0.5)ej(π/2,π/2,π/2))	(([0.20,0.81]ej[π/2,π/2], [0.19,0.59]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.50,0.4,0.45)ej(π/2,π/2,π/2))
O2	F5	(([0.07,0.65]ej[π/2,π], [0.33,0.72]ej[π/2,π/2], [0.31,0.68]ej[π/2,π/2]), (0.36,0.52,0.41)ej(π/2,π/2,π/2))	(([0.3,0.81]ej[π/2,π/2], [0.21,0.57]ej[π/2,π/2], [0.21,0.62]ej[π/2,π/2]), (0.56,0.42,0.38)ej(π/2,π/2,π/2))
	F6	(([0.04,0.81]ej[π/2,π], [0.15,0.53]ej[π/2,π/2], [0.2,0.5]ej[π/2,π/2]), (0.45,0.34,0.3)ej(π/2,π/2,π/2))	(([0.28,0.79]ej[π/2,π/2], [0.21,0.66]ej[π/2,π/2], [0.31,0.63]ej[π/2,π/2]), (0.47,0.42,0.48)ej(π/2,π/2,π/2))
	F7	(([0.02,0.63]ej[π/2,π/2], [0.31,0.72]ej[π/2,π/2], [0.26,0.69]ej[π/2,π/2]), (0.28,0.34,0.4)ej(π/2,π/2,π/2))	(([0.23,0.82]ej[π/2,π/2], [0.14,0.47]ej[π/2,π/2], [0.15,0.45]ej[π/2,π/2]), (0.57,0.3,0.38)ej(π/2,π/2,π/2))
	F8	(([0.04,0.63]ej[π/2,π], [0.37,0.77]ej[π/2,π/2], [0.33,0.68]ej[π/2,π/2]), (0.49,0.54,0.4)ej(π/2,π/2,π/2))	(([0.18,0.81]ej[π/3,π/2], [0.18,0.56]ej[π/2,π/2], [0.2,0.6]ej[π/2,π/2]), (0.49,0.37,0.43)ej(π/2,π/2,π))
O3	F9	(([0.07,0.67]ej[π/2,π/2], [0.2,0.57]ej[π/2,π/2], [0.21,0.69]ej[π/2,π/2]), (0.46,0.38,0.22)ej(π/2,π/2,π/2))	(([0.1,0.73]ej[π/2,π/2], [0.23,0.63]ej[π/2,π/2], [0.26,0.63]ej[π/2,π/2]), (0.41,0.41,0.32)ej(π/2,π/2,π/2))
	F10	(([0.10,0.81]ej[π/2,π], [0.16,0.59]ej[π/2,π/2], [0.21,0.55]ej[π/2,π/2]), (0.44,0.54,0.4)ej(π/2,π/2,π/2))	(([0.2,0.80]ej[π/2,π/2], [0.16,0.42]ej[π/2,π/2], [0.22,0.58]ej[π/2,π/2]), (0.54,0.39,0.43)ej(π/2,π/2,π/2))
	F11	(([0.06,0.7]ej[π/2,π/2], [0.22,0.6]ej[π/2,π/2], [0.3,0.54]ej[π/2,π/2]), (0.4,0.47,0.39)ej(π/2,π/2,π/2))	(([0.22,0.81]ej[π/2,π], [0.14,0.5]ej[π/2,π/2], [0.21,0.63]ej[π/2,π/2]), (0.54,0.4,0.41)ej(π/2,π/2,π/2))
	F12	(([0.12,0.5]ej[π/2,π], [0.41,0.81]ej[π/2,π/2], [0.41,0.56]ej[π/2,π/2]), (0.31,0.61,0.45)ej(π/2,π/2,π/2))	(([0.11,0.71]ej[π/2,π/2], [0.45,0.63]ej[π/2,π/2], [0.41,0.61]ej[π/2,π/2]), (0.45,0.43,0.36)ej(π/2,π,π))
O4	F13	(([0.05,0.76]ej[π/3,π/2], [0.24,0.61]ej[π/2,π/2], [0.4,0.7]ej[π/2,π/2]), (0.40,0.36,0.3)ej(π/2,π/2,π/2))	(([0.2,0.8]ej[π/2,π/2], [0.2,0.5]ej[π/2,π/2], [0.3,0.5]ej[π/2,π/2]), (0.52,0.3,0.49)ej(π/2,π/2,π/2))
	F14	(([0.05,0.67]ej[π/2,π/2], [0.21,0.61]ej[π/2,π/2], [0.20,0.62]ej[π/2,π/2]), (0.43,0.40,0.43)ej(π/2,π/2,π/2))	(([0.19,0.8]ej[π/2,π/2], [0.34,0.45]ej[π/2,π/2], [0.22,0.75]ej[π/2,π/2]), (0.4,0.3,0.37)ej(π/2,π/2,π/2))
	F15	(([0.07,0.81]ej[π/2,π/2], [0.17,0.52]ej[π/2,π/2], [0.21,0.68]ej[π/2,π/2]), (0.5,0.3,0.4)ej(π/2,π/2,π/2))	(([0.36,0.71]ej[π/2,π/2], [0.25,0.61]ej[π/2,π/2], [0.31,0.7]ej[π/2,π/2]), (0.45,0.46,0.35)ej(π/2,π/2,π/2))
	F16	(([0.08,0.5]ej[π/3,π/2], [0.5,0.78]ej[π/2,π/2], [0.51,0.67]ej[π/2,π/2]), (0.3,0.71,0.7)ej(π/2,π/2,π/2))	(([0.23,0.41]ej[π/2,π/2], [0.51,0.8]ej[π/2,π/2], [0.49,0.7]ej[π/2,π/2]), (0.28,0.71,0.4)ej(π/2,π,π))

).

6.1. Assessment Process

Step 1: Normalize assessment matrix of experts. Since the indicators applied in this case are all benefit indicators, the normalized matrix is the same as the original matrix.

Step 2: The weight vector of experts is $\omega =\left\{0.3,0.25,0.2,0.25\right\}$. Suppose parameter $\lambda =2,\kappa =2$; parameter vector $P=\left({\displaystyle \frac{1}{4}}\right.,{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},\left.{\displaystyle \frac{1}{4}}\right)$ The group assessment matrix $M={\left({\Omega }_{ij}\right)}_{m\times n}$ is presented in

Table A5. Group Assessment matrix.

Partition	Attribute	Z1	Z2
O1	F1	(([0.24,0.84]ej2π[0.25,0.3], [0.18,0.49]ej2π[0.06,0.1], [0.17,0.46]ej2π[0.06,0.06]), (0.63,0.31,0.26)ej2π(0.25,0.11,0.13))	(([0.26,0.85]ej2π[0.25,0.25], [0.19,0.48]ej2π[0.06,0.06], [0.2,0.51]ej2π[0.06,0.06]), (0.66,0.3,0.2)ej2π(0.25,0.1,0.1))
	F2	(([0.43,0.9]ej2π[0.25,0.25], [0.12,0.37]ej2π[0.06,0.06], [0.11,0.43]ej2π[0.06,0.06]), (0.71,0.22,0.31)ej2π(0.25,0.1,0.06))	(([0.46,0.91]ej2π[0.25,0.25], [0.11,0.38]ej2π[0.06,0.06], [0.15,0.42]ej2π[0.06,0.06]), (0.76,0.23,0.23)ej2π(0.25,0.06,0.09))
	F3	(([0.48,0.86]ej2π[0.18,0.25], [0.14,0.37]ej2π[0.06,0.06], [0.07,0.43]ej2π[0.06,0.06]), (0.85,0.14,0.2)ej2π(0.25,0.1,0.1))	(([0.380.91]ej2π[0.25,0.25], [0.12,0.38]ej2π[0.06,0.06], [0.15,0.42]ej2π[0.06,0.06]), (0.77,0.22,0.68)ej2π(0.25,0.06,0.06))
	F4	(([0.34,0.95]ej2π[0.25,0.25], [0.1,0.35]ej2π[0.06,0.06], [0.1,0.33]ej2π[0.06,0.06]), (0.77,0.21,0.18)ej2π(0.25,0.06,0.06))	(([0.5,0.94]ej2π[0.25,0.26], [0.1,0.35]ej2π[0.06,0.06], [0.1,0.34]ej2π[0.06,0.06]), (0.8,0.22,0.21)ej2π(0.25,0.06,0.06))
O2	F5	(([0.11,0.91]ej2π[0.25,0.25], [0.13,0.42]ej2π[0.06,0.06], [0.1,0.4]ej2π[0.06,0.1]), (0.68,0.26,0.24)ej2π(0.25,0.06,0.13))	(([0.13,0.91]ej2π[0.25,0.25], [0.13,0.4]ej2π[0.06,0.06], [0.15,0.42]ej2π[0.06,0.1]), (0.65,0.25,0.25)ej2π(0.25,0.06,0.1))
	F6	(([0.06,0.95]ej2π[0.18,0.25], [0.07,0.26]ej2π[0.06,0.06], [0.11,0.24]ej2π[0.06,0.06]), (0.72,0.18,0.22)ej2π(0.25,0.06,0.06))	(([0.10.96]ej2π[0.25,0.25], [0.06,0.22]ej2π[0.06,0.06], [0.1,0.27]ej2π[0.06,0.06]), (0.76,0.14,0.14)ej2π(0.25,0.06,0.1))
	F7	(([0.08,0.87]ej2π[0.25,0.25], [0.19,0.51]ej2π[0.06,0.06], [0.2,0.51]ej2π[0.06,0.1]), (0.62,0.3,0.3)ej2π(0.25,0.06,0.06))	(([0.03,0.89]ej2π[0.25,0.25], [0.15,0.43]ej2π[0.06,0.06], [0.14,0.34]ej2π[0.06,0.06]), (0.65,0.27,0.2)ej2π(0.25,0.06,0.06))
	F8	(([0.06,0.95]ej2π[0.25,0.4], [0.11,0.32]ej2π[0.06,0.06], [0.09,0.33]ej2π[0.06,0.1]), (0.76,0.21,0.21)ej2π(0.25,0.06,0.06))	(([0.06,0.91]ej2π[0.25,0.25], [0.15,0.42]ej2π[0.06,0.06], [0.15,0.35]ej2π[0.06,0.1]), (0.76,0.26,0.21)ej2π(0.25,0.06,0.06))
O3	F9	(([0.08,0.93]ej2π[0.25,0.4], [0.1,0.33]ej2π[0.06,0.06], [0.1,0.32]ej2π[0.06,0.1]), (0.71,0.2,0.15)ej2π(0.25,0.06,0.06))	(([0.2,0.96]ej2π[0.25,0.4], [0.07,0.27]ej2π[0.06,0.06], [0.1,0.28]ej2π[0.06,0.06]), (0.76,0.15,0.2)ej2π(0.25,0.06,0.06))
	F10	(([0.05,0.94]ej2π[0.25,0.25], [0.09,0.26]ej2π[0.05,0.06], [0.1,0.26]ej2π[0.04,0.06]), (0.7,0.17,0.2)ej2π(0.25,0.1,0.1))	(([0.12,0.95]ej2π[0.25,0.25], [0.06,0.22]ej2π[0.06,0.06], [0.1,0.25]ej2π[0.06,0.06]), (0.75,0.18,0.16)ej2π(0.25,0.1,0.1))
	F11	(([0.09,0.94]ej2π[0.25,0.25], [0.08,0.3]ej2π[0.06,0.06], [0.1,0.25]ej2π[0.06,0.06]) (0.75,0.19,0.19)ej2π(0.25,0.13,0.13))	(([0.21,0.95]ej2π[0.22,0.25], [0.06,0.23]ej2π[0.04,0.06], [0.11,0.27]ej2π[0.04,0.06]) (0.74,0.16,0.14)ej2π(0.25,0.06,0.06))
	F12	(([0.15,0.92]ej2π[0.25,0.4], [0.12,0.36]ej2π[0.06,0.06], [0.1,0.33]ej2π[0.06,0.06]), (0.77,0.23,0.16)ej2π(0.25,0.06,0.06))	(([0.32,0.92]ej2π[0.25,0.25], [0.1,0.35]ej2π[0.06,0.06], [0.15,0.28]ej2π[0.06,0.06]), (0.81,0.19,0.15)ej2π(0.25,0.06,0.1))
O4	F13	(([0.07,0.92]ej2π[0.25,0.25], [0.1,0.36]ej2π[0.06,0.06], [0.16,0.31]ej2π[0.06,0.06]) (0.71,0.21,0.17)ej2π(0.25,0.13,0.13))	(([0.08,0.92]ej2π[0.25,0.25], [0.11,0.29]ej2π[0.06,0.06], [0.1,0.33]ej2π[0.06,0.06]) (0.72,0.2,0.14)ej2π(0.25,0.06,0.06))
	F14	(([0.1,0.96]ej2π[0.25,0.25], [0.07,0.31]ej2π[0.06,0.06], [0.1,0.29]ej2π[0.06,0.06]), (0.72,0.19,0.2)ej2π(0.25,0.06,0.06))	(([0.1,0.89]ej2π[0.25,0.25], [0.06,0.25]ej2π[0.06,0.06], [0.1,0.27]ej2π[0.06,0.06]), (0.75,0.17,0.19)ej2π(0.25,0.13,0.13))
	F15	(([0.12,0.93]ej2π[0.25,0.25], [0.07,0.27]ej2π[0.06,0.06], [0.09,0.32]ej2π[0.06,0.06]), (0.71,0.18,0.14)ej2π(0.25,0.06,0.06))	(([0.24,0.94]ej2π[0.25,0.25], [0.07,0.23]ej2π[0.06,0.06], [0.1,0.26]ej2π[0.06,0.06]), (0.75,0.17,0.17)ej2π(0.25,0.06,0.06))
	F16	(([0.22,0.88]ej2π[0.25,0.25], [0.08,0.27]ej2π[0.06,0.06], [0.1,0.29]ej2π[0.06,0.06]), (0.78,0.18,0.24)ej2π(0.25,0.13,0.13))	(([0.15,0.84]ej2π[0.25,0.25], [0.19,0.49]ej2π[0.06,0.06], [0.2,0.49]ej2π[0.06,0.06]), (0.63,0.36,0.28)ej2π(0.25,0.06,0.06))
Partition	Attribute	Z3	Z4
O1	F1	(([0.11,0.81]ej2π[0.21,0.25], [0.21,0.48]ej2π[0.06,0.06], [0.17,0.41]ej2π[0.06,0.06]), (0.53,0.34,0.23)ej2π(0.18,0.06,0.06))	(([0.48,0.95]ej2π[0.19,0.25], [0.08,0.28]ej2π[0.06,0.06], [0.1,0.28]ej2π[0.06,0.06]), (0.84,0.16,0.19)ej2π(0.25,0.06,0.06))
	F2	(([0.21,0.87]ej2π[0.18,0.25], [0.13,0.42]ej2π[0.06,0.06], [0.13,0.38]ej2π[0.06,0.06]), (0.73,0.25,0.22)ej2π(0.18,0.06,0.1))	(([0.44,0.9]ej2π[0.25,0.25], [0.09,0.32]ej2π[0.06,0.06], [0.11,0.31]ej2π[0.06,0.06]), (0.79,0.19,0.19)ej2π(0.25,0.06,0.06))
	F3	(([0.08,0.92]ej2π[0.25,0.4], [0.12,0.37]ej2π[0.06,0.06], [0.14,0.41]ej2π[0.06,0.06]), (0.67,0.23,0.21)ej2π(0.25,0.06,0.06))	(([0.5,0.92]ej2π[0.22,0.25], [0.13,0.41]ej2π[0.06,0.06], [0.13,0.43]ej2π[0.06,0.06]), (0.76,0.24,0.26)ej2π(0.25,0.06,0.06))
	F4	(([0.15,0.92]ej2π[0.25,0.25], [0.12,0.37]ej2π[0.06,0.06], [0.14,0.4]ej2π[0.06,0.06]), (0.76,0.23,0.21)ej2π(0.25,0.06,0.06))	(([0.44,0.95]ej2π[0.25,0.25], [0.09,0.32]ej2π[0.06,0.06], [0.1,0.32]ej2π[0.06,0.06]), (0.79,0.19,0.21)ej2π(0.25,0.06,0.09))
O2	F5	(([0.11,0.89]ej2π[0.25,0.31], [0.18,0.46]ej2π[0.06,0.06], [0.16,0.4]ej2π[0.06,0.06]), (0.64,0.27,0.21)ej2π(0.25,0.06,0.06))	(([0.52,0.94]ej2π[0.25,0.25], [0.1,0.31]ej2π[0.06,0.06], [0.1,0.34]ej2π[0.06,0.06]), (0.83,0.21,0.18)ej2π(0.22,0.06,0.06))
	F6	(([0.1,0.95]ej2π[0.25,0.4], [0.09,0.28]ej2π[0.06,0.06], [0.1,0.25]ej2π[0.06,0.06]), (0.74,0.16,0.14)ej2π(0.25,0.06,0.06))	(([0.44,0.94]ej2π[0.22,0.25], [0.12,0.38]ej2π[0.06,0.06], [0.14,0.34]ej2π[0.06,0.06]), (0.76,0.21,0.2)ej2π(0.25,0.06,0.06))
	F7	(([0.03,0.88]ej2π[0.25,0.25], [0.16,0.4]ej2π[0.06,0.06], [0.19,0.37]ej2π[0.06,0.06]), (0.59,0.25,0.21)ej2π(0.25,0.06,0.1))	(([0.37,0.97]ej2π[0.25,0.25], [0.12,0.23]ej2π[0.06,0.06], [0.08,0.23]ej2π[0.06,0.06]), (0.81,0.14,0.16)ej2π(0.25,0.06,0.06))
	F8	(([0.1,0.87]ej2π[0.25,0.4], [0.17,0.45]ej2π[0.06,0.06], [0.15,0.4]ej2π[0.06,0.06]), (0.7,0.31,0.23)ej2π(0.25,0.06,0.1))	(([0.37,0.95]ej2π[0.18,0.25], [0.09,0.29]ej2π[0.06,0.06], [0.1,0.32]ej2π[0.06,0.06]), (0.81,0.14,0.16)ej2π(0.25,0.06,0.12))
O3	F9	(([0.15,0.92]ej2π[0.25,0.25], [0.09,0.3]ej2π[0.06,0.06], [0.1,0.35]ej2π[0.06,0.06]), (0.7,30.18,0.11)ej2π(0.25,0.06,0.06))	(([0.18,0.94]ej2π[0.25,0.25], [0.12,0.34]ej2π[0.06,0.06], [0.11,0.34]ej2π[0.06,0.06]), (0.7,0.2,0.21)ej2π(0.25,0.06,0.09))
	F10	(([0.23,0.95]ej2π[0.25,0.4], [0.07,0.28]ej2π[0.06,0.06], [0.09,0.27]ej2π[0.06,0.06]), (0.74,0.24,0.23)ej2π(0.25,0.1,0.1))	(([0.41,0.95]ej2π[0.25,0.25], [0.09,0.33]ej2π[0.06,0.06], [0.11,0.36]ej2π[0.06,0.06]), (0.7,0.2,0.21)ej2π(0.25,0.1,0.1))
	F11	(([0.1,0.92]ej2π[0.25,0.25], [0.11,0.36]ej2π[0.06,0.06], [0.15,0.36]ej2π[0.06,0.06]), (0.65,0.24,0.16)ej2π(0.25,0.06,0.06))	(([0.1,0.96]ej2π[0.25,0.4], [0.17,0.27]ej2π[0.06,0.06], [0.15,0.33]ej2π[0.06,0.06]), (0.83,0.18,0.21)ej2π(0.25,0.06,0.06))
	F12	(([0.21,0.8]ej2π[0.25,0.4], [0.2,0.53]ej2π[0.06,0.06], [0.2,0.45]ej2π[0.06,0.06]), (0.61,0.35,0.25)ej2π(0.25,0.06,0.06))	(([0.3,0.91]ej2π[0.25,0.25], [0.17,0.39]ej2π[0.06,0.06], [0.16,0.4]ej2π[0.06,0.06]), (0.73,0.24,0.17)ej2π(0.25,0.13,0.13))
O4	F13	(([0.11,0.92]ej2π[0.18,0.25], [0.12,0.36]ej2π[0.06,0.06], [0.17,0.36]ej2π[0.06,0.06]), (0.7,0.23,0.23)ej2π(0.25,0.06,0.06))	(([0.19,0.95]ej2π[0.25,0.25], [0.08,0.24]ej2π[0.06,0.06], [0.12,0.22]ej2π[0.06,0.06]), (0.8,0.14,0.21)ej2π(0.25,0.06,0.06))
	F14	(([0.11,0.91]ej2π[0.25,0.25], [0.1,0.33]ej2π[0.06,0.06], [0.1,0.34]ej2π[0.06,0.06]), (0.66,0.19,0.18)ej2π(0.25,0.06,0.06))	(([0.19,0.94]ej2π[0.25,0.25], [0.12,0.24]ej2π[0.06,0.06], [0.11,0.37]ej2π[0.06,0.06]), (0.74,0.14,0.16)ej2π(0.25,0.06,0.06))
	F15	(([0.16,0.95]ej2π[0.25,0.25], [0.07,0.3]ej2π[0.06,0.06], [0.11,0.35]ej2π[0.06,0.06]), (0.76,0.17,0.21)ej2π(0.25,0.06,0.06))	(([0.28,0.92]ej2π[0.25,0.25], [0.12,0.37]ej2π[0.06,0.06], [0.14,0.4]ej2π[0.06,0.06]), (0.74,0.22,0.17)ej2π(0.25,0.06,0.06))
	F16	(([0.18,0.75]ej2π[0.18,0.25], [0.26,0.56]ej2π[0.06,0.06], [0.25,0.41]ej2π[0.06,0.06]), (0.51,0.42,0.43)ej2π(0.25,0.06,0.06))	(([0.22,0.73]ej2π[0.25,0.25], [0.25,0.56]ej2π[0.06,0.06], [0.2,20.37]ej2π[0.06,0.06]), (0.45,0.41,0.21)ej2π(0.25,0.13,0.13))

.

Step 3: According to Equations (59)–(61), the weight obtained using the entropy method $w_j^s$ is shown as follows:

$$\begin{array}{l}{w}_1^s=0.1823,w_2^s=0.0335,w_3^s=0.0372,w_4^s=0.0067,\\ w_5^s=0.0469,w_6^s=0.0089,w_7^s=0.1082,w_8^s=0.0148,\\ w_9^s=0.0048,w_{10}^s=0.0061,w_{11}^s=0.0442,w_{12}^s=0.0493,\\ w_{13}^s=0.0201,w_{14}^s=0.0047,w_{15}^s=0.0022,w_{16}^s=0.4302\end{array}$$

Step 4: According to Equations (62)–(64), the weight obtained using CILOS method’s weight $w_j^t$ is presented as follows:

$$\begin{array}{l}{w}_1^t=0.0304,w_2^t=0.0754,w_3^t=0.0420,w_4^t=0.0660,\\ w_5^t=0.0438,w_6^t=0.1037,w_7^t=0.0350,w_8^t=0.0504,\\ w_9^t=0.0227,w_{10}^t=0.1163,w_{11}^t=0.0523,w_{12}^t=0.0252,\\ w_{13}^t=0.0724,w_{14}^t=0.1900,w_{15}^t=0.0344,w_{16}^t=0.0084\end{array}$$

Step 5: The calculated comprehensive weight value $\varpi$ is shown in Equation (65). The results are shown in Figure 2.

$$\begin{array}{l}{\varpi }_1=0.1980,{\varpi }_2=0.0902,{\varpi }_3=0.0558,{\varpi }_4=0.0158,\\ {\varpi }_5=0.0734,{\varpi }_6=0.0330,{\varpi }_7=0.1353,{\varpi }_8=0.0266,\\ {\varpi }_9=0.0039,{\varpi }_{10}=0.0253,{\varpi }_{11}=0.0826,{\varpi }_{12}=0.0444,\\ {\varpi }_{13}=0.0520,{\varpi }_{14}=0.0319,{\varpi }_{15}=0.0027,{\varpi }_{16}=0.1291\end{array}$$

Step 6: Apply the CCFOFDWPPMM operator to aggregate the attributes of each alternative. The combined value of each alternative is as follows:

$$\begin{array}{l}CCFOFDWPPMM\left(Z_1\right)=\\ \left(\left(\left[0.2,0.89\right]e^{j2\pi \left[0.22,0.25\right]},\left[0.13,0.4\right]e^{j2\pi \left[0.07,0.07\right]},\left[0.13,0.39\right]e^{j2\pi \left[0.07,0.08\right]}\right),\left(0.67,0.25,0.23\right)e^{j2\pi \left(0.23,0.08,0.11\right)}\right)\\ CCFOFDWPPMM\left(Z_2\right)=\\ \left(\left(\left[0.23,0.9\right]e^{j2\pi \left[0.23,0.23\right]},\left[0.12,0.4\right]e^{j2\pi \left[0.07,0.07\right]},\left[0.16,0.38\right]e^{j2\pi \left[0.07,0.07\right]}\right),\left(0.68,0.24,0.24\right)e^{j2\pi \left(0.23,0.07,0.09\right)}\right)\\ CCFOFDWPPMM\left(Z_3\right)=\\ \left(\left(\left[0.11,0.84\right]e^{j2\pi \left[0.22,0.25\right]},\left[0.19,0.48\right]e^{j2\pi \left[0.07,0.07\right]},\left[0.18,0.43\right]e^{j2\pi \left[0.07,0.07\right]}\right),\left(0.58,0.32,0.25\right)e^{j2\pi \left(0.22,0.07,0.08\right)}\right)\\ CCFOFDWPPMM\left(Z_4\right)=\\ \left(\left(\left[0.37,0.92\right]e^{j2\pi \left[0.21,0.23\right]},\left[0.13,0.37\right]e^{j2\pi \left[0.07,0.07\right]},\left[0.13,0.37\right]e^{j2\pi \left[0.07,0.07\right]}\right),\left(0.75,0.22,0.21\right)e^{j2\pi \left(0.23,0.07,0.09\right)}\right)\end{array}$$

Step 7: According to Equation (20), the combined score value of each alternative can be determined as follows:

$$\begin{array}{l}S\left(Z_1\right)=0.1481,S\left(Z_2\right)=0.1577\\ S\left(Z_3\right)=0.1024,S\left(Z_4\right)=0.1872\end{array}$$

Considering the overall score value of each alternative, the outcome of the ranking is as follows:

$$Z_4\succ Z_2\succ Z_1\succ Z_3$$

6.2. Sensitivity Analysis

The advantage of the CCFOFDWPPMM operator is that it adopts Dombi operational rules, and different parameters in the Dombi operational rules yield different ranking results. Therefore, we first conducted a sensitivity analysis on parameter $\lambda$ in the Dombi operator to observe the ranking of the first-class discipline construction in four industry-characteristic universities under different parameters. The results are presented in

Table 4. Comparison of scores.

Parameter	Score Function Result	Ranking Result
$\lambda =2$	$S_4>S_2>S_1>S_3$	$Z_4\succ Z_2\succ Z_1\succ Z_3$
$\lambda =4$	$S_4>S_2>S_1>S_3$	$Z_4\succ Z_2\succ Z_1\succ Z_3$
$\lambda =6$	$S_4>S_2>S_1>S_3$	$Z_4\succ Z_2\succ Z_1\succ Z_3$
$\lambda =8$	$S_4>S_2>S_1>S_3$	$Z_4\succ Z_2\succ Z_1\succ Z_3$
$\lambda =10$	$S_4>S_1>S_2>S_3$	$Z_4\succ Z_1\succ Z_2\succ Z_3$

and Figure 3. From Table 4, we can see that as the parameter varies, the optimal university is always Z₄. This indicates that Beijing University of Posts and Telecommunications maintains its leading position due to its high-level industry innovation platform, profound industry think-tank influence, significant contributions to industry-critical technological breakthroughs and high-value patents, and deep involvement in industry ecosystem co-construction. When the parameter is less than or equal to 8, the ranking results tend to be stable. When $\lambda$ is greater than 8, the rankings of Z₁ and Z₂ change, with Beijing Jiaotong University’s advantages in industry target alignment and industry resource coordination density becoming more evident, propelling its overall ranking ahead of North China Electric Power University. These results indicate that the flexibility of the parameter can alter the corresponding rankings of universities, making the results more flexible.

In practical applications, parameter $\lambda$ is a core tool for dynamic adjustment and assessment orientation. When experts give priority to breakthroughs in core disciplines, they can choose a smaller $\lambda$ value ($\lambda \le 8$). At this time, the assessment mechanism pays close attention to the performance of top disciplines, and Beijing University of Posts and Telecommunications ranks first with its overwhelmingly superior disciplines. This model is suitable for strategic goals that require rapid improvements in core competitiveness. If experts attach more importance to the balance of the discipline ecosystem, they should choose a larger $\lambda$ value ($\lambda >8$). Under this condition, the assessment criteria shift to the system’s risk resistance: Beijing University of Posts and Telecommunications continues to lead in terms of overall stability, while Beijing Jiaotong University effectively improved its overall performance with significant advantages in the two key indicators of “industry target alignment” and “industry resource coordination density,” demonstrating stronger discipline synergy and resistance to weak discipline risks and thereby surpassing North China Electric Power University to rank second. This model not only validates Beijing University of Posts and Telecommunications’ long-term stability but also highlights Beijing Jiaotong University’s advantages in sustainable development dimensions, making it suitable for decision-making scenarios focused on the long-term healthy development of higher education.

6.3. Comparative Analysis

To demonstrate the effectiveness and superiority of our presented method, we compare the CCFOFDWPPMM operator with the CCFOFDWA operator, CCFOFDWG operator, CCFOFDWPAM operator, CCFOFDWPGM operator, CCFOFDWPBM operator, and the CCFOFDWPMSM operators (assuming $\lambda =2,\kappa =2$, parameter vector $P=\left({\displaystyle \frac{1}{4}}\right.,{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}}\left.,{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}}\right)$).

Table 5. Quantitative comparative analysis of various operators.

Aggregation Operator	Ranking of Score Values	Ranking Result
CCFOFDWA	$S_1>S_4>S_2>S_3$	$Z_1\succ Z_4\succ Z_2\succ Z_3$
CCFOFDWG	$S_4>S_1>S_3>S_2$	$Z_4\succ Z_1\succ Z_3\succ Z_2$
CCFOFDWPAM	$S_2>S_4>S_1>S_3$	$Z_2\succ Z_4\succ Z_1\succ Z_3$
CCFOFDWPGM	$S_2>S_4>S_1>S_3$	$Z_2\succ Z_4\succ Z_1\succ Z_3$
CCFOFDWPBM	$S_2>S_4>S_1>S_3$	$Z_2\succ Z_4\succ Z_1\succ Z_3$
CCFOFDWPMSM	$S_2>S_4>S_1>S_3$	$Z_2\succ Z_4\succ Z_1\succ Z_3$
CCFOFDWPPMM	$S_4>S_2>S_1>S_3$	$Z_4\succ Z_2\succ Z_1\succ Z_3$

and

Table 6. Qualitative comparative analysis of various operators.

Aggregation Operator	Whether to Enhance the Flexibility	Whether to Consider Interrelationship of Both Attributes	Whether to Consider Interrelationship of Multiple Attributes	Whether to Reduce the Negative Effect	Whether to Consider Partitioning of Attributes
CCFOFDWA	Yes	No	No	No	No
CCFOFDWG	Yes	No	No	No	No
CCFOFDWPAM	Yes	No	No	No	Yes
CCFOFDWPGM	Yes	No	No	No	Yes
CCFOFDWPBM	Yes	Yes	No	No	Yes
CCFOFDWPMSM	Yes	Yes	Yes	No	Yes
CCFOFDWPPMM	Yes	Yes	Yes	Yes	Yes

show the results for quantitative comparative analysis and qualitative comparative analysis, respectively.

From Table 5, it can be seen that the best alternative obtained by applying the proposed CCFOFDWPPMM operator and the CCFOFDWG operator is $Z_4$, which validates the effectiveness of the proposed operator. Meanwhile, the worst alternative obtained by applying the other five operators except the CCFOFDWG operator is $Z_2$, which further verifies the applicability of the CCFOFDWPPMM operator proposed in this paper.

By analyzing the results of Table 5 and Table 6, we can summarize the reasons for the different ranking results as follows: (1) The reason why the ranking results obtained by applying the CCFOFDWA operator and the CCFOFDWG operator are different from those obtained by applying the CCFOFDWPPMM operator is that they do not take into account the interrelationship among attributes, and fail to eliminate the effect of the singularities on the aggregation results. (2) Compared with the CCFOFDWPAM operator and the CCFOFDWPGM operator, we find that the rankings of alternatives $Z_1$ and $Z_3$ are the same as the results obtained by applying the operator proposed in this paper, both of which are located in the third and fourth places. We can see that although these two operators improve the flexibility of the operation, they fail to effectively consider the interrelationships among the attributes, which makes a small difference in the sorting results of alternatives $Z_2$ and $Z_4$. (3) Although the CCFOFDWPBM operator considers the interrelationships between two attributes and the effect of the division of attributes, it is unable to consider the interrelationships among multiple attributes, making the ranking results slightly different from those obtained when applying the CCFOFDWPPMM operator proposed in this paper. (4) When compared with the CCFOFDWPMSM operator, the worst alternative is consistent with that of the CCFOFDWPPMM operator. However, the optimal alternative is different. The reason for this is that the CCFOFDWPMSM operator ignores the effect of singularities on the final aggregation result during the operation. Therefore, the CCFOFDWPPMM operator has greater superiority compared to the CCFOFDWPMSM.

As a result, through the above qualitative and quantitative analyses, we can conclude that the CCFOFDWPPMM operator presented in this paper has greater applicability and effectiveness in complex multi-attribute assessment problems. It can not only consider the interrelationships among multiple attributes in the process of information aggregation but can also effectively avoid the effect of extreme values on the aggregation results. At the same time, it can also significantly improve the flexibility of the operation. In summary, the novel multi-attribute assessment method has greater flexibility, adaptability, and applicability, and more effectively addresses complex assessment problems than the existing operators.

6.4. Discussion

As an important force serving national strategic needs and specific industry development, the accurate assessment of first-class discipline construction in industry-characteristic universities is of vital significance for optimizing discipline layout, guiding resource investment, and improving service capabilities. This paper proposes an effective assessment model that not only fully characterizes the uncertainty of expert information, effectively aggregates assessment information, and scientifically determines attribute weights, but also proposes a targeted assessment indicator system for first-class discipline construction in industry-characteristic universities. The results of the sensitivity analysis and comparative analysis show that the method proposed in this paper has significant advantages in the assessment of first-class discipline construction in industry-characteristic universities. We will elaborate on the innovative value of this paper from the perspectives of theoretical value and practical value.

Firstly, the expert information representation mechanism achieves a dimensional leap by creatively integrating the advantages of FOFS and CCFS, proposing a brand-new CCFOFS. This novel fuzzy set structure represents a significant theoretical breakthrough, enabling the precise characterization of multiple dimensions of decision-making information within a unified framework. These dimensions include experts’ degree of affirmation, hesitation, and negation toward alternative, potential fractional-order orthogonal relationships among alternative attributes, and even complex information with the periodic fluctuation characteristics inherent in expert judgments. This comprehensive representation capability greatly enhances accuracy when describing complex and fuzzy uncertain information in the real world, providing a more powerful assessment tool for assessing first-class discipline construction in industry-characteristic universities. Secondly, the information aggregation theory completed a paradigm shift, innovatively proposing the CCFOFDPMM operator and the CCFOFDWPPMM operator. These operators cleverly integrate three key theoretical components. The first is the parameterization flexibility provided by the Dombi operation, which allows for the flexible adjustment of aggregation behavior based on expert risk preferences; the nonlinear weighting characteristics of the PA operator, which automatically reduce the negative impact of outliers on overall results and enhance the robustness of aggregation. The second is the powerful capabilities of the PMM operator, which can simultaneously handle strong correlations within attribute groups and weak correlations or independence between groups. This integration significantly reduces information distortion during the aggregation process, ensuring that the final assessment results more accurately reflect the comprehensive level of the assessed object and are more suitable for assessment scenarios with complex attribute correlations, such as the assessment of first-class discipline construction in industry-characteristic universities. The third component is the innovation in decision-making methods: the IDOCRIW method achieves the fusion of dual objectivities. It not only considers the objective dispersion of the indicator data itself but also deeply analyzes the conflicting influences between indicators. This dual consideration avoids the one-sidedness of single objective weighting methods. The IDOCRIW method operates under the CCFOFS framework, and the objective weighting results it generates can make better use of the objective information contained in the assessment data, providing a strong objective basis for the subjective judgment link in the framework, effectively overcoming the arbitrariness that pure subjective weighting may bring, thereby generating more reasonable and credible attribute weights and significantly enhancing the scientific nature of the entire assessment process and the persuasiveness of the results. Finally, in terms of the innovation of the assessment index system, this paper comprehensively sorts out and deeply analyzes relevant research and practical developments at home and abroad in the specific field of “first-class discipline construction in industry-characteristic universities,” breaks through the constraints of the comprehensive university assessment system framework, closely combines the essential characteristics and development needs of industry-characteristic universities, and constructs, for the first time, an index system that is suitable for assessing first-class discipline construction. The system clearly defines the four core dimensions of “talent cultivation,” “scientific research,” “social services,” and “open education.” These four dimensions not only cover the key functional aspects of first-class discipline construction, but also highlight the special mission of industry-characteristic universities in serving national strategies and specific industries. We further refined and established 16 specific assessment indicators that are highly targeted and operational. These indicators closely focus on the goals and pain points of first-class discipline construction in industry-characteristic universities and can effectively capture the key elements and unique value of their development. The establishment of this indicator system provides a solid theoretical foundation and practical assessment framework for scientifically and systematically assessing first-class discipline construction in industry-characteristic universities.

This study thoroughly analyzes the core elements of assessing first-class discipline construction in industry-characteristic universities and provides practical guidance for first-class discipline construction in industry-characteristic universities. (1) In terms of talent cultivation, the competitiveness of graduates in core industry positions has the greatest weight, which is the core orientation and effectiveness benchmark for first-class discipline construction in industry-characteristic universities. This weight distribution reveals the direction of the return to the essence of higher education and reflects that the effectiveness of first-class discipline talent cultivation primarily depends on the fit between graduates and core industry positions. Improving the competitiveness of students in core industry positions can enhance the social recognition and reputation of disciplines, thereby helping to attract high-quality students and positively promoting enrollment. A good disciplinary reputation can attract outstanding industries to actively seek cooperation and strengthen the ties between industry, academia, and research. In the future, planners of industry-characteristic universities need to optimize the layout of disciplines and restructure professional courses with a focus on the competitiveness of graduates in core industry positions. They should actively predict talent gaps and integrate core industry skill standards into the curriculum system and build practical training platforms to effectively transform knowledge transfer into innovative capabilities. (2) In the dimension of scientific research, contributions to breakthroughs in key technologies in the industry have the greatest weight, and are an important driving force for the development of first-class discipline construction in industry-characteristic universities. This high weight indicates that contributions to key technological breakthroughs in the industry have become the core breakthrough point for breaking through disciplinary bottlenecks, meaning that the level of scientific research in first-class disciplines in industry-characteristic universities depends primarily on their contribution to key technological breakthroughs in the industry. Enhancing this indicator will significantly enhance the academic influence of disciplines, facilitate theoretical innovation and technological breakthroughs, accelerate the conversion of research results, attract top talent, and ultimately comprehensively enhance the core competitiveness of disciplines. Future university planners need to closely align with national strategies, concentrate resources to support core technological breakthroughs, ensure young scholars have autonomy in breakthrough research, and highlight their direct contributions to breakthroughs in industry-critical technologies. (3) In the social services dimension, the effectiveness of technology transfer has the greatest weight, which is the channel for value creation and the transformation of first-class disciplines in industry-characteristic universities. The effectiveness of technology transfer directly reflects the ultimate value of disciplinary knowledge in promoting industry and improving people’s livelihoods. The improvement of this indicator will significantly strengthen the disciplinary voice in industry and feed back into basic research funding through transfer gains; the industry standards it forms will further enhance the status and academic influence of the discipline. University planners need to make solving industrial and livelihood issues their core goal, improve their technology maturity assessment mechanism, strengthen R&D incentives, focus on cultivating core technology clusters, and allocate resources in a targeted manner to break through conversion bottlenecks. (4) In terms of open education, the weight of the indicator of industry resource coordination density is much higher than other indicators. This indicator determines the status of first-class disciplines in the industrial ecosystem and is the core representation of the effectiveness of open education in first-class disciplines in industry-characteristic universities, indicating that in-depth industry collaboration is a key indicator for measuring the effectiveness of discipline construction. Close industry resource collaboration increases in the discipline’s voice in the industrial ecosystem, which is conducive to the discipline’s deep integration into the industrial innovation network, grasping the initiative in technological development, enhancing the industry’s reputation, and expanding the discipline’s development space. In order to improve this indicator, future planners of industry-characteristic universities need to strengthen strategic coordination with core industry partners, take the lead in building a platform for the deep integration of industry, academia, and research, cultivate urgently needed talent for the industry, focus on research and the development of results with common technological value for the industry, promote the integration of the teacher structure and curriculum system with the forefront of the industry, and improve students’ ability to solve practical industry problems.

We acknowledge that this study has several limitations: first, it was primarily conducted by academic researchers, whose perspectives and focus are centered on theoretical development and academic value. Throughout the research process, there was a lack of in-depth, sustained participation from industry experts, university administrators, or policymakers. This may have resulted in the proposed assessment model and the interpretation of the final assessment results failing to fully integrate the latest perspectives on industry development, core pain points in management practices, and practical considerations for policy implementation, which, to a certain extent, limited the efficiency of translating the research results into actual management decisions. Second, there is a lack of integration between subjective and objective factors in the weight determination method. Although we incorporated expert opinions through the Delphi method during the indicator selection stage and innovatively adopted the IDOCRIW method to determine the objective weights of the final attributes, we failed to systematically incorporate domain experts’ subjective preferences regarding the strategic importance of different dimensions or specific indicators into the final weight calculation during the core weight determination process. The current weight distribution is completely data-driven, which is objective, but when assessing complex issues such as first-class discipline construction in industry-characteristic universities, which involve strategic orientation and characteristic development, it may not fully reflect the priority development requirements of certain aspects in a specific context. Finally, this study mainly focuses on the static assessment framework. The method we propose is essentially an assessment of the status of first-class discipline construction in industry-characteristic universities at a specific point in time. However, first-class discipline construction is a long-term, dynamically evolving process. The current assessment system and methods are unable to effectively track the changing trends, development speed, policy lag effects, or the long-term cumulative effects of first-class discipline construction over time, nor can they predict future development potential or identify key development turning points.

7. Conclusions

The main contribution of this paper is to propose a novel multi-attribute assessment method and apply it to solve the assessment problem of first-class discipline construction in industry-characteristic universities. First, we combine FOFS and CCFS to propose a new CCFOFS. This fuzzy set can more comprehensively characterize uncertain information in complex decision-making environments, while representing membership, hesitation, non-membership, score orthogonality, and complex information with phase cycle characteristics. Second, for complex assessment scenarios in which attributes are correlated and can be grouped in the CCFOFS environment, we innovatively propose the CCFOFDPMM operator and the CCFOFDWPPMM operator. These operators combine the flexibility of Dombi operations, the nonlinear weighting characteristics of PA operators, and the ability of PMM operators to handle associations within and between attribute groups. They effectively reduce distortion during the information aggregation process, flexibly handle heterogeneous associations between attributes, minimize the negative impact of outliers on aggregation results, and significantly enhance the robustness of aggregation outcomes. Furthermore, we employ the IDOCRIW method to objectively determine attribute weights in the CCFOFS environment. This method comprehensively considers the objective dispersion of data and the conflicting influences between indicators, achieving the effective integration of objective information and enhancing the reliability of weight determination. Finally, based on a systematic review of domestic and international research and practical developments, we constructed an assessment indicator system for first-class discipline construction in industry-characteristic universities, covering 16 specific indicators under the four core dimensions of “talent cultivation,” “scientific research,” “social services,” and “open education.” On this basis, we integrated the above-mentioned theoretical tools to propose a complete set of multi-attribute assessment methods for assessing first-class discipline construction in industry-characteristic universities, and applied it to solve the problem of assessing first-class discipline construction in industry-characteristic universities. The research results fully verify the rationality, scientificity, and effectiveness of the proposed method. In summary, the method proposed in this paper has significant advantages in portraying complex expert judgments, handling fuzzy and uncertain information, integrating complementary double objective weights, and scientifically assessing first-class discipline construction in industry-characteristic universities.

First-class discipline construction in industry-characteristic universities should take the competitiveness of graduates in core industry positions as its fundamental goal, be driven by breakthroughs in key industry technologies, realize value creation through effective technology transfer, and build a closed-loop system based on the close coordination of industry resources. Additionally, attention should be paid to other important indicators with non-maximal weights (such as industry target alignment, industry ecosystem co-construction participation rate, and the adequacy of emergency service response), and efforts should be made to comprehensively and coordinately advance improvements in these areas to ensure the comprehensiveness and coordination of first-class discipline construction. Universities are the core entities promoting the innovative development of industries, and this is especially true for industry-characteristic universities, which shoulder a dual mission: to break through key technologies at the forefront of the industry and accelerate the transformation of results, while also deeply integrating industry standards to cultivate innovative talent. This study provides a practical framework for improving the first-class discipline construction of industry-characteristic universities and enhancing their disciplinary competitiveness. In the future, as industry-characteristic universities improve their first-class discipline construction, they need to internalize industry services as a core characteristic of disciplinary organization, balance educational value and industrial contribution, fulfill their social responsibilities, cultivate knowledgeable and technically outstanding talents for society, and ultimately achieve a sustainable cycle of education empowering industry progress and industry driving educational innovation.

However, while this study achieved certain innovative progress, we also identified the limitations of the methods proposed in this paper during the discussion. In the future, our key research efforts will focus on addressing these limitations. First, we will actively expand the scope of research participants to achieve a deeper integration of diverse perspectives. Future work should strive to establish a broader industry–academia–research collaborative research team, with collaboration throughout the entire research process. This kind of in-depth cooperation can ensure that the model and results are not only theoretically rigorous, but also accurately meet industrial needs and management practices, significantly enhancing the application value of the results. Second, deepen the integration of subjectivity and objectivity in weight determination. Future research needs to build on the existing foundation, we will aim to systematically integrate subjective preference information into the core steps of weight calculation, select appropriate subjective weight determination methods based on the characteristics of first-class discipline construction in industry-characteristic universities, and use scientific combination weighting methods to integrate the two to form the final comprehensive weight. This will not only preserve the objective facts reflected in the data, but also incorporate experts’ value judgments on strategic direction and characteristic development, making the weighting distribution more reasonable, more contextually adaptive, and more supportive of decision-making. Finally, we will focus on developing dynamic assessment capabilities. One of the core challenges for future research is upgrading the static assessment framework to a dynamic monitoring and prediction system. This requires the introduction of time series analysis techniques to the methodology, using assessment data from consecutive years to analyze the change trajectory, growth rate, and stability of key indicators for first-class discipline construction, constructing a dynamic comprehensive assessment model, regularly tracking and assessing disciplines, and generating development indices or trend charts. In addition, machine learning prediction models can be used to attempt to predict future trends in disciplines, assess the effects of policy interventions, or identify potential bottlenecks, providing a scientific basis for long-term planning and the dynamic adjustment of first-class discipline construction.