Some Complex Intuitionistic Uncertain Linguistic Heronian Mean Operators and Their Application in Multiattribute Group Decision Making

Abstract

In this paper, a new decision-making algorithm has been presented in the context of a complex intuitionistic uncertain linguistic set (CIULS) environment. CIULS integrates the concept the complex of a intuitionistic fuzzy set (CIFS) and uncertain linguistic set (ULS) to deal with uncertain and imprecise information in a more proactive manner. To investigate the interrelation between the pairs of CIULSs, we combine the concept of the Heronian mean (HM) and the complex intuitionistic uncertain linguistic (CIUL) to describe some new operators, namely, CIUL arithmetic HM (CIULAHM), CIUL weighted arithmetic HM (CIULWAHM), CIUL geometric HM (CIULGHM), and CIUL weighted geometric HM (CIULWGHM). The main advantage of these suggested operators is that they considered the interaction between pairs of objects during the formulation process. Also, a number of distinct brief cases and properties of the operators are analyzed. In addition, based on these operators, we have stated a MAGDM (“multiattribute group decision-making”) problem-solving algorithm. The consistency of the algorithm is illustrated by a computational example that compares the effects of the algorithm with a number of well-known existing methods.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

MAGDM issues are the critical exploration aspects of the current judgement philosophy to deal with questionable and incorrect facts in time complications. If the reasons remain fuzzy, the signature values involved in decision-making problems are not continuously seen to be crisp artefacts, and some of them are extensively sufficient to be identified by a number of hypotheses. The fuzzy set (FS) theory is one of those that Zadeh [1] has built to handle with awkward and difficult facts. FS applies only to the term of the degree of truth limited to the unit interval. FS has gained a great deal of interest from various academics and has been exploited by a number of scientists in the nature of separate fields. For example, L-FS was investigated by Goguen [2]. L-FS is essentially a mixture of two theories, such as FS and lattice’s ordered series, which is a useful strategy for dealing with difficult facts. In addition, Torra [3] reworked the FS theorem to explain the hesitant FS (HFS) principle, which covers the degree of truth in the form of the finite subset of the unit interval. Pawlak [4] looked at the rough sets and the FSs. Zhang [5] introduced the concept of bipolar FS (BFS) containing two degrees with a law that is the degree of truth belonging to [0, 1] and the degree of falsehood belonging to [−1, 0]. BFS has gained considerable attention from separate intellectuals and has been extensively used by many scientists in the world of various fields. For instance, the theory of bipolar soft set was developed by Mahmood [6].

FS is a major apparatus for dealing with troublesome and complex information in day-to-day natural life problems, and a number of researchers have made extensive use of it in different fields. However, in some cases, the theory of FS is not capable of dealing with such a kind of concern, for example, if an individual gives certain sources of knowledge, including the degree of truth and falsehood, then the theory of FS has failed. To deal with such problems, Atanassov [7] used the principle of intuitionistic FS (IFS) with the law that the totality of the degrees of each other lies inside the unit interval. IFS is a simplified version of FS to deal with uncomfortable experience of natural life problems. IFS has gained considerable recognition from various academics and has been employed by a number of scientists in distinct neighbourhoods. For example, Beg and Rashid [8] discussed the principle of intuitionistic HFS (IHFS) holding the degree of truth and the degree of falsehood in the form of a finite unit interval subset. The law of IHFS is that the absolute maximum (also for the least) of the truth and the minimum (also for the maximum) falsity is limited to the unit interval. In addition, Atanassov [9] introduced the principle of interval-valued IFS (IVIFS), which is the extension of the interval-valued FS (IVFS). IVIFS refers to the degree of truth and falsehood in the shape of a subinterval of the unit interval. The IFS and IVIFS have received large concentrations from separate intellectuals and have been extensively used by many scientists in the world in various fields [10–14].

Complex FS (CFS) theory is one of the most proficient techniques developed by Ramot et al. [15] to manage uncomfortable and difficult details. CFS covers only the term of the degree of truth in the structure of complex numbers relevant to the complex plane in the unit disc with a restriction that the true and imaginary portions of the degree of truth are limited to the unit interval. CFS has attracted considerable interest from a variety of researchers and has been exploited by a number of scientists in distinct fields. For example, the neuro fuzzy architecture used was investigated by Chen et al. [16]. Ramot et al. [17] has studied a dynamic fuzzy logic. Zhang et al. [18] investigated the activity properties of CFSs. The CFS theory has also been established by Nguyen et al. [19], Dick [20], and Tamir et al. [21]. Tamir et al. [22] presented a concept of generalized complex fuzzy propositional logic. The aggregation operators on the complex fuzzy information have been defined by the researchers in [23–25].

CFS is an important apparatus for dealing with troublesome and complex information in day-to-day natural life problems, and a number of researchers have made extensive use of it in different fields. However, in some cases, the theory of CFS is not capable of dealing with this kind of concern, for example, if an individual gives certain sources of knowledge, including the degree of truth and falsehood, then the theory of CFS has failed. To handle with such sort of troubles, Alkouri and Salleh [26] used the theory of complex IFS (CIFS) with a requirement that the totality of the real parts (also for imaginary parts) of both degrees is inside the unit interval. CIFS is a modified form of CFS to deal with awkward and convoluted awareness of natural world problems. The CIFS has attracted considerable interest from various academics and has been exploited by a number of scientists in separate fields. For example, Al-Qudah et al. [27] presented a decision-making approach under the complex multifuzzy soft set environment. Kumar and Bajaj [28] used the CIF concept in the soft set environment to investigate the dynamic intuitive fuzzy soft set. Garg and Rani [29] have established a number of knowledge measures for the CYPSs. Ngan et al. [30] looked at the quaternion number depending on the CIFS. Rani and Garg [31] presented preference relation for the complex intuitionistic fuzzy set in individual and group decision-making process. Ali et al. [32] studied the complex intuitionistic fuzzy groups. Garg and Rani have established the theory of aggregation operators for IFCS [33]. In addition, Rahman et al. [34] developed the hybrid model of the hypersoft set with complex fuzzy set and complex intuitionistic fuzzy set and neurtrosophic set. CIFS has received considerable attention from separate intellectuals and has been widely used by many scientists in the world in various fields [35–37].

However, in different real difficulties, it is not easy for decision makers to express their views in quantitative representations. For example, as a professional considering the applicant’s degree of advanced expertise, the use of linguistic expressions, such as linguistic phrases, “very good,” “good,” or “medium” may be considered for being additionally suitable or familiar to convey his or her opinion. To handle such sorts of concerns, Zadeh [38] investigated the linguistic variable theory (LV) in order to describe the interests of decision makers. In addition, the principle of the two-fold linguistic set was established by Herrera and Martinez [39]. Liu and Jin [40] have studied the uncertain LV (ULV). Heronian mean operators based on the intuitionistic uncertain linguistic set (IULS) were developed in [41]. Liu and Liu [42] studied the partitioned Bonferroni mean IULS operators. In addition, Liu et al. [43] investigated the weighted Bonferroni order weighted average operators for IULS. Liu et al. [44] used the concept of Hamy as a mean operator for IULSs. The theory of Bonferroni mean IULS operators has been established by Liu and Zhang [45]. But, to date, no one has used these concepts in the CIULS setting, and to discover the interrelationship between some numbers of CIULS, HM operators are very useful for dealing with uncomfortable and troublesome knowledge in everyday difficulties.

(1) To investigate the CIULS and discuss their operational laws.

(2) To explore the CIULAHM, CIULWAHM, CIULGHM, and CIULWGHM operators and discuss their special cases with some properties.

(3) A MAGDM procedure is developed by using the explored operators based on CIULSs.

(4) Some numerical examples are illustrated with the help of investigated approaches.

(5) In order to determine the efficiency and competence of the developed operators, comparative analysis and graphic expressions are often used to demonstrate the superiority of the methods developed.

The remainder of the paper is presented as follows. In Section 2, we refer to some basic concepts, such as the CIFS and their operating rules. The current idea of LSs, ULVs, and their operations is also updated in this report. In addition, the definition of HM with parameters and without parameters is discussed. In Section 3, we investigated the CIULS and examined their operating rules. In Section 4, we examined the CIULAHM, the CIULWAHM, the CIULGHM, and the CIULWGHM operators and addressed their specific cases with those properties. Section 5 develops a MAGDM procedure by using CIULS-based explored operators. Some numerical examples are illustrated with the help of investigated approaches. To discover the consistency and expertise of the developed operators, comparative analysis and graphic expressions are often used to show the superiority of the methods developed. The end of the script is explored in Section 6.

2. Preliminaries

For better describing the investigated ideas, we recall some fundamental notions such as CIFSs and their operational laws. The existing idea of LSs, ULVs, and their operations is also revised in this study. Moreover, the idea of HMO with parameters and without parameters is also discussed. Throughout the article, the symbol $\overbrace{{\mathcal{X}}_{\text{UNI}}}$ is used for fixed sets and the terms ${\mathcal{J}}_{{\mathfrak{W}}_{\text{CI}}}$ and ${\mathcal{K}}_{{\mathfrak{W}}_{\text{CI}}}$ are shown the grade of positive and the grade of negative.

Definition 1.

(see [26]). A CIFS ${\mathfrak{W}}_{\text{CI}}$ is demonstrated by$$\begin{array}{}\text{(1)}& {\mathfrak{W}}_{\text{CI}}={\mathcal{J}}_{{\mathfrak{W}}_{\text{CI}}}\breve{\mathfrak{k}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{CI}}}\breve{\mathfrak{k}}:\breve{\mathfrak{k}}\in \overbrace{{\mathcal{X}}_{\text{UNI}}},\end{array}$$where ${\mathcal{J}}_{{\mathfrak{W}}_{\text{CI}}}\breve{\mathfrak{k}}={\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{IP}}\breve{\mathfrak{k}}}$ and ${\mathcal{K}}_{{\mathfrak{W}}_{\text{CI}}}\breve{\mathfrak{k}}={\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}}}\breve{\mathfrak{k}}}$ with the rules such that $0\le {\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}+{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}\le 1$ and $0\le {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}}}\breve{\mathfrak{k}}+{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}}}\breve{\mathfrak{k}}\le 1$. Furthermore, the refusal grade is demonstrated in the form of ${\mathrm{ℒ}}_{{\mathfrak{W}}_{\text{CI}}}\breve{\mathfrak{k}}={\mathrm{ℒ}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathrm{ℒ}}_{{\mathfrak{W}}_{\text{IP}}}\breve{\mathfrak{k}}}=1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}{e}^{i2\pi 1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}}}\breve{\mathfrak{k}}-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}}}\breve{\mathfrak{k}}}$. In this paper, the complex intuitionistic fuzzy numbers (CIFNs) are represented by ${\mathfrak{W}}_{\text{CI}-i}={\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},i=\mathrm{1,2},\dots ,\overbrace{\mathrm{\Xi }}$.

Definition 2.

(see [33]). Based on any two CIFNs ${\mathfrak{W}}_{\text{CI}-i}={\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},i=\mathrm{1,2}$, then

(1) ${\mathfrak{W}}_{\text{CI}-1}\oplus {\mathfrak{W}}_{\text{CI}-2}=\begin{array}{c}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}+{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}+{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}},\\ {\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}}\end{array}$.

(2) ${\mathfrak{W}}_{\text{CI}-1}\otimes {\mathfrak{W}}_{\text{CI}-2}=\begin{array}{c}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}},\\ {\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}+{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}+{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}}\end{array}$.

(3) ${\Phi }_{\text{SC}}{\mathfrak{W}}_{\text{CI}-1}=1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}}^{{\Phi }_{\text{SC}}}{e}^{i2\pi 1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}}^{{\Phi }_{\text{SC}}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}^{{\Phi }_{\text{SC}}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}^{{\Phi }_{\text{SC}}}\breve{\mathfrak{k}}}$.

(4) ${\mathfrak{W}}_{\text{CI}-1}^{{\Phi }_{\text{SC}}}={\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}^{{\Phi }_{\text{SC}}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}^{{\Phi }_{\text{SC}}}\breve{\mathfrak{k}}},1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}}^{{\Phi }_{\text{SC}}}{e}^{i2\pi 1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}}^{{\Phi }_{\text{SC}}}}$.

Definition 3.

(see [33]). For two CIFNs ${\mathfrak{W}}_{\text{CI}-i}={\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},i=\mathrm{1,2}$, the score value and accuracy value are demonstrated by$$\begin{array}{c}\text{(2)}& \overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CI}-1}={\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}+{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}},\overline{\overline{\mathfrak{F}}}{\mathfrak{W}}_{\text{CI}-1}={\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}+{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}+{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}+{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}},\end{array}$$where $\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CI}-1}\in -\mathrm{1,1}$ and $\overline{\overline{\mathfrak{F}}}{\mathfrak{W}}_{\text{CI}-1}\in \mathrm{0,1}$. To find the relationships between any two CIFNs, we use the following rules:

(1) If $\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CI}-1}>\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CI}-2}$, then ${\mathfrak{W}}_{\text{CI}-1}>{\mathfrak{W}}_{\text{CI}-2}$.

(2) If $\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CI}-1}<\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CI}-2}$, then ${\mathfrak{W}}_{\text{CI}-1}<{\mathfrak{W}}_{\text{CI}-2}$.

(3) If $\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CI}-1}=\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CI}-2}$, then

(1) If $\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CI}-1}>\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CI}-2}$, then ${\mathfrak{W}}_{\text{CI}-1}>{\mathfrak{W}}_{\text{CI}-2}$.

(2) If $\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CI}-1}<\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CI}-2}$, then ${\mathfrak{W}}_{\text{CI}-1}<{\mathfrak{W}}_{\text{CI}-2}$.

(3) If $\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CI}-1}=\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CI}-2}$, then ${\mathfrak{W}}_{\text{CI}-1}={\mathfrak{W}}_{\text{CI}-2}$.

Definition 4.

(see [38]). A LS is demonstrated by$$\begin{array}{}\text{(3)}& \psi ={\psi }_0,{\psi }_1,{\psi }_2,\dots ,{\psi }_{\overline{\overline{k_{\text{SC}}}}-1},\end{array}$$where $\overline{\overline{k_{\text{SC}}}}$ should be odd, which grips the ensuing circumstances:

(1) If $\overline{\overline{k_{\text{SC}}}}>{\overline{\overline{k_{\text{SC}}}}}^{\prime }$, then ${\psi }_{\overline{\overline{k_{\text{SC}}}}}>{\psi }_{{\overline{\overline{k_{\text{SC}}}}}^{\prime }}$.

(2) The negative operator $\text{neg}{\psi }_{\overline{\overline{k_{\text{SC}}}}}={\psi }_{{\overline{\overline{k_{\text{SC}}}}}^{\prime }}$ with a rule $\overline{\overline{k_{\text{SC}}}}+{\overline{\overline{k_{\text{SC}}}}}^{\prime }=\overline{\overline{k_{\text{SC}}}}+1$.

(3) If $\overline{\overline{k_{\text{SC}}}}\ge {\overline{\overline{k_{\text{SC}}}}}^{\prime }$, $\max {\psi }_{\overline{\overline{k_{\text{SC}}}}},{\psi }_{{\overline{\overline{k_{\text{SC}}}}}^{\prime }}={\psi }_{\overline{\overline{k_{\text{SC}}}}}$, and if $\overline{\overline{k_{\text{SC}}}}\le {\overline{\overline{k_{\text{SC}}}}}^{\prime }$, $\max {\psi }_{\overline{\overline{k_{\text{SC}}}}},{\psi }_{{\overline{\overline{k_{\text{SC}}}}}^{\prime }}={\psi }_{\overline{\overline{k_{\text{SC}}}}}$.

Likewise, $\widehat{\psi }={\psi }_i:i\in R$ conveyed the LSs. A set $\psi ={\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}},{\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}}\in \widehat{\psi }i\le \mathfrak{s}$, where ${\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}}$ characterize the upper and lower limits of $\psi$ is called ULVs [40]. By utilizing any two ULVs ${\psi }_1={\psi }_{{\phi }_1},{\psi }_{{\varphi }_1}$ and ${\psi }_2={\psi }_{{\phi }_2},{\psi }_{{\varphi }_2}$ belonging to ${\widehat{\psi }}_{0,h}$,

(1) ${\psi }_1\oplus {\psi }_2={\psi }_{{\phi }_1},{\psi }_{{\varphi }_1}\oplus {\psi }_{{\phi }_2},{\psi }_{{\varphi }_2}={\psi }_{{\phi }_1+{\phi }_2-{\phi }_1{\phi }_2/h},{\psi }_{{\varphi }_1+{\varphi }_2-{\varphi }_1{\varphi }_2/h}$.

(2) ${\psi }_1\otimes {\psi }_2={\psi }_{{\phi }_1},{\psi }_{{\varphi }_1}\otimes {\psi }_{{\phi }_2},{\psi }_{{\varphi }_2}={\psi }_{{\phi }_1\times {\phi }_2/h},{\psi }_{{\varphi }_1\times {\varphi }_2/h}$.

(3) ${\Phi }_{\text{SC}}{\psi }_1={\Phi }_{\text{SC}}{\psi }_{{\phi }_1},{\psi }_{{\varphi }_1}={\psi }_{h1-{1-{\phi }_1/h}^{{\Phi }_{\text{SC}}}},{\psi }_{h1-{1-{\varphi }_1/h}^{{\Phi }_{\text{SC}}}}$.

(4) ${\psi }_1^{{\Phi }_{\text{SC}}}={\psi }_{h{{\phi }_1/h}^{{\Phi }_{\text{SC}}}},{\psi }_{h{{\varphi }_1/h}^{{\Phi }_{\text{SC}}}}$.

Definition 5.

(see [41]). The HM operator ${\text{HM}}^{P_{\text{SC}},q_{\text{SC}}}:{\Theta }^{\overbrace{\mathrm{\Xi }}}⟶\Theta$ is demonstrated by$$\begin{array}{}\text{(4)}& {\text{HM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CI}-1},{\mathfrak{W}}_{\text{CI}-2},\dots ,{\mathfrak{W}}_{\text{CI}-\overbrace{\mathrm{\Xi }}}={\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CI}-i}^{p_{\text{SC}}}{\mathfrak{W}}_{\text{CI}-\mathfrak{s}}^{q_{\text{SC}}}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}.\end{array}$$

If we define the HM operator without parameter, it is demonstrated by:$$\begin{array}{c}\text{(5)}& {\text{HM}}^{p_{\text{SC}},q_{\text{SC}}}:{\Theta }^{\overbrace{\mathrm{\Xi }}}⟶\Theta ,\text{by}\text{HM}{\mathfrak{W}}_{\text{CI}-1},{\mathfrak{W}}_{\text{CI}-2},\dots ,{\mathfrak{W}}_{\text{CI}-\overbrace{\mathrm{\Xi }}}=\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CI}-i}{\mathfrak{W}}_{\text{CI}-\mathfrak{s}}}}.\end{array}$$

3. Complex Intuitionistic Uncertain Linguistic Variables

In this study, we elaborate the fundamental notions of CIULVs and their related principles by utilizing the remaining theories of ULVs and CIFSs.

Definition 6.

A CIULV ${\mathfrak{W}}_{\text{CIU}}$ is demonstrated by$$\begin{array}{}\text{(6)}& {\mathfrak{W}}_{\text{CIU}}={\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}},{\mathcal{J}}_{{\mathfrak{W}}_{\text{CIU}}}\breve{\mathfrak{k}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{CIU}}}\breve{\mathfrak{k}}:\breve{\mathfrak{k}}\in \overbrace{{\mathcal{X}}_{\text{UNI}}},\end{array}$$where ${\mathcal{J}}_{{\mathfrak{W}}_{\text{CIU}}}\breve{\mathfrak{k}}={\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}}}\breve{\mathfrak{k}}}$ and ${\mathcal{K}}_{{\mathfrak{W}}_{\text{CIU}}}\breve{\mathfrak{k}}={\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}}}\breve{\mathfrak{k}}}$ with the rules such that $0\le {\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}+{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}\le 1$ and $0\le {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}}}\breve{\mathfrak{k}}+{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}}}\breve{\mathfrak{k}}\le 1$ with ${\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}}\in \widehat{\psi }i\le \mathfrak{s}$. Furthermore, the refusal grade is demonstrated in the form of ${\mathrm{ℒ}}_{{\mathfrak{W}}_{\text{CIU}}}\breve{\mathfrak{k}}={\mathrm{ℒ}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathrm{ℒ}}_{{\mathfrak{W}}_{IP}}\breve{\mathfrak{k}}}=1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}{e}^{i2\pi 1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}}}\breve{\mathfrak{k}}-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}}}\breve{\mathfrak{k}}}$. In this paper, the complex intuitionistic uncertain linguistic numbers (CIULNs) are represented by$$\begin{array}{}\text{(7)}& {\mathfrak{W}}_{\text{CIU}-i}={\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},\hspace{1em}i,\mathfrak{s}=\mathrm{1,2},\dots ,\overbrace{\mathrm{\Xi }}.\end{array}$$

Definition 7.

For two CIULNs ${\mathfrak{W}}_{\text{CIU}-i}={\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},i=\mathrm{1,2}$, some operational laws are stated as

(1) ${\mathfrak{W}}_{\text{CIU}-1}\oplus {\mathfrak{W}}_{\text{CIU}-2}=\begin{array}{c}{\psi }_{{\phi }_1+{\phi }_2-{\phi }_1{\phi }_2/h},{\psi }_{{\varphi }_1+{\varphi }_2-{\varphi }_1{\varphi }_2/h},\\ \begin{array}{c}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}+{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}+{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}},\\ {\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}}\end{array}\end{array}$.

(2) ${\mathfrak{W}}_{\text{CIU}-1}\otimes {\mathfrak{W}}_{\text{CIU}-2}=\begin{array}{c}{\psi }_{{\phi }_1\times {\phi }_2/h},{\psi }_{{\varphi }_1\times {\varphi }_2/h},\\ \begin{array}{c}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}},\\ {\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}+{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}+{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}}\end{array}\end{array}$.

(3) ${\Phi }_{\text{SC}}{\mathfrak{W}}_{\text{CIU}-1}=\begin{array}{c}{\psi }_{h1-{1-{\phi }_1/h}^{{\Phi }_{\text{SC}}}},{\psi }_{h1-{1-{\varphi }_1/h}^{{\Phi }_{\text{SC}}}},\\ 1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}}^{{\Phi }_{\text{SC}}}{e}^{i2\pi 1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}}^{{\Phi }_{\text{SC}}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}^{{\Phi }_{\text{SC}}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}^{{\Phi }_{\text{SC}}}\breve{\mathfrak{k}}}\end{array}$.

(4) ${\mathfrak{W}}_{\text{CIU}-1}^{{\Phi }_{\text{SC}}}=\begin{array}{c}{\psi }_{h{{\phi }_1/h}^{{\Phi }_{\text{SC}}}},{\psi }_{h{{\varphi }_1/h}^{{\Phi }_{\text{SC}}}},\\ {\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}^{{\Phi }_{\text{SC}}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}^{{\Phi }_{\text{SC}}}\breve{\mathfrak{k}}},1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}}^{{\Phi }_{\text{SC}}}{e}^{i2\pi 1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}}^{{\Phi }_{\text{SC}}}}\end{array}$.

Proposition 1.

For two CIULNs ${\mathfrak{W}}_{\text{CIU}-i}={\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},$ the operations defined in Definition 7 are also CIULNs.

Proof.

For any two CIULNs ${\mathfrak{W}}_{\text{CIU}-1}={\psi }_{{\phi }_1},{\psi }_{{\varphi }_1},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}}$ and ${\mathfrak{W}}_{\text{CIU}-2}={\psi }_{{\phi }_2},{\psi }_{{\varphi }_2},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}}$, then by using the idea of T-norm and T-conorm such that$$\begin{array}{}\text{(8)}& T:\mathrm{0,1}\times \mathrm{0,1}⟶\mathrm{0,1}.\end{array}$$

is called T-norm, if $T$ holds the following conditions:

(1) Commutativity

(2) Monotonicity

(3) Associativity

(4) $Tx,1=x$

And similarly, for T-conorm, we defined a function such that$$\begin{array}{}\text{(9)}& S:\mathrm{0,1}\times \mathrm{0,1}⟶\mathrm{0,1}\end{array}$$is called T-conorm, if $S$ holds the following conditions:

(1) Commutativity

(2) Monotonicity

(3) Associativity

(4) $Sx,0=x$

Then, we prove that the above four conditions.

(1) The addition of two linguistic number is again linguistic number such that ${\phi }_1+{\phi }_2-{\phi }_1{\phi }_2/h$ and ${\varphi }_1+{\varphi }_2-{\varphi }_1{\varphi }_2/h$ are also T-conorm, the real and imiginary parts of the truth are T-conorm which indicates that these two satisfy the conditions of T-conorm such that ${\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}+{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}$ and ${\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}+{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}$, the real and imaginary parts of the truth are T-conorm which means that these two function satisfy the conditions of T-conorm such that $S{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}},0={\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}+{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}={\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}+0-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}0={\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}$, and thus, by using the definition of T-conorm, the values of the two should be in unit interval. Similarly, the function ${\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}$ and ${\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}$ are in the form of T-conorm, which means that these two satisfy the conditions of T-norm such that $T{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}},1={\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}={\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}1={\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}$. Therefore, from the above analysis, we get the result such that

$$\begin{array}{}\text{(10)}& {\mathfrak{W}}_{\text{CIU}-1}\oplus {\mathfrak{W}}_{\text{CIU}-2}=\begin{array}{c}{\psi }_{{\phi }_1+{\phi }_2-{\phi }_1{\phi }_2/h},{\psi }_{{\varphi }_1+{\varphi }_2-{\varphi }_1{\varphi }_2/h},\\ \begin{array}{c}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}+{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}+{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}},\\ {\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-2}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-2}}\breve{\mathfrak{k}}}\end{array}\end{array}.\end{array}$$

The points 2–4 are similar.

Definition 8.

For CIULN ${\mathfrak{W}}_{\text{CIU}-1}={\psi }_{{\phi }_1},{\psi }_{{\varphi }_1},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}}$, the score and accuracy functions are defined as$$\begin{array}{c}\text{(11)}& \overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CIU}-1}=\frac{1}{4}{\phi }_1+{\varphi }_1\times {\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}+{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}},\overline{\overline{\mathfrak{F}}}{\mathfrak{W}}_{\text{CIU}-1}=\frac{1}{4}{\phi }_1+{\varphi }_1\times {\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}+{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-1}}\breve{\mathfrak{k}}+{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}}+{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-1}}\breve{\mathfrak{k}},\end{array}$$where $\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CIU}-1}\in -\mathrm{1,1}$ and $\overline{\overline{\mathfrak{F}}}{\mathfrak{W}}_{\text{CIU}-1}\in \mathrm{0,1}$.

An order relation between pairs of two CIULNs is stated as

(1) If $\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CIU}-1}>\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CIU}-2}$, then ${\mathfrak{W}}_{\text{CIU}-1}>{\mathfrak{W}}_{\text{CIU}-2}$.

(2) If $\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CIU}-1}<\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CIU}-2}$, then ${\mathfrak{W}}_{\text{CIU}-1}<{\mathfrak{W}}_{\text{CIU}-2}$.

(3) If $\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CIU}-1}=\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CIU}-2}$, then

(1) If $\overline{\overline{\mathfrak{F}}}{\mathfrak{W}}_{\text{CIU}-1}>\overline{\overline{\mathfrak{F}}}{\mathfrak{W}}_{\text{CIU}-2}$, then ${\mathfrak{W}}_{\text{CIU}-1}>{\mathfrak{W}}_{\text{CIU}-2}$.

(2) If $\overline{\overline{\mathfrak{F}}}{\mathfrak{W}}_{\text{CIU}-1}<\overline{\overline{\mathfrak{F}}}{\mathfrak{W}}_{\text{CIU}-2}$, then ${\mathfrak{W}}_{\text{CIU}-1}<{\mathfrak{W}}_{\text{CIU}-2}$.

(3) If $\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CIU}-1}=\overline{\overline{\zeta }}{\mathfrak{W}}_{\text{CIU}-2}$, then ${\mathfrak{W}}_{\text{CIU}-1}={\mathfrak{W}}_{\text{CIU}-2}$.

4. Complex Intuitionistic Uncertain Linguistic Heronian Mean Operators

In this study, we investigate the ideas of the CIULAHM operator, CIULWAHM operator, CIULGHM operator, and CIULWGHM operator and discuss their particular cases with the help of parameters. Some properties for investigated operators are developed such that idempotency, monotonicity, and boundedness are also explored.

Definition 9.

For the families of CIULNs ${\mathfrak{W}}_{\text{CIU}-i}={\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},i,\mathfrak{s}=\mathrm{1,2},\dots ,\overbrace{\mathrm{\Xi }}$, the CIULAHM operator is mapping ${\text{CIULAHM}}^{p_{\text{SC}},q_{\text{SC}}}:{\Theta }^{\overbrace{\mathrm{\Xi }}}⟶\Theta ,$ defined by$$\begin{array}{}\text{(12)}& {\text{CIULAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}={\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}-i}^{p_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}^{q_{\text{SC}}}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}.\end{array}$$

By using Definition 9, we investigate the following result.

Theorem 1.

For the families of CIULNs ${\mathfrak{W}}_{\text{CIU}-i}={\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},i,\mathfrak{s}=\mathrm{1,2},\dots ,\overbrace{\mathrm{\Xi }}$, and by using Definitions 7 and 9, we obtain$$\begin{array}{c}\text{(13)}& {\text{CIULAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\mathrm{\Xi }}=\begin{array}{c}{\psi }_{{1-{{\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{\phi }_i^{p_{\text{SC}}}{\phi }_S^{q_{\text{SC}}}/h}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}},{\psi }_{{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{\varphi }_i^{p_{\text{SC}}}{\varphi }_S^{q_{\text{SC}}}/h}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}}\\ \begin{array}{c}{1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{S=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}^{p_{\text{SC}}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-S}}^{q_{\text{SC}}}\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}\\ e^{i2\pi {1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}^{p_{\text{SC}}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-S}}^{q_{\text{SC}}}\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}},\\ 1-{1-{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{S=1}^{\overbrace{\mathrm{\Xi }}}{1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-S}}\breve{\mathfrak{k}}}^{q_{\text{SC}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}\\ e^{i2\pi {1-{1-{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-S}}\breve{\mathfrak{k}}}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}}\end{array}\end{array},\end{array}$$

Proof.

By using Definition 7, we obtain$$\begin{array}{c}\text{(14)}& {\mathfrak{W}}_{\text{CIU}-i}^{p_{\text{SC}}}=\begin{array}{c}{\psi }_{h{{\phi }_i/h}^{p_{\text{SC}}}},{\psi }_{h{{\varphi }_i/h}^{p_{\text{SC}}}},\\ \begin{array}{c}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}^{p_{\text{SC}}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}^{p_{\text{SC}}}\breve{\mathfrak{k}}},\\ 1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}{e}^{i2\pi 1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}}\end{array}\end{array},{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}^{p_{\text{SC}}}=\begin{array}{c}{\psi }_{h{{\phi }_s/h}^{p_{\text{SC}}}},{\psi }_{h{{\varphi }_s/h}^{p_{\text{SC}}}},\\ \begin{array}{c}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-\mathfrak{s}}}^{p_{\text{SC}}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-\mathfrak{s}}}^{p_{\text{SC}}}\breve{\mathfrak{k}}},\\ 1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-\mathfrak{s}}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}{e}^{i2\pi 1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-\mathfrak{s}}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}}\end{array}\end{array}.\end{array}$$

By using the above information, we obtain$$\begin{array}{}\text{(15)}& {\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}-i}^{p_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}^{q_{\text{SC}}}}}=\begin{array}{c}{\psi }_{1-{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{\phi }_i^{p_{\text{SC}}}{\phi }_{\mathfrak{s}}^{q_{\text{SC}}}/h}}},{\psi }_{1-{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{\varphi }_i^{p_{\text{SC}}}{\varphi }_{\mathfrak{s}}^{q_{\text{SC}}}/h}}},\\ \begin{array}{c}1-{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}^{p_{\text{SC}}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-\mathfrak{s}}}^{q_{\text{SC}}}\breve{\mathfrak{k}}}}{e}^{i2\pi 1-{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}^{p_{\text{SC}}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-\mathfrak{s}}}^{q_{\text{SC}}}\breve{\mathfrak{k}}}}},\\ {\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-\mathfrak{s}}}\breve{\mathfrak{k}}}^{q_{\text{SC}}}}}{e}^{i2\pi {\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-\mathfrak{s}}}\breve{\mathfrak{k}}}^{q_{\text{SC}}}}}}\end{array}\end{array},\end{array}$$and then$$\begin{array}{c}\text{(16)}& \frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}-i}^{p_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}^{q_{\text{SC}}}}}=\begin{array}{c}{\psi }_{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{\phi }_i^{p_{\text{SC}}}{\phi }_{\mathfrak{s}}^{q_{\text{SC}}}/h}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}},{\psi }_{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{\varphi }_i^{p_{\text{SC}}}{\varphi }_{\mathfrak{s}}^{q_{\text{SC}}}/h}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}},\\ \begin{array}{c}1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}\mathrm{-}i}}^{p_{\text{SC}}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-S}}^{q_{\text{SC}}}\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}\\ e^{i2\pi 1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}^{p_{\text{SC}}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-\mathfrak{s}}}^{q_{\text{SC}}}\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}},\\ {{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}\mathrm{-}i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-S}}\breve{\mathfrak{k}}}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}\\ e^{i2\pi {{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}\mathrm{-}i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-S}}\breve{\mathfrak{k}}}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}\end{array}\end{array},{\text{CIULAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}={\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}-i}^{p_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}^{q_{\text{SC}}}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}\\ =\begin{array}{c}{\psi }_{{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{\phi }_i^{p_{\text{SC}}}{\phi }_S^{q_{\text{SC}}}/h}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}},{\psi }_{{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{\varphi }_i^{p_{\text{SC}}}{\varphi }_S^{q_{\text{SC}}}/h}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}},,\\ \begin{array}{c}{1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{S=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}^{p_{\text{SC}}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-S}}^{q_{\text{SC}}}\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}\\ e^{i2\pi {1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}^{p_{\text{SC}}}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-S}}^{q_{\text{SC}}}\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}},\\ 1-{1-{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{S=1}^{\overbrace{\mathrm{\Xi }}}{1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-S}}\breve{\mathfrak{k}}}^{q_{\text{SC}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}\\ e^{i2\pi {1-{1-{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}\mathrm{-}i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-S}}\breve{\mathfrak{k}}}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}}\end{array}\end{array}.\end{array}$$

Moreover, by using Definition 7, we prove that certain properties for investigated ideas are similar to idempotency, monotonicity, and boundedness, which are stated below.

Property 1.

For the families of CIULNs ${\mathfrak{W}}_{\text{CIU}-i}={\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},i,\mathfrak{s}=\mathrm{1,2},\dots ,\overbrace{\mathrm{\Xi }}$, we have

(1) If ${\mathfrak{W}}_{\text{CIU}-i}={\mathfrak{W}}_{\text{CIU}}$, then$$\begin{array}{}\text{(17)}& {\text{CIULAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}={\mathfrak{W}}_{\text{CIU}}.\end{array}$$

(2) For CIULN ${\mathfrak{W}}_{\text{CIU}-i}^{\prime }={\psi }_{{\phi }_i}^{\prime },{\psi }_{{\varphi }_{\mathfrak{s}}}^{\prime },{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}^{\prime }\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}^{\prime }\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}^{\prime }\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}^{\prime }\breve{\mathfrak{k}}},i,\mathfrak{s}=\mathrm{1,2},\dots ,\overbrace{\mathrm{\Xi }}$, such that ${\psi }_{{\phi }_i}^{\prime }\le {\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}}^{\prime }\le {\psi }_{{\varphi }_{\mathfrak{s}}},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}^{\prime }\le {\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}$, ${\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}^{\prime }\le {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}$, ${\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}^{\prime }\ge {\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}$, and ${\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}^{\prime }\ge {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}$, we have$$\begin{array}{}\text{(18)}& {\text{CIULAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1}^{\prime },{\mathfrak{W}}_{\text{CIU}-2}^{\prime },\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}^{\prime }\le {\text{CIULAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}.\end{array}$$

(3) If ${\mathfrak{W}}_{\text{CIU}-A}=\min {\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}$ and ${\mathfrak{W}}_{\text{CIU}-B}=\max {\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}$, then

$$\begin{array}{}\text{(19)}& {\mathfrak{W}}_{\text{CIU}-A}\le {\text{CIULAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}\le {\mathfrak{W}}_{\text{CIU}-B}\end{array}$$

Proof.

We prove the above three equations, such that

(1) If ${\mathfrak{W}}_{\text{CIU}-i}={\mathfrak{W}}_{\text{CIU}},i=\mathrm{1,2},\dots ,\overbrace{\mathrm{\Xi }}$, then$$\begin{array}{c}\text{(20)}& {\text{CIULAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}={\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}-i}^{p_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}^{q_{\text{SC}}}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}={\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}}^{p_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}}^{q_{\text{SC}}}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}={\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}}^{p_{\text{SC}}+q_{\text{SC}}}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}\\ ={{\mathfrak{W}}_{\text{CIU}}^{p_{\text{SC}}+q_{\text{SC}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}={\mathfrak{W}}_{\text{CIU}}.\end{array}$$

(2) When ${\psi }_{{\phi }_i}^{\prime }\le {\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}}^{\prime }\le {\psi }_{{\varphi }_{\mathfrak{s}}},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}^{\prime }\le {\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}$, ${\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}^{\prime }\le {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}$, ${\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}^{\prime }\ge {\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}$, and ${\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}^{\prime }\ge {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}},\text{then\hspace{0.17em}we\hspace{0.17em}have\hspace{0.17em}}{\mathfrak{W}}_{\text{CIU}-i}^{\prime }\le {\mathfrak{W}}_{\text{CIU}-i},i=\mathrm{1,2},\dots ,\overbrace{\mathrm{\Xi }}$.

  Thus, ${\mathfrak{W}}_{\text{CIU}-i}^{\prime }\le {\mathfrak{W}}_{\text{CIU}-i}$ and ${\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}^{\prime }\le {\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}$, such that$$\begin{array}{c}\text{(21)}& {\mathfrak{W}}_{\text{CIU}-i}^{\prime p_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}^{\prime q_{\text{SC}}}\le {\mathfrak{W}}_{\text{CIU}-i}^{p_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}^{q_{\text{SC}}}⟹\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}-i}^{\prime p_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}^{\prime q_{\text{SC}}}}}\le \frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}-i}^{p_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}^{q_{\text{SC}}}}}\\ ⟹{\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}-i}^{\prime p_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}^{\prime q_{\text{SC}}}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}\le {\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}-i}^{p_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}^{q_{\text{SC}}}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}.\end{array}$$

   Then, we obtain$$\begin{array}{}\text{(22)}& {\text{CIULAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1}^{\prime },{\mathfrak{W}}_{\text{CIU}-2}^{\prime },\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}^{\prime }\le {\text{CIULAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}.\end{array}$$

(3) If ${\mathfrak{W}}_{\text{CIU}-A}=\min {\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}$ and ${\mathfrak{W}}_{\text{CIU}-B}=\max {\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}$, then by using Property 1, we get

$$\begin{array}{c}\text{(23)}& {\mathfrak{W}}_{\text{CIU}-A}\le {\text{CIULAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}},{\text{CIULAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}\le {\mathfrak{W}}_{\text{CIU}-B},\end{array}$$and then$$\begin{array}{}\text{(24)}& {\mathfrak{W}}_{\text{CIU}-A}\le {\text{CIULAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}\le {\mathfrak{W}}_{\text{CIU}-B}.\end{array}$$

Moreover, by using the investigated operators, we discuss some cases of the explored operators, which are discussed below.

(1) For $q_{\text{SC}}⟶0$, the idea of CIULAHM operator is converted to CIUL generalized linear descending weighted mean (CIULGLDWM) operator, such that$$\begin{array}{c}\text{(25)}& {\text{CIULAHM}}^{p_{\text{SC}},0}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}=\underset{q_{\text{SC}}⟶0}{\lim }{\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}-i}^{p_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}^{q_{\text{SC}}}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}\\ ={\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}-i}^{p_{\text{SC}}}}}^{1/p_{\text{SC}}}\\ =\begin{array}{c}{\psi }_{{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{1-{\phi }_i^{p_{\text{SC}}}/h}^{\overbrace{\mathrm{\Xi }}+1-i}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}}},{\psi }_{{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{1-{\varphi }_i^{p_{\text{SC}}}/h}^{\overbrace{\mathrm{\Xi }}+1-i}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}}},\\ \begin{array}{c}{1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}^{p_{\text{SC}}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}+1-i}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}}\\ e^{i2\pi {1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}^{p_{\text{SC}}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}+1-i}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}}},\\ 1-{1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}}^{\overbrace{\mathrm{\Xi }}+1-i}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}}\\ e^{i2\pi 1-{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}}^{\overbrace{\mathrm{\Xi }}+1-i}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}}}\end{array}\end{array}.\end{array}$$

(2) For $p_{\text{SC}}⟶0$, the idea of CIULAHM operator is converted to CIUL generalized linear ascending weighted mean (CIULGLAWM) operator, such that$$\begin{array}{c}\text{(26)}& CIULAH{M}^{0,q_{SC}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}=\underset{p_{\text{SC}}⟶0}{\lim }{\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}-i}^{p_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}^{q_{\text{SC}}}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}={\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}-i}^{q_{\text{SC}}}}}^{1/q_{\text{SC}}}\\ =\begin{array}{c}{\psi }_{{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{1-{\phi }_i^{q_{\text{SC}}}/h}^{\overbrace{\mathrm{\Xi }}+1-i}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/q_{\text{SC}}}},{\psi }_{{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{1-{\varphi }_i^{q_{\text{SC}}}/h}^{\overbrace{\mathrm{\Xi }}+1-i}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/q_{\text{SC}}}},\\ \begin{array}{c}{1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}^{q_{\text{SC}}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}+1-i}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/q_{\text{SC}}}\\ e^{i2\pi {1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}^{q_{\text{SC}}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}+1-i}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/q_{\text{SC}}}},\\ 1-{1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}}^{q_{\text{SC}}}}^{\overbrace{\mathrm{\Xi }}+1-i}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/q_{\text{SC}}}\\ e^{i2\pi 1-{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}}^{q_{\text{SC}}}}^{\overbrace{\mathrm{\Xi }}+1-i}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/q_{\text{SC}}}}\end{array}\end{array}.\end{array}$$

(3) For $p_{\text{SC}}=q_{\text{SC}}=1/2$, the idea of CIULAHM operator is converted to CIUL basic HM (CIULBHM) operator, such that$$\begin{array}{c}\text{(27)}& {\text{CIULAHM}}^{1/2,1/2}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}=\begin{array}{c}{\psi }_{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-\frac{{\phi }_i^{1/2}{\phi }_{\mathfrak{s}}^{1/2}}{h}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}},{\psi }_{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-\frac{{\varphi }_i^{1/2}{\varphi }_{\mathfrak{s}}^{1/2}}{h}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}},\\ \begin{array}{c}1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}^{1/2}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-\mathfrak{s}}}^{1/2}\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}\\ e^{i2\pi 1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}^{1/2}\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-\mathfrak{s}}}^{1/2}\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}},\\ {{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}}^{1/2}{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-\mathfrak{s}}}\breve{\mathfrak{k}}}^{1/2}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}\\ e^{i2\pi {{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}}^{1/2}{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-\mathfrak{s}}}\breve{\mathfrak{k}}}^{1/2}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}\end{array}\end{array}.\end{array}$$

(4) For $p_{\text{SC}}=q_{\text{SC}}=1$, the idea of CIULAHM operator is converted to CIUL basic HM (CIULBHM) operator, such that

$$\begin{array}{c}\text{(28)}& {\text{CIULAHM}}^{\mathrm{1,1}}{\mathfrak{W}}_{CIU-1},{\mathfrak{W}}_{CIU-2},\dots ,{\mathfrak{W}}_{CIU-\overbrace{\mathrm{\Xi }}}=\begin{array}{c}{\psi }_{{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-\frac{{\phi }_i^1{\phi }_{\mathfrak{s}}^1}{h}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/2}},{\psi }_{{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-\frac{{\varphi }_i^1{\varphi }_{\mathfrak{s}}^1}{h}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/2}},\\ \begin{array}{c}{1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}^1\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-\mathfrak{s}}}^1\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/2}\\ e^{i2\pi {1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}^1\breve{\mathfrak{k}}{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-\mathfrak{s}}}^1\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/2}},\\ 1-{1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}}^1{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-\mathfrak{s}}}\breve{\mathfrak{k}}}^1}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/2}\\ e^{i2\pi 1-{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}}^1{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-\mathfrak{s}}}\breve{\mathfrak{k}}}^1}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/2}}\end{array}\end{array}.\end{array}$$

Definition 10.

Based on any families of CIULNs ${\mathfrak{W}}_{\text{CIU}-i}={\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},i,\mathfrak{s}=\mathrm{1,2},\dots ,\overbrace{\mathrm{\Xi }}$, the CIULWAHM operator is a mapping ${\text{CIULWAHM}}^{p_{\text{SC}},q_{\text{SC}}}:{\Theta }^{\overbrace{\mathrm{\Xi }}}⟶\Theta$ defined by$$\begin{array}{}\text{(29)}& {\text{CIULWAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{CIU-1},{\mathfrak{W}}_{CIU-2},\dots ,{\mathfrak{W}}_{CIU-\overbrace{\mathrm{\Xi }}}={\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-i}{\mathfrak{W}}_{\text{CIU}-i}}^{p_{\text{SC}}}{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-\mathfrak{s}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}}^{q_{\text{SC}}}}}}^{1/p_{\text{SC}}+q_{\text{SC}}},\end{array}$$where ${\widehat{\Omega }}_W={\widehat{\Omega }}_{W-1},{\widehat{\Omega }}_{W-2},\dots ,{\widehat{\Omega }}_{W-\overbrace{\mathrm{\Xi }}}$ expresses the weight vector with a condition that is ${\displaystyle {\sum }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\widehat{\Omega }}_{W-i}}=1,{\widehat{\Omega }}_{W-i}\in \mathrm{0,1}$. By using Definition 10, we investigate the following result.

Theorem 2.

For families of CIULNs ${\mathfrak{W}}_{\text{CIU}-i}={\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},i,\mathfrak{s}=\mathrm{1,2},\dots ,\overbrace{\mathrm{\Xi }}$, by using Definitions 7 and 10, we obtain$$\begin{array}{c}\text{(30)}& {\text{CIULGHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}=\begin{array}{c}\begin{array}{c}{\psi }_{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}{1-1-{1-{{\phi }_i/h}^{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-i}}}^{p_{\text{SC}}}{1-{1-{\phi }_s/h}^{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-S}}}^{q_{\text{SC}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}}}^{1/p_{\text{SC}}+q_{\text{SC}}},}\\ {\psi }_{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}{1-1-{1-{{\phi }_i/h}^{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-i}}}^{p_{\text{SC}}}{1-{1-{\phi }_s/h}^{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-S}}}^{q_{\text{SC}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}}\end{array},\\ \begin{array}{c}1-{1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{S=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-i}}}^{p_{\text{SC}}}{1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-S}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-S}}}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}\\ e^{i2\pi 1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-i}}}^{p_{\text{SC}}}{1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-S}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-S}}}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{}^{1/p_{\text{SC}}+q_{\text{SC}}}}\\ 1-{1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{S=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}^{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-S}}^{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-S}}\breve{\mathfrak{k}}}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}\\ e^{i2\pi 1-{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}^{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-S}}^{q_{\text{SC}}}\breve{\mathfrak{k}}}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}}\end{array}\end{array}.\end{array}$$

Proof.

Trivial.

Theorem 3.

The CIULAHM operator is a certain brief case of CIULWAHM operator.

Proof.

We know that$$\begin{array}{}\text{(31)}& {\text{CIULWAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{CIU-1},{\mathfrak{W}}_{CIU-2},\dots ,{\mathfrak{W}}_{CIU-\overbrace{\mathrm{\Xi }}}={\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-i}{\mathfrak{W}}_{\text{CIU}-i}}^{p_{\text{SC}}}{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-\mathfrak{s}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}}^{q_{\text{SC}}}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}.\end{array}$$

If ${\widehat{\Omega }}_W=1/\overbrace{\mathrm{\Xi }},1/\overbrace{\mathrm{\Xi }},\dots ,1/\overbrace{\mathrm{\Xi }}$, then$$\begin{array}{c}\text{(32)}& {\text{CIULWAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}={\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-i}{\mathfrak{W}}_{\text{CIU}-i}}^{p_{\text{SC}}}{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-\mathfrak{s}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}}^{q_{\text{SC}}}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}={\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{\overbrace{\mathrm{\Xi }}\frac{1}{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}-i}}^{p_{\text{SC}}}{\overbrace{\mathrm{\Xi }}\frac{1}{\overbrace{\mathrm{\Xi }}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}}^{q_{\text{SC}}}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}\\ ={\frac{2}{\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}{\displaystyle \sum _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \sum _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{{\mathfrak{W}}_{\text{CIU}-i}}^{p_{\text{SC}}}{{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}}^{q_{\text{SC}}}}}}^{1/p_{\text{SC}}+q_{\text{SC}}}\\ =CIULAH{M}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}.\end{array}$$

Definition 11.

For the families of CIULNs, the CIULGHM operator is mapping ${\text{CIULGHM}}^{p_{\text{SC}},q_{\text{SC}}}:{\Theta }^{\overbrace{\mathrm{\Xi }}}⟶\Theta$defined by$$\begin{array}{}\text{(33)}& {\text{CIULGHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}=\frac{1}{p_{\text{SC}}+q_{\text{SC}}}{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{p}_{\text{SC}}{\mathfrak{W}}_{\text{CIU}-i}+q_{\text{SC}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}.\end{array}$$

By using Definition 11, we investigate the following result.

Theorem 4.

For families of CIULNs ${\mathfrak{W}}_{\text{CIU}-i}={\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},i,\mathfrak{s}=\mathrm{1,2},\dots ,\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1$, and by using Definitions 7 and 11, we obtain$$\begin{array}{c}\text{(34)}& {\text{CIULWAHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}=\begin{array}{c}\begin{array}{c}{\psi }_{1-{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\phi }_i/h}^{p_{\text{SC}}}{1-{\phi }_S/h}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}},}\\ {\psi }_{1-{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\varphi }_i/h}^{p_{\text{SC}}}{1-{\varphi }_S/h}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}},}\end{array},\\ \begin{array}{c}1-{1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{S=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-S}}\breve{\mathfrak{k}}}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}\\ e^{i2\pi 1-{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-S}}{\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}}\\ {1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{S=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}^{p_{\text{SC}}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-S}}^{q_{\text{SC}}}\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}\\ e^{i2\pi {1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}^{p_{\text{SC}}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}^{q_{\text{SC}}}\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}}\end{array}\end{array}.\end{array}$$

Proof.

Trivial.

Moreover, by using the investigated operators, we discuss some cases of the explored operators, which are discussed below.

(1) For $q_{\text{SC}}⟶0$, the idea of CIULGHM operator is converted to CIUL generalized geometric linear descending weighted mean (CIULGGLDWM) operator, such that$$\begin{array}{c}\text{(35)}& {\text{CIULGHM}}^{p_{\text{SC}},0}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}=\underset{q_{\text{SC}}⟶0}{\lim }\frac{1}{p_{\text{SC}}+q_{\text{SC}}}{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{p}_{\text{SC}}{\mathfrak{W}}_{\text{CIU}-i}+q_{\text{SC}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}=\frac{1}{p_{\text{SC}}}{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{p}_{\text{SC}}{\mathfrak{W}}_{\text{CIU}-i}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}\\ =\begin{array}{c}\begin{array}{c}{\psi }_{1-{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\phi }_i/h}^{p_{\text{SC}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}}},\\ {\psi }_{1-{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\varphi }_i/h}^{p_{\text{SC}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}}}\end{array},\\ \begin{array}{c}1-{1-{{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}}}^{\overbrace{\mathrm{\Xi }}+1-i}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}}\\ e^{i2\pi 1-{1-{{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}}^{p_{\text{SC}}}}}^{\overbrace{\mathrm{\Xi }}+1-i}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}}},\\ {1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}^{p_{\text{SC}}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}+1-i}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}}\\ e^{i2\pi {1-{{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}^{p_{\text{SC}}}\breve{\mathfrak{k}}}}^{\overbrace{\mathrm{\Xi }}+1-i}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}}}\end{array}\end{array}.\end{array}$$

(2) For $p_{\text{SC}}⟶0$, the idea of CIULGHM operator is converted to CIUL generalized geometric linear ascending weighted mean (CIULGGLAWM) operator, such that$$\begin{array}{c}\text{(36)}& {\text{CIULGHM}}^{0,q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}=\underset{p_{\text{SC}}⟶0}{\lim }\frac{1}{p_{\text{SC}}+q_{\text{SC}}}{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{p}_{\text{SC}}{\mathfrak{W}}_{\text{CIU}-i}+q_{\text{SC}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}=\frac{1}{q_{\text{SC}}}{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{q}_{\text{SC}}{\mathfrak{W}}_{\text{CIU}-i}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}\\ =\begin{array}{c}\begin{array}{c}{\psi }_{1-{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\phi }_i/h}^{q_{\text{SC}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/q_{\text{SC}}}},\\ {\psi }_{1-{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\varphi }_i/h}^{q_{\text{SC}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/q_{\text{SC}}}}\end{array},\\ \begin{array}{c}1-{1-{{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}}^{q_{\text{SC}}}}}^{\overbrace{\mathrm{\Xi }}+1-i}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/q_{\text{SC}}}\\ e^{i2\pi 1-{1-{{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}}^{q_{\text{SC}}}}}^{\overbrace{\mathrm{\Xi }}+1-i}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/q_{\text{SC}}}},\\ {1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}^{q_{\text{SC}}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}+1-i}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/q_{\text{SC}}}\\ e^{i2\pi {1-{{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}^{q_{\text{SC}}}\breve{\mathfrak{k}}}}^{\overbrace{\mathrm{\Xi }}+1-i}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/q_{\text{SC}}}}\end{array}\end{array}.\end{array}$$

(3) For $p_{\text{SC}}=q_{\text{SC}}=1/2$, the idea of CIULGHM operator is converted to CIUL basic geometric HM (CIULBGHM) operator, such that$$\begin{array}{c}\text{(37)}& {\text{CIULGHM}}^{1/2,1/2}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}=\begin{array}{c}\begin{array}{c}{\psi }_{1-1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\phi }_i/h}^{1/2}{1-{\phi }_{\mathfrak{s}}/h}^{1/2}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}},\\ {\psi }_{1-1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\varphi }_i/h}^{1/2}{1-{\varphi }_{\mathfrak{s}}/h}^{1/2}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}},\end{array},\\ \begin{array}{c}{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}}^{1/2}{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-\mathfrak{s}}}\breve{\mathfrak{k}}}^{1/2}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}\\ e^{i2\pi {{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}}^{1/2}{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-\mathfrak{s}}}\breve{\mathfrak{k}}}^{1/2}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}},\\ 1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}^{1/2}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-\mathfrak{s}}}^{1/2}\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}\\ e^{i2\pi 1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}^{1/2}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-\mathfrak{s}}}^{1/2}\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}\end{array}\end{array}.\end{array}$$

(4) For $p_{\text{SC}}=q_{\text{SC}}=1$, the idea of CIULGHM operator is converted to CIUL geometric line HM (CIULGLHM) operator, such that

$$\begin{array}{c}\text{(38)}& {\text{CIULGHM}}^{\mathrm{1,1}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}=\begin{array}{c}\begin{array}{c}{\psi }_{1-{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\phi }_i/h}^1{1-{\phi }_{\mathfrak{s}}/h}^1}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/2}},\\ {\psi }_{1-{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{\varphi }_i/h}^1{1-{\varphi }_{\mathfrak{s}}/h}^1}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/2}}\end{array},\\ \begin{array}{c}{1-{1-{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-\mathfrak{s}}}\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/2}\\ e^{i2\pi {1-{1-{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-\mathfrak{s}}}\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/2}},\\ {1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-\mathfrak{s}}}\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/2}\\ e^{i2\pi {1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-\mathfrak{s}}}\breve{\mathfrak{k}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/2}}\end{array}\end{array}.\end{array}$$

Property 2.

For families of CIULNs ${\mathfrak{W}}_{\text{CIU}-i}={\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},i,\mathfrak{s}=\mathrm{1,2},\dots ,\overbrace{\mathrm{\Xi }}$, then

(1) If ${\mathfrak{W}}_{\text{CIU}-i}={\mathfrak{W}}_{\text{CIU}}$, then$$\begin{array}{}\text{(39)}& {\text{CIULGHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}={\mathfrak{W}}_{\text{CIU}}.\end{array}$$

(2) For CIULN ${\mathfrak{W}}_{\text{CIU}-i}^{\prime }={\psi }_{{\phi }_i}^{\prime },{\psi }_{{\varphi }_{\mathfrak{s}}}^{\prime },{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}^{\prime }\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}^{\prime }\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}^{\prime }\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}^{\prime }\breve{\mathfrak{k}}},i,\mathfrak{s}=\mathrm{1,2},\dots ,\overbrace{\mathrm{\Xi }}$, such that ${\psi }_{{\phi }_i}^{\prime }\le {\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}}^{\prime }\le {\psi }_{{\varphi }_{\mathfrak{s}}},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}^{\prime }\le {\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}$, ${\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}^{\prime }\le {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}$, ${\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}^{\prime }\ge {\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}$, and ${\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}^{\prime }\ge {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}$, we have$$\begin{array}{}\text{(40)}& {\text{CIULGHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1}^{\prime },{\mathfrak{W}}_{\text{CIU}-2}^{\prime },\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}^{\prime }\le {\text{CIULGHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}.\end{array}$$

(3) If ${\mathfrak{W}}_{\text{CIU}-A}=\min {\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}$ and ${\mathfrak{W}}_{\text{CIU}-B}=\max {\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}$, then

$$\begin{array}{}\text{(41)}& {\mathfrak{W}}_{\text{CIU}-A}\le {\text{CIULGHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}\le {\mathfrak{W}}_{\text{CIU}-B}.\end{array}$$

Proof.

Follows from the proof similar to Property 1.

Definition 12.

For the families of CIULNs, the CIULWGHM operator is mapping ${\text{CIULWGHM}}^{p_{\text{SC}},q_{\text{SC}}}:{\Theta }^{\overbrace{\mathrm{\Xi }}}⟶\Theta$stated by$$\begin{array}{}\text{(42)}& {\text{CIULWGHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}=\frac{1}{p_{\text{SC}}+q_{\text{SC}}}{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{\mathfrak{s}=1}^{\overbrace{\mathrm{\Xi }}}{p_{\text{SC}}{\mathfrak{W}}_{\text{CIU}-i}}^{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-i}}+{q_{\text{SC}}{\mathfrak{W}}_{\text{CIU}-\mathfrak{s}}}^{\overbrace{\mathrm{\Xi }}{\widehat{\Omega }}_{W-\mathfrak{s}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1},\end{array}$$where ${\widehat{\Omega }}_W={\widehat{\Omega }}_{W-1},{\widehat{\Omega }}_{W-2},\dots ,{\widehat{\Omega }}_{W-\overbrace{\mathrm{\Xi }}}$ expresses the weight vector with a condition that is ${\displaystyle {\sum }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\widehat{\Omega }}_{W-i}}=1,{\widehat{\Omega }}_{W-i}\in \mathrm{0,1}$. By using Definition 12, we investigate the following result.

Theorem 5.

Based on any families of CIULNs ${\mathfrak{W}}_{\text{CIU}-i}={\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}},{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}},i,\mathfrak{s}=\mathrm{1,2},\dots ,\overbrace{\mathrm{\Xi }}$, then by using Definitions 7 and 12, we obtain$$\begin{array}{c}\text{(43)}& {\text{CIULGHM}}^{p_{\text{SC}},q_{\text{SC}}}{\mathfrak{W}}_{\text{CIU}-1},{\mathfrak{W}}_{\text{CIU}-2},\dots ,{\mathfrak{W}}_{\text{CIU}-\overbrace{\mathrm{\Xi }}}=\begin{array}{c}\begin{array}{c}{\psi }_{{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{1-{{\phi }_i/h}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}}^{p_{\text{SC}}}{1-{1-{{\phi }_s/h}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}}\\ {\psi }_{{1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{1-{{\varphi }_i/h}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}}^{p_{\text{SC}}}{1-{1-{{\varphi }_s/h}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}}\end{array},\\ \begin{array}{c}{1-{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{S=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-i}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-i}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-i}}}^{p_{\text{SC}}}{1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}-S}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}\\ e^{i2\pi {1-{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-i}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-i}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-i}}}^{p_{\text{SC}}}{1-{1-{\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}-S}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}}\\ 1-{{{\displaystyle \prod _{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle \prod _{S=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-i}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}}^{p_{\text{SC}}}{1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}-S}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}\\ e^{i2\pi 1-{{{\displaystyle {\prod }_{i=1}^{\overbrace{\mathrm{\Xi }}}{\displaystyle {\prod }_{S=1}^{\overbrace{\mathrm{\Xi }}}1-{1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-i}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}}^{p_{\text{SC}}}{1-{1-{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}-S}}\breve{\mathfrak{k}}}^{\overbrace{\mathrm{\Xi }}{\stackrel{⌢}{\Omega }}_{W-S}}}^{q_{\text{SC}}}}}}^{2/\overbrace{\mathrm{\Xi }}\overbrace{\mathrm{\Xi }}+1}}^{1/p_{\text{SC}}+q_{\text{SC}}}}\end{array}\end{array}.\end{array}$$

Proof.

Trivial.

Theorem 6.

The CIULGHM operator is a certain brief case of CIULWGHM operator.

Proof.

Trivial.

5. MADM Procedure Based on CIULSHM Operators

In genuine decision troubles, there occur the exchanges among the attributes. At the similar moment, due to the ambiguity of the attributes, they can be certainly shown by the CIULSs. So, by using the CIUL information (CIULI), it is essential to utilize various decision-making processes to sort out the exchanges among the characteristics.

In this analysis, we shall investigate a methodology to MAGDM procedure by using the CIULI by CIULWAHM operator or CIULWGHM operator. Reflect a MAGDM procedure by using the CIULI: let ${\overline{\mathrm{\Phi }}}_{\text{Al}}={\overline{\mathrm{\Phi }}}_{\text{Al}-1},{\overline{\mathrm{\Phi }}}_{\text{Al}-2},\dots ,{\overline{\mathrm{\Phi }}}_{\text{Al}-\overbrace{\mathrm{\Xi }}}$ be the family of alternatives and their attributes ${\overline{\mathrm{ℒ}}}_{\text{At}}={\overline{\mathrm{ℒ}}}_{\text{At}-1},{\overline{\mathrm{ℒ}}}_{\text{At}-2},\dots ,{\overline{\mathrm{ℒ}}}_{\text{At}-\overbrace{m}}$. For this, we choose the weight vectors ${\widehat{\mathrm{\Omega }}}_W={\widehat{\mathrm{\Omega }}}_{W-1},{\widehat{\mathrm{\Omega }}}_{W-2},\dots ,{\widehat{\mathrm{\Omega }}}_{W-\overbrace{n}}$ with a rule that is ${\displaystyle {\sum }_{i=1}^{\overbrace{n}}{\widehat{\Omega }}_{W-i}}=1$. Moreover, we choose the family of decision makers such that ${\overline{\mathcal{D}}}_{Dm}={\overline{\mathcal{D}}}_{Dm-1},{\overline{\mathcal{D}}}_{Dm-2},\dots ,{\overline{\mathcal{D}}}_{Dm-\overbrace{\mathrm{\Xi }}}$, and ${\widehat{\Omega }}_W^{\prime }={\widehat{\Omega }}_{W-1}^{\prime },{\widehat{\Omega }}_{W-2}^{\prime },\dots ,{\widehat{\Omega }}_{W-\overbrace{n}}^{\prime }$ with a rule that is ${\displaystyle {\sum }_{i=1}^{\overbrace{n}}{\widehat{\Omega }}_{W-i}^{\prime }}=1$ are expressing the weight vectors of decision makers. To resolve the above discussed issues, we choose the decision matrix ${\breve{\mathrm{ℛ}}}^i,i=\mathrm{1,2},\dots ,\overbrace{n}$, whose every term in the form of CIULNs such that ${\mathfrak{W}}_{\text{CIU}}={\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}},{\mathcal{J}}_{{\mathfrak{W}}_{\text{CIU}}}\breve{\mathfrak{k}},{\mathcal{K}}_{{\mathfrak{W}}_{\text{CIU}}}\breve{\mathfrak{k}}$, where ${\mathcal{J}}_{{\mathfrak{W}}_{\text{CIU}}}\breve{\mathfrak{k}}={\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}}}\breve{\mathfrak{k}}}$ and ${\mathcal{K}}_{{\mathfrak{W}}_{\text{CIU}}}\breve{\mathfrak{k}}={\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}{e}^{i2\pi {\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}}}\breve{\mathfrak{k}}}$ with the rules such that $0\le {\mathcal{J}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}+{\mathcal{K}}_{{\mathfrak{W}}_{\text{RP}}}\breve{\mathfrak{k}}\le 1$ and $0\le {\mathcal{J}}_{{\mathfrak{W}}_{\text{IP}}}\breve{\mathfrak{k}}+{\mathcal{K}}_{{\mathfrak{W}}_{\text{IP}}}\breve{\mathfrak{k}}\le 1$ with ${\psi }_{{\phi }_i},{\psi }_{{\varphi }_{\mathfrak{s}}}\in \widehat{\psi }i\le \mathfrak{s}$. For resolving the aforementioned issues, we use the following MAGDM procedures:

  Step 1: utilize the CIULWAHM operator to total the choice matrices which are given by decision makers with weighted vectors.

  Step 2: utilize the CIULAHM operator, CIULWAHM operator, CIULGHM operator, and CIULWGHM operator to collect the choice matrices which are in Step 1.

  Step 3: by using the score function, we analyze the score principles of the accumulated values in Step 2.

  Step 4: rank all the options and discover the most excellent one.

Example 1.

The MAGDM issue is cited from Ref. [41]. There is a speculation organization, which plans to pick the most ideal interest in some options. There are four potential alternatives for the speculation organization to browse: (1) a vehicle organization ${\overline{\mathrm{\Phi }}}_{\text{Al}-1}$; (2) a food organization ${\overline{\mathrm{\Phi }}}_{\text{Al}-2}$; (3) a PC organization${\overline{\mathrm{\Phi }}}_{\text{Al}-3}$; and (4) a mobile organization ${\overline{\mathrm{\Phi }}}_{\text{Al}-4}$. The venture organization considers four criteria to settle on decisions: (1) the hazard investigation ${\overline{\mathrm{ℒ}}}_{\text{At}-1}$; (2) the development examination ${\overline{\mathrm{ℒ}}}_{\text{At}-2}$; (3) the natural impact investigation ${\overline{\mathrm{ℒ}}}_{\text{At}-3}$; and (4) social impact ${\overline{\mathrm{ℒ}}}_{\text{At}-4}$, where all criteria values are benefit type. The weight vector of the criteria is ${\widehat{\Omega }}_W^{\prime }={0.5,0.4,0.1}^T,{\widehat{\Omega }}_W={0.4,0.3,0.2,0.1}^T$. The four potential options are assessed regarding the four rules by the type of CIULNs, and complex intuitionistic uncertain linguistic decision matrices ${\breve{\mathrm{ℛ}}}^i,i=\mathrm{1,2,3}$ are developed and listed in the form of Tables 1–3 , respectively.

For resolving the aforementioned issues, we use the following MAGDM procedures:

  Step 1: by utilizing the CIULWAHM operator, we aggregated the decision matrices which are given by decision makers with weighted vectors. The aggregated decision matrix is discussed in the form of Table 4 for $p_{\text{SC}},q_{\text{SC}}=1$.

  Step 2: utilize the CIULAHM operator, CIULWAHM operator, CIULGHM operator, and CIULWGHM operator to aggregate the decision matrices which are in Step 1, which are discussed in the form of Table 5 for $p_{\text{SC}},q_{\text{SC}}=0.4$.

  Step 3: by using the score function, we compute their values which are listed in Table 6.

  Step 4: rank all the options and invent the superlative one, which are discussed in the form of Table 7.

From the above analysis, we obtain different results by using the investigated operators such as CIULAHM operator, CIULWAHM operator, CIULGHM operator, and CIULWGHM operator. The best options are ${\overline{\mathrm{\Phi }}}_{\text{AI}-1},{\overline{\mathrm{\Phi }}}_{\text{AI}-2}$, and ${\overline{\mathrm{\Phi }}}_{\text{AI}-4}$ by using different operators. The graphical interpretations of the information of Table 6 are discussed in the form of Figure 1.

Table 1

Decision matrix in terms of CIULNs provided by expert ${\breve{\mathrm{ℛ}}}^1$.

Alternatives/attributes	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-1}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-2}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-3}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-4}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}$	$\begin{array}{c}{\psi }_5,{\psi }_5,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.7e^{i2\pi 0.7}\end{array}\end{array}$	$\begin{array}{c}{\psi }_2,{\psi }_3,\\ \begin{array}{c}0.4e^{i2\pi 0.4},\\ 0.6e^{i2\pi 0.6}\end{array}\end{array}$	$\begin{array}{c}{\psi }_5,{\psi }_6,\\ \begin{array}{c}0.5e^{i2\pi 0.5},\\ 0.5e^{i2\pi 0.5}\end{array}\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.6e^{i2\pi 0.6}\end{array}\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.6e^{i2\pi 0.6}\end{array}\end{array}$	$\begin{array}{c}{\psi }_5,{\psi }_5,\\ \begin{array}{c}0.4e^{i2\pi 0.4},\\ 0.5e^{i2\pi 0.5}\end{array}\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ \begin{array}{c}0.1e^{i2\pi 0.1},\\ 0.8e^{i2\pi 0.8}\end{array}\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_4,\\ \begin{array}{c}0.5e^{i2\pi 0.5},\\ 0.5e^{i2\pi 0.5}\end{array}\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.7e^{i2\pi 0.7}\end{array}\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_4,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.7e^{i2\pi 0.7}\end{array}\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ \begin{array}{c}0.3e^{i2\pi 0.3},\\ 0.7e^{i2\pi 0.7}\end{array}\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.7e^{i2\pi 0.7}\end{array}\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}$	$\begin{array}{c}{\psi }_6,{\psi }_6,\\ \begin{array}{c}0.5e^{i2\pi 0.5},\\ 0.4e^{i2\pi 0.4}\end{array}\end{array}$	$\begin{array}{c}{\psi }_2,{\psi }_3,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.8e^{i2\pi 0.8}\end{array}\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.6e^{i2\pi 0.6}\end{array}\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_3,\\ \begin{array}{c}0.3e^{i2\pi 0.3},\\ 0.6e^{i2\pi 0.6}\end{array}\end{array}$

Table 2

Decision matrix in terms of CIULNs provided by expert ${\breve{\mathrm{ℛ}}}^2$.

Alternatives/attributes	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-1}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-2}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-3}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-4}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}$	$\begin{array}{c}{\psi }_5,{\psi }_5,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.5e^{i2\pi 0.5}\end{array}\end{array}$	$\begin{array}{c}{\psi }_2,{\psi }_3,\\ \begin{array}{c}0.4e^{i2\pi 0.4},\\ 0.5e^{i2\pi 0.1}\end{array}\end{array}$	$\begin{array}{c}{\psi }_5,{\psi }_6,\\ \begin{array}{c}0.5e^{i2\pi 0.5},\\ 0.3e^{i2\pi 0.2}\end{array}\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.1e^{i2\pi 0.1}\end{array}\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.5e^{i2\pi 0.3}\end{array}\end{array}$	$\begin{array}{c}{\psi }_5,{\psi }_6,\\ \begin{array}{c}0.5e^{i2\pi 0.5},\\ 0.3e^{i2\pi 0.2}\end{array}\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ \begin{array}{c}0.1e^{i2\pi 0.1},\\ 0.6e^{i2\pi 0.5}\end{array}\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_4,\\ \begin{array}{c}0.5e^{i2\pi 0.5},\\ 0.3e^{i2\pi 0.3}\end{array}\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.4e^{i2\pi 0.7}\end{array}\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ \begin{array}{c}0.1e^{i2\pi 0.1},\\ 0.6e^{i2\pi 0.5}\end{array}\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ \begin{array}{c}0.3e^{i2\pi 0.3},\\ 0.4e^{i2\pi 0.7}\end{array}\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.4e^{i2\pi 0.7}\end{array}\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}$	$\begin{array}{c}{\psi }_6,{\psi }_6,\\ \begin{array}{c}0.5e^{i2\pi 0.5},\\ 0.4e^{i2\pi 0.4}\end{array}\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ \begin{array}{c}0.3e^{i2\pi 0.3},\\ 0.4e^{i2\pi 0.7}\end{array}\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.3e^{i2\pi 0.6}\end{array}\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_3,\\ \begin{array}{c}0.3e^{i2\pi 0.3},\\ 0.3e^{i2\pi 0.6}\end{array}\end{array}$

Table 3

Decision matrix in terms of CIULNs provided by expert ${\breve{\mathrm{ℛ}}}^3$.

Alternatives/attributes	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-1}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-2}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-3}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-4}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}$	$\begin{array}{c}{\psi }_5,{\psi }_5,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.7e^{i2\pi 0.7}\end{array}\end{array}$	$\begin{array}{c}{\psi }_2,{\psi }_3,\\ \begin{array}{c}0.4e^{i2\pi 0.4},\\ 0.6e^{i2\pi 0.6}\end{array}\end{array}$	$\begin{array}{c}{\psi }_5,{\psi }_6,\\ \begin{array}{c}0.5e^{i2\pi 0.5},\\ 0.5e^{i2\pi 0.5}\end{array}\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.6e^{i2\pi 0.6}\end{array}\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}$	$\begin{array}{c}{\psi }_5,{\psi }_5,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.5e^{i2\pi 0.5}\end{array}\end{array}$	$\begin{array}{c}{\psi }_2,{\psi }_3,\\ \begin{array}{c}0.4e^{i2\pi 0.4},\\ 0.5e^{i2\pi 0.1}\end{array}\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ \begin{array}{c}0.1e^{i2\pi 0.1},\\ 0.8e^{i2\pi 0.8}\end{array}\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_4,\\ \begin{array}{c}0.5e^{i2\pi 0.5},\\ 0.5e^{i2\pi 0.5}\end{array}\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.5e^{i2\pi 0.3}\end{array}\end{array}$	$\begin{array}{c}{\psi }_5,{\psi }_6,\\ \begin{array}{c}0.5e^{i2\pi 0.5},\\ 0.3e^{i2\pi 0.2}\end{array}\end{array}$	$\begin{array}{c}{\psi }_2,{\psi }_3,\\ \begin{array}{c}0.4e^{i2\pi 0.4},\\ 0.6e^{i2\pi 0.6}\end{array}\end{array}$	$\begin{array}{c}{\psi }_5,{\psi }_6,\\ \begin{array}{c}0.5e^{i2\pi 0.5},\\ 0.5e^{i2\pi 0.5}\end{array}\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ \begin{array}{c}0.2e^{i2\pi 0.2},\\ 0.4e^{i2\pi 0.7}\end{array}\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ \begin{array}{c}0.1e^{i2\pi 0.1},\\ 0.6e^{i2\pi 0.5}\end{array}\end{array}$	$\begin{array}{c}{\psi }_2,{\psi }_3,\\ \begin{array}{c}0.4e^{i2\pi 0.4},\\ 0.5e^{i2\pi 0.1}\end{array}\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ \begin{array}{c}0.1e^{i2\pi 0.1},\\ 0.8e^{i2\pi 0.8}\end{array}\end{array}$

Table 4

Aggregated decision matrix of the experts by CIULWAHM operator.

	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-1}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-2}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-3}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-4}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}$	$\begin{array}{c}{\psi }_{1.6489},{\psi }_{1.6489},\\ \begin{array}{c}0.6665e^{i2\pi 0.6665},\\ 0.2189e^{i2\pi 0.2189}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{0.6595},{\psi }_{0.9893},\\ \begin{array}{c}0.7947e^{i2\pi 0.7947},\\ 0.1834e^{i2\pi 0.1393}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{1.6489},{\psi }_{1.9786},\\ \begin{array}{c}0.8474e^{i2\pi 0.8474},\\ 0.1188e^{i2\pi 0.1089}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{0.9893},{\psi }_{1.3191},\\ \begin{array}{c}0.6665e^{i2\pi 0.6665},\\ 0.1393e^{i2\pi 0.1393}\end{array}\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}$	$\begin{array}{c}{\psi }_{1.3693},{\psi }_{1.6489},\\ \begin{array}{c}0.6665e^{i2\pi 0.6665},\\ 0.1769e^{i2\pi 0.141}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{1.4874},{\psi }_{1.6737},\\ \begin{array}{c}0.8166e^{i2\pi 0.8166},\\ 0.1188e^{i2\pi 0.0804}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{0.9893},{\psi }_{1.3191},\\ \begin{array}{c}0.5891e^{i2\pi 0.5891},\\ 0.2803e^{i2\pi 0.2596}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{1.3191},{\psi }_{1.3191},\\ \begin{array}{c}0.8474e^{i2\pi 0.8474},\\ 0.1188e^{i2\pi 0.1188}\end{array}\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$	$\begin{array}{c}{\psi }_{1.0392},{\psi }_{1.3693},\\ \begin{array}{c}0.6665e^{i2\pi 0.6665},\\ 0.1823e^{i2\pi 0.2493}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{1.2247},{\psi }_{1.4177},\\ \begin{array}{c}0.6488e^{i2\pi 0.6488},\\ 0.2217e^{i2\pi 0.1942}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{1.2124},{\psi }_{1.5431},\\ \begin{array}{c}0.7409e^{i2\pi 0.7409},\\ 0.191e^{i2\pi 0.2591}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{1.3693},{\psi }_{1.6993},\\ \begin{array}{c}0.6824e^{i2\pi 0.6824},\\ 0.1823e^{i2\pi 0.2544}\end{array}\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}$	$\begin{array}{c}{\psi }_{1.8187},{\psi }_{1.8735},\\ \begin{array}{c}0.8212e^{i2\pi 0.8212},\\ 0.107e^{i2\pi 0.1313}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{0.9447},{\psi }_{1.2845},\\ \begin{array}{c}0.6854e^{i2\pi 0.6854},\\ 0.2186e^{i2\pi 0.2867}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{0.9367},{\psi }_{1.2669},\\ \begin{array}{c}0.6786e^{i2\pi 0.6786},\\ 0.141e^{i2\pi 0.1916}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{0.9893},{\psi }_{1.0392},\\ \begin{array}{c}0.7167e^{i2\pi 0.7167},\\ 0.1765e^{i2\pi 0.2221}\end{array}\end{array}$

Table 5

Aggregated values of the alternatives by CIULAHM, CIULWAHM, CIULGHM, and CIULWGHM operators.

	$\mathbf{C}\mathbf{I}\mathbf{U}\mathbf{L}\mathbf{A}\mathbf{H}\mathbf{M}$	$\mathbf{C}\mathbf{I}\mathbf{U}\mathbf{L}\mathbf{W}\mathbf{A}\mathbf{H}\mathbf{M}$	$\mathbf{C}\mathbf{I}\mathbf{U}\mathbf{L}\mathbf{G}\mathbf{H}\mathbf{M}$	$\mathbf{C}\mathbf{I}\mathbf{U}\mathbf{L}\mathbf{W}\mathbf{G}\mathbf{H}\mathbf{M}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}$	$\begin{array}{c}{\psi }_{1.2043},{\psi }_{1.5056},\\ \begin{array}{c}0.745e^{i2\pi 0.745},\\ 0.0808e^{i2\pi 0.0723}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{0.9328},{\psi }_{1.1553},\\ \begin{array}{c}0.9873e^{i2\pi 0.9873},\\ 0.0014e^{i2\pi 0.001}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{1.147},{\psi }_{1.4558},\\ \begin{array}{c}0.6082e^{i2\pi 0.6082},\\ 0.1622e^{i2\pi 0.1482}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{1.0985},{\psi }_{1.246},\\ \begin{array}{c}0.1072e^{i2\pi 0.1072},\\ 0.7938e^{i2\pi 0.7828}\end{array}\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}$	$\begin{array}{c}{\psi }_{1.2632},{\psi }_{1.4188},\\ \begin{array}{c}0.732e^{i2\pi 0.732},\\ 0.0852e^{i2\pi 0.0699}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{0.9789},{\psi }_{1.1316},\\ \begin{array}{c}0.9853e^{i2\pi 0.9853},\\ 0.0015e^{i2\pi 0.0009}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{1.2462},{\psi }_{1.4112},\\ \begin{array}{c}0.5974e^{i2\pi 0.5974},\\ 0.1663e^{i2\pi 0.1409}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{1.1493},{\psi }_{1.2644},\\ \begin{array}{c}0.0959e^{i2\pi 0.0959},\\ 0.7799e^{i2\pi 0.7576}\end{array}\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$	$\begin{array}{c}{\psi }_{1.2662},{\psi }_{1.5712},\\ \begin{array}{c}0.6847e^{i2\pi 0.6847},\\ 0.1001e^{i2\pi 0.1307}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{0.9406},{\psi }_{1.1606},\\ \begin{array}{c}0.9786e^{i2\pi 0.9786},\\ 0.0019e^{i2\pi 0.0033}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{1.2638},{\psi }_{1.5664},\\ \begin{array}{c}0.5324e^{i2\pi 0.5324},\\ 0.1939e^{i2\pi 0.2383}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{1.1212},{\psi }_{1.2707},\\ \begin{array}{c}0.0796e^{i2\pi 0.0796},\\ 0.8017e^{i2\pi 0.8218}\end{array}\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}$	$\begin{array}{c}{\psi }_{1.0178},{\psi }_{1.223},\\ \begin{array}{c}0.7258e^{i2\pi 0.7258},\\ 0.078e^{i2\pi 0.1085}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{0.7861},{\psi }_{0.9838},\\ \begin{array}{c}0.9877e^{i2\pi 0.9877},\\ 0.001e^{i2\pi 0.0022}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{1.0313},{\psi }_{1.2197},\\ \begin{array}{c}0.582e^{i2\pi 0.582},\\ 0.1572e^{i2\pi 0.203}\end{array}\end{array}$	$\begin{array}{c}{\psi }_{1.0612},{\psi }_{1.1872},\\ \begin{array}{c}0.1009e^{i2\pi 0.1009},\\ 0.7749e^{i2\pi 0.8036}\end{array}\end{array}$

Table 6

Score values of each alternative from the aggregated values by different operators.

	$\mathrm{CIULAHM}$	$\mathrm{CIULWAHM}$	$\mathrm{CIULGHM}$	$\mathrm{CIULWGHM}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}$	$0.1971$	$1.0359$	$1.0944$	$-0.896$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}$	$0.1797$	$1.045$	$1.0732$	$-0.934$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$	$-0.0260$	$1.0353$	$1.0701$	$-0.9700$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}$	$0.0295$	$0.8747$	$0.8849$	$-0.888$

Table 7

Ranking values of the alternatives based on score values by different operators.

Methods	Ranking values
CIULAHM operator	${\overline{\mathrm{\Phi }}}_{\text{AI}-1}\ge {\overline{\mathrm{\Phi }}}_{\text{AI}-2}\ge {\overline{\mathrm{\Phi }}}_{\text{AI}-4}\ge {\overline{\mathrm{\Phi }}}_{\text{AI}-3}$
CIULWAHM operator	${\overline{\mathrm{\Phi }}}_{\text{AI}-2}\ge {\overline{\mathrm{\Phi }}}_{\text{AI}-1}\ge {\overline{\mathrm{\Phi }}}_{\text{AI}-3}\ge {\overline{\mathrm{\Phi }}}_{\text{AI}-4}$
CIULGHM operator	${\overline{\mathrm{\Phi }}}_{\text{AI}-1}\ge {\overline{\mathrm{\Phi }}}_{\text{AI}-2}\ge {\overline{\mathrm{\Phi }}}_{\text{AI}-3}\ge {\overline{\mathrm{\Phi }}}_{\text{AI}-4}$
CIULWGHM operator	${\overline{\mathrm{\Phi }}}_{\text{AI}-4}\ge {\overline{\mathrm{\Phi }}}_{\text{AI}-1}\ge {\overline{\mathrm{\Phi }}}_{\text{AI}-2}\ge {\overline{\mathrm{\Phi }}}_{\text{AI}-3}$

[figure omitted; refer to PDF]

5.1. Influence of Parameters

To demonstrate the stability and validity of the investigated operators with the help of the parameters $p_{\text{SC}}$ and $q_{\text{SC}}$ are discussed by using the information of Example 1. The stability of the parameters by using the information of Example 1 is discussed in the form of Tables 8–10 by using the investigated CIULAHM, CIULWAHM, CIULGHM, and CIULWGHM operators with the help of parameter $q_{\text{SC}}=0.4$.

Table 8

Influence of the parameters $p_{\text{SC}}$ on the ranking of the alternatives with $q_{\text{SC}}=0.4$.

$p_{\text{SC}}$	Operators	Score values	Ranking values
$0.1$	AHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.8313,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.8962$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.8114,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.4714$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WAHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.3543,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.4224$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.427,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.3226$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	GHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.1441,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.1348$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.034,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.0354$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WGHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=-0.233,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=-0.295$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.331,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=-0.254$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
$0.2$	AHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.4547,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.5372$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.4055,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.3124$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WAHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.4426,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.5227$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.4544,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.4137$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	GHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.3201,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.3139$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.1428,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.2552$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WGHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=-0.21,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=-0.251$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.248,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=-0.235$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
$0.5$	AHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.4353,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.5036$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.1744,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=1.5009$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WAHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.1718,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.2607,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.1246,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.1527$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	GHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.211,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.233$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.0433,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.121$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WGHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=-0.314,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=-0.352$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.335,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=-0.303$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
$1$	AHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.0441,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.1156$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.4113,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.0221$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WAHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.4053,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.4044$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.3418,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.4539$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	GHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.5304,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=1.0322$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.3434,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.4446$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WGHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=-0.312,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=-0.344$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.323,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=-0.413$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
$2$	AHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.1345,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.1343$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.041,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.1431$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WAHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=1.0332,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=1.1143$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.4513,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=1.2151$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	GHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=1.0352,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=1.2243$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.5315,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=1.0533$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WGHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=-0.341,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=-0.442\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.342,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=-0.404$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$

Table 9

Influence of the parameter $q_{\text{SC}}$ on the ranking of the alternatives with $p_{\text{SC}}=0.4$.

$q_{\text{SC}}$	Operators	Score values	Ranking values
$0.1$	AHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.535,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=1.0244$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.432,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=1.0034$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WAHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.4201,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.4541$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.3452,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.4557$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	GHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.1039,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.1034$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.042,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.0533$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WGHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=-0.333,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=-0.321$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.345,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=-0.332$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
$0.2$	AHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.3007,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.5451$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.4005,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.4342$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WAHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.5032,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.5695$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.4581,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.5416$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	GHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.4981,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.5053$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.1559,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.4139$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WGHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=-0.441,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=-0.506$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.463,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=-0.529$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
$0.5$	AHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.5501,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.6017$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.2114,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.213$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WAHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.1353,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.2332$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.1344,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.1244$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	GHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.3308,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.3387$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.0881,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.1706$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WGHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=-0.372,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=-0.436$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.418,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=-0.434$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
$1$	AHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.0311\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.0204$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.6622,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.6731$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WAHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.5365,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.6144$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.5654,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.4057$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	GHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.8633,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.8637$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.6737,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.5025$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WGHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=-0.316,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=-0.380$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.388,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=-0.351$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
$2$	AHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.2822,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.2485$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.042,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.049$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WAHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=1.0817,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=1.08$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=1.0606,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.6393$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	GHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=1.0444,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=1.0233$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=1.0203,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.4354$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$
	WGHM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=-0.379,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=-0.442$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.475,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=-0.386$	${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}\ge {\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$

Table 10

Comparative analysis of the proposed and existing operators for Example 1.

Methods	Operators	Score values	Ranking values
Liu et al. [41]	HM	$\text{Failed}$	$\text{Failed}$
Liu and Liu [42]	PBM	$\text{Failed}$	$\text{Failed}$
Liu et al. [43]	WBOWA	$\text{Failed}$	$\text{Failed}$
Liu et al. [44]	HaM	$\text{Failed}$	$\text{Failed}$
Liu and Zhang [45]	BM	$\text{Failed}$	$\text{Failed}$
Proposed operators	CIULAHM operator	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.1971,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.1797$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.026,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.0295$	${\overline{\mathrm{\Phi }}}_{Al-1}\ge {\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-4}\ge {\overline{\mathrm{\Phi }}}_{Al-3}$
	CIULWAHM operator	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=1.0359,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=1.045$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=1.0353,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.8747$	${\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-1}\ge {\overline{\mathrm{\Phi }}}_{Al-3}\ge {\overline{\mathrm{\Phi }}}_{Al-4}$
	CIULGHM operator	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=1.0944,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=1.0732$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=1.0701,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.8849$	${\overline{\mathrm{\Phi }}}_{Al-1}\ge {\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-3}\ge {\overline{\mathrm{\Phi }}}_{Al-4}$
	CIULWGHM operator	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=-0.896,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=-0.934$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.97,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=-0.888$	${\overline{\mathrm{\Phi }}}_{Al-4}\ge {\overline{\mathrm{\Phi }}}_{Al-1}\ge {\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-3}$

5.2. Comparative Analysis

In addition, we want to enhance the excellence and quantity of the investigated operators centered on CIULSs with the help of comparative analysis between explored operators with certain prevailing operators to find the validity and capability of the investigated operators. The information related to existing ideas are discussed as follows: Heronian mean operators based on intuitionistic uncertain linguistic set (IULS) were developed by Liu et al. [41]. Liu and Liu [42] investigated the partitioned Bonferroni mean (PBM) operators for IULS. Moreover, Liu et al. [43] explored weighted Bonferroni ordered weighted averaging (WBOWA) operators for IULS. Liu et al. [44] utilized the idea of Hamy mean (HaM) operators for IULSs. The theory of Bonferroni mean (BM) operators for IULS was developed by Liu and Zhang [45]. The comparative analyses of the investigated operators with certain remaining operators are discussed in the form of Table 10, by using the information of Example 1.

The graphical interpretations of the information of Table 10 are discussed in the form of Figure 2.

[figure omitted; refer to PDF]

From the obtained results, we acquire the effect; if we choose the complex intuitionistic uncertain linguistic type of information, then the existing operators centered on IULVs are not able to cope with it. But, if we prefer the intuitionistic uncertain linguistic type of knowledge, then the proposed operators centered on IUL variables can cope with it. For this, we choose the intuitionistic uncertain linguistic type of knowledge and resolve it by utilizing scrutinized and accessible operators to discover the consistency and efficiency of the offered approaches.

Example 2.

The information of this example is taken from Ref. [41]. There is a speculation organization, which plans to pick the most ideal interest in some options. There are four potential alternatives for the speculation organization to browse: (1) a vehicle organization ${\overline{\mathrm{\Phi }}}_{\text{Al}-1}$; (2) a food organization ${\overline{\mathrm{\Phi }}}_{\text{Al}-2}$; (3) a PC organization${\overline{\mathrm{\Phi }}}_{\text{Al}-3}$; and (4) a mobile organization ${\overline{\mathrm{\Phi }}}_{\text{Al}-4}$. The venture organization considers four criteria to settle on decisions: (1) the hazard investigation ${\overline{\mathrm{ℒ}}}_{\text{At}-1}$; (2) the development examination ${\overline{\mathrm{ℒ}}}_{\text{At}-2}$; (3) the natural impact investigation ${\overline{\mathrm{ℒ}}}_{\text{At}-3}$; and (4) social impact ${\overline{\mathrm{ℒ}}}_{\text{At}-4}$, where all criteria values are benefit type. The weight vector of the criteria is ${\widehat{\Omega }}_W^{\prime }={0.4,0.32,0.28}^T,{\widehat{\Omega }}_W={0.32,0.26,0.18,0.24}^T$. The four potential options are assessed regarding the four rules by the type of CIULNs, and complex intuitionistic uncertain linguistic decision matrices ${\breve{\mathrm{ℛ}}}^i,i=\mathrm{1,2,3}$ are developed and listed in the form of Tables 11–13 .

For resolving the aforementioned issues, we use the following MAGDM procedures:

  Step 1: by utilizing the CIULWAHM operator, we aggregated the decision matrices which are given by decision makers with weighted vectors. The aggregated decision matrix is discussed in the form of Table 14 for $p_{\text{SC}},q_{\text{SC}}=1$.

  Step 2: utilize the CIULAHM operator, CIULWAHM operator, CIULGHM operator, and CIULWGHM operator to aggregate the decision matrices which are in Step 1, which are discussed in the form of Table 15 for $p_{\text{SC}},q_{\text{SC}}=1$.

  Step 3: the score values of the given alternatives are computed and results are listed in Table 16.

  Step 4: rank all the alternatives and find the best one, which are discussed in the form of Table 17.

From the above analysis, we obtain different results by using the investigated operators such as CIULAHM operator, CIULWAHM operator, CIULGHM operator, and CIULWGHM operator. The best options are ${\overline{\mathrm{\Phi }}}_{\text{Al}-2}$ and ${\overline{\mathrm{\Phi }}}_{\text{Al}-4}$. The graphical interpretations of the information of Table 16 are discussed in the form of Figure 3.

The comparative analysis of the investigated operators with some existing operators is discussed in the form of Table 18 by using the information of Example 2.

From this, we acquire the result; if we choose the complex intuitionistic uncertain linguistic type of knowledge, then the existing operators grounded on IULVs are not able to cope with it. But, if we choose the intuitionistic uncertain linguistic type of information, then the proposed operators based on CIUL variables can cope with it. Therefore, the proposed operators are extensively powerful and more reliable than the existing ideas [41–45]. The graphical interpretations of the information of Table 18 are discussed in the form of Figure 4.

Table 11

Decision matrix provided by expert ${\breve{\mathrm{ℛ}}}^1$ in terms of CIULNs.

Alternatives/attributes	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-1}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-2}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-3}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-4}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}$	$\begin{array}{c}{\psi }_5,{\psi }_5,\\ 0.2,0.7\end{array}$	$\begin{array}{c}{\psi }_2,{\psi }_3,\\ 0.4,0.6\end{array}$	$\begin{array}{c}{\psi }_5,{\psi }_6,\\ 0.5,0.5\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ 0.2,0.6\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ 0.4,0.6\end{array}$	$\begin{array}{c}{\psi }_5,{\psi }_5,\\ 0.4,0.5\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ 0.1,0.8\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_4,\\ 0.5,0.5\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ 0.2,0.7\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_4,\\ 0.2,0.7\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ 0.3,0.7\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ 0.2,0.7\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}$	$\begin{array}{c}{\psi }_6,{\psi }_6,\\ 0.5,0.4\end{array}$	$\begin{array}{c}{\psi }_2,{\psi }_3,\\ 0.2,0.8\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ 0.2,0.6\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_3,\\ 0.3,0.6\end{array}$

Table 12

Decision matrix provided by expert ${\breve{\mathrm{ℛ}}}^2$ in terms of CIULNs.

Alternatives/attributes	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-1}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-2}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-3}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-4}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}$	$\begin{array}{c}{\psi }_4,{\psi }_4,\\ 0.1,0.7\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ 0.2,0.7\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ 0.2,0.8\end{array}$	$\begin{array}{c}{\psi }_6,{\psi }_6,\\ 0.4,0.5\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}$	$\begin{array}{c}{\psi }_5,{\psi }_6,\\ 0.4,0.5\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ 0.3,0.6\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ 0.2,0.6\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ 0.2,0.7\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ 0.2,0.6\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_4,\\ 0.2,0.7\end{array}$	$\begin{array}{c}{\psi }_2,{\psi }_3,\\ 0.4,0.6\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ 0.3,0.7\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}$	$\begin{array}{c}{\psi }_5,{\psi }_5,\\ 0.3,0.6\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ 0.4,0.5\end{array}$	$\begin{array}{c}{\psi }_2,{\psi }_3,\\ 0.3,0.6\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_4,\\ 0.2,0.6\end{array}$

Table 13

Decision matrix provided by expert ${\breve{\mathrm{ℛ}}}^3$ in terms of CIULNs.

Alternatives/attributes	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-1}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-2}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-3}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-4}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}$	$\begin{array}{c}{\psi }_5,{\psi }_5,\\ 0.2,0.6\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ 0.3,0.7\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ 0.4,0.5\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_4,\\ 0.2,0.7\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ 0.3,0.7\end{array}$	$\begin{array}{c}{\psi }_5,{\psi }_5,\\ 0.3,0.6\end{array}$	$\begin{array}{c}{\psi }_2,{\psi }_3,\\ 0.1,0.8\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ 0.4,0.6\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$	$\begin{array}{c}{\psi }_4,{\psi }_4,\\ 0.2,0.7\end{array}$	$\begin{array}{c}{\psi }_5,{\psi }_5,\\ 0.3,0.6\end{array}$	$\begin{array}{c}{\psi }_1,{\psi }_3,\\ 0.1,0.8\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_4,\\ 0.2,0.7\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ 0.2,0.7\end{array}$	$\begin{array}{c}{\psi }_3,{\psi }_4,\\ 0.1,0.7\end{array}$	$\begin{array}{c}{\psi }_4,{\psi }_5,\\ 0.3,0.6\end{array}$	$\begin{array}{c}{\psi }_5,{\psi }_5,\\ 0.4,0.5\end{array}$

Table 14

Aggregated decision matrix of the expert by CIULWAHM operator.

	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-1}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-2}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-3}$	${\overline{\mathbf{ℒ}}}_{\mathbf{A}\mathbf{t}-4}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}$	$\begin{array}{c}{\psi }_{1.6377},{\psi }_{1.6377},\\ 0.6394,0.2512\end{array}$	$\begin{array}{c}{\psi }_{0.9178},{\psi }_{1.2718},\\ 0.7743,0.2101\end{array}$	$\begin{array}{c}{\psi }_{1.4011},{\psi }_{1.7561},\\ 0.6645,0.249\end{array}$	$\begin{array}{c}{\psi }_{1.4746},{\psi }_{1.6219},\\ 0.7514,0.1921\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}$	$\begin{array}{c}{\psi }_{1.5179},{\psi }_{1.8703},\\ 0.7318,0.2452\end{array}$	$\begin{array}{c}{\psi }_{1.508},{\psi }_{1.6377},\\ 0.7548,0.1879\end{array}$	$\begin{array}{c}{\psi }_{1.0473},{\psi }_{1.4029},\\ 0.6832,0.2512\end{array}$	$\begin{array}{c}{\psi }_{1.1758},{\psi }_{1.4063},\\ 0.6785,0.2613\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$	$\begin{array}{c}{\psi }_{1.2718},{\psi }_{1.5179},\\ 0.7682,0.2164\end{array}$	$\begin{array}{c}{\psi }_{1.51},{\psi }_{1.51},\\ 0.6125,0.2971\end{array}$	$\begin{array}{c}{\psi }_{0.7869},{\psi }_{1.2855},\\ 0.7058,0.2701\end{array}$	$\begin{array}{c}{\psi }_{1.285},{\psi }_{1.5289},\\ 0.7056,0.2076\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}$	$\begin{array}{c}{\psi }_{1.6432},{\psi }_{1.7664},\\ 0.7029,0.2101\end{array}$	$\begin{array}{c}{\psi }_{1.0194},{\psi }_{1.377},\\ 0.7682,0.2084\end{array}$	$\begin{array}{c}{\psi }_{1.03},{\psi }_{1.3856},\\ 0.6857,0.2678\end{array}$	$\begin{array}{c}{\psi }_{1.3704},{\psi }_{1.3704},\\ 0.7254,0.193\end{array}$

Table 15

Aggregated values of the alternatives by CIULAHM, CIULWAHM, CIULGHM, and CIULWGHM operators.

	$\mathrm{CIULAHM}$	$\mathrm{CIULWAHM}$	$\mathrm{CIULGHM}$	$\mathrm{CIULWGHM}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}$	$\begin{array}{c}{\psi }_{1.349},{\psi }_{1.5684},\\ 0.7112,0.6388\end{array}$	$\begin{array}{c}{\psi }_{1.0074},{\psi }_{1.1614},\\ 0.9854,0.0032\end{array}$	$\begin{array}{c}{\psi }_{1.3489},{\psi }_{1.5684},\\ 0.9575,0.0305\end{array}$	$\begin{array}{c}{\psi }_{1.2581},{\psi }_{1.3935},\\ 0.0893,0.8493\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}$	$\begin{array}{c}{\psi }_{1.3068},{\psi }_{1.5753},\\ 0.729,0.6321\end{array}$	$\begin{array}{c}{\psi }_{0.9926},{\psi }_{1.1902},\\ 0.9898,0.0029\end{array}$	$\begin{array}{c}{\psi }_{1.3068},{\psi }_{1.5753},\\ 0.9628,0.0247\end{array}$	$\begin{array}{c}{\psi }_{1.2501},{\psi }_{1.4363},\\ 0.1025,0.838\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$	$\begin{array}{c}{\psi }_{1.2038},{\psi }_{1.4594},\\ 0.6848,0.672\end{array}$	$\begin{array}{c}{\psi }_{0.9192},{\psi }_{1.0962},\\ 0.9833,0.0044\end{array}$	$\begin{array}{c}{\psi }_{1.2038},{\psi }_{1.4594},\\ 0.9491,0.2595\end{array}$	$\begin{array}{c}{\psi }_{1.1739},{\psi }_{1.3417},\\ 0.0807,0.859\end{array}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}$	$\begin{array}{c}{\psi }_{1.2568},{\psi }_{1.4716},\\ 0.7154,0.6171\end{array}$	$\begin{array}{c}{\psi }_{0.9543},{\psi }_{1.1079},\\ 0.9875,0.0025\end{array}$	$\begin{array}{c}{\psi }_{1.2567},{\psi }_{1.4716},\\ 0.9587,0.213\end{array}$	$\begin{array}{c}{\psi }_{1.216},{\psi }_{1.3588},\\ 0.0934,0.8385\end{array}$

Table 16

Score values of the given alternatives.

	$\mathrm{CIULAHM}$	$\mathrm{CIULWAHM}$	$\mathrm{CIULGHM}$	$\mathrm{CIULWGHM}$
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}$	0.0528	0.5325	0.5302	−0.5038
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}$	0.0698	0.5386	0.5318	−0.494
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}$	0.0085	0.4932	0.4591	−0.4895
${\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}$	0.067	0.5078	0.5087	−0.4796

Table 17

Ordering of the given alternatives.

Methods	Ranking values
CIULAHM operator	${\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-4}\ge {\overline{\mathrm{\Phi }}}_{Al-1}\ge {\overline{\mathrm{\Phi }}}_{Al-3}$
CIULWAHM operator	${\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-1}\ge {\overline{\mathrm{\Phi }}}_{Al-4}\ge {\overline{\mathrm{\Phi }}}_{Al-3}$
CIULGHM operator	${\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-1}\ge {\overline{\mathrm{\Phi }}}_{Al-4}\ge {\overline{\mathrm{\Phi }}}_{Al-3}$
CIULWGHM operator	${\overline{\mathrm{\Phi }}}_{Al-4}\ge {\overline{\mathrm{\Phi }}}_{Al-3}\ge {\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-1}$

[figure omitted; refer to PDF]

Table 18

Comparative analysis of the proposed and existing operators for Example 2.

Methods	Operators	Score values	Ranking values
Liu et al. [41]	HM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.0033,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.0304$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.0141,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.0231$	${\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-4}\ge {\overline{\mathrm{\Phi }}}_{Al-1}\ge {\overline{\mathrm{\Phi }}}_{Al-3}$
Liu and Liu [42]	PBM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.0133,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.0354$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.0241,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.0321$	${\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-4}\ge {\overline{\mathrm{\Phi }}}_{Al-1}\ge {\overline{\mathrm{\Phi }}}_{Al-3}$
Liu et al. [43]	WBOWA	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.0023,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.0244$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.0031,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.0221$	${\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-4}\ge {\overline{\mathrm{\Phi }}}_{Al-1}\ge {\overline{\mathrm{\Phi }}}_{Al-3}$
Liu et al. [44]	HaM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.0036,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.0277$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.0041,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.0211$	${\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-4}\ge {\overline{\mathrm{\Phi }}}_{Al-1}\ge {\overline{\mathrm{\Phi }}}_{Al-3}$
Liu and Zhang [45]	BM	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.1016,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.1357\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.0131,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.1301$	${\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-4}\ge {\overline{\mathrm{\Phi }}}_{Al-1}\ge {\overline{\mathrm{\Phi }}}_{Al-3}$
Proposed operators	CIULAHM operator	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.0528,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.0698$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.0085,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.067$	${\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-4}\ge {\overline{\mathrm{\Phi }}}_{Al-1}\ge {\overline{\mathrm{\Phi }}}_{Al-3}$
	CIULWAHM operator	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.5325,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.5386$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.4932,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.5078$	${\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-1}\ge {\overline{\mathrm{\Phi }}}_{Al-4}\ge {\overline{\mathrm{\Phi }}}_{Al-3}$
	CIULGHM operator	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=0.5302,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=0.5318$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=0.4591,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=0.5087$	${\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-1}\ge {\overline{\mathrm{\Phi }}}_{Al-4}\ge {\overline{\mathrm{\Phi }}}_{Al-3}$
	CIULWGHM operator	$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-1}=-0.5038,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-2}=-0.494$,$\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-3}=-0.4895,\overline{\overline{\zeta }}{\overline{\mathrm{\Phi }}}_{\mathbf{A}\mathbf{l}-4}=-0.4796$	${\overline{\mathrm{\Phi }}}_{Al-4}\ge {\overline{\mathrm{\Phi }}}_{Al-3}\ge {\overline{\mathrm{\Phi }}}_{Al-2}\ge {\overline{\mathrm{\Phi }}}_{Al-1}$

[figure omitted; refer to PDF]

6. Conclusion

The idea of CIULS is developed, and their fundamental laws are discussed. CIULS covers the uncertain linguistic terms; the degree of truth and the degree of falsity are in the form of complex number, whose sum of the real parts (Imaginary parts) is restarted to unit interval. In addition, to analyze the interrelation between any numbers of CIULS, we use the concept of CIULS and HM operators being formed by CIULAHM operator, CIULWAHM operator, CIULGHM operator, and CIULWGHM operator. The major advantages of utilizing the HM operator in the given pairs of CIULNs are that it can interact the different pairs of the argument at the same time. Also, the stated operators have well handled the pairs of the linguistic values along with their membership degrees. Certain higher accidents and the characteristics of the operators under investigation are often illustrated by the use of parameters. In comparison, the MAGDM procedure is built through the use of CIULS-based explored operators. A number of numerical representations are demonstrated with the aid of the methods examined. In order to discover the continuity and experience of the operator’s generated, comparative analysis and graphic expressions are often used to show the predominance of residential approaches. Based on the different pairs of the stated operators and their associated parameters, a decision maker can select their required task as per their preferences. Also, they can analyze their decision impact on the optimal alternatives by varying the parameters used in the decision-making process. Therefore, the suggested decision-making approach is beneficial for an expert to handle the decision-making problem in an uncertain and vague environment. Future work can focus on extending the proposed approach in different fuzzy environments to solve the problems related to decision making, medical diagnosis, pattern recognition, and so on [46–50].

Acknowledgments

The authors are grateful for the financial help provided by Taif University Researchers Supporting Project (TURSP-2020/73), Taif University, Taif, Saudi Arabia. This research was also supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (grant no. NRF-2020R1I1A3074141), the Brain Research Program through the NRF funded by the Ministry of Science, ICT and Future Planning (grant no. NRF-2019M3C7A1020406), and “Regional Innovation Strategy (RIS)” through the NRF funded by the Ministry of Education.

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