Ulam-Hyers stability and existence results for a coupled sequential Hilfer-Hadamard-type integrodifferential system

1.   Introduction

A multi-attribute decision-making (MADM) tool plays an important role in the environment of fuzzy set theory and to evaluate unreliable and awkward information. According to Yukalov and Sornette ^[1], it is sometimes very much difficult to handle a huge part of data or information because of either the design of the data or due to the uncertainty involved in the data or information. Human beings have faced such issues for the last few decades. To oversee such issues, Zadeh ^[2] initiated the notion of fuzzy sets (FS), which takes the truth grade from [0, 1]. Moreover, Atanassove ^[3] modified the technique of FS and initiated the principle of intuitionistic FS (IFS) by including the falsity grade in FS, with the condition that the sum of both the grades must be less than or equal to 1. Sometimes, when a decision maker provides information, the principle of IFS failed, in which the sum of both the grades exceeds 1. For this situation, Yager ^[4] introduced the principle of Pythagorean FS (PFS), with the condition that instead of sum of both the grades, the sum of squares of both the grades must not exceed 1. Although with the introduction of PFS, decision makers have gained much more flexibility to give the values of both the grades but yet there can arise many situations, where PFS failed as even the sum of squares of both the grades exceeds 1. To handle this ambiguity, Yager ^[5] improved the rule of PFS and initiated the notion of q-rung orthopair FS (QROFS), with the rule that the sum of the q-powers (q $\ge$ 1) of the duplet must not exceeds 1. The QROFS has received massive attraction from many researchers and has utilized in different areas ^{[6,7,8,9,10,11,12,13,14]}. The mathematical portrayal of the IFS and their generalizations are clarified with the assistance of Figure 1.

Ramot et al. ^[15] investigated the idea of complex FS (CFS) by including the unreal part in the truth grade, called complex truth grade, whose real and unreal parts belong to [0, 1]. Due to certain complications, Alkouri and Salleh ^[16] modified the principle of CFS by including the falsity grade in the CFS to elaborate the complex IFS (CIFS), with the rule that the sum of the real and unreal parts of the duplet must be limited to [0, 1]. Moreover, Ullah ^[17] utilized the idea of complex PFS (CPFS) by improving the rule of IFS to initiate a new rule for CPFS, with the condition that the sum of the squares of the real and unreal parts of the duplet must be limited to [0, 1]. Akram and Naz initiated decision making for CPFS ^[18]. Although CPFS has proved its effectiveness, there are many situations where it fails to perform, for instance, when a decision maker provides such sorts of information whose sum of the squares of the real and unreal parts exceeds from [0, 1]; for this, the complex QROFS (CQROFS) was developed by Liu et al. ^[19,20]. Furthermore, the CQROFS is portrayed by the interest degree and the non-participation degree, who's total q-powers of the genuine part (likewise for nonexistent part) are not by and large or comparable to 1. The CQROFS is much more generalized and helpful to the researchers of the field as compared to CPFS and CIFS. Mahmood and Ali ^[21,22] enhanced the effectiveness of the notion of CQROFS by studying the TOPSIS technique, correlation coefficients, and Maclaurin operators in the environment of CQROFS.

In terms of the real-life importance of linguistic terms (LT), Zadeh ^[23] introduced the theory of LT sets (LTS) for evaluating the awkward and unreliable problems in real-life decisions. Furthermore, Herrera and Martinez ^[24] pioneered the theory of the 2-tuple LT (2TLT) set, which is more modified than the LTSs. Herrera and Martinez ^[25,26] utilized the theory of LTS in different fields. Li and Liu ^[27] initiated the theory of Heronian mean operators for 2TLT.

DTRS is one of the capable methods to assess abnormal and convoluted data. Different researchers have adjusted it to various ways like loss function (LF) ^[28], trait decrease ^[29], and further developed models utilizing DTRS ^[30]. Yao ^[31] introduced the three-ways decision (3WD), which is changed from DTRS to adapt to practical choice issues. 3WD can isolate all widespread sets into three distinct parts: the positive area (POS(C)), the negative district (NEG(C)), and the limit locale (BND(C)). As a mix of DTRSs and the Bayesian choice method, the procedures of 3WD has successfully overseen numerous request issues. This speculation has been applied in various fields ^[32,33]. The graphical portrayal of the 3WD dependent on the harsh set is examined with the assistance of Figure 2.

Liu and Yang ^[34] introduced the misfortune work in 3WDs dependent on intuitionistic fuzzy linguistic sets. In view of the above examination, we initiated the following: we generalize the massive important techniques of three-way decisions (3WD) ^[35] and DTRS with Complex q-rung orthopair 2-tuple linguistic variable (CQRO2-TLV), to elaborate certain important properties. Moreover, the generalized Maclurin symmetric mean (GMSM) ^[36] is a dominant and more flexible method to determine the accuracy and dominancy of real life issues. Therefore, by considering the complex q-rung orthopair 2-tuple linguistic (CQRO2-TL) information and generalized Maclurin symmetric mean (GMSM), we present the CQRO2-TL GMSM (CQRO2-TLGMSM) operator and weighted CQRO2-TL GMSM (WCQRO2-TLGMSM) operator, and demonstrated their effective properties. Finally, a model is applied to exhaustively elucidate the proposed procedure, and the effects of different contingent probabilities on choice results are examined. The investigation of the elaborated approach is likewise discussed regarding certain ways to deal with the capability and capacity of the presented approach. For simplicity, the explored work for this manuscript is explained with the help of Figure 3.

This manuscript is outlined as follows. In Section 2, we briefly audit some helpful ideas of 2-TLF ^[24], converse 2-TLF ^[24], CQROFS ^{[19,20,21,22,35]}, and their functional laws. Furthermore, the GMSM operators ^[36] are discussed. In Section 3, we explored CQRO2-TLV and their algebraic laws. In Section 4, we generalize 3WD and DTRS with CQRO2-TLV to elaborate certain important properties. GMSM is a dominant and more flexible way to determine the accuracy and dominancy of real life issues. Therefore, by considering the CQRO2-TL information and GMSM, in section 5 we present the CQRO2-TLGMSM operator and the WCQRO2-TLGMSM operator and demonstrated their effective properties. In Section 6, a model is applied to exhaustively elucidate the proposed procedure, and the effects of different contingent probabilities on choice results are examined. The conclusion of this manuscript is discussed in Section 7.

2.   Preliminaries

For the convenience and improved understanding for the reader, in this section we will recall some basic notions which will be used in the manuscript. The symbols ${Ữ}_{UNI}, {\mu }_{{\mathcal{Q}}_{CQ}}\left(ử\right)$, and ${\eta }_{{\mathcal{Q}}_{CQ}}\left(ử\right)$ will denote the universal set, the truth grade, and the falsity grade, respectively, where ${\alpha }_{SC}, {\delta }_{SC}, {q}_{SC}\ge 1$.

Definition 2.1. ^[24] For a LS ${S}_{LT} = \left\{{s}_{{S}_{LT-0}}, {s}_{{S}_{LT-1}}, {s}_{{S}_{LT-2}}, {s}_{{S}_{LT-3}}, \dots., {s}_{{S}_{LT-g}}\right\}$ with ${\beta }_{SC}\in \left[\mathrm{0, 1}\right]$, the 2-TL function ${\Delta }_{LT}$ is given by:

The 2-TL inverse function ${\Delta }_{LT}^{-1}$ is given by:

Definition 2.2. ^[19] A CQROFS is given by

where ${\mu }_{{\mathcal{Q}}_{CQ}}\left(\mathrm{ử}\right) = {\mu }_{{\mathcal{Q}}_{CQRP}}\left(\mathrm{ử}\right){e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQIP}}\left(\mathrm{ử}\right)\right)}$ and ${\eta }_{{\mathcal{Q}}_{CQ}}\left(\mathrm{ử}\right) = {\eta }_{{\mathcal{Q}}_{CQRP}}\left(\mathrm{ử}\right){e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQIP}}\left(\mathrm{ử}\right)\right)}$,

with the conditions: $0\le {\mu }_{{\mathcal{Q}}_{CQRP}}^{{q}_{SC}}\left(\mathrm{ử}\right)+{\eta }_{{\mathcal{Q}}_{CQRP}}^{{q}_{SC}}\left(\mathrm{ử}\right)\le \mathrm{1, 0}\le {\mu }_{{\mathcal{Q}}_{CQIP}}^{{q}_{SC}}\left(\mathrm{ử}\right)+{\eta }_{{\mathcal{Q}}_{CQIP}}^{{q}_{SC}}\left(\mathrm{ử}\right)\le 1$.

Moreover, ${\zeta }_{{\mathcal{Q}}_{CQ}}\left(\mathrm{ử}\right) = \left(1-{\left({\mu }_{{\mathcal{Q}}_{CQRP}}^{{q}_{SC}}\left(\mathrm{ử}\right)+{\eta }_{{\mathcal{Q}}_{CQRP}}^{{q}_{SC}}\left(\mathrm{ử}\right)\right)}^{\frac{1}{{q}_{SC}}}\right){e}^{i2\pi \left(1-{\left({\mu }_{{\mathcal{Q}}_{CQIP}}^{{q}_{SC}}\left(\mathrm{ử}\right)+{\eta }_{{\mathcal{Q}}_{CQIP}}^{{q}_{SC}}\left(\mathrm{ử}\right)\right)}^{\frac{1}{{q}_{SC}}}\right)}$ is called the refusal grade. The QROFN is given as

Definition 2.3. ^[20] For any two CQROFNs

${\mathcal{Q}}_{CQ-1} = \left({\mu }_{{\mathcal{Q}}_{CQRP-1}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQIP-1}}\right)}, {\eta }_{{\mathcal{Q}}_{CQRP-1}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQIP-1}}\right)}\right)$ and

${\mathcal{Q}}_{CQ-2} = \left({\mu }_{{\mathcal{Q}}_{CQRP-2}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQIP-2}}\right)}, {\eta }_{{\mathcal{Q}}_{CQRP-2}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQIP-2}}\right)}\right)$, then

(1). ${\mathcal{Q}}_{CQ-1}{ \oplus }_{CQ}{\mathcal{Q}}_{CQ-2} = \left(\begin{array}{c}{\left(\begin{array}{c}{\mu }_{{\mathcal{Q}}_{CQRP-1}}^{{q}_{CQ}}+{\mu }_{{\mathcal{Q}}_{CQRP-2}}^{{q}_{CQ}}-\\ {\mu }_{{\mathcal{Q}}_{CQRP-1}}^{{q}_{CQ}}{\mu }_{{\mathcal{Q}}_{CQRP-2}}^{{q}_{CQ}}\end{array}\right)}^{\frac{1}{{q}_{CQ}}}{e}^{i2\pi {\left(\begin{array}{c}{\mu }_{{\mathcal{Q}}_{CQIP-1}}^{{q}_{CQ}}+{\mu }_{{\mathcal{Q}}_{CQIP-2}}^{{q}_{CQ}}-\\ {\mu }_{{\mathcal{Q}}_{CQIP-1}}^{{q}_{CQ}}{\mu }_{{\mathcal{Q}}_{CQIP-2}}^{{q}_{CQ}}\end{array}\right)}^{\frac{1}{{q}_{CQ}}}}, \\ {\eta }_{{\mathcal{Q}}_{CQRP-1}}{\eta }_{{\mathcal{Q}}_{CQRP-2}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQIP-1}}{\eta }_{{\mathcal{Q}}_{CQIP-2}}\right)}\end{array}\right)$;

(2). ${\mathcal{Q}}_{CQ-1}{\otimes }_{CQ}{\mathcal{Q}}_{CQ-2} = \left(\begin{array}{c}{\mu }_{{\mathcal{Q}}_{CQRP-1}}{\mu }_{{\mathcal{Q}}_{CQRP-2}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQIP-1}}{\mu }_{{\mathcal{Q}}_{CQIP-2}}\right)}, \\ {\left(\begin{array}{c}{\eta }_{{\mathcal{Q}}_{CQRP-1}}^{{q}_{CQ}}+{\eta }_{{\mathcal{Q}}_{CQRP-2}}^{{q}_{CQ}}-\\ {\eta }_{{\mathcal{Q}}_{CQRP-1}}^{{q}_{CQ}}{\eta }_{{\mathcal{Q}}_{CQRP-2}}^{{q}_{CQ}}\end{array}\right)}^{\frac{1}{{q}_{CQ}}}{e}^{i2\pi {\left(\begin{array}{c}{\eta }_{{\mathcal{Q}}_{CQIP-1}}^{{q}_{CQ}}+{\eta }_{{\mathcal{Q}}_{CQIP-2}}^{{q}_{CQ}}-\\ {\eta }_{{\mathcal{Q}}_{CQIP-1}}^{{q}_{CQ}}{\eta }_{{\mathcal{Q}}_{CQIP-2}}^{{q}_{CQ}}\end{array}\right)}^{\frac{1}{{q}_{CQ}}}}\end{array}\right)$;

(3). ${\mathcal{Q}}_{CQ-1}^{{\delta }_{SC}} = \left(\begin{array}{c}{\mu }_{{\mathcal{Q}}_{CQRP-1}}^{{\delta }_{SC}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQIP-1}}^{{\delta }_{SC}}\right)}, \\ {\left(1-{\left(1-{\eta }_{{\mathcal{Q}}_{CQRP-1}}^{{q}_{CQ}}\right)}^{{\delta }_{SC}}\right)}^{\frac{1}{{q}_{CQ}}}{e}^{i2\pi {\left(1-{\left(1-{\eta }_{{\mathcal{Q}}_{CQIP-1}}^{{q}_{CQ}}\right)}^{{\delta }_{SC}}\right)}^{\frac{1}{{q}_{CQ}}}}\end{array}\right)$;

(4). ${\delta }_{SC}{\mathcal{Q}}_{CQ-1} = \left(\begin{array}{c}{\left(1-{\left(1-{\mu }_{{\mathcal{Q}}_{CQRP-1}}^{{q}_{CQ}}\right)}^{{\delta }_{SC}}\right)}^{\frac{1}{{q}_{CQ}}}{e}^{i2\pi {\left(1-{\left(1-{\mu }_{{\mathcal{Q}}_{CQIP-1}}^{{q}_{CQ}}\right)}^{{\delta }_{SC}}\right)}^{\frac{1}{{q}_{CQ}}}}, \\ {\eta }_{{\mathcal{Q}}_{CQRP-1}}^{{\delta }_{SC}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQIP-1}}^{{\delta }_{SC}}\right)}\end{array}\right)$.

Definition 2.4. ^[21] For any two CQROFNs

${\mathcal{Q}}_{CQ-1} = \left({\mu }_{{\mathcal{Q}}_{CQRP-1}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQIP-1}}\right)}, {\eta }_{{\mathcal{Q}}_{CQRP-1}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQIP-1}}\right)}\right)$ and

${\mathcal{Q}}_{CQ-2} = \left({\mu }_{{\mathcal{Q}}_{CQRP-2}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQIP-2}}\right)}, {\eta }_{{\mathcal{Q}}_{CQRP-2}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQIP-2}}\right)}\right)$, the score and accuracy functions are given by:

Based on the two above notions, the compassion between two CQROFNs is given by:

(1). If $Ş\left({\mathcal{Q}}_{CQ-1}\right) > Ş\left({\mathcal{Q}}_{CQ-2}\right)$, then ${\mathcal{Q}}_{CQ-1} > {\mathcal{Q}}_{CQ-2}$;

(2). If $Ş\left({\mathcal{Q}}_{CQ-1}\right) = Ş\left({\mathcal{Q}}_{CQ-2}\right)$, then:

ⅰ) if $Ȟ\left({\mathcal{Q}}_{CQ-1}\right) > Ȟ\left({\mathcal{Q}}_{CQ-2}\right)$, then ${\mathcal{Q}}_{CQ-1} > {\mathcal{Q}}_{CQ-2}$;

ⅱ) if $Ȟ\left({\mathcal{Q}}_{CQ-1}\right) = Ȟ\left({\mathcal{Q}}_{CQ-2}\right)$, then ${\mathcal{Q}}_{CQ-1} = {\mathcal{Q}}_{CQ-2}$.

Definition 2.5. ^[36] For the family of ${\mathcal{Q}}_{CQ-j}\left(j = \mathrm{1, 2}, 3, \dots, n\right)$, the MSM operator is given by:

where ${K}_{SC} = \mathrm{1, 2}, \dots, n, \left({j}_{1}, {j}_{2}, \dots, {j}_{{K}_{SC}}\right)$ denotes the K-tuple kind of $(\mathrm{1, 2}, .., n),$ and the symbol ${C}_{n}^{{K}_{SC}}$ designates the binomial co-efficient (BCO). Furthermore:

(1). $M{SM}^{{K}_{SC}}\left({\mathcal{Q}}_{CQ}, {\mathcal{Q}}_{CQ}, \dots, {\mathcal{Q}}_{CQ}\right) = {\mathcal{Q}}_{CQ}$;

(2). $M{SM}^{{K}_{SC}}\left({\mathcal{Q}}_{CQ-1}, {\mathcal{Q}}_{CQ-2}, \dots, {\mathcal{Q}}_{CQ-n}\right)\le M{SM}^{{K}_{SC}}\left({\mathcal{Q}}_{CQ-*1}, {\mathcal{Q}}_{CQ-*2}, \dots, {\mathcal{Q}}_{CQ-*n}\right)$, if ${\mathcal{Q}}_{CQ-j}\le {\mathcal{Q}}_{CQ-*j}$ for all $j$.

(3). $\underset{j}{\mathrm{min}}{\mathcal{Q}}_{CQ-j}\le M{SM}^{{K}_{SC}}\left({\mathcal{Q}}_{CQ-1}, {\mathcal{Q}}_{CQ-2}, \dots, {\mathcal{Q}}_{CQ-n}\right)\le \underset{j}{\mathrm{max}}{\mathcal{Q}}_{CQ-j}$.

Definition 2.6. ^[36] For the family of ${\mathcal{Q}}_{CQ-j}\left(j = \mathrm{1, 2}, 3, \dots, n\right)$, the GMSM operator is given by:

where ${K}_{SC} = \mathrm{1, 2}, \dots, n, {\alpha }_{SC-1}, {\alpha }_{SC-2}, \dots, {\alpha }_{SC-{K}_{SC}}\ge 0, \left({j}_{1}, {j}_{2}, \dots, {j}_{{K}_{SC}}\right)$ signifies the K-tuple collection of $\left(\mathrm{1, 2}, .., n\right).$ Furthermore:

(1). $GM{SM}^{\left({K}_{SC}, {\alpha }_{SC-1}, {\alpha }_{SC-2}, \dots, {\alpha }_{SC-{K}_{SC}}\right)}\left({\mathcal{Q}}_{CQ}, {\mathcal{Q}}_{CQ}, \dots, {\mathcal{Q}}_{CQ}\right)$ $= {\mathcal{Q}}_{CQ}, GM{SM}^{\left({K}_{SC}, {\alpha }_{SC-1}, {\alpha }_{SC-2}, \dots, {\alpha }_{SC-{K}_{SC}}\right)}\left(\mathrm{0, 0}, \dots, 0\right) = 0$;

(2). $GM{SM}^{\left({K}_{SC}, {\alpha }_{SC-1}, {\alpha }_{SC-2}, \dots, {\alpha }_{SC-{K}_{SC}}\right)}\left({\mathcal{Q}}_{CQ-1}, {\mathcal{Q}}_{CQ-2}, \dots, {\mathcal{Q}}_{CQ-n}\right)\le$ $GM{SM}^{\left({K}_{SC}, {\alpha }_{SC-1}, {\alpha }_{SC-2}, \dots, {\alpha }_{SC-{K}_{SC}}\right)}\left({\mathcal{Q}}_{CQ-*1}, {\mathcal{Q}}_{CQ-*2}, \dots, {\mathcal{Q}}_{CQ-*n}\right)$, if ${\mathcal{Q}}_{CQ-j}\le {\mathcal{Q}}_{CQ-*j}$ for all $j$;

(3). $\underset{j}{\mathrm{min}}{\mathcal{Q}}_{CQ-j}\le GM{SM}^{\left({K}_{SC}, {\alpha }_{SC-1}, {\alpha }_{SC-2}, \dots, {\alpha }_{SC-{K}_{SC}}\right)}$ $\left({\mathcal{Q}}_{CQ-1}, {\mathcal{Q}}_{CQ-2}, \dots, {\mathcal{Q}}_{CQ-n}\right)\le \underset{j}{\mathrm{max}}{\mathcal{Q}}_{CQ-j}$.

2.1.   Review of the three-way decisions based on DTRSs

As discussed in ^[34], DTRS contains two status and three actions, whose information: ${\mathrm{\Omega }}_{S} = \left\{{\mathcal{F}}_{B}, \sim {\mathcal{F}}_{NB}\right\}$ and ${A}_{AC} = \left\{{\chi }_{{P}_{AC}}, {\chi }_{{B}_{AC}}, {\chi }_{{N}_{AC}}\right\}$, where ${\mathrm{\Omega }}_{S}$ is utilized for having a place and for to be not having a place and ${A}_{AC}$ is utilized for +ve, limit and -ve. Table 1 shows the lose capacities and their portrayals.

Form Table 1, the inequality is given as follows:

Given the Bayesian risk choice hypothesis ^[34], we have:

By using the Eq (2.12), the expected losses ${Y}_{EL}\left({\chi }_{{j}_{AC}}\left|\right[ử]\right), j = P, B, N$, and the different actions are expressed below:

From the above information, we have:

3.   Complex q-rung orthopair 2-tuple linguistic sets

Based on the above analysis, we have explored the novel approach of CQRO2-TLS and its fundamental properties. These properties are also explained with the help of some numerical examples.

Definition 3.1. A CQRO2-TLS is given by

where ${\mu }_{{\mathcal{Q}}_{CQTL}}\left(ử\right) = {\mu }_{{\mathcal{Q}}_{CQTLRP}}\left(ử\right){e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQTLIP}}\left(ử\right)\right)}$ and ${\eta }_{{\mathcal{Q}}_{CQTL}}\left(ử\right) = {\eta }_{{\mathcal{Q}}_{CQTLRP}}\left(ử\right){e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQTLIP}}\left(ử\right)\right)}$, with the conditions: $0\le {\mu }_{{\mathcal{Q}}_{CQTLRP}}^{{q}_{SC}}\left(ử\right)+{\eta }_{{\mathcal{Q}}_{CQTLRP}}^{{q}_{SC}}\left(ử\right)\le \mathrm{1, 0}\le {\mu }_{{\mathcal{Q}}_{CQTLIP}}^{{q}_{SC}}\left(ử\right)+{\eta }_{{\mathcal{Q}}_{CQTLIP}}^{{q}_{SC}}\left(ử\right)\le 1$, where ${q}_{SC}$ expresses the rational numbers and the pair $\left({s}_{{S}_{LT}\left(ử\right)}, {\alpha }_{SC}\right)$ is called 2-TLV with ${\alpha }_{SC}\in \left[-\frac{1}{2g}, \frac{1}{2g}\right\}$ and ${s}_{{S}_{LT}\left(ử\right)}\in {S}_{LT}$.

Moreover,

${\zeta }_{{\mathcal{Q}}_{CQTL}}\left(ử\right) = \left(1-{\left({\mu }_{{\mathcal{Q}}_{CQTLRP}}^{{q}_{SC}}\left(ử\right)+{\eta }_{{\mathcal{Q}}_{CQTLRP}}^{{q}_{SC}}\left(ử\right)\right)}^{\frac{1}{{q}_{SC}}}\right){e}^{i2\pi \left(1-{\left({\mu }_{{\mathcal{Q}}_{CQTLIP}}^{{q}_{SC}}\left(ử\right)+{\eta }_{{\mathcal{Q}}_{CQTLIP}}^{{q}_{SC}}\left(ử\right)\right)}^{\frac{1}{{q}_{SC}}}\right)}$ is called the refusal grade. The CQRO2T-LN is shown by:

Definition 3.2. For any two CQRO2-TLNs

${\mathcal{Q}}_{CQTL-1} = \left(\left({s}_{{S}_{LT-1}}, {\alpha }_{SC-1}\right), \left({\mu }_{{\mathcal{Q}}_{CQTLRP-1}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQTLIP-1}}\right)}, {\eta }_{{\mathcal{Q}}_{CQTLRP-1}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQTLIP-1}}\right)}\right)\right)$ and ${\mathcal{Q}}_{CQTL-2} = \left(\left({s}_{{S}_{LT-2}}, {\alpha }_{SC-2}\right), \left({\mu }_{{\mathcal{Q}}_{CQTLRP-2}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQTLIP-2}}\right)}, {\eta }_{{\mathcal{Q}}_{CQTLRP-2}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQTLIP-2}}\right)}\right)\right)$,

(1). ${\mathcal{Q}}_{CQTL-1}{ \oplus }_{CQTL}{\mathcal{Q}}_{CQTL-2} = \left(\begin{array}{c}{\Delta }_{LT}\left({\Delta }_{LT}^{-1}\left({s}_{{S}_{LT-1}}, {\alpha }_{SC-1}\right)+{\Delta }_{LT}^{-1}\left({s}_{{S}_{LT-2}}, {\alpha }_{SC-2}\right)\right).\\ \left(\begin{array}{c}{\left(\begin{array}{c}{\mu }_{{\mathcal{Q}}_{CQTLRP-1}}^{{q}_{CQ}}+{\mu }_{{\mathcal{Q}}_{CQTLRP-2}}^{{q}_{CQ}}-\\ {\mu }_{{\mathcal{Q}}_{CQTLRP-1}}^{{q}_{CQ}}{\mu }_{{\mathcal{Q}}_{CQTLRP-2}}^{{q}_{CQ}}\end{array}\right)}^{\frac{1}{{q}_{CQ}}}{e}^{i2\pi {\left(\begin{array}{c}{\mu }_{{\mathcal{Q}}_{CQTLIP-1}}^{{q}_{CQ}}+{\mu }_{{\mathcal{Q}}_{CQTLIP-2}}^{{q}_{CQ}}-\\ {\mu }_{{\mathcal{Q}}_{CQTLIP-1}}^{{q}_{CQ}}{\mu }_{{\mathcal{Q}}_{CQTLIP-2}}^{{q}_{CQ}}\end{array}\right)}^{\frac{1}{{q}_{CQ}}}}, \\ {\eta }_{{\mathcal{Q}}_{CQTLRP-1}}{\eta }_{{\mathcal{Q}}_{CQTLRP-2}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQTLIP-1}}{\eta }_{{\mathcal{Q}}_{CQTLIP-2}}\right)}\end{array}\right)\end{array}\right)$;

(2). ${\mathcal{Q}}_{CQTL-1}{\otimes }_{CQTL}{\mathcal{Q}}_{CQTL-2} = \left(\begin{array}{c}{\Delta }_{LT}\left({\Delta }_{LT}^{-1}\left({s}_{{S}_{LT-1}}, {\alpha }_{SC-1}\right)\times {\Delta }_{LT}^{-1}\left({s}_{{S}_{LT-2}}, {\alpha }_{SC-2}\right)\right).\\ \left(\begin{array}{c}{\mu }_{{\mathcal{Q}}_{CQTLRP-1}}{\mu }_{{\mathcal{Q}}_{CQTLRP-2}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQTLIP-1}}{\mu }_{{\mathcal{Q}}_{CQTLIP-2}}\right)}, \\ {\left(\begin{array}{c}{\eta }_{{\mathcal{Q}}_{CQTLRP-1}}^{{q}_{CQ}}+{\eta }_{{\mathcal{Q}}_{CQTLRP-2}}^{{q}_{CQ}}-\\ {\eta }_{{\mathcal{Q}}_{CQTLRP-1}}^{{q}_{CQ}}{\eta }_{{\mathcal{Q}}_{CQTLRP-2}}^{{q}_{CQ}}\end{array}\right)}^{\frac{1}{{q}_{CQ}}}{e}^{i2\pi {\left(\begin{array}{c}{\eta }_{{\mathcal{Q}}_{CQTLIP-1}}^{{q}_{CQ}}+{\eta }_{{\mathcal{Q}}_{CQTLIP-2}}^{{q}_{CQ}}-\\ {\eta }_{{\mathcal{Q}}_{CQTLIP-1}}^{{q}_{CQ}}{\eta }_{{\mathcal{Q}}_{CQTLIP-2}}^{{q}_{CQ}}\end{array}\right)}^{\frac{1}{{q}_{CQ}}}}\end{array}\right)\end{array}\right)$;

(3). ${\mathcal{Q}}_{CQTL-1}^{{\delta }_{SC}} = \left(\begin{array}{c}{\Delta }_{LT}\left({\Delta }_{LT}^{-1}{\left({s}_{{S}_{LT-1}}, {\alpha }_{SC-1}\right)}^{{\delta }_{SC}}\right), {\mu }_{{\mathcal{Q}}_{CQTLRP-1}}^{{\delta }_{SC}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQTLIP-1}}^{{\delta }_{SC}}\right)}, \\ \left({\left(1-{\left(1-{\eta }_{{\mathcal{Q}}_{CQTLRP-1}}^{{q}_{CQ}}\right)}^{{\delta }_{SC}}\right)}^{\frac{1}{{q}_{CQ}}}{e}^{i2\pi {\left(1-{\left(1-{\eta }_{{\mathcal{Q}}_{CQTLIP-1}}^{{q}_{CQ}}\right)}^{{\delta }_{SC}}\right)}^{\frac{1}{{q}_{CQ}}}}\right)\end{array}\right)$;

(4). ${\delta }_{SC}{\mathcal{Q}}_{CQTL-1} = \left(\begin{array}{c}{\Delta }_{LT}\left({\delta }_{SC}\times {\Delta }_{LT}^{-1}\left({s}_{{S}_{LT-1}}, {\alpha }_{SC-1}\right)\right), \\ \left(\begin{array}{c}{\left(1-{\left(1-{\mu }_{{\mathcal{Q}}_{CQTLRP-1}}^{{q}_{CQ}}\left(ử\right)\right)}^{{\delta }_{SC}}\right)}^{\frac{1}{{q}_{CQ}}}{e}^{i2\pi {\left(1-{\left(1-{\mu }_{{\mathcal{Q}}_{CQTLIP-1}}^{{q}_{CQ}}\left(ử\right)\right)}^{{\delta }_{SC}}\right)}^{\frac{1}{{q}_{CQ}}}}, \\ {\eta }_{{\mathcal{Q}}_{CQTLRP-1}}^{{\delta }_{SC}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQTLIP-1}}^{{\delta }_{SC}}\right)}\end{array}\right)\end{array}\right)$.

Definition 3.3. For any two QROF2-TLNs

${\mathcal{Q}}_{CQTL-1} = \left(\left({s}_{{S}_{LT-1}}, {\alpha }_{SC-1}\right), \left({\mu }_{{\mathcal{Q}}_{CQTLRP-1}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQTLIP-1}}\right)}, {\eta }_{{\mathcal{Q}}_{CQTLRP-1}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQTLIP-1}}\right)}\right)\right)$ and ${\mathcal{Q}}_{CQTL-2} = \left(\left({s}_{{S}_{LT-2}}, {\alpha }_{SC-2}\right), \left({\mu }_{{\mathcal{Q}}_{CQTLRP-2}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQTLIP-2}}\right)}, {\eta }_{{\mathcal{Q}}_{CQTLRP-2}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQTLIP-2}}\right)}\right)\right)$, the score and accuracy functions are given by:

Based on the two above notions, the compassion between two QROF2-TLNs is given by:

(1). If $Ş\left({\mathcal{Q}}_{CQTL-1}\right) > Ş\left({\mathcal{Q}}_{CQTL-2}\right)$, then ${\mathcal{Q}}_{CQTL-1} > {\mathcal{Q}}_{CQTL-2}$;

(2). If $Ş\left({\mathcal{Q}}_{CQTL-1}\right) = Ş\left({\mathcal{Q}}_{CQTL-2}\right)$, then:

ⅰ) $Ȟ\left({\mathcal{Q}}_{CQTL-1}\right) > Ȟ\left({\mathcal{Q}}_{CQTL-2}\right)$, implies ${\mathcal{Q}}_{CQTL-1} > {\mathcal{Q}}_{CQTL-2}$;

ⅱ) $Ȟ\left({\mathcal{Q}}_{CQTL-1}\right) = Ȟ\left({\mathcal{Q}}_{CQTL-2}\right)$, implies ${\mathcal{Q}}_{CQTL-1} = {\mathcal{Q}}_{CQTL-2}$.

4.   A novel DTRS model based on CQRO2-TLVs

The purpose of this section is to explore the novel DTRS model by using the CQROF2-TLV based on three-way decisions. The LF matrix for CQROF2-TLV is presented in Table 2.

Form Table 2, the inequality is given as:

Furthermore, by using the Eq (4.2), we explain the expected losses ${Y}_{EL}\left({\chi }_{{j}_{AC}}\left|\right[ử]\right), j = P, B, N$ and the different actions are expressed below:

Assumed that $\mathrm{Pr}\left({\mathcal{F}}_{B}\left|\right[ử]\right) = {\delta }_{B}$ and $\mathrm{Pr}\left({\sim \mathcal{F}}_{NB}\left|\right[ử]\right) = {\delta }_{NB}$, then the Eq (4.2) is given as: ${\delta }_{B}+{\delta }_{NB} = 1$.

Theorem 4.1. With usual meanings

Proof. First, we prove that the Eq (4.6) holds. The proofs of Eq (4.7) and of Eq (4.8) are similar.

Moreover, we diagnose the expected values, such that

To handle whenever the normal qualities failed to discover the connections between any of two predictable losses, we investigate the ideas of precision work, which is expressed as follows:

Furthermore, we diagnosed the three-way decision rules, such that

Therefore,

5.   Certain GMSM operators for CQRO2-TLVs

The interrelationship among the different attributes in real decision theory is ever-present. The MSM and GMSM operators are efficient techniques to perfectly evaluate the interrelation between the characteristics. The purpose of this communication is to explore the GMSM and weighted GMSM based on CQRO2-TLVs.

Definition 5.1. For the family of CQRO2-TLVs ${\mathcal{Q}}_{CQTL-j}\left(j = \mathrm{1, 2}, 3, \dots, n\right)$, the CQRO2-TLGMSM operator is given by:

where ${K}_{SC} = \mathrm{1, 2}, \dots, n, {\alpha }_{SC-1}, {\alpha }_{SC-2}, \dots, {\alpha }_{SC-{K}_{SC}}\ge 0, \left({j}_{1}, {j}_{2}, \dots, {j}_{{K}_{SC}}\right)$ designates the K-tuple collection of $\left(\mathrm{1, 2}, .., n\right).$

Theorem 5.2. For the family of CQRO2-TLVs

${\mathcal{Q}}_{CQTL-j} = \left(\left({s}_{{S}_{LT-j}}, {\alpha }_{SC-j}\right), \left({\mu }_{{\mathcal{Q}}_{CQTLRP-j}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQTLIP-j}}\right)}, {\eta }_{{\mathcal{Q}}_{CQTLRP-j}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQTLIP-j}}\right)}\right)\right)\left(j = \mathrm{1, 2}, 3, \dots, n\right)$,

Proof. Let

Furthermore, by using the investigated operators, we discuss certain properties such as idempotency, commutativity, monotonicity, and boundedness, which are very important for the proposed work.

Theorem 5.3. For the family of CQRO2-TLVs

${\mathcal{Q}}_{CQTL-j} = \left(\left({s}_{{S}_{LT-j}}, {\alpha }_{SC-j}\right), \left({\mu }_{{\mathcal{Q}}_{CQTLRP-j}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQTLIP-j}}\right)}, {\eta }_{{\mathcal{Q}}_{CQTLRP-j}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQTLIP-j}}\right)}\right)\right)\left(j = \mathrm{1, 2}, 3, \dots, n\right)$, then

(1). ${\mathcal{Q}}_{CQTL-j} = {\mathcal{Q}}_{CQTL} = \left(\left({s}_{{S}_{LT}}, {\alpha }_{SC}\right), \left({\mu }_{{\mathcal{Q}}_{CQTLRP}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQTLIP}}\right)}, {\eta }_{{\mathcal{Q}}_{CQTLRP}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQTLIP}}\right)}\right)\right)$, implies $CQRO2-TLGM{SM}^{\left({K}_{SC}, {\alpha }_{SC-1}, {\alpha }_{SC-2}, \dots, {\alpha }_{SC-{K}_{SC}}\right)}\left({\mathcal{Q}}_{CQTL-1}, {\mathcal{Q}}_{CQTL-2}, \dots, {\mathcal{Q}}_{CQTL-n}\right)$ $= {\mathcal{Q}}_{CQTL}$.

(2). if ${\mathcal{Q}}_{CQTL-j}$ is a collection of CQRO2-TLVs and ${\mathcal{Q}}_{CQTL-j}^{\text{'}}$ is any replacement of ${\mathcal{Q}}_{CQTL-j}$, then $CQRO2-TLGM{SM}^{\left({K}_{SC}, {\alpha }_{SC-1}, {\alpha }_{SC-2}, \dots, {\alpha }_{SC-{K}_{SC}}\right)}\left({\mathcal{Q}}_{CQTL-1}, {\mathcal{Q}}_{CQTL-2}, \dots, {\mathcal{Q}}_{CQTL-n}\right) = CQRO2-TLGM{SM}^{\left({K}_{SC}, {\alpha }_{SC-1}, {\alpha }_{SC-2}, \dots, {\alpha }_{SC-{K}_{SC}}\right)}\left({\mathcal{Q}}_{CQTL-1}^{\text{'}}, {\mathcal{Q}}_{CQTL-2}^{\text{'}}, \dots, {\mathcal{Q}}_{CQTL-n}^{\text{'}}\right)$.

(3). if ${\mathcal{Q}}_{CQTL-j}$ and ${\mathcal{Q}}_{CQTL-j}^{\text{'}}$ are any two collections of CQRO2-TLVs with a situations s.t.${\mathcal{Q}}_{CQTL-j}\le {\mathcal{Q}}_{CQTL-j}^{\text{'}}$,

then $CQRO2-TLGM{SM}^{\left({K}_{SC}, {\alpha }_{SC-1}, {\alpha }_{SC-2}, \dots, {\alpha }_{SC-{K}_{SC}}\right)}\left({\mathcal{Q}}_{CQTL-1}, {\mathcal{Q}}_{CQTL-2}, \dots, {\mathcal{Q}}_{CQTL-n}\right)\le CQRO2-TLGM{SM}^{\left({K}_{SC}, {\alpha }_{SC-1}, {\alpha }_{SC-2}, \dots, {\alpha }_{SC-{K}_{SC}}\right)}\left({\mathcal{Q}}_{CQTL-1}^{\text{'}}, {\mathcal{Q}}_{CQTL-2}^{\text{'}}, \dots, {\mathcal{Q}}_{CQTL-n}^{\text{'}}\right)$.

(4). if ${\mathcal{Q}}_{CQTL-j} = \left(\left({s}_{{S}_{LT-j}}, {\alpha }_{SC-j}\right), \left({\mu }_{{\mathcal{Q}}_{CQTLRP-j}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQTLIP-j}}\right)}, {\eta }_{{\mathcal{Q}}_{CQTLRP-j}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQTLIP-j}}\right)}\right)\right)$ is a family of the CQRO2-TLVs, then

Proof.

(1). If ${\mathcal{Q}}_{CQTL} = \left(\left({s}_{{S}_{LT}}, {\alpha }_{SC}\right), \left({\mu }_{{\mathcal{Q}}_{CQTLRP}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQTLIP}}\right)}, {\eta }_{{\mathcal{Q}}_{CQTLRP}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQTLIP}}\right)}\right)\right)$, then we have

(2). By supposition, we have ${\mathcal{Q}}_{CQTL-j},$ a collection CQRO2-TLVs and ${\mathcal{Q}}_{CQTL-j}^{\text{'}}$ is a slight substitution of ${\mathcal{Q}}_{CQTL-j}$, then

(3). By hypothesis, we have if ${\mathcal{Q}}_{CQTL-j}$ and ${\mathcal{Q}}_{CQTL-j}^{\text{'}}$ are any two collectionss of CQRO2-TLVs with the settings s.t. ${\mathcal{Q}}_{CQTL-j}\le {\mathcal{Q}}_{CQTL-j}^{\text{'}}$, then

Hence

(4). By hypothesis, if we have

a collection of the CQRO2-TLVs, ${\mathcal{Q}}_{CQTL-j}^{-} = min\left({\mathcal{Q}}_{CQTL-1}, {\mathcal{Q}}_{CQTL-2}, \dots, {\mathcal{Q}}_{CQTL-n}\right)$ and ${\mathcal{Q}}_{CQTL-j}^{+} = max\left({\mathcal{Q}}_{CQTL-1}, {\mathcal{Q}}_{CQTL-2}, \dots, {\mathcal{Q}}_{CQTL-n}\right)$, then by means of the properties (1) & (3), we have

Hence

Definition 5.4. For the family of CQRO2-TLVs ${\mathcal{Q}}_{QTL-j}\left(j = \mathrm{1, 2}, 3, \dots, n\right)$, the WGMSM operator is given by:

where ${K}_{SC} = \mathrm{1, 2}, \dots, n, {\alpha }_{SC-1}, {\alpha }_{SC-2}, \dots, {\alpha }_{SC-{K}_{SC}}\ge 0, \left({j}_{1}, {j}_{2}, \dots, {j}_{{K}_{SC}}\right)$ represents the K-tuple collection of $(\mathrm{1, 2}, .., n),$ and ${\mathrm{\Omega }}_{\mathcal{W}\mathcal{V}} = {\left({\mathrm{\Omega }}_{\mathcal{W}\mathcal{V}-1}, {\mathrm{\Omega }}_{\mathcal{W}\mathcal{V}-2}, \dots, {\mathrm{\Omega }}_{\mathcal{W}\mathcal{V}-n}\right)}^{T}$ with the condition $\sum _{j = 1}^{n}{\mathrm{\Omega }}_{\mathcal{W}\mathcal{V}-j} = 1$.

Theorem 5.5. For the family of CQRO2-TLVs

Proof. Straightforward.

Theorem 5.6. For the family of CQROF2-TLVs

(1). if ${\mathcal{Q}}_{CQTL-j} = {\mathcal{Q}}_{CQTL} = \left(\left({s}_{{S}_{LT}}, {\alpha }_{SC}\right), \left({\mu }_{{\mathcal{Q}}_{CQTLRP}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQTLIP}}\right)}, {\eta }_{{\mathcal{Q}}_{CQTLRP}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQTLIP}}\right)}\right)\right)$, then $WCQRO2-TLGM{SM}^{\left({K}_{SC}, {\alpha }_{SC-1}, {\alpha }_{SC-2}, \dots, {\alpha }_{SC-{K}_{SC}}\right)}\left({\mathcal{Q}}_{CQTL-1}, {\mathcal{Q}}_{CQTL-2}, \dots, {\mathcal{Q}}_{CQTL-n}\right) = {\mathcal{Q}}_{CQTL}$.

(2). if ${\mathcal{Q}}_{CQTL-j}$ is a collection CQRO2-TLVs & ${\mathcal{Q}}_{CQTL-j}^{\text{'}}$ is a replacement for ${\mathcal{Q}}_{CQTL-j}$, then

(3). if ${\mathcal{Q}}_{CQTL-j}$ and ${\mathcal{Q}}_{CQTL-j}^{\text{'}}$ are two collections of CQRO2-TLVs with the settings s.t. ${\mathcal{Q}}_{CQTL-j}\le {\mathcal{Q}}_{CQTL-j}^{\text{'}}$, then

(4). if ${\mathcal{Q}}_{CQTL-j} = \left(\left({s}_{{S}_{LT-j}}, {\alpha }_{SC-j}\right), \left({\mu }_{{\mathcal{Q}}_{CQTLRP-j}}{e}^{i2\pi \left({\mu }_{{\mathcal{Q}}_{CQTLIP-j}}\right)}, {\eta }_{{\mathcal{Q}}_{CQTLRP-j}}{e}^{i2\pi \left({\eta }_{{\mathcal{Q}}_{CQTLIP-j}}\right)}\right)\right)$ is a family of the CQRO2-TLVs, then

Proof. Straightforward.

6.   Three-way decisions based on the CQRO2-TLVs

We explored the LF based on CQRO2-TLVs and 3WD for CQRO2-TLVs, where the probability vector is represented by: ${A}_{AC} = \left\{{\chi }_{{P}_{AC}}, {\chi }_{{B}_{AC}}, {\chi }_{{N}_{AC}}\right\}$, ${\mathrm{\Omega }}_{S} = \left\{{\mathcal{F}}_{B}, \sim {\mathcal{F}}_{NB}\right\}$ and

$\mathcal{D} = \left\{\mathrm{Pr}\left({\mathcal{F}}_{B}\left|\right[ử]\right), \mathrm{Pr}\left({\sim \mathcal{F}}_{NB}\left|\right[ử]\right)\right\}$ with $\mathrm{Pr}\left({\mathcal{F}}_{B}\left|\right[ử]\right)+\mathrm{Pr}\left({\sim \mathcal{F}}_{NB}\left|\right[ử]\right) = 1$. Then, we have the following:

Step 1: By using the Eqs (6.1)–(6.3), we examine the LFs, which are stated below:

Step 2: By using the Eq (6.4), we aggregate the decision matrix which is construct by decision experts.

Step 3: By using the Eqs (6.5)–(6.7), we examine the expected losses ${Y}_{EL}\left({\chi }_{{j}_{AC}}\left|\right[ử]\right), j = P, B, N$ and the different actions are expressed below:

Step 4: By using the Eqs (6.8)–(6.10), we examine the expected values, which are stated below:

Moreover, we investigate the accuracy function, which is stated below:

Step 5: By using the Eqs (6.14)–(6.16), we examine the three-way decision rules, which are listed below:

Step 6: This ends the proof.

Example 6.1. Three decision experts $\left\{{ử}_{1}, {ử}_{2}, {ử}_{3}\right\}$ and ${D}_{DE-k}(k = \mathrm{1, 2}, 3)$ with weight vectors ${\mathrm{\Omega }}_{\mathcal{W}\mathcal{V}} = {\left(\mathrm{0.4, 0.35, 0.25}\right)}^{\boldsymbol{T}}$ are given. Then, $\mathrm{Pr}\left({\mathcal{F}}_{B}\left|\right[{ử}_{j}]\right) = 0.3, j = 1, 2, 3, 4$.

Step 1: By using the Eqs (6.1)–(6.3), we examine the LFs of the Tables 3, 4, and 5, which are stated below:

Step 2: Consider Eq (6.4), for ${K}_{SC} = 3$, and the values of ${\alpha }_{SC-1} = {\alpha }_{SC-2} = {\alpha }_{SC-3} = 1$, we have Table 6.

Step 3: By using the Eqs (6.5)–(6.7), we examine the expected losses ${Y}_{EL}\left({\chi }_{{j}_{AC}}\left|\right[ử]\right), j = P, B, N$, for ${\delta }_{B-j} = 0.4, j = \mathrm{1, 2}, 3$ and ${q}_{SC} = 1$, and the separate actions are expressed and discussed in Table 7.

Step 4: By using the Eqs (6.8)–(6.10), we examine the expected values, which are stated in Table 8.

When the predictable values are unsuccessful in catching the relationships amongst any two predictable losses, we discover the concepts of accuracy function, which is quantified in Table 9.

Step 5: By using the Eqs (6.11)–(6.13), we examine the three-way decision rules, as discussed in Table 10.

Step 6: This ends the proof.

The obtained result states that all these alternatives belong to positive opinions, which are ${P}_{AC-1}$. Furthermore, the comparison of the explored work with some existing work is discussed in Table 11.

From the above analysis, we conclude that all approaches have provided similar consequences and are exposed in Table 11; additionally, all alternatives belong to the +ve regions. The graphical interpretation for the information's score and the expected positive, boundary, and negative values of Table 8 and Table 9 are shown in Figure 4 and Figure 5, respectively.

In the explored work, if we choose the values of $q = 1, 2$ and the imaginary part is zero, then the explored work is reduced for intuitionistic 2-tuple and Pythagorean 2-tuple linguistic sets. Similarly, if we choose the values of $q =$1, 2, then the explored will be reduced for complex intuitionistic 2-tuple and complex Pythagorean 2-tuple linguistic sets. The presented approach is more powerful and more proficient than the existing ones, as given in ^{[34,35,36,37]}.

7.   Conclusions

We modified the notions of 3WD and DTRS in the environment of CQRO2-TLV and elaborated certain important properties. Moreover, GMSM is a dominant and more flexible method to determine the accuracy and dominancy of real life issues. Therefore, by considering the CQRO2-TL information and GMSM, we presented CQRO2-TLGMSM operator and the WCQRO2-TLGMSM operator, and demonstrated their effective properties. We also elaborated a q-rung orthopair 2-tuple linguistic DTRS model and discussed its applications. Furthermore, we discussed some important and well known properties of the defined aggregation operators like idempotency, commutativity, monotonicity, and boundedness. We discussed some examples to explain our proposed methods and techniques. To prove the authenticity, workability, effectiveness, and supremacy of our proposed methods, techniques, and notions, we initiated a comparative analysis and proved that our initiated notions are much better as compared to certain existing notions.

In the future, we will discuss the proposed notions for CQRO2-TLVSs in the framework of complex q-rung orthopair fuzzy sets ^[38], picture hesitant fuzzy sets ^[39,40], neutrosophic sets ^[41,42], and some more useful frameworks and notions, as given in ^{[43,44,45,46,47,48,49]}.

Acknowledgments

Researchers Supporting Project number (RSP-2021/244), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare no conflicts of interest about the publication of the research article.