Interval-Valued p,q-Rung Orthopair Fuzzy Exponential TODIM Approach and Its Application to Green Supplier Selection

Abstract

An interval-valued q-rung orthopair fuzzy set ($IV{q}^{r}OFS$) is a robust and esteemed model in managing imprecise information, utilizing a controllable parameter $q\ge 1$. Expanding its applicability, we propose a refinement to it, termed the interval-valued p,q-rung orthopair fuzzy set, denoted $IVp,q^{r}OFS$. This advancement enables the handling of scenarios where varied assessment levels for membership and non-membership grades are necessary, a challenge unaddressed by existing extensions of interval-valued intuitionistic fuzzy sets. We establish the fundamental mathematical operations for the $IVp,q^{r}OFS$ and explore their key properties. To compare interval-valued p,q-rung orthopair fuzzy ($IVp,q^{r}OF$) numbers, a novel score function is put forward and compared to the known $IV{q}^{r}OFS$-based score functions to determine its viability. Moreover, we present $IVp,q^{r}OF$ aggregation operators (AOs), accompanied by rigorous proof of their essential characteristics, including boundedness, monotonicity, and symmetry. This study introduces the interaction of $IVp,q^{r}OF$ criteria through the inter-criteria correlation (CRITIC) approach. This method determines criteria weights based on the proposed distance measurements and the score function. The CRITIC method is integrated with the exponential TODIM approach (exp-TODIM), resulting in an innovative decision-making framework that is less susceptible to parameter fluctuations and showcases a high stability. An illustrative example of selecting a suitable supplier is provided to demonstrate the practical application of the developed exp-TODIM approach. Finally, a sensitivity analysis is conducted to exhibit the method’s stability, and a comparative analysis is performed to elucidate its strengths and advantages.

1. Introduction

The process of decision making (DM) plays a vital part in various everyday scenarios, as it involves the systematic evaluation of alternatives by decision experts (DEs) in order to determine the optimal choice. During the early stages of DM’s development, DEs employed real numerical values to allocate to the assessed information. As the number of criteria or parameters in DM problems increases, so does their complexity. This complexity can result in experts assigning incorrect numerical values when evaluating the alternatives. The uncertainty and vagueness inherent in human judgments underscore the insufficiency of classical set theory. Thereby, Zadeh [1] introduced the theory of fuzzy sets (FSs) as an attempt to address imprecise data, allowing DEs to express their level of satisfaction/membership grade (MG) regarding the quality of an alternative within the bounded unit interval, specifically [0, 1]. Undoubtedly, FS theory has proven to be effective in addressing evaluations that are characterized by uncertainty. However, this theory was found to be insufficient in providing an explanation for the level of dissatisfaction/non-membership grade (NMG). In order to mitigate the limitations associated with FSs, Atanassov [2] initiated the paradigm of intuitionistic FSs (IFSs). This framework is designed to address situations where the necessity measure of the NMG does not necessarily originate from the MG. FSs have garnered significant attention from researchers who have examined their fundamental characteristics and illustrated their practical applications [3,4].

After establishing the independence of MGs and NMGs, numerous authors undertook the endeavor of expanding the permissible combinations of MGs and NMGs while ensuring mathematical tractability. Yager [5] and Senapati and Yager [6] investigated two novel types of sets that extend the concept of IFSs. These sets are referred to as Pythagorean fuzzy sets (PyFSs) and Fermatean fuzzy sets (FFSs). The effectiveness of their contribution relies on the application of powers, specifically two in PyFS or three in FFS, for the purpose of adjusting the assessments of MGs and NMGs. q-rung orthopair fuzzy sets ($q^{r}OFS$), as introduced by Yager [7], encapsulate a broader spectrum of situations that result in a satisfactory generalization. The previously mentioned extensions of FSs can be classified as specific instances of $q^{r}OFS$. In this classification, IFSs, PyFSs, and FFSs are assigned the values of one, two, and three, respectively, for the parameter q. Following the emergence of this influential concept, several scholars were drawn to the study of $q^{r}OFS$s and their applications in DM [8,9,10,11].

Interval depiction of fuzzy information offers DEs an additional and adaptable approach to modeling uncertainty. The proposal of the interval-valued fuzzy sets (IVFSs) was first coined by Gorzlczany [12]. The main inspiration for formulating this innovative concept is the inadequacy of conventional FSs in effectively representing linguistic expression within the framework of linguistic modeling of a given phenomenon. To date, many interval extensions for various frameworks have been laid out. Atanassov [13] introduced the concept of interval-valued IFSs (IVIFSs), outlining two functions represented by interval numbers: MGs and NMGs. It is stipulated that the combined total of their upper bounds must be confined within the unit interval. Peng and Yang [14] pioneered the paradigm of interval-valued PyFS theory as a logical progression of PyFSs. In study [15], Joshi et al. proposed the introduction of a novel hybrid model called interval-valued $q^{r}OFS$s ($IV{q}^{r}OFS$s). This model combines the concepts of IVFSs [12] with $q^{r}OFS$s. Wang et al. [16] incorporated the concept of $IV{q}^{r}OFS$s into aggregation operators (AOs) and demonstrated various valuable results and implications of their diagnosed methodologies. Gao and Xu [17] presented the concept of interval-valued q-rung orthopair fuzzy $\left(IV{q}^{r}OF\right)$ functions based on the theoretical foundation of $IV{q}^{r}OFS$s. They further examined the continuity, derivatives, and differentials of these functions. Subsequently, Garg [18] dispatched a novel measure of possibility degree between $IV{q}^{r}OFSs$ and then employed it in the context of multi-criteria group DM (MCGDM). Jin et al. [19] proposed a scientific risk assessment technique to mitigate the occurrence of failure under the background of $IV{q}^{r}OFS$s. Yang et al. [20] conducted a comprehensive analysis of existing research and put forth a set of continuous AOs for $IV{q}^{r}OFS$ data. In addition, significant advancements have been made in the field of researching and exploring $IV{q}^{r}OFS$s and their diverse range of applications across various domains.

However, certain instances and circumstances may necessitate the assessment with distinct term levels for the upper bounds of MGs $\overline{\Omega }$ and NMGs $\overline{\mho }$. The aforementioned $IV{q}^{r}OFS$ is unable to fulfill this requirement due to the fact that, in $IV{q}^{r}OFS$s, the term levels of $\overline{\Omega }$ and $\overline{\mho }$ are considered identical, i.e., $0\le {\overline{\Omega }}^q+{\overline{\mho }}^q\le 1$; $q\ge 1.$ For example, if we consider $\overline{\Omega }=0.7$ and $\overline{\mho }=0.8$, then clearly ${0.7}^2+{0.8}^2=1.13>1$. Therefore, we next check ${0.7}^3+{0.8}^3=0.855<1$. However, ${0.7}^3+{0.8}^2=0.983<1$. In order to more effectively represent the situation, it is advantageous to permit variations in the levels of $\overline{\Omega }$ and $\overline{\mho }$. Therefore, it is imperative to introduce a new fuzzy tool that incorporates the constraint $0\le {\overline{\Omega }}^p+{\overline{\mho }}^q\le 1;p,q\ge 1.$

MCGDM covers a widely recognized domain within the realm of decision-support mechanisms, encompassing various methodologies, such as the technique for order preference by similarity to ideal solutions (TOPSIS) method researched by Hwang and Yoon [21]; the vlsekriterijumska optimizacija i kompromisno resenje (VIKOR) method probed by Opricovic [22]; combinative distance-based assessment (CODAS) provided by Ghorabaee and their collaborators [23]; the elimination et choice translating reality (ELECTRE) method, developed by Benayoun, Roy, and Sussman [24]; and the tomada de decisao interativa e multicriterio (TODIM) method, pioneered by Gomes and Lima [25], have been widely explored in the literature. Among these different MCGDM methods, TODIM, which encounters DEs’ psychological behavior, has achieved a broad applicability. Divsalar et al. [26] introduced the TODIM approach based on the proposed Choquet integral for MCGDM in probabilistic hesitant FSs. Zhao et al. [27] proposed an extension of the TODIM approach to PyFSs in order to cope with MCGDM problems. Liu and Shen [28] put forth an enhanced Choquet-TODIM approach that integrates fuzzy measures and the TODIM method. In study [29], Wang et al. expanded the application of the TODIM approach to incorporate two-tuple linguistic neutrosophic FSs. However, until now, the TODIM technique remains an unresolved matter in addressing MCGDM problems within the context of $IV{q}^{r}OFS$s. In addition, the assignment of weights to criteria plays a substantial role in influencing the outcomes of evaluations for various schemes. Objective weighting approaches are utilized to extract criteria weight information from the data structure in the initial decision matrix. This approach mitigates the impact of subjective factors, such as the knowledge, experience, and preferences of different experts, on criteria sets. Consequently, the rationality of weight information acquisition is enhanced. The CRITIC approach [30] is widely recognized as a prominent objective weighting method due to its ability to incorporate both the contrast intensity of each criteria and the conflict relationship between criteria. Presently, the CRITIC method has been extensively utilized in various decision-making contexts [31,32]. Hence, in situations where the weights of the criteria are entirely unknown, this study incorporates the CRITIC approach into the TODIM model for the proposed FS in order to augment the rationality of the decision outcomes.

In addition to the aforementioned studies, several scholars have directed their attention towards enhancing the theory of $IV{q}^{r}OFS$s through the development of $IV{q}^{r}OFS$ score functions [33,34,35,36,37,38]. Initially, Ju et al. [33] introduced an $IV{q}^{r}OFS$ score function, employing it within an algorithm designed for ranking purposes. Rani and Mishra [34] contributed a novel score function intended to address instances of failure when comparing interval-valued fermatean fuzzy numbers using a weighted aggregated sum product assessment-based method. Moreover, Jeevaraj [35] introduced various score functions within the class of interval-valued fermatean fuzzy sets and compared different ranking methods by employing these proposed score functions, demonstrating their effectiveness. Gong and Ma [38] introduced a new scoring function and exact function by leveraging the principles of probability theory and the total probability formula, highlighting drawbacks within existing score functions. Rani et al. [36] further enhanced the interval-valued fermatean fuzzy score function and studied certain Einstein-operation-based operators along with their relevant characteristics. However, at times, these advancements resulted in inconsistencies with real-world applications. It was observed that the existing score functions often fail to adequately differentiate between pairs of $IV{q}^{r}OFS$ numbers in specific scenarios (refer to Section 3.3 for more details).

The motivating factors behind the proposed theory and methodology can be elucidated as follows:

☞

The existing $IV{q}^{r}OFS$ is founded on the principle that the qth sum of the upper bounds of the MG and NMG must be constrained by 1, i.e., $0\le {\overline{\Omega }}^q+{\overline{\mho }}^q\le 1$; $q\ge 1.$ To achieve acceptable outcomes, DEs must select the smallest integer q satisfying the inequality $0\le {\overline{\Omega }}^q+{\overline{\mho }}^q\le 1$ because, in the majority of problems, a greater value of q influences the results [39]. In practical applications, however, we may encounter scenarios where the aforementioned inequality holds for various powers of $\overline{\Omega }$ and $\overline{\mho }$. For example, if we consider $\overline{\Omega }=0.7$ and $\overline{\mho }=0.8$, then according to $IV{q}^{r}OFS$s, $q=3$ is the smallest integer that meets the condition $0\le {\overline{\Omega }}^q+{\overline{\mho }}^q\le 1$, but it is also fulfilled if we set the powers of $\overline{\Omega }$ and $\overline{\mho }$ to 3 and 2, respectively. Due to the limitation that the $IV{q}^{r}OFS$ only has one parameter, q, the specification of distinct powers for the MG and NMG is restricted. To address this constraint, it becomes imperative to augment the current $IV{q}^{r}OFS$ by incorporating an additional parameter.

☞

In $IV{q}^{r}OF$-based MCGDM problems, the score function is the central idea. However, currently available score functions [33,34,35,36,37,38] cannot distinguish between the $IV{q}^{r}OFSNs$ in numerous cases (refer to Section 3.3). Thus, it is essential to design an improved and capable score function for $IV{q}^{r}OFS$s.

☞

In recent years, the CRITIC method has been proven to be an efficient approach in MCGDM problems for weighting criteria. It determines objective weights by considering the differences and correlations between multiple criteria simultaneously [40]. Though several extensions of this strategy have been devised in the literature, its expansion to the $IV{q}^{r}OF$ setting is still missing.

☞

$IV{q}^{r}OF$ data ranking systems have come a long way from their inception. However, the psychological actions of experts have not been adequately explored, even though they reflect the inherent irrationality of humans. Most current approaches assume that DEs are purely rational and want to maximize their own interests, which overlooks important psychological elements like reference dependency and loss aversion [41]. In fact, specialists are limited in their ability to think rationally by a combination of psychological and cognitive factors [42]. To close this gap in solving $IV{q}^{r}OF$ MCGDM problems, the development of the TODIM approach is needed.

In light of the above viewpoints, this work makes the following contributions:

✍

To initiate a new fuzzy tool, namely the $IVp,q^{r}OFS$, and its theoretical foundations, including mathematical operations, distance measure, and their desirable characteristics.

✍

To formulate a novel score function with its pioneering axioms in order to conquer the drawbacks of the prevailing sore functions.

✍

To introduce some $IVp,q^{r}OF$ AOs and also verify their associated results.

✍

To lay down the CRITIC method to assess criteria weights utilizing the presented score function and divergence measure within the $IVp,q^{r}OF$ setting.

✍

To develop the $IVp,q^{r}OF$ TODIM strategy by integrating the described AOs, distance measure, and CRITIC approach for addressing MCGDM issues with unknown weight information.

✍

To present an application case of green supplier selection, together with parameter analyses and a comparative study, to exemplify the effectiveness and stability of the acquired results.

The remaining sections of this work are formatted as follows. Section 2 briefly covers the relevant definitions of IVqROFSs for the upcoming sections. Section 3 introduces the concept of $IVp,q^{r}OFS$s, along with their underlying theory, including basic operational laws and the comparison strategy. Section 4 defines $IVp,q^{r}OF$ AOs and delivers their well-known results. In addition, Section 4 describes $IVp,q^{r}OF$ distance measures and their requisite properties. Section 5 first establishes the CRITIC approach and then builds a novel exp-TODIM method that solves MCGDM problems with data in $IVp,q^{r}OF$ format. Section 6 provides a case study of feasible carbon storage site selection to exemplify the application of the proposed method. Further, a sensitivity analysis of various ratings of the decision strategic coefficient is offered to evaluate the presented method’s performance. The comparative analysis and concluding remarks are described in Section 7 and Section 8, respectively.

2. Basic Concepts

Before we begin our introductions, let us recap some related topics ahead of them.

Definition 1

([33]). Let Ł be a universe of discourse. An $IV{q}^{r}OFS$ $S$ on Ł is defined as

$$S=\left\{\left(g,\mu \left(g\right),v\left(g\right)\right)|g\in S\right\},$$

(1)

where $\mu \left(g\right)=\left[\underline{\Omega }\left(g\right),\overline{\Omega }\left(g\right)\right]\subseteq \left[0,1\right]$ denotes the interval of the MG and $v\left(g\right)=\left[\underline{\mho }\left(g\right),\overline{\mho }\left(g\right)\right]\subseteq \left[0,1\right]$ denotes the interval of the NMG of $g\in S$ such that $0\le {\left(\overline{\Omega }\left(g\right)\right)}^q+{\left(\overline{\mho }\left(g\right)\right)}^q\le 1$; $q\ge 1.$

Remark 1.

If $\underline{\Omega }\left(g\right)=\overline{\Omega }\left(g\right)$ and $\underline{\mho }\left(g\right)=\overline{\mho }\left(g\right)$, an $IV{q}^{r}OFS$ will reduce to a $q^{r}OFS$.

For convenience, $\left(\left[\underline{\Omega }\left(g\right),\overline{\Omega }\left(g\right)\right],\left[\underline{\mho }\left(g\right),\overline{\mho }\left(g\right)\right]\right)$ is called the $IV{q}^{r}OF$ number ($IV{q}^{r}OFN$) and symbolized by $\mathfrak{N}=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right).$

Definition 2.

Let $\mathfrak{N}=\left(\left[{\underline{\Omega }}_{⟆},{\overline{\Omega }}_{⟆}\right],\left[{\underline{\mho }}_{⟆},{\overline{\mho }}_{⟆}\right]\right)$ be an $IV{q}^{r}OFN$; its score functions formulated by Ju et al. [33] is given in Equation (2)

$$S_J\left(\mathfrak{N}\right)=\frac{1}{4}\left(2+{\underline{\Omega }}_{}^q+{\overline{\Omega }}_{}^q-{\underline{\mho }}_{}^q-{\overline{\mho }}_{}^q\right),$$

(2)

Definition 3

([33]). Let $\mathfrak{N}=\left(\left[{\underline{\Omega }}_{},{\overline{\Omega }}_{}\right],\left[{\underline{\mho }}_{},{\overline{\mho }}_{}\right]\right)$ be an $IV{q}^{r}OFN$; its accuracy function is given by

$$A\left(\mathfrak{N}\right)=\frac{1}{2}\left({\underline{\Omega }}_{}^q+{\overline{\Omega }}_{}^q+{\underline{\mho }}_{}^q+{\overline{\mho }}_{}^q\right).$$

(3)

Based on the score and accuracy function of $IV{q}^{r}OFNs$, Ju et al. [33] explored the following laws to compare them.

Definition 4.

Let ${\mathfrak{N}}_1=\left(\left[{\underline{\Omega }}_1,{\overline{\Omega }}_1\right],\left[{\underline{\mho }}_1,{\overline{\mho }}_1\right]\right)$ and ${\mathfrak{N}}_2=\left(\left[{\underline{\Omega }}_2,{\overline{\Omega }}_2\right],\left[{\underline{\mho }}_2,{\overline{\mho }}_2\right]\right)$ be two $IV{q}^{r}OFNs$, and let $S\left({\mathfrak{N}}_1\right)\left(A\left({\mathfrak{N}}_1\right)\right)$ and $S\left({\mathfrak{N}}_2\right)\left(A\left({\mathfrak{N}}_2\right)\right)$ be the score values (accuracy values) of ${\mathfrak{N}}_1$ and ${\mathfrak{N}}_2$, respectively. Then,

1. 

If $S\left({\mathfrak{N}}_1\right)>S\left({\mathfrak{N}}_2\right),$ then ${\mathfrak{N}}_1$ is larger than ${\mathfrak{N}}_2$, marked by ${\mathfrak{N}}_1>{\mathfrak{N}}_2;$

2. 

If $S\left({\mathfrak{N}}_1\right)<S\left({\mathfrak{N}}_2\right),$ then ${\mathfrak{N}}_1$ is smaller than ${\mathfrak{N}}_2$, marked by ${\mathfrak{N}}_1<{\mathfrak{N}}_2;$

3. 

If $S\left({\mathfrak{N}}_1\right)=S\left({\mathfrak{N}}_2\right),$ then we need to compare their accuracies:

i. 

If $A\left({\mathfrak{N}}_1\right)>A\left({\mathfrak{N}}_2\right),$ then ${\mathfrak{N}}_1$ is larger than ${\mathfrak{N}}_2$, marked by ${\mathfrak{N}}_1>{\mathfrak{N}}_2;$

ii. 

If $A\left({\mathfrak{N}}_1\right)<A\left({\mathfrak{N}}_2\right),$ then ${\mathfrak{N}}_1$ is smaller than ${\mathfrak{N}}_2$, marked by ${\mathfrak{N}}_1<{\mathfrak{N}}_2;$

iii. 

If $A\left({\mathfrak{N}}_1\right)=A\left({\mathfrak{N}}_2\right),$ then ${\mathfrak{N}}_1$ is equal to ${\mathfrak{N}}_2$, marked by ${\mathfrak{N}}_1={\mathfrak{N}}_2.$

Definition 5

([33]). Let ${\mathfrak{N}}_1=\left(\left[{\underline{\Omega }}_1,{\overline{\Omega }}_1\right],\left[{\underline{\mho }}_1,{\overline{\mho }}_1\right]\right)$ and ${\mathfrak{N}}_2=\left(\left[{\underline{\Omega }}_2,{\overline{\Omega }}_2\right],\left[{\underline{\mho }}_2,{\overline{\mho }}_2\right]\right)$ be two $IV{q}^{r}OFNs$, $\left(⟆\ge 0,q\ge 1\right)$. Then, the basic operations on them are listed as:

1. 

${\mathfrak{N}}_1⨁{\mathfrak{N}}_2=\left(\left[\sqrt[q]{{\underline{\Omega }}_1^q+{\underline{\Omega }}_2^q-{\underline{\Omega }}_1^q{\underline{\Omega }}_2^q},\sqrt[q]{{\overline{\Omega }}_1^q+{\overline{\Omega }}_2^q-{\overline{\Omega }}_1^q{\overline{\Omega }}_2^q}\right],\left[{\underline{\mho }}_1{\underline{\mho }}_2,{\overline{\mho }}_1{\overline{\mho }}_2\right]\right)$;

2. 

${\mathfrak{N}}_1⨂{\mathfrak{N}}_2=\left(\left[{\underline{\Omega }}_1{\underline{\Omega }}_2,{\overline{\Omega }}_1{\overline{\Omega }}_2\right],\left[\sqrt[q]{{\underline{\mho }}_1^q+{\underline{\mho }}_2^q-{\underline{\mho }}_1^q{\underline{\mho }}_2^q},\sqrt[q]{{\overline{\mho }}_1^q+{\overline{\mho }}_2^q-{\overline{\mho }}_1^q{\overline{\mho }}_2^q}\right]\right)$;

3. 

${\mathfrak{N}}_1^{⟆}=\left(\left[{\left({\underline{\Omega }}_1\right)}^{⟆},{\left({\overline{\Omega }}_1\right)}^{⟆}\right],\left[\sqrt[q]{1-{\left(1-{\underline{\mho }}_1^q\right)}^{⟆}},\sqrt[q]{1-{\left(1-{\overline{\mho }}_1^q\right)}^{⟆}}\right]\right)$;

4. 

$⟆{\mathfrak{N}}_1=\left(\left[\sqrt[q]{1-{\left(1-{\underline{\Omega }}_1^q\right)}^{⟆}},\sqrt[q]{1-{\left(1-{\overline{\Omega }}_1^q\right)}^{⟆}}\right],\left[{\left({\underline{\mho }}_1\right)}^{⟆},{\left({\overline{\mho }}_1\right)}^{⟆}\right]\right)$;

5. 

${\mathfrak{N}}_1^c=\left(\left[{\underline{\mho }}_1,{\overline{\mho }}_1\right],\left[{\underline{\Omega }}_1,{\overline{\Omega }}_1\right]\right).$

3. Interval-Valued p,q-Rung Orthopair Fuzzy Sets

3.1. Notion of $IVp,q^{r}OFS$s

Definition 6.

Let Ł be a universe of discourse. An $IVp,q^{r}OFS$ $\mathtt{I}$ on Ł is defined as

$$I=\left\{\left(g,\mu \left(g\right),v\left(g\right)\right)g|\in I\right\},$$

(4)

where $\mu \left(g\right)=\left[\underline{\Omega }\left(g\right),\overline{\Omega }\left(g\right)\right]\subseteq \left[0,1\right]$ denotes the interval of the MG and $v\left(g\right)=\left[\underline{\mho }\left(g\right),\overline{\mho }\left(g\right)\right]\subseteq \left[0,1\right]$ denotes the interval of the NMG of $g\in I$ such that $0\le {\left(\overline{\Omega }\left(g\right)\right)}^p+{\left(\overline{\mho }\left(g\right)\right)}^q\le 1$; $p,q\ge 1.$

The degree of uncertainty π(g) can be disclosed as follows:

$$\pi \left(g\right)=\left[\underline{\pi }\left(g\right),\overline{\pi }\left(g\right)\right]=\left[\sqrt[\ell ]{1-\left({\left(\overline{\Omega }\left(g\right)\right)}^p+{\left(\overline{\mho }\left(g\right)\right)}^q\right)},\sqrt[\ell ]{1-\left({\left(\underline{\Omega }\left(g\right)\right)}^p+{\left(\underline{\mho }\left(g\right)\right)}^q\right)}\right],$$

(5)

where ℓ is the least common multiple (LCM) of p and q.

For clarity, $\left(\left[\underline{\mho }\left(g\right),\overline{\mho }\left(g\right)\right],\left[\underline{\mho }\left(g\right),\overline{\mho }\left(g\right)\right]\right)$ is called the $IVp,q^{r}OF$ number $\left(IVp,q^{r}OFN\right)$ and denoted as $ß=\left(\left[\underline{\mho },\overline{\mho }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$.

Remark 2.

Consider the case where we have to find the minimum value of $p,q\ge 1$ for a given $IVp,q^{r}OFN$ $\left(\left[\underline{\mho },\overline{\mho }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$ so that $0\le {\overline{\Omega }}^p+{\overline{\mho }}^q\le 1$. It is always possible to find a unique solution to these problems using some iterative computing techniques, even though they have no closed-form solution. We shall refer to the minimum values of p and q satisfying $0\le {\overline{\Omega }}^p+{\overline{\mho }}^q\le 1$ as the $p,q$-niche of $\left(\left[\underline{\mho },\overline{\mho }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$. We note that if $p_1,q_1$ is the $p,q$-niche of $\left(\left[\underline{\mho },\overline{\mho }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$, then $\left(\left[\underline{\mho },\overline{\mho }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$ is valid for all $p\ge p_1$, and $q\ge q_1$.

Remark 3.

Consider the case where ${\overline{\Omega }}^p+{\overline{\mho }}^q>1,$ and we have to find $p,q$-niche. Suppose $p^{*}$ and $q^{*}$ are the minimum values for which ${\overline{\Omega }}^{p^{*}}+{\overline{\mho }}^q\le 1,$ and ${\overline{\Omega }}^p+{\overline{\mho }}^{q^{*}}\le 1.$ In such situations, we should refer to the minimum value that meets the condition with a larger grade.

Suppose $Ł=\left\{g_1,g_2,\dots ,g_n\right\}$ is a given data set and $\mathbb{F}$ is some fuzzy concept. Suppose a DE expresses their preference as an $IVp,q^{r}OFN$ $\left(\left[{\underline{\mho }}_j,{\overline{\mho }}_j\right],\left[{\underline{\mho }}_j,{\overline{\mho }}_j\right]\right)$ for each g ∈ Ł. Now, the challenge is how to estimate the correct values of p and q to represent the information appropriately. Here, we can proceed as follows:

(I). For each $IVp,q^{r}OFN$ $\left(\left[{\underline{\mho }}_j,{\overline{\mho }}_j\right],\left[{\underline{\mho }}_j,{\overline{\mho }}_j\right]\right)$, determine its $p,q$-niche, say $p_j,q_j$.

(II). Determine the $p^{*},q^{*}$-niche such that $p^{*}=\underset{j}{max}\left\{p_j\right\}$ and $q^{*}=\underset{j}{max}\left\{q_j\right\}$.

(III). Represent $I$ as an $IV{p}^{*},{q^{*}}^{r}OFS$.

Deduction: Some specific forms of $IVp,q^{r}OFS$s are set forth as follows:

(i). Definition 6 degenerates to an $IV{q}^{r}OFS$ [15] if we set $p=q$.

(ii). Definition 6 degenerates to an $IVFFS$ [34] if we set $p=q=3$.

(iii). Definition 6 degenerates to an $IVPyFS$ [14] if we set $p=q=2$.

(iv). Definition 6 degenerates to an $IVIFS$ [13] if we set $p=q=1$.

(v). Definition 6 degenerates to a $p,q^{r}OFS$ [43] if we set $\overline{\Omega }=\underline{\Omega },$ and $\overline{\mho }=\underline{\mho }$.

(vi). Definition 6 degenerates to an $q^{r}OFS$ [7] if we set $\overline{\Omega }=\underline{\Omega },$ $\overline{\mho }=\underline{\mho }$ and $p=q$.

(vii). Definition 6 degenerates to an $FFS$ [44] if we set $\overline{\Omega }=\underline{\Omega },$ $\overline{\mho }=\underline{\mho }$ and $p=q=3$.

(viii). Definition 6 degenerates to a $PyFS$ [5] if we set $\overline{\Omega }=\underline{\Omega },$ $\overline{\mho }=\underline{\mho }$ and $p=q=2$.

(ix). Definition 6 degenerates to an $IFS$ [2] if we set $\overline{\Omega }=\underline{\Omega },$ $\overline{\mho }=\underline{\mho }$ and $p=q=1$.

Thus, the proposed $IVp,q^{r}OFS$ is the generalization of an $IV{q}^{r}OFS$ [15], $IVFFS$ [34], $IVPyFS$ [14], $IVIFS$ [13], $p,q^{r}OFS$ [43], $q^{r}OFS$ [7], $FFS$ [44]$PyFS$ [5] and IFS [2].

Definition 7.

Let ${ß}_1=\left(\left[{\underline{\Omega }}_1,{\overline{\Omega }}_1\right],\left[{\underline{\mho }}_1,{\overline{\mho }}_1\right]\right)$ and ${ß}_2=\left(\left[{\underline{\Omega }}_2,{\overline{\Omega }}_2\right],\left[{\underline{\mho }}_2,{\overline{\mho }}_2\right]\right)$ be any two $IVp,q^{r}OFNs$. Then, the relation between these two $IVp,q^{r}OFNs$ is given in the following:

1. 

${ß}_1={ß}_2$ if ${\underline{\Omega }}_1={\underline{\Omega }}_2$, ${\overline{\Omega }}_1={\overline{\Omega }}_2$, ${\underline{\mho }}_1={\underline{\mho }}_2$ and ${\overline{\Omega }}_1={\overline{\Omega }}_2$.

2. 

${ß}_1\subseteq {ß}_2$ if ${\underline{\Omega }}_1\le {\underline{\Omega }}_2$, ${\overline{\Omega }}_1\le {\overline{\Omega }}_2$, ${\underline{\mho }}_1\ge {\underline{\mho }}_2$ and ${\overline{\Omega }}_1\ge {\overline{\Omega }}_2$.

3.2. Operational Laws of $IVp,q^{r}OFNs$

Definition 8.

Let ${ß}_1=\left(\left[{\underline{\Omega }}_1,{\overline{\Omega }}_1\right],\left[{\underline{\mho }}_1,{\overline{\mho }}_1\right]\right)$ and ${ß}_2=\left(\left[{\underline{\Omega }}_2,{\overline{\Omega }}_2\right],\left[{\underline{\mho }}_2,{\overline{\mho }}_2\right]\right)$ be two $IVp,q^{r}OFNs$, $\left(⟆\ge 0,p,q\ge 1\right)$. Then, the basic operations on them are listed as:

1. 

${ß}_1⨁{ß}_2=\left(\left[\sqrt[p^{*}]{{\underline{\Omega }}_1^{p^{*}}+{\underline{\Omega }}_2^{p^{*}}-{\underline{\Omega }}_1^{p^{*}}{\underline{\Omega }}_2^{p^{*}}},\sqrt[p^{*}]{{\overline{\Omega }}_1^{p^{*}}+{\overline{\Omega }}_2^{p^{*}}-{\overline{\Omega }}_1^{p^{*}}{\overline{\Omega }}_2^{p^{*}}}\right],\left[{\underline{\mho }}_1{\underline{\mho }}_2,{\overline{\mho }}_1{\overline{\mho }}_2\right]\right)$;

2. 

${ß}_1⨂{ß}_2=\left(\left[{\underline{\Omega }}_1{\underline{\Omega }}_2,{\overline{\Omega }}_1{\overline{\Omega }}_2\right],\left[\sqrt[q^{*}]{{\underline{\mho }}_1^{q^{*}}+{\underline{\mho }}_2^{q^{*}}-{\underline{\mho }}_1^{q^{*}}{\underline{\mho }}_2^{q^{*}}},\sqrt[q^{*}]{{\overline{\mho }}_1^{q^{*}}+{\overline{\mho }}_2^{q^{*}}-{\overline{\mho }}_1^{q^{*}}{\overline{\mho }}_2^{q^{*}}}\right]\right)$;

3. 

${ß}_1^{⟆}=\left(\left[{\left({\underline{\Omega }}_1\right)}^{⟆},{\left({\overline{\Omega }}_1\right)}^{⟆}\right],\left[\sqrt[q^{*}]{1-{\left(1-{\underline{\mho }}_1^{q^{*}}\right)}^{⟆}},\sqrt[q^{*}]{1-{\left(1-{\overline{\mho }}_1^{q^{*}}\right)}^{⟆}}\right]\right)$;

4. 

$⟆{ß}_1=\left(\left[\sqrt[p^{*}]{1-{\left(1-{\underline{\Omega }}_1^{p^{*}}\right)}^{⟆}},\sqrt[p^{*}]{1-{\left(1-{\overline{\Omega }}_1^{p^{*}}\right)}^{⟆}}\right],\left[{\left({\underline{\mho }}_1\right)}^{⟆},{\left({\overline{\mho }}_1\right)}^{⟆}\right]\right)$;

5. 

${ß}_1^c=\left(\left[{\underline{\mho }}_1,{\overline{\mho }}_1\right],\left[{\underline{\Omega }}_1,{\overline{\Omega }}_1\right]\right).$

Theorem 1.

Let ${ß}_j=\left(\left[{\underline{\Omega }}_j,{\overline{\Omega }}_j\right],\left[{\underline{\mho }}_j,{\overline{\mho }}_j\right]\right)(j=1,2,3)$ be any three $IVp,q^{r}OFNs$, and $⟆,{⟆}_1,{⟆}_2>0$. Then,

1. 

${ß}_1\oplus {ß}_2={ß}_2\oplus {ß}_1$;

2. 

${ß}_1\otimes {ß}_2={ß}_2\otimes {ß}_1$;

3. 

$\left({ß}_1\oplus {ß}_2\right)\oplus {ß}_3={ß}_1\oplus \left({ß}_2\oplus {ß}_3\right)$;

4. 

$\left({ß}_1\otimes {ß}_2\right)\otimes {ß}_3={ß}_1\otimes \left({ß}_2\otimes {ß}_3\right)$;

5. 

$⟆\left({ß}_1\oplus {ß}_2\right)=⟆{ß}_1\oplus ⟆{ß}_2$;

6. 

${\left({ß}_1\otimes {ß}_2\right)}^{⟆}={ß}_1^{⟆}\otimes {ß}_2^{⟆}$;

7. 

${⟆}_1{ß}_1\oplus {⟆}_2{ß}_1=\left({⟆}_1+{⟆}_2\right){ß}_1$;

8. 

${ß}_1^{{⟆}_1}\otimes {ß}_1^{{⟆}_2}={ß}_1^{{⟆}_1+{⟆}_2}$;

9. 

$\left({⟆}_1{⟆}_2\right){ß}_1={⟆}_1\left({⟆}_2{ß}_1\right)$.

Proof. 

We only verify components 1, 3, 5, 7, and 9, and the rest follow suit.

(i). As ${ß}_1⨁{ß}_2=\left(\left[\sqrt[p^{*}]{{\underline{\Omega }}_1^{p^{*}}+{\underline{\Omega }}_2^{p^{*}}-{\underline{\Omega }}_1^{p^{*}}{\underline{\Omega }}_2^{p^{*}}},\sqrt[p^{*}]{{\overline{\Omega }}_1^{p^{*}}+{\overline{\Omega }}_2^{p^{*}}-{\overline{\Omega }}_1^{p^{*}}{\overline{\Omega }}_2^{p^{*}}}\right],\left[{\underline{\mho }}_1{\underline{\mho }}_2,{\overline{\mho }}_1{\overline{\mho }}_2\right]\right)=\left(\left[\sqrt[p^{*}]{{\underline{\Omega }}_2^{p^{*}}+{\underline{\Omega }}_1^{p^{*}}-{\underline{\Omega }}_2^{p^{*}}{\underline{\Omega }}_1^{p^{*}}},\sqrt[p^{*}]{{\overline{\Omega }}_2^{p^{*}}+{\overline{\Omega }}_1^{p^{*}}-{\overline{\Omega }}_2^{p^{*}}{\overline{\Omega }}_1^{p^{*}}}\right],\left[{\underline{\mho }}_2{\underline{\mho }}_1,{\overline{\mho }}_2{\overline{\mho }}_1\right]\right)={ß}_2⨁{ß}_1.$

(iii). $\left({ß}_1\oplus {ß}_2\right)\oplus {ß}_3=\left(\left[\sqrt[p^{*}]{{\underline{\Omega }}_1^{p^{*}}+{\underline{\Omega }}_2^{p^{*}}-{\underline{\Omega }}_1^{p^{*}}{\underline{\Omega }}_2^{p^{*}}},\sqrt[p^{*}]{{\overline{\Omega }}_1^{p^{*}}+{\overline{\Omega }}_2^{p^{*}}-{\overline{\Omega }}_1^{p^{*}}{\overline{\Omega }}_2^{p^{*}}}\right],\left[{\underline{\mho }}_1{\underline{\mho }}_2,{\overline{\mho }}_1{\overline{\mho }}_2\right]\right)$ $\oplus \left(\left[{\underline{\Omega }}_3,{\overline{\Omega }}_3\right],\left[{\underline{\mho }}_3,{\overline{\mho }}_3\right]\right)=\left(\begin{array}{c}\left[\begin{array}{c}\sqrt[p^{*}]{\left({\underline{\Omega }}_1^{p^{*}}+{\underline{\Omega }}_2^{p^{*}}-{\underline{\Omega }}_1^{p^{*}}{\underline{\Omega }}_2^{p^{*}}\right)+{\underline{\Omega }}_3^{p^{*}}-\left({\underline{\Omega }}_1^{p^{*}}+{\underline{\Omega }}_2^{p^{*}}-{\underline{\Omega }}_1^{p^{*}}{\underline{\Omega }}_2^{p^{*}}\right){\underline{\Omega }}_3^{p^{*}}},\\ \sqrt[p^{*}]{\left({\overline{\Omega }}_1^{p^{*}}+{\overline{\Omega }}_2^{p^{*}}-{\overline{\Omega }}_1^{p^{*}}{\overline{\Omega }}_2^{p^{*}}\right)+{\overline{\Omega }}_3^{p^{*}}-\left({\overline{\Omega }}_1^{p^{*}}+{\overline{\Omega }}_2^{p^{*}}-{\overline{\Omega }}_1^{p^{*}}{\overline{\Omega }}_2^{p^{*}}\right){\overline{\Omega }}_3^{p^{*}}}\end{array}\right],\\ \left[\left({\underline{\mho }}_1{\underline{\mho }}_2\right){\underline{\mho }}_3,\left({\overline{\mho }}_1{\overline{\mho }}_2\right){\overline{\mho }}_3\right]\end{array}\right)=\left(\begin{array}{c}\left[\begin{array}{c}\sqrt[p^{*}]{{\underline{\Omega }}_1^{p^{*}}+\left({\underline{\Omega }}_2^{p^{*}}+{\underline{\Omega }}_3^{p^{*}}-{\underline{\Omega }}_2^{p^{*}}{\underline{\Omega }}_3^{p^{*}}\right)-{\underline{\Omega }}_1^{p^{*}}\left({\underline{\Omega }}_2^{p^{*}}+{\underline{\Omega }}_3^{p^{*}}-{\underline{\Omega }}_2^{p^{*}}{\underline{\Omega }}_3^{p^{*}}\right)},\\ \sqrt[p^{*}]{{\overline{\Omega }}_1^{p^{*}}+\left({\overline{\Omega }}_2^{p^{*}}+{\overline{\Omega }}_3^{p^{*}}-{\overline{\Omega }}_2^{p^{*}}{\overline{\Omega }}_3^{p^{*}}\right)-{\overline{\Omega }}_1^{p^{*}}\left({\overline{\Omega }}_2^{p^{*}}+{\overline{\Omega }}_3^{p^{*}}-{\overline{\Omega }}_2^{p^{*}}{\overline{\Omega }}_3^{p^{*}}\right)}\end{array}\right],\\ \left[{\underline{\mho }}_1\left({\underline{\mho }}_2{\underline{\mho }}_3\right),{\overline{\mho }}_1\left({\overline{\mho }}_2{\overline{\mho }}_3\right)\right]\end{array}\right)={ß}_1\oplus \left({ß}_2\oplus {ß}_3\right).$

(v). $⟆\left({ß}_1\oplus {ß}_2\right)=⟆\left(\left(\left[\sqrt[p^{*}]{{\underline{\Omega }}_1^{p^{*}}+{\underline{\Omega }}_2^{p^{*}}-{\underline{\Omega }}_1^{p^{*}}{\underline{\Omega }}_2^{p^{*}}},\sqrt[p^{*}]{{\overline{\Omega }}_1^{p^{*}}+{\overline{\Omega }}_2^{p^{*}}-{\overline{\Omega }}_1^{p^{*}}{\overline{\Omega }}_2^{p^{*}}}\right],\left[{\underline{\mho }}_1{\underline{\mho }}_2,{\overline{\mho }}_1{\overline{\mho }}_2\right]\right)\right)=\left(\left[\sqrt[p^{*}]{1-{\left(1-{\underline{\Omega }}_1^{p^{*}}-{\underline{\Omega }}_2^{p^{*}}+{\underline{\Omega }}_1^{p^{*}}{\underline{\Omega }}_2^{p^{*}}\right)}^{⟆}},\sqrt[p^{*}]{1-{\left(1-{\overline{\Omega }}_1^{p^{*}}-{\overline{\Omega }}_2^{p^{*}}+{\overline{\Omega }}_1^{p^{*}}{\overline{\Omega }}_2^{p^{*}}\right)}^{⟆}}\right],\left[{\left({\underline{\mho }}_1{\underline{\mho }}_2\right)}^{⟆},{\left({\overline{\mho }}_1{\overline{\mho }}_2\right)}^{⟆}\right]\right)$

$⟆{ß}_1\oplus ⟆{ß}_2=\left(\left[\sqrt[p^{*}]{1-{\left(1-{\underline{\Omega }}_1^{p^{*}}\right)}^{⟆}},\sqrt[p^{*}]{1-{\left(1-{\overline{\Omega }}_1^{p^{*}}\right)}^{⟆}}\right],\left[{\left({\underline{\mho }}_1\right)}^{⟆},{\left({\overline{\mho }}_1\right)}^{⟆}\right]\right)\oplus \left(\left[\sqrt[p^{*}]{1-{\left(1-{\underline{\Omega }}_2^{p^{*}}\right)}^{⟆}},\sqrt[p^{*}]{1-{\left(1-{\overline{\Omega }}_2^{p^{*}}\right)}^{⟆}}\right],\left[{\left({\underline{\mho }}_2\right)}^{⟆},{\left({\overline{\mho }}_2\right)}^{⟆}\right]\right)=\left(\left[\sqrt[p^{*}]{1-{\left(1-{\underline{\Omega }}_1^{p^{*}}-{\underline{\Omega }}_2^{p^{*}}+{\underline{\Omega }}_1^{p^{*}}{\underline{\Omega }}_2^{p^{*}}\right)}^{⟆}},\sqrt[p^{*}]{1-{\left(1-{\overline{\Omega }}_1^{p^{*}}-{\overline{\Omega }}_2^{p^{*}}+{\overline{\Omega }}_1^{p^{*}}{\overline{\Omega }}_2^{p^{*}}\right)}^{⟆}}\right],\left[{\left({\underline{\mho }}_1{\underline{\mho }}_2\right)}^{⟆},{\left({\overline{\mho }}_1{\overline{\mho }}_2\right)}^{⟆}\right]\right)=⟆\left({ß}_1\oplus {ß}_2\right).$

(vii). ${⟆}_1{ß}_1\oplus {⟆}_2{ß}_1=\left(\left[\sqrt[p^{*}]{1-{\left(1-{\underline{\Omega }}_1^{p^{*}}\right)}^{{⟆}_1}},\sqrt[p^{*}]{1-{\left(1-{\overline{\Omega }}_1^{p^{*}}\right)}^{{⟆}_1}}\right],\left[{\left({\underline{\mho }}_1\right)}^{{⟆}_1},{\left({\overline{\mho }}_1\right)}^{{⟆}_1}\right]\right)\oplus \left(\left[\sqrt[p^{*}]{1-{\left(1-{\underline{\Omega }}_1^{p^{*}}\right)}^{{⟆}_2}},\sqrt[p^{*}]{1-{\left(1-{\overline{\Omega }}_1^{p^{*}}\right)}^{{⟆}_2}}\right],\left[{\left({\underline{\mho }}_1\right)}^{{⟆}_2},{\left({\overline{\mho }}_1\right)}^{{⟆}_2}\right]\right)=\left(\left[\sqrt[p^{*}]{1-{\left(1-{\underline{\Omega }}_1^{p^{*}}\right)}^{\left({⟆}_1+{⟆}_2\right)}},\sqrt[p^{*}]{1-{\left(1-{\overline{\Omega }}_1^{p^{*}}\right)}^{\left({⟆}_1+{⟆}_2\right)}}\right],\left[{\left({\underline{\mho }}_1\right)}^{\left({⟆}_1+{⟆}_2\right)},{\left({\overline{\mho }}_1\right)}^{\left({⟆}_1+{⟆}_2\right)}\right]\right)=\left({⟆}_1+{⟆}_2\right){ß}_1.$

(ix). $\left({⟆}_1{⟆}_2\right){ß}_1=\left(\left[\sqrt[p^{*}]{1-{\left(1-{\underline{\Omega }}_1^{p^{*}}\right)}^{\left({⟆}_1{⟆}_2\right)}},\sqrt[p^{*}]{1-{\left(1-{\overline{\Omega }}_1^{p^{*}}\right)}^{\left({⟆}_1{⟆}_2\right)}}\right],\left[{\left({\underline{\mho }}_1\right)}^{\left({⟆}_1{⟆}_2\right)},{\left({\overline{\mho }}_1\right)}^{\left({⟆}_1{⟆}_2\right)}\right]\right)$

${⟆}_1\left({⟆}_2{ß}_1\right)=\left(\left[\sqrt[p^{*}]{1-{\left(1-{\underline{\Omega }}_1^{p^{*}}\right)}^{\left({⟆}_2\right)}},\sqrt[p^{*}]{1-{\left(1-{\overline{\Omega }}_1^{p^{*}}\right)}^{\left({⟆}_2\right)}}\right],\left[{\left({\underline{\mho }}_1\right)}^{\left({⟆}_2\right)},{\left({\overline{\mho }}_1\right)}^{\left({⟆}_2\right)}\right]\right)=\left(\left[\sqrt[p^{*}]{1-{\left(1-{\underline{\Omega }}_1^{p^{*}}\right)}^{\left({⟆}_2{⟆}_1\right)}},\sqrt[p^{*}]{1-{\left(1-{\overline{\Omega }}_1^{p^{*}}\right)}^{\left({⟆}_2{⟆}_1\right)}}\right],\left[{\left({\underline{\mho }}_1\right)}^{\left({⟆}_2{⟆}_1\right)},{\left({\overline{\mho }}_1\right)}^{\left({⟆}_2{⟆}_1\right)}\right]\right)=\left({⟆}_1{⟆}_2\right){ß}_1.$□

3.3. Comparison Rule of $IVp,q^{r}OFNs$

The present part initially goes through the existing score functions for scoring arbitrary $IV{q}^{r}OFNs$ and explores some of their counter-intuitive instances. Furthermore, a novel score function for $IVp,q^{r}OFNs$ with enviable characteristics is proposed to overcome the inadequacies of existing functions.

Definition 9

([33]). Let $\mathfrak{N}=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$ be an $IV{q}^{r}OFN$; its score function $S_J\left(\mathfrak{N}\right)$ is given in Equation (6)

$$S_J\left(\mathfrak{N}\right)=\frac{1}{4}\left(2+{\underline{\Omega }}^q+{\overline{\Omega }}^q-{\underline{\mho }}^q-{\overline{\mho }}^q\right),$$

(6)

Example 1.

Let ${\mathfrak{N}}_1=\left(\left[0.30,0.60\right],\left[0.22,0.42\right]\right)$ and ${\mathfrak{N}}_2=\left(\left[0.35,0.55\right],\left[0.27,0.37\right]\right)\in IV{q}^{r}OFS$. If we employ Equation (6) to ${\mathfrak{N}}_1$ and ${\mathfrak{N}}_2$, then $S_J\left({\mathfrak{N}}_1\right)=S_J\left({\mathfrak{N}}_2\right)=0.5650$. This implies ${\mathfrak{N}}_1={\mathfrak{N}}_2$ but ${\mathfrak{N}}_1\ne {\mathfrak{N}}_2$. Thus, the function $S_J$ fails to rank the $IV{q}^{r}OFNs$ ${\mathfrak{N}}_1$ and ${\mathfrak{N}}_2$.

Definition 10

([34]). Let ${\mathfrak{N}}_{IVFFN}=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$ be an $IVFFN$; its score function $S_R\left({\mathfrak{N}}_{IVFFN}\right)$ is given in Equation (7)

$$S_R\left({\mathfrak{N}}_{IVFFN}\right)=\frac{1}{2}\left({\underline{\Omega }}^3+{\overline{\Omega }}^3-{\underline{\mho }}^3-{\overline{\mho }}^3\right),$$

(7)

This score function is further extended to the $IV{q}^{r}OFN$ $\mathfrak{N}$ as given in Equation (8)

$$S_{Rg}\left(\mathfrak{N}\right)=\frac{1}{2}\left({\underline{\Omega }}^q+{\overline{\Omega }}^q-{\underline{\mho }}^q-{\overline{\mho }}^q\right),$$

(8)

Example 2.

Let ${\mathfrak{N}}_3=\left(\left[0.50,0.80\right],\left[0.50,0.80\right]\right)$ and ${\mathfrak{N}}_4=\left(\left[0.66,0.78\right],\left[0.66,0.78\right]\right)\in IV{q}^{r}OFS$. If we employ Equation (8) to ${\mathfrak{N}}_3$ and ${\mathfrak{N}}_4$, then $S_{Rg}\left({\mathfrak{N}}_3\right)=S_{Rg}\left({\mathfrak{N}}_4\right)=0$. This implies ${\mathfrak{N}}_3={\mathfrak{N}}_4$ but ${\mathfrak{N}}_3\ne {\mathfrak{N}}_4$. Thus, the function $S_{Rg}$ fails to rank the $IV{q}^{r}OFNs$ ${\mathfrak{N}}_3$ and ${\mathfrak{N}}_4$.

Definition 11

([35]). Let ${\mathfrak{N}}_{IVFFN}=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$ be an $IVFFN$; its score function $S_{Ja}\left({\mathfrak{N}}_{IVFFN}\right)$ is given in Equation (9)

$$S_{Ja}\left({\mathfrak{N}}_{IVFFN}\right)=\frac{1}{2}\left(-{\underline{\Omega }}^3+{\overline{\Omega }}^3+{\underline{\mho }}^3-{\overline{\mho }}^3\right),$$

(9)

This score function is further extended to the $IV{q}^{r}OFN$ $\mathfrak{N}$ as given in Equation (10)

$$S_{Jag}\left(\mathfrak{N}\right)=\frac{1}{2}\left(-{\underline{\Omega }}^q+{\overline{\Omega }}^q+{\underline{\mho }}^q-{\overline{\mho }}^q\right),$$

(10)

Example 3.

Let${\mathfrak{N}}_5=\left(\left[\sqrt[4]{0.25},\sqrt[4]{0.32}\right],\left[\sqrt[4]{0.35},\sqrt[4]{0.50}\right]\right)$ and ${\mathfrak{N}}_6=\left(\left[\sqrt[4]{0.15},\sqrt[4]{0.42}\right],\left[\sqrt[4]{0.25},\sqrt[4]{0.60}\right]\right)\in IV{q}^{r}OFS$. If we employ Equation (10) to ${\mathfrak{N}}_5$ and ${\mathfrak{N}}_6$, then $S_{Jag}\left({\mathfrak{N}}_5\right)=S_{Jag}\left({\mathfrak{N}}_6\right)=-0.0400$. This implies ${\mathfrak{N}}_5={\mathfrak{N}}_6$ but ${\mathfrak{N}}_5\ne {\mathfrak{N}}_6$. Thus, the function $S_{Jag}$ fails to rank the $IV{q}^{r}OFNs$ ${\mathfrak{N}}_5$ and ${\mathfrak{N}}_6$.

Definition 12

([35]). Let ${\mathfrak{N}}_{IVFFN}=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$ be an $IVFFN$; its score function $S_{Jb}\left({\mathfrak{N}}_{IVFFN}\right)$ is given in Equation (11)

$$S_{Jb}\left({\mathfrak{N}}_{IVFFN}\right)=\frac{1}{2}\left(-{\underline{\Omega }}^3+{\overline{\Omega }}^3-{\underline{\mho }}^3+{\overline{\mho }}^3\right),$$

(11)

This score function is further extended to the $IV{q}^{r}OFN$ $\mathfrak{N}$ as given in Equation (12)

$$S_{Jbg}\left(\mathfrak{N}\right)=\frac{1}{2}\left(-{\underline{\Omega }}^q+{\overline{\Omega }}^q-{\underline{\mho }}^q+{\overline{\mho }}^q\right),$$

(12)

Example 4.

Let${\mathfrak{N}}_7=\left(\left[\sqrt[4]{0.25},\sqrt[4]{0.30}\right],\left[\sqrt[4]{0.35},\sqrt[4]{0.60}\right]\right)$ and ${\mathfrak{N}}_8=\left(\left[\sqrt[4]{0.20},\sqrt[4]{0.40}\right],\left[\sqrt[4]{0.40},\sqrt[4]{0.50}\right]\right)\in IV{q}^{r}OFS$. If we employ Equation (12) to ${\mathfrak{N}}_7$ and ${\mathfrak{N}}_8$, then $S_{Jbg}\left({\mathfrak{N}}_7\right)=S_{Jbg}\left({\mathfrak{N}}_8\right)=0.1500$. This implies ${\mathfrak{N}}_7={\mathfrak{N}}_2$ but ${\mathfrak{N}}_1\ne {\mathfrak{N}}_8$. Thus, the function $S_{Jbg}$ fails to rank the $IV{q}^{r}OFNs$ ${\mathfrak{N}}_7$ and ${\mathfrak{N}}_8$.

Definition 13

([36]). Let ${\mathfrak{N}}_{IVFFN}=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$ be an $IVFFN$; its score function $S_R\left({\mathfrak{N}}_{IVFFN}\right)$ is given in Equation (13)

$$S_R\left({\mathfrak{N}}_{IVFFN}\right)=\frac{1}{2}\left(\left({\underline{\Omega }}^3-{\underline{\mho }}^3\right)\left(1+\sqrt[3]{1-{\underline{\Omega }}^3-{\underline{\mho }}^3}\right)+\left({\overline{\Omega }}^3-{\overline{\mho }}^3\right)\left(1+\sqrt[3]{{1-\overline{\Omega }}^3-{\overline{\mho }}^3}\right)\right),$$

(13)

This score function is further extended to the $IV{q}^{r}OFN$ $\mathfrak{N}$ as given in Equation (14)

$$S_{Rg}\left(\mathfrak{N}\right)=\frac{1}{2}\left(\left({\underline{\Omega }}^q-{\underline{\mho }}^q\right)\left(1+\sqrt[q]{1-{\underline{\Omega }}^q-{\underline{\mho }}^q}\right)+\left({\overline{\Omega }}^q-{\overline{\mho }}^q\right)\left(1+\sqrt[q]{{1-\overline{\Omega }}^q-{\overline{\mho }}^q}\right)\right),$$

(14)

Example 5.

Let${\mathfrak{N}}_9=\left(\left[\sqrt[5]{0.35},\sqrt[5]{0.50}\right],\left[\sqrt[5]{0.35},\sqrt[5]{0.50}\right]\right)$ and ${\mathfrak{N}}_{10}=\left(\left[\sqrt[5]{0.30},\sqrt[5]{0.60}\right],\left[\sqrt[5]{0.30},\sqrt[5]{0.60}\right]\right)\in IV{q}^{r}OFS$. If we employ Equation (14) to ${\mathfrak{N}}_9$ and ${\mathfrak{N}}_{10}$, then $S_{Rg}\left({\mathfrak{N}}_1\right)=S_{Rg}\left({\mathfrak{N}}_2\right)=0$. This implies ${\mathfrak{N}}_9={\mathfrak{N}}_{10}$ but ${\mathfrak{N}}_9\ne {\mathfrak{N}}_{10}$. Thus, the function $S_{Rg}$ fails to rank the $IV{q}^{r}OFNs$ ${\mathfrak{N}}_9$ and ${\mathfrak{N}}_{10}$.

Definition 14

([37]). Let ${\mathfrak{N}}_{IVFFN}=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$ be an $IVIFN$; its score function $S_W\left({\mathfrak{N}}_{IVIFN}\right)$ is given in Equation (15)

$$S_W\left({\mathfrak{N}}_{IVIFN}\right)=\frac{1}{2}\left(\left(\underline{\Omega }+\overline{\Omega }\right)\left(\underline{\Omega }+\underline{\mho }\right)-\left(\underline{\mho }+\overline{\mho }\right)\left(\overline{\Omega }+\overline{\mho }\right)\right),$$

(15)

This score function is further extended to the $IV{q}^{r}OFN$ $\mathfrak{N}$ as given in Equation (16)

$$S_{Wg}\left(\mathfrak{N}\right)=\frac{1}{2}\left(\left({\underline{\Omega }}^q+{\overline{\Omega }}^q\right)\left({\underline{\Omega }}^q+{\underline{\mho }}^q\right)-\left({\underline{\mho }}^q+{\overline{\mho }}^q\right)\left({\overline{\Omega }}^q+{\overline{\mho }}^q\right)\right),$$

(16)

Example 6.

Let${\mathfrak{N}}_{11}=\left(\left[\sqrt[3]{0.46},\sqrt[3]{0.46}\right],\left[\sqrt[3]{0.12},\sqrt[3]{0.46}\right]\right)$ and ${\mathfrak{N}}_{12}=\left(\left[\sqrt[3]{0.12},\sqrt[3]{0.79}\right],\left[\sqrt[3]{0.08},\sqrt[3]{0.12}\right]\right)\in IV{q}^{r}OFS$. If we employ Equation (16) to ${\mathfrak{N}}_{11}$ and ${\mathfrak{N}}_{12}$, then $S_{Wg}\left({\mathfrak{N}}_{11}\right)=S_{Wg}\left({\mathfrak{N}}_{12}\right)=0$. This implies ${\mathfrak{N}}_{11}={\mathfrak{N}}_{12}$ but ${\mathfrak{N}}_{11}\ne {\mathfrak{N}}_{12}$. Thus, the function $S_{Wg}$ fails to rank the $IV{q}^{r}OFNs$ ${\mathfrak{N}}_{11}$ and ${\mathfrak{N}}_{12}$.

Definition 15

([38]). Let ${\mathfrak{N}}_{IVIFN}=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$ be an $IVIFN$; its score function $S_G\left({\mathfrak{N}}_{IVIFN}\right)$ is given in Equation (17)

$$S_G\left({\mathfrak{N}}_{IVIFN}\right)=\frac{1}{2}\left(\overline{\mho }+\underline{\mho }-\overline{\Omega }-\underline{\Omega }\right)+\frac{\overline{\Omega }+\underline{\Omega }+2\left(\overline{\Omega }\underline{\Omega }-\overline{\mho }\underline{\mho }\right)}{\overline{\mho }+\underline{\mho }+\overline{\Omega }+\underline{\Omega }},$$

(17)

This score function is further extended to the $IV{q}^{r}OFN$ $\mathfrak{N}$ as given in Equation (18)

$$S_{Gg}\left(\mathfrak{N}\right)=\frac{1}{2}\left({\overline{\mho }}^q+{\underline{\mho }}^q-{\overline{\Omega }}^q-{\underline{\Omega }}^q\right)+\frac{{\overline{\Omega }}^q+{\underline{\Omega }}^q+2\left({\overline{\Omega }}^q{\underline{\Omega }}^q-{\overline{\mho }}^q{\underline{\mho }}^q\right)}{{\overline{\mho }}^q+{\underline{\mho }}^q+{\overline{\Omega }}^q+{\underline{\Omega }}^q},$$

(18)

Example 7.

Let${\mathfrak{N}}_{13}=\left(\left[0.11,0.25\right],\left[0.29,0.37\right]\right)$ and ${\mathfrak{N}}_{14}=\left(\left[0.18,0.28\right],\left[0.36,0.54\right]\right)\in IV{q}^{r}OFS$. If we employ Equation (17) to ${\mathfrak{N}}_{13}$ and ${\mathfrak{N}}_{14}$, then $S_G\left({\mathfrak{N}}_{13}\right)=S_G\left({\mathfrak{N}}_{14}\right)=0.3465$. This implies ${\mathfrak{N}}_{13}={\mathfrak{N}}_{14}$ but ${\mathfrak{N}}_{13}\ne {\mathfrak{N}}_{14}$. Thus, the function $S_G$ fails to rank the $IV{q}^{r}OFNs$ ${\mathfrak{N}}_{13}$ and ${\mathfrak{N}}_{14}$.

Definition 16.

Let $ß=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$ be an $IVp,q^{r}OFN$; its score function is given by

$$S^{⏜}\left(ß\right)=\frac{1}{3}\left(1+\frac{{\underline{\Omega }}^p+{\overline{\Omega }}^p-{\underline{\mho }}^q-{\overline{\mho }}^q}{2}+\frac{{\underline{\Omega }}^p\left(1-{\underline{\mho }}^q\right)+{\overline{\Omega }}^p\left(1-{\overline{\mho }}^q\right)}{2+{\underline{\mho }}^q+{\overline{\mho }}^q}\right)$$

(19)

where $S^{⏜}\left(ß\right)\in \left[0,1\right]$. The greater value of $S^{⏜}\left(ß\right)$ determines the better option.

Definition 17.

Let $ß=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$ be an $IVp,q^{r}OFN$; its accuracy function is given by

$$A\left(ß\right)=\frac{1}{2}\left({\underline{\Omega }}^p+{\overline{\Omega }}^p+{\underline{\mho }}^q+{\overline{\mho }}^q\right).$$

(20)

Proposition 1.

(Maximum value) If $ß=\left(\left[1,1\right],\left[0,0\right]\right)\in IVp,q^{r}OFS$, then $S^{⏜}\left(ß\right)=1$.

Proposition 2.

(Minimum value) If $ß=\left(\left[0,0\right],\left[1,1\right]\right)\in IVp,q^{r}OFS$, then $S^{⏜}\left(ß\right)=0$.

Theorem 2.

For an $IVp,q^{r}OFN$ $ß=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$, the score function $S^{⏜}\left(ß\right)$ increases monotonically with respect to $\underline{\Omega }\left(\overline{\Omega }\right)$ and decreases monotonically with respect to $\underline{\mho }\left(\overline{\mho }\right)$.

Proof. 

The partial derivatives of Equation (19) with respect to $\underline{\Omega }$ and $\overline{\Omega }$ are given as

$$\frac{\partial S^{⏜}\left(ß\right)}{\partial \underline{\Omega }}=\frac{1}{3}p{\underline{\Omega }}^{p-1}\left(\frac{1}{2}+\frac{1-{\underline{\mho }}^q}{2+{\underline{\mho }}^q+{\overline{\mho }}^q}\right)\ge 0,$$

and

$$\frac{\partial S^{⏜}\left(ß\right)}{\partial \overline{\Omega }}=\frac{1}{3}p{\overline{\Omega }}^{p-1}\left(\frac{1}{2}+\frac{1-{\overline{\mho }}^q}{2+{\underline{\mho }}^q+{\overline{\mho }}^q}\right)\ge 0.$$

Analogously, the partial derivatives of Equation (19) with respect to $\underline{\mho }$ and $\overline{\mho }$ are given as

$$\frac{\partial S^{⏜}\left(ß\right)}{\partial \underline{\mho }}=\frac{-1}{3}q{\underline{\mho }}^{q-1}\left(\frac{1}{2}+\frac{{\underline{\Omega }}^p}{2+{\underline{\mho }}^q+{\overline{\mho }}^q}\right)\le 0,$$

and

$$\frac{\partial S^{⏜}\left(ß\right)}{\partial \overline{\mho }}=\frac{-1}{3}q{\overline{\mho }}^{q-1}\left(\frac{1}{2}+\frac{{\overline{\Omega }}^p}{2+{\underline{\mho }}^q+{\overline{\mho }}^q}\right)\le 0.$$

Therefore, the proof is shown. □

Theorem 3.

Let ${ß}_1$, ${ß}_2\in IVp,q^{r}OFS$. If ${ß}_1\subseteq {ß}_2,$ then $S^{⏜}\left({ß}_1\right)\le S^{⏜}\left({ß}_2\right)$.

Proof. 

The proof follows from Theorem 2. □

Theorem 4.

Let ${ß}_1$, ${ß}_2$, ${ß}_3\in IVp,q^{r}OFS$. If ${ß}_1\subseteq {ß}_2,$ then $S^{⏜}\left({ß}_1\oplus {ß}_3\right)\le S^{⏜}\left({ß}_2\oplus {ß}_3\right)$.

Proof. 

From Definition 8, we have ${ß}_1\oplus {ß}_3=\left(\right[\sqrt[p^{*}]{{\underline{\Omega }}_1^{p^{*}}+{\underline{\Omega }}_3^{p^{*}}-{\underline{\Omega }}_1^{p^{*}}{\underline{\Omega }}_3^{p^{*}}},\sqrt[p^{*}]{{\overline{\Omega }}_1^{p^{*}}+{\overline{\Omega }}_3^{p^{*}}-{\overline{\Omega }}_1^{p^{*}}{\overline{\Omega }}_3^{p^{*}}}],$ $[{\underline{\mho }}_1{\underline{\mho }}_3,{\overline{\mho }}_1{\overline{\mho }}_3]),$ and ${ß}_2\oplus {ß}_3=\left(\left[\sqrt[p^{*}]{{\underline{\Omega }}_2^{p^{*}}+{\underline{\Omega }}_3^{p^{*}}-{\underline{\Omega }}_2^{p^{*}}{\underline{\Omega }}_3^{p^{*}}},\sqrt[p^{*}]{{\overline{\Omega }}_2^{p^{*}}+{\overline{\Omega }}_3^{p^{*}}-{\overline{\Omega }}_2^{p^{*}}{\overline{\Omega }}_3^{p^{*}}}\right],\left[{\underline{\mho }}_2{\underline{\mho }}_3,{\overline{\mho }}_2{\overline{\mho }}_3\right]\right).$

Since ${ß}_1\subseteq {ß}_2,$ it implies that ${\underline{\Omega }}_1\le {\underline{\Omega }}_2$, ${\overline{\Omega }}_1\le {\overline{\Omega }}_2$, ${\underline{\mho }}_1\ge {\underline{\mho }}_2$ and ${\overline{\mho }}_1\ge {\overline{\mho }}_2$. Using these inequalities, we have

$\sqrt[p^{*}]{{\underline{\Omega }}_1^{p^{*}}+{\underline{\Omega }}_3^{p^{*}}-{\underline{\Omega }}_1^{p^{*}}{\underline{\Omega }}_3^{p^{*}}}\le \sqrt[p^{*}]{{\underline{\Omega }}_2^{p^{*}}+{\underline{\Omega }}_3^{p^{*}}-{\underline{\Omega }}_2^{p^{*}}{\underline{\Omega }}_3^{p^{*}}},$ $\sqrt[p^{*}]{{\overline{\Omega }}_1^{p^{*}}+{\overline{\Omega }}_3^{p^{*}}-{\overline{\Omega }}_1^{p^{*}}{\overline{\Omega }}_3^{p^{*}}}$ $\le \sqrt[p^{*}]{{\overline{\Omega }}_2^{p^{*}}+{\overline{\Omega }}_3^{p^{*}}-{\overline{\Omega }}_2^{p^{*}}{\overline{\Omega }}_3^{p^{*}}},$ ${\underline{\mho }}_1{\underline{\mho }}_3\ge {\underline{\mho }}_2{\underline{\mho }}_3,$ ${\overline{\mho }}_1{\overline{\mho }}_3\ge {\overline{\mho }}_2{\overline{\mho }}_3.$ This implies that ${ß}_1\oplus {ß}_3\subseteq {ß}_2\oplus {ß}_3$. Thus, according to Theorem 3, we get $S^{⏜}\left({ß}_1\oplus {ß}_3\right)\le S^{⏜}\left({ß}_2\oplus {ß}_3\right)$. □

Theorem 5.

Let ${ß}_1$, ${ß}_2$ $\in IVp,q^{r}OFS$ and $\omega >0$. If ${ß}_1\subseteq {ß}_2,$ then $S^{⏜}\left(\omega {ß}_1\right)\le S^{⏜}\left(\omega {ß}_2\right)$.

Proof. 

Similar to the proof of Theorem 4. □

Definition 18.

Let${ß}_j=\left(\left[{\underline{\Omega }}_j,{\overline{\Omega }}_j\right],\left[{\underline{\mho }}_j,{\overline{\mho }}_j\right]\right)(j=1,2)$be any two$IVp,q^{r}OFNs$; the comparison scheme is described as follows:

(i) 

If $S^{⏜}\left({ß}_1\right)<S^{⏜}\left({ß}_2\right)$, then ${ß}_1\prec {ß}_2;$

(ii) 

If $S^{⏜}\left({ß}_1\right)>S^{⏜}\left({ß}_2\right)$, then ${ß}_1\succ {ß}_2;$

(iii) 

If $S^{⏜}\left({ß}_1\right)=S^{⏜}\left({ß}_2\right)$, and

(a) 

If $A\left({ß}_1\right)<A\left({ß}_2\right)$, then ${ß}_1\prec {ß}_2;$

(b) 

If $A\left({ß}_1\right)>A\left({ß}_2\right)$, then ${ß}_1\succ {ß}_2;$

(c) 

If $A\left({ß}_1\right)=A\left({ß}_2\right)$, then ${ß}_1\simeq {ß}_2.$

To exemplify the suitability of the devised score function $S^{⏜}$ in ranking $IV{q}^{r}OFNs$, we solve Examples 1–7 employing the new score function and the existing ones [33,34,35,36,37,38], and the results are reported in

Table 1. Comparison results of score values.

Score Function	Example 1	Example 2	Example 3	Example 4	Example 5	Example 6	Example 7
$S_J$ [33]	${\mathfrak{N}}_1={\mathfrak{N}}_2$	${\mathfrak{N}}_3={\mathfrak{N}}_4$	${\mathfrak{N}}_5={\mathfrak{N}}_6$	${\mathfrak{N}}_7<{\mathfrak{N}}_8$	${\mathfrak{N}}_9={\mathfrak{N}}_{10}$	${\mathfrak{N}}_{11}>{\mathfrak{N}}_{12}$	${\mathfrak{N}}_{13}>{\mathfrak{N}}_{14}$
$S_{Rg}$ [34]	${\mathfrak{N}}_1={\mathfrak{N}}_2$	${\mathfrak{N}}_3={\mathfrak{N}}_4$	${\mathfrak{N}}_5={\mathfrak{N}}_6$	${\mathfrak{N}}_7<{\mathfrak{N}}_8$	${\mathfrak{N}}_9={\mathfrak{N}}_{10}$	${\mathfrak{N}}_{11}>{\mathfrak{N}}_{12}$	${\mathfrak{N}}_{13}>{\mathfrak{N}}_{14}$
$S_{Jag}$ [35]	${\mathfrak{N}}_1={\mathfrak{N}}_2$	${\mathfrak{N}}_3={\mathfrak{N}}_4$	${\mathfrak{N}}_5={\mathfrak{N}}_6$	${\mathfrak{N}}_7<{\mathfrak{N}}_8$	${\mathfrak{N}}_9={\mathfrak{N}}_{10}$	${\mathfrak{N}}_{11}<{\mathfrak{N}}_{12}$	${\mathfrak{N}}_{13}>{\mathfrak{N}}_{14}$
$S_{Jbg}$ [35]	${\mathfrak{N}}_1>{\mathfrak{N}}_2$	${\mathfrak{N}}_3>{\mathfrak{N}}_4$	${\mathfrak{N}}_5<{\mathfrak{N}}_6$	${\mathfrak{N}}_7={\mathfrak{N}}_8$	${\mathfrak{N}}_9<{\mathfrak{N}}_{10}$	${\mathfrak{N}}_{11}<{\mathfrak{N}}_{12}$	${\mathfrak{N}}_{13}<{\mathfrak{N}}_{14}$
$S_{Rg}$ [36]	${\mathfrak{N}}_1<{\mathfrak{N}}_2$	${\mathfrak{N}}_3={\mathfrak{N}}_4$	${\mathfrak{N}}_5={\mathfrak{N}}_6$	${\mathfrak{N}}_7<{\mathfrak{N}}_8$	${\mathfrak{N}}_9={\mathfrak{N}}_{10}$	${\mathfrak{N}}_{11}<{\mathfrak{N}}_{12}$	${\mathfrak{N}}_{11}>{\mathfrak{N}}_{12}$
$S_{Wg}$ [37]	${\mathfrak{N}}_1<{\mathfrak{N}}_2$	${\mathfrak{N}}_3<{\mathfrak{N}}_4$	${\mathfrak{N}}_5>{\mathfrak{N}}_6$	${\mathfrak{N}}_7<{\mathfrak{N}}_8$	${\mathfrak{N}}_9>{\mathfrak{N}}_{10}$	${\mathfrak{N}}_{11}={\mathfrak{N}}_{12}$	${\mathfrak{N}}_{13}>{\mathfrak{N}}_{14}$
$S_{Gg}$ [38]	${\mathfrak{N}}_1<{\mathfrak{N}}_2$	${\mathfrak{N}}_3={\mathfrak{N}}_4$	${\mathfrak{N}}_5<{\mathfrak{N}}_6$	${\mathfrak{N}}_7<{\mathfrak{N}}_8$	${\mathfrak{N}}_9={\mathfrak{N}}_{10}$	${\mathfrak{N}}_{11}<{\mathfrak{N}}_{12}$	${\mathfrak{N}}_{13}={\mathfrak{N}}_{14}$
${\mathrm{S}}^{⏜}$	${\mathfrak{N}}_1<{\mathfrak{N}}_2$	${\mathfrak{N}}_3>{\mathfrak{N}}_4$	${\mathfrak{N}}_5>{\mathfrak{N}}_6$	${\mathfrak{N}}_7<{\mathfrak{N}}_8$	${\mathfrak{N}}_9>{\mathfrak{N}}_{10}$	${\mathfrak{N}}_{11}<{\mathfrak{N}}_{12}$	${\mathfrak{N}}_{13}>{\mathfrak{N}}_{14}$

.

The rankings presented in Table 1 clearly illustrate that the devised score function $S^{⏜}$ can rank the $IV{q}^{r}OFNs$ effectively and also successfully overcome the inadequacies of the previous score functions. Hence, it is concluded that the devised score function surpasses the existing ones in practical settings.

3.4. Distance Measure

Definition 19.

Let ${ß}_j=\left(\left[{\underline{\Omega }}_j,{\overline{\Omega }}_j\right],\left[{\underline{\mho }}_j,{\overline{\mho }}_j\right]\right)(j=1,2)$ be any two $IVp,q^{r}OFNs$; the distance measure is described as follows:

$$d\left({ß}_1,{ß}_2\right)=\frac{1}{4}\left(\left|{\underline{\Omega }}_1^{p^{*}}-{\underline{\Omega }}_2^{p^{*}}\right|+\left|{\overline{\Omega }}_1^{p^{*}}-{\overline{\Omega }}_2^{p^{*}}\right|+\left|{\underline{\mho }}_1^{q^{*}}-{\underline{\mho }}_2^{q^{*}}\right|+\left|{\overline{\mho }}_1^{q^{*}}-{\overline{\mho }}_2^{q^{*}}\right|+\left|{\underline{\omega }}_1^{{\ell }^{*}}-{\underline{\omega }}_2^{{\ell }^{*}}\right|+\left|{\overline{\omega }}_1^{{\ell }^{*}}-{\overline{\omega }}_2^{{\ell }^{*}}\right|\right).$$

(21)

Theorem 6.

Let ${ß}_j=\left(\left[{\underline{\Omega }}_j,{\overline{\Omega }}_j\right],\left[{\underline{\mho }}_j,{\overline{\mho }}_j\right]\right)(j=1,2,3)$ be any three $IVp,q^{r}OFNs$; then, the following items hold:

1. 

$0\le d\left({ß}_1,{ß}_2\right)\le 1;$

2. 

$d\left({ß}_1,{ß}_2\right)=0$ iff ${ß}_1={ß}_2$;

3. 

$d\left({ß}_1,{ß}_2\right)=d\left({ß}_2,{ß}_1\right)$;

4. 

$d\left({ß}_1,{ß}_3\right)\le d\left({ß}_1,{ß}_2\right)+d\left({ß}_2,{ß}_3\right)$.

Proof. 

It is evident that (2) and (3) hold according to the formulation of the distance measure; we can then show (1) and (4) as follows:

(1). $d\left({ß}_1,{ß}_2\right)=\frac{1}{4}(\left|{\underline{\Omega }}_1^{p^{*}}-{\underline{\Omega }}_2^{p^{*}}\right|+\left|{\overline{\Omega }}_1^{p^{*}}-{\overline{\Omega }}_2^{p^{*}}\right|+\left|{\underline{\mho }}_1^{q^{*}}-{\underline{\mho }}_2^{q^{*}}\right|+\left|{\overline{\mho }}_1^{q^{*}}-{\overline{\mho }}_2^{q^{*}}\right|+\left|{\underline{\omega }}_1^{{\ell }^{*}}-{\underline{\omega }}_2^{{\ell }^{*}}\right|+\left|{\overline{\omega }}_1^{{\ell }^{*}}-{\overline{\omega }}_2^{{\ell }^{*}}\right|)\le \frac{1}{4}\left({\underline{\Omega }}_1^{p^{*}}+{\underline{\mho }}_1^{q^{*}}+{\underline{\omega }}_1^{{\ell }^{*}}+{\underline{\Omega }}_2^{p^{*}}+{\underline{\mho }}_2^{q^{*}}+{\underline{\omega }}_2^{{\ell }^{*}}+{\overline{\Omega }}_1^{p^{*}}+{\overline{\mho }}_1^{q^{*}}+{\overline{\omega }}_1^{{\ell }^{*}}+{\overline{\Omega }}_2^{p^{*}}+{\overline{\mho }}_2^{q^{*}}+{\overline{\omega }}_2^{{\ell }^{*}}\right)$. As $\left|x-y\right|\le \left|x\right|+\left|y\right|$. $\le \frac{1}{4}\left(1+1+1+1\right)\le 1.$

Using Definition 19, it is simple to derive $0\le d\left({ß}_1,{ß}_2\right)$.

(4). $d\left({ß}_1,{ß}_3\right)=\frac{1}{4}(\left|{\underline{\Omega }}_1^{p^{*}}-{\underline{\Omega }}_3^{p^{*}}\right|+\left|{\overline{\Omega }}_1^{p^{*}}-{\overline{\Omega }}_3^{p^{*}}\right|+\left|{\underline{\mho }}_1^{q^{*}}-{\underline{\mho }}_3^{q^{*}}\right|+\left|{\overline{\mho }}_1^{q^{*}}-{\overline{\mho }}_3^{q^{*}}\right|+\left|{\underline{\omega }}_1^{{\ell }^{*}}-{\underline{\omega }}_3^{{\ell }^{*}}\right|+\left|{\overline{\omega }}_1^{{\ell }^{*}}-{\overline{\omega }}_3^{{\ell }^{*}}\right|)=\frac{1}{4}\left(\begin{array}{c}\left|{\underline{\Omega }}_1^{p^{*}}-{\underline{\Omega }}_2^{p^{*}}+{\underline{\Omega }}_2^{p^{*}}-{\underline{\Omega }}_3^{p^{*}}\right|+\left|{\overline{\Omega }}_1^{p^{*}}-{\overline{\Omega }}_2^{p^{*}}+{\overline{\Omega }}_2^{p^{*}}-{\overline{\Omega }}_3^{p^{*}}\right|+\left|{\underline{\mho }}_1^{q^{*}}-{\underline{\mho }}_2^{q^{*}}+{\underline{\mho }}_2^{q^{*}}-{\underline{\mho }}_3^{q^{*}}\right|+\\ \left|{\overline{\mho }}_1^{q^{*}}-{\overline{\mho }}_2^{q^{*}}+{\overline{\mho }}_2^{q^{*}}-{\overline{\mho }}_3^{q^{*}}\right|+\left|{\underline{\omega }}_1^{{\ell }^{*}}-{\underline{\omega }}_2^{{\ell }^{*}}+{\underline{\omega }}_2^{{\ell }^{*}}-{\underline{\omega }}_3^{{\ell }^{*}}\right|+\left|{\overline{\omega }}_1^{{\ell }^{*}}-{\overline{\omega }}_2^{{\ell }^{*}}+{\overline{\omega }}_2^{{\ell }^{*}}-{\overline{\omega }}_3^{{\ell }^{*}}\right|\end{array}\right)\frac{1}{4}\left(\begin{array}{c}\left|{\underline{\Omega }}_1^{p^{*}}-{\underline{\Omega }}_2^{p^{*}}\right|+\left|{\underline{\Omega }}_2^{p^{*}}-{\underline{\Omega }}_3^{p^{*}}\right|+\left|{\overline{\Omega }}_1^{p^{*}}-{\overline{\Omega }}_2^{p^{*}}\right|+\left|{\overline{\Omega }}_2^{p^{*}}-{\overline{\Omega }}_3^{p^{*}}\right|+\left|{\underline{\mho }}_1^{q^{*}}-{\underline{\mho }}_2^{q^{*}}\right|+\left|{\underline{\mho }}_2^{q^{*}}-{\underline{\mho }}_3^{q^{*}}\right|+\\ \left|{\overline{\mho }}_1^{q^{*}}-{\overline{\mho }}_2^{q^{*}}\right|+\left|{\overline{\mho }}_2^{q^{*}}-{\overline{\mho }}_3^{q^{*}}\right|+\left|{\underline{\omega }}_1^{{\ell }^{*}}-{\underline{\omega }}_2^{{\ell }^{*}}\right|+\left|{\underline{\omega }}_2^{{\ell }^{*}}-{\underline{\omega }}_3^{{\ell }^{*}}\right|+\left|{\overline{\omega }}_1^{{\ell }^{*}}-{\overline{\omega }}_2^{{\ell }^{*}}\right|+\left|{\overline{\omega }}_2^{{\ell }^{*}}-{\overline{\omega }}_3^{{\ell }^{*}}\right|\end{array}\right)$ $\le d\left({ß}_1,{ß}_2\right)+d\left({ß}_2,{ß}_3\right).$ □

4. Interval-Valued p,q-Rung Orthopair Fuzzy AOs

4.1. Interval-Valued p,q-Rung Orthopair Fuzzy Averaging AOs

Definition 20.

Let ${ß}_j=\left(j=1\left(1\right)n\right)$ be a collection of some$IVp,q^{r}OFNs$and$={\left({\omega }_1,{\omega }_2,\dots ,{\omega }_n\right)}^T$be the weight vector of ${ß}_j=\left(j=1\left(1\right)n\right)$, with ${\omega }_j\in \left[0,1\right],$ ${\displaystyle \sum _{j=1}^n}j=1.$ Then, the $IVp,q^{r}OF$ weighted averaging ($IVp,q^{r}OFWA$) operator is formulated as follows:

$$IVp,q^{r}OFWA\left({ß}_1,{ß}_2,\dots ,{ß}_n\right)={\omega }_1{ß}_1\oplus {\omega }_2{ß}_2\oplus \dots {\omega }_n{ß}_n.$$

(22)

In the light of $IVp,q^{r}OF$ operational laws, we provide the following result.

Theorem 7.

Let ${ß}_j=\left(j=1\left(1\right)n\right)$ be a collection of some $IVp,q^{r}OFNs$; then, the aggregated result of the $IVp,q^{r}OFWA$ operator is given as:

$$IVp,q^{r}OFWA\left({ß}_1,{ß}_2,\dots ,{ß}_n\right)=\left(\left[\sqrt[p^{*}]{1-\prod _{j=1}^n{\left(1-{\underline{\Omega }}_j^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-\prod _{j=1}^n{\left(1-{\overline{\Omega }}_j^{p^{*}}\right)}^{{\omega }_j}}\right],\left[\prod _{j=1}^n{\underline{\mho }}_j^{{\omega }_j},\prod _{j=1}^n{\overline{\mho }}_j^{{\omega }_j}\right]\right).$$

(23)

Proof. 

We show Equation (23) using mathematical indication.

For $n=2$, we have

$IVp,q^{r}OFWA\left({ß}_1,{ß}_2\right)={\omega }_1{ß}_1\oplus {\omega }_2{ß}_2=\left(\right[\sqrt[p^{*}]{1-{\left(1-{\underline{\Omega }}_1^{p^{*}}\right)}^{{\omega }_1}{\left(1-{\underline{\Omega }}_2^{p^{*}}\right)}^{{\omega }_2}},\sqrt[p^{*}]{1-{\left(1-{\overline{\Omega }}_1^{p^{*}}\right)}^{{\omega }_1}{\left(1-{\overline{\Omega }}_2^{p^{*}}\right)}^{{\omega }_2}}],$$\left[{\underline{\mho }}_1^{{\omega }_1}{\underline{\mho }}_2^{{\omega }_2},{\overline{\mho }}_1^{{\omega }_1}{\overline{\mho }}_2^{{\omega }_2}\right])=(\left[\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^2}{\left(1-{\underline{\Omega }}_j^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^2}{\left(1-{\overline{\Omega }}_j^{p^{*}}\right)}^{{\omega }_j}}\right],\left[{\displaystyle \prod _{j=1}^2}{\underline{\mho }}_j^{{\omega }_j},{\displaystyle \prod _{j=1}^2}{\overline{\mho }}_j^{{\omega }_j}\right]).$

Thus, Equation (23) is kept for $j=2$.

Suppose Equation (23) holds for $j=k$. Then, we have

$IVp,q^{r}OFWA\left({ß}_1,{ß}_2,\dots ,{ß}_k\right)=\left(\left[\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^k}{\left(1-{\underline{\Omega }}_j^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^k}{\left(1-{\overline{\Omega }}_j^{p^{*}}\right)}^{{\omega }_j}}\right],\left[{\displaystyle \prod _{j=1}^k}{\underline{\mho }}_j^{{\omega }_j},{\displaystyle \prod _{j=1}^k}{\overline{\mho }}_j^{{\omega }_j}\right]\right)$.

Now, we check for $j=k+1$

$IVp,q^{r}OFWA\left({ß}_1,{ß}_2,\dots ,{ß}_{k+1}\right)={\oplus }_{j=1}^{k+1}\left({\omega }_j{ß}_j\right)={\oplus }_{j=1}^k\left({\omega }_j{ß}_j\right)\oplus {\omega }_{k+1}{ß}_{k+1}=\left(\left[\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^k}{\left(1-{\underline{\Omega }}_j^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^k}{\left(1-{\overline{\Omega }}_j^{p^{*}}\right)}^{{\omega }_j}}\right],\left[{\displaystyle \prod _{j=1}^k}{\underline{\mho }}_j^{{\omega }_j},{\displaystyle \prod _{j=1}^k}{\overline{\mho }}_j^{{\omega }_j}\right]\right)\oplus \left(\left[\sqrt[p^{*}]{1-{\left(1-{\underline{\Omega }}_{k+1}^{p^{*}}\right)}^{{\omega }_{k+1}}},\sqrt[p^{*}]{1-{\left(1-{\overline{\Omega }}_{k+1}^{p^{*}}\right)}^{{\omega }_{k+1}}}\right],\left[{\underline{\mho }}_{k+1}^{{\omega }_j},{\overline{\mho }}_{k+1}^{{\omega }_{k+1}}\right]\right)=\left(\left[\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^{k+1}}{\left(1-{\underline{\Omega }}_j^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^{k+1}}{\left(1-{\overline{\Omega }}_j^{p^{*}}\right)}^{{\omega }_j}}\right],\left[{\displaystyle \prod _{j=1}^{k+1}}{\underline{\mho }}_j^{{\omega }_j},{\displaystyle \prod _{j=1}^{k+1}}{\overline{\mho }}_j^{{\omega }_j}\right]\right).$

As a result, the proof is finished. □

Theorem 8.

Let ${ß}_j=\left(\left[{\underline{\Omega }}_j,{\overline{\Omega }}_j\right],\left[{\underline{\mho }}_j,{\overline{\mho }}_j\right]\right)\left(j=1\left(1\right)n\right)$ be a collection of some $IVp,q^{r}OFNs$ such that ${ß}_j=ß=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$ for all j. Then,

$$IVp,q^{r}OFWA\left({ß}_1,{ß}_2,\dots ,{ß}_n\right)=ß.$$

(24)

Proof. 

${ß}_j=ß=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$ for all j. Therefore,

$IVp,q^{r}OFWA\left({ß}_1,{ß}_2,\dots ,{ß}_n\right)=\left(\left[\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^n}{\left(1-{\underline{\Omega }}_j^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^n}{\left(1-{\overline{\Omega }}_j^{p^{*}}\right)}^{{\omega }_j}}\right],\left[{\displaystyle \prod _{j=1}^n}{\underline{\mho }}_j^{{\omega }_j},{\displaystyle \prod _{j=1}^n}{\overline{\mho }}_j^{{\omega }_j}\right]\right)=\left(\left[\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^n}{\left(1-{\underline{\Omega }}^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^n}{\left(1-{\overline{\Omega }}^{p^{*}}\right)}^{{\omega }_j}}\right],\left[{\displaystyle \prod _{j=1}^n}{\underline{\mho }}^{{\omega }_j},{\displaystyle \prod _{j=1}^n}{\overline{\mho }}^{{\omega }_j}\right]\right)=\left(\left[\sqrt[p^{*}]{1-{\left(1-{\underline{\Omega }}^{p^{*}}\right)}^{{\displaystyle {\displaystyle \sum _{j=1}^n}}{\omega }_j}},\sqrt[p^{*}]{1-{\left(1-{\overline{\Omega }}^{p^{*}}\right)}^{{\displaystyle {\displaystyle \sum _{j=1}^n}}{\omega }_j}}\right],\left[{\underline{\mho }}^{{\displaystyle {\displaystyle \sum _{j=1}^n}}{\omega }_j},{\overline{\mho }}^{{\displaystyle \sum _{j=1}^n}{\omega }_j}\right]\right)=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right)=ß$ □

Theorem 9.

Let ${ß}_j=\left(\left[{\underline{\Omega }}_j,{\overline{\Omega }}_j\right],\left[{\underline{\mho }}_j,{\overline{\mho }}_j\right]\right)$ and ${\dot{ß}}_j=\left(\left[{\underline{\dot{\Omega }}}_j,{\overline{\dot{\Omega }}}_j\right],\left[{\underline{\dot{\mho }}}_j,{\overline{\dot{\mho }}}_j\right]\right)\left(j=1\left(1\right)n\right)$ be any two collections of some $IVp,q^{r}OFNs$ such that ${\underline{\Omega }}_j\le {\underline{\dot{\Omega }}}_j$, ${\overline{\Omega }}_j\le {\overline{\dot{\Omega }}}_j$, ${\underline{\mho }}_j\ge {\underline{\dot{\mho }}}_j$, and ${\overline{\mho }}_j\ge {\overline{\dot{\mho }}}_j$ for all j. Then,

$$IVp,q^{r}OFWA\left({ß}_1,{ß}_2,\dots ,{ß}_n\right)\le IVp,q^{r}OFWA\left({\dot{ß}}_1,{\dot{ß}}_2,\dots ,{\dot{ß}}_n\right).$$

(25)

Proof. 

As ${\underline{\Omega }}_j\le {\underline{\dot{\Omega }}}_j$, ${\overline{\Omega }}_j\le {\overline{\dot{\Omega }}}_j$, ${\underline{\mho }}_j\ge {\underline{\dot{\mho }}}_j$, and ${\overline{\mho }}_j\ge {\overline{\dot{\mho }}}_j$ for all $j=1\left(1\right)n$, then

$1-{\displaystyle \prod _{j=1}^n}{\left(1-{\underline{\Omega }}_j^{p^{*}}\right)}^{{\omega }_j}\le 1-{\displaystyle \prod _{j=1}^n}{\left(1-{\underline{\dot{\Omega }}}_j^{p^{*}}\right)}^{{\omega }_j},$ $1-{\displaystyle \prod _{j=1}^n}{\left(1-{\overline{\Omega }}_j^{p^{*}}\right)}^{{\omega }_j}\le 1-{\displaystyle \prod _{j=1}^n}{\left(1-{\overline{\dot{\Omega }}}_j^{p^{*}}\right)}^{{\omega }_j},$ ${\displaystyle \prod _{j=1}^n}{\underline{\mho }}_j^{{\omega }_j}\ge {\displaystyle \prod _{j=1}^n}{\underline{\dot{\mho }}}_j^{{\omega }_j}$, and ${\displaystyle \prod _{j=1}^n}{\overline{\mho }}_j^{{\omega }_j}\ge {\displaystyle \prod _{j=1}^n}{\overline{\dot{\mho }}}_j^{{\omega }_j}$. Therefore,

$\left(\left[\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^n}{\left(1-{\underline{\Omega }}_j^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^n}{\left(1-{\overline{\Omega }}_j^{p^{*}}\right)}^{{\omega }_j}}\right],\left[{\displaystyle \prod _{j=1}^n}{\underline{\mho }}_j^{{\omega }_j},{\displaystyle \prod _{j=1}^n}{\overline{\mho }}_j^{{\omega }_j}\right]\right)\le \left(\right[\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^n}{\left(1-{\underline{\dot{\Omega }}}_j^{p^{*}}\right)}^{{\omega }_j}},$$\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^n}{\left(1-{\overline{\dot{\Omega }}}_j^{p^{*}}\right)}^{{\omega }_j}}],[{\displaystyle \prod _{j=1}^n}{\underline{\dot{\mho }}}_j^{{\omega }_j},{\displaystyle \prod _{j=1}^n}{\overline{\dot{\mho }}}_j^{{\omega }_j}\left]\right).$

Hence, $IVp,q^{r}OFWA\left({ß}_1,{ß}_2,\dots ,{ß}_n\right)\le IVp,q^{r}OFWA\left({\dot{ß}}_1,{\dot{ß}}_2,\dots ,{\dot{ß}}_n\right).$ □

Theorem 10.

Let ${ß}_j=(\left[{\underline{\Omega }}_j,{\overline{\Omega }}_j\right],\left[{\underline{\mho }}_j,{\overline{\mho }}_j\right])$ be a collection of some $IVp,q^{r}OFNs$; if

${ß}^{+}=\left(\left[\underset{j}{max}\left\{{\underline{\Omega }}_j\right\},\underset{j}{max}\left\{{\overline{\Omega }}_j\right\}\right],\left[\underset{j}{min}\left\{{\underline{\mho }}_j\right\},\underset{j}{min}\left\{{\overline{\mho }}_j\right\}\right]\right)$ and${ß}^{-}=(\left[\underset{j}{min}\left\{{\underline{\Omega }}_j\right\},\underset{j}{min}\left\{{\overline{\Omega }}_j\right\}\right],[\underset{j}{max}\left\{{\underline{\mho }}_j\right\},$$\underset{j}{max}\left\{{\overline{\mho }}_j\right\}\left]\right)$, then

$${ß}^{-}\le IVp,q^{r}OFWA\left({ß}_1,{ß}_2,\dots ,{ß}_n\right)\le {ß}^{+}.$$

(26)

Proof. 

According to Theorem 8, $IVp,q^{r}OFWA\left({ß}^{-},{ß}^{-},\dots ,{ß}^{-}\right)={ß}^{-},$ $IVp,q^{r}OFWA\left({ß}^{+},{ß}^{+},\dots ,{ß}^{+}\right)$ = ${ß}^{+}.$

Next, in line with Theorem 9,

${ß}^{-}\le IVp,q^{r}OFWA\left({ß}_1,{ß}_2,\dots ,{ß}_n\right)\le {ß}^{+}.$ □

4.2. Interval-Valued p,q-Rung Orthopair Fuzzy Geometric AOs

Definition 21.

Let ${ß}_j=\left(j=1\left(1\right)n\right)$ be a collection of some$IVp,q^{r}OFNs$ and $={\left({\omega }_1{,}_2,\dots ,{\omega }_n\right)}^T$be the weight vector of ${ß}_j=\left(j=1\left(1\right)n\right)$, with ${\omega }_j\in \left[0,1\right],$ ${\displaystyle \sum _{j=1}^n}{\omega }_j=1.$ Then, the $IVp,q^{r}OF$ weighted geometric ($IVp,q^{r}OFWG$) operator is formulated as follows:

$$IVp,q^{r}OFWG\left({ß}_1,{ß}_2,\dots ,{ß}_n\right)={ß}_1^{{\omega }_1}\otimes {ß}_2^{{\omega }_2}\otimes \dots \otimes {ß}_n^{{\omega }_n}.$$

(27)

In the light of $IVp,q^{r}OF$ operational laws, we provide the following result.

Theorem 11.

Let ${ß}_j=\left(j=1\left(1\right)n\right)$ be a collection of some $IVp,q^{r}OFNs$; then, the aggregated result of the $IVp,q^{r}OFWG$ operator is given as:

$$IVp,q^{r}OFWG\left({ß}_1,{ß}_2,\dots ,{ß}_n\right)=\left(\left[\prod _{j=1}^n{\underline{\Omega }}_j^{{\omega }_j},\prod _{j=1}^n{\overline{\Omega }}_j^{{\omega }_j}\right],\left[\sqrt[p^{*}]{1-\prod _{j=1}^n{\left(1-{\underline{\mho }}_j^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-\prod _{j=1}^n{\left(1-{\overline{\mho }}_j^{p^{*}}\right)}^{{\omega }_j}}\right]\right).$$

(28)

Proof. 

We show Equation (28) using mathematical induction.

For $n=2$, we have

$IVp,q^{r}OFWG\left({ß}_1,{ß}_2\right)={ß}_1^{{\omega }_1}\otimes {ß}_2^{{\omega }_2}=\left(\left[{\underline{\Omega }}_1^{{\omega }_1}{\underline{\Omega }}_2^{{\omega }_2},{\overline{\Omega }}_1^{{\omega }_1}{\overline{\Omega }}_2^{{\omega }_2}\right],\left[\sqrt[p^{*}]{1-{\left(1-{\underline{\mho }}_1^{p^{*}}\right)}^{{\omega }_1}{\left(1-{\underline{\mho }}_2^{p^{*}}\right)}^{{\omega }_2}},\sqrt[p^{*}]{1-{\left(1-{\overline{\mho }}_1^{p^{*}}\right)}^{{\omega }_1}{\left(1-{\overline{\mho }}_2^{p^{*}}\right)}^{{\omega }_2}}\right]\right)=\left(\left[{\displaystyle \prod _{j=1}^2}{\underline{\Omega }}_j^{{\omega }_j},{\displaystyle \prod _{j=1}^2}{\overline{\Omega }}_j^{{\omega }_j}\right],\left[\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^2}{\left(1-{\underline{\mho }}_j^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^2}{\left(1-{\overline{\mho }}_j^{p^{*}}\right)}^{{\omega }_j}}\right]\right).$

Thus, Equation (28) is kept for $j=2$.

Suppose Equation (28) holds for $j=k$. Then, we have

$IVp,q^{r}OFWG\left({ß}_1,{ß}_2,\dots ,{ß}_k\right)=\left(\left[\left[{\displaystyle \prod _{j=1}^k}{\underline{\Omega }}_j^{{\omega }_j},{\displaystyle \prod _{j=1}^k}{\overline{\Omega }}_j^{{\omega }_j}\right],\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^k}{\left(1-{\underline{\mho }}_j^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^k}{\left(1-{\overline{\mho }}_j^{p^{*}}\right)}^{{\omega }_j}}\right]\right)$.

Now, we check for $j=k+1$

$IVp,q^{r}OFWG\left({ß}_1,{ß}_2,\dots ,{ß}_{k+1}\right)={\otimes }_{j=1}^{k+1}\left({ß}_j^{{\omega }_j}\right)={\otimes }_{j=1}^k\left({ß}_j^{{\omega }_j}\right)\otimes {ß}_{k+1}^{{\omega }_{k+1}}=\left(\left[{\displaystyle \prod _{j=1}^k}{\underline{\Omega }}_j^{{\omega }_j},{\displaystyle \prod _{j=1}^k}{\overline{\Omega }}_j^{{\omega }_j}\right],\left[\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^k}{\left(1-{\underline{\mho }}_j^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^k}{\left(1-{\overline{\mho }}_j^{p^{*}}\right)}^{{\omega }_j}}\right]\right)\otimes \left(\left[{\underline{\Omega }}_{k+1}^{{\omega }_j},{\overline{\Omega }}_{k+1}^{{\omega }_{k+1}}\right],\left[\sqrt[p^{*}]{1-{\left(1-{\underline{\mho }}_{k+1}^{p^{*}}\right)}^{{\omega }_{k+1}}},\sqrt[p^{*}]{1-{\left(1-{\overline{\mho }}_{k+1}^{p^{*}}\right)}^{{\omega }_{k+1}}}\right]\right)=\left(\left[{\displaystyle \prod _{j=1}^{k+1}}{\underline{\Omega }}_j^{{\omega }_j},{\displaystyle \prod _{j=1}^{k+1}}{\overline{\Omega }}_j^{{\omega }_j}\right],\left[\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^{k+1}}{\left(1-{\underline{\mho }}_j^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^{k+1}}{\left(1-{\overline{\mho }}_j^{p^{*}}\right)}^{{\omega }_j}}\right]\right).$

As a result, the proof is finished. □

Theorem 12.

Let ${ß}_j=\left(\left[{\underline{\Omega }}_j,{\overline{\Omega }}_j\right],\left[{\underline{\mho }}_j,{\overline{\mho }}_j\right]\right)\left(j=1\left(1\right)n\right)$ be a collection of some $IVp,q^{r}OFNs$ such that ${ß}_j=ß=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$ for all j. Then,

$$IVp,q^{r}OFWG\left({ß}_1,{ß}_2,\dots ,{ß}_n\right)=ß.$$

(29)

Proof. 

${ß}_j=ß=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right)$ for all j. Therefore,

$IVp,q^{r}OFWG\left({ß}_1,{ß}_2,\dots ,{ß}_n\right)=\left(\left[{\displaystyle \prod _{j=1}^n}{\underline{\Omega }}_j^{{\omega }_j},{\displaystyle \prod _{j=1}^n}{\overline{\Omega }}_j^{{\omega }_j}\right],\left[\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^n}{\left(1-{\underline{\mho }}_j^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^n}{\left(1-{\overline{\mho }}_j^{p^{*}}\right)}^{{\omega }_j}}\right]\right)=\left(\left[{\displaystyle \prod _{j=1}^n}{\underline{\Omega }}^{{\omega }_j},{\displaystyle \prod _{j=1}^n}{\overline{\Omega }}^{{\omega }_j}\right],\left[\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^n}{\left(1-{\underline{\mho }}^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^n}{\left(1-{\overline{\mho }}^{p^{*}}\right)}^{{\omega }_j}}\right]\right)=\left(\left[{\underline{\Omega }}^{{\displaystyle \sum _{j=1}^n}j},{\overline{\Omega }}^{{\displaystyle \sum _{j=1}^n}j}\right],\left[\sqrt[p^{*}]{1-{\left(1-{\underline{\mho }}^{p^{*}}\right)}^{{\displaystyle \sum _{j=1}^n}j}},\sqrt[p^{*}]{1-{\left(1-{\overline{\mho }}^{p^{*}}\right)}^{{\displaystyle \sum _{j=1}^n}j}}\right]\right)=\left(\left[\underline{\Omega },\overline{\Omega }\right],\left[\underline{\mho },\overline{\mho }\right]\right)=ß$□

Theorem 13.

Let ${ß}_j=\left(\left[{\underline{\Omega }}_j,{\overline{\Omega }}_j\right],\left[{\underline{\mho }}_j,{\overline{\mho }}_j\right]\right)$ and ${\dot{ß}}_j=\left(\left[{\underline{\dot{\Omega }}}_j,{\overline{\dot{\Omega }}}_j\right],\left[{\underline{\dot{\mho }}}_j,{\overline{\dot{\mho }}}_j\right]\right)\left(j=1\left(1\right)n\right)$ be any two collections of some $IVp,q^{r}OFNs$ such that ${\underline{\Omega }}_j\le {\underline{\dot{\Omega }}}_j$, ${\overline{\Omega }}_j\le {\overline{\dot{\Omega }}}_j$, ${\underline{\mho }}_j\ge {\underline{\dot{\mho }}}_j$, and ${\overline{\mho }}_j\ge {\overline{\dot{\mho }}}_j$ for all j. Then,

$$IVp,q^{r}OFWG\left({ß}_1,{ß}_2,\dots ,{ß}_n\right)\le IVp,q^{r}OFWG\left({\dot{ß}}_1,{\dot{ß}}_2,\dots ,{\dot{ß}}_n\right).$$

(30)

Proof. 

As ${\underline{\Omega }}_j\le {\underline{\dot{\Omega }}}_j$, ${\overline{\Omega }}_j\le {\overline{\dot{\Omega }}}_j$, ${\underline{\mho }}_j\ge {\underline{\dot{\mho }}}_j$, and ${\overline{\mho }}_j\ge {\overline{\dot{\mho }}}_j$ for all $j=1\left(1\right)n$, then

${\displaystyle \prod _{j=1}^n}{\underline{\Omega }}_j^{{\omega }_j}\ge {\displaystyle \prod _{j=1}^n}{\underline{\dot{\Omega }}}_j^{{\omega }_j}$, ${\displaystyle \prod _{j=1}^n}{\overline{\Omega }}_j^{{\omega }_j}\ge {\displaystyle \prod _{j=1}^n}{\overline{\dot{\Omega }}}_j^{{\omega }_j}$, ${\displaystyle \prod _{j=1}^n}{\left(1-{\underline{\mho }}_j^{p^{*}}\right)}^{{\omega }_j}\le {\displaystyle \prod _{j=1}^n}{\left(1-{\underline{\dot{\mho }}}_j^{p^{*}}\right)}^{{\omega }_j},$ and ${\displaystyle \prod _{j=1}^n}{\left(1-{\overline{\mho }}_j^{p^{*}}\right)}^{{\omega }_j}$ $\le {\displaystyle \prod _{j=1}^n}{\left(1-{\overline{\dot{\mho }}}_j^{p^{*}}\right)}^{{\omega }_j}.$ Therefore,

$\left(\left[{\displaystyle \prod _{j=1}^n}{\underline{\Omega }}_j^{{\omega }_j},{\displaystyle \prod _{j=1}^n}{\overline{\Omega }}_j^{{\omega }_j}\right],\left[\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^n}{\left(1-{\underline{\mho }}_j^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^n}{\left(1-{\overline{\mho }}_j^{p^{*}}\right)}^{{\omega }_j}}\right]\right)\le \left(\left[{\displaystyle \prod _{j=1}^n}{\underline{\dot{\Omega }}}_j^{{\omega }_j},{\displaystyle \prod _{j=1}^n}{\overline{\dot{\Omega }}}_j^{{\omega }_j}\right],\left[\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^n}{\left(1-{\underline{\dot{\mho }}}_j^{p^{*}}\right)}^{{\omega }_j}},\sqrt[p^{*}]{1-{\displaystyle \prod _{j=1}^n}{\left(1-{\overline{\dot{\mho }}}_j^{p^{*}}\right)}^{{\omega }_j}}\right]\right).$

Hence, $IVp,q^{r}OFWG\left({ß}_1,{ß}_2,\dots ,{ß}_n\right)\le IVp,q^{r}OFWG\left({\dot{ß}}_1,{\dot{ß}}_2,\dots ,{\dot{ß}}_n\right).$ □

Theorem 14.

Let ${ß}_j=\left(\left[{\underline{\Omega }}_j,{\overline{\Omega }}_j\right],\left[{\underline{\mho }}_j,{\overline{\mho }}_j\right]\right)$ be a collection of some $IVp,q^{r}OFNs$; if

${ß}^{+}=\left(\left[\underset{j}{max}\left\{{\underline{\Omega }}_j\right\},\underset{j}{max}\left\{{\overline{\Omega }}_j\right\}\right],\left[\underset{j}{min}\left\{{\underline{\mho }}_j\right\},\underset{j}{min}\left\{{\overline{\mho }}_j\right\}\right]\right)$ and${ß}^{-}=(\left[\underset{j}{min}\left\{{\underline{\Omega }}_j\right\},\underset{j}{min}\left\{{\overline{\Omega }}_j\right\}\right],[\underset{j}{max}\left\{{\underline{\mho }}_j\right\},$$\underset{j}{max}\left\{{\overline{\mho }}_j\right\}\left]\right)$, then

$${ß}^{-}\le IVp,q^{r}OFWG({ß}_1,{ß}_2,\dots ,{ß}_n)\le {ß}^{+}.$$

(31)

Proof. 

According to Theorem 12, $IVp,q^{r}OFWG({ß}^{-},{ß}^{-},\dots ,{ß}^{-})={ß}^{-},$ $IVp,q^{r}OFWG({ß}^{+},{ß}^{+},\dots ,{ß}^{+})={ß}^{+}.$

Next, in line with Theorem 13,

${ß}^{-}\le IVp,q^{r}OFWG\left({ß}_1,{ß}_2,\dots ,{ß}_n\right)\le {ß}^{+}.$ □

5. MCGDM Based on CRITIC and Exp-TODIM

This section is dedicated to building the CRITIC technique and exp-TODIM framework for handling MCDM problems with $IVp,q^{r}OF$ data. In general, the CRITIC technique assigns weights to the DM problem based on the level of contrast and conflict evaluation [45]. It is designed to prevent the subjectivity that makes choosing one criterion above others a challenging task when many opinions have been provided [46,47]. As a result, the CRITIC technique was chosen to determine the weight of each criterion in order to avoid conceptual synchronization of alternatives influencing the outcome of the present study. The exp-TODIM methodology suggested by [48] is a novel form of the TODIM approach that is appropriate for multi-criteria techniques with minimal application complexity and a high prediction capability since it is based on a value function that is more consistent with prospect theory. Considering the said importance of exp-TODIM, it was further decided to extend the exp-TODIM framework in the setting of an $IVp,q^{r}OFS$.

5.1. Interval-Valued p,q-Rung Orthopair Fuzzy CRITIC Approach

It is often essential in multi-criteria decision issues to assign different weights to distinct criteria for the following two reasons. (i). For different study areas and assessment goals, different weights are usually allocated to each criterion because of differences in focus and emphasis. For instance, technology is given the most weight when evaluating high-end manufacturing suppliers such as gas turbines [49]. However, environmental factors will be given a higher weight in the assessment of suppliers in industries that necessitate precautions to protect the environment, such as paper-making firms [50]. (ii). Making full use of the details contained in the findings of the assessment can help to distinguish the evaluation outcomes. Assessment data typically incorporate mathematical properties such as variance, extreme deviation, information entropy, etc. Using information entropy as a case in point, the higher the criterion’s information entropy, the greater its data dispersion; hence, it should be allocated more weight. Weighting methods that are commonly employed comprise subjective methods, objective methods, and approaches that mix subjective and objective weights. Subjective weighting strategies that are relevant include AHP [51], DEMATEL [52], and others. The subjective technique can reflect DEs’ understanding and expertise, but it is extremely personal. Objective weighing approaches, like the entropy weight method [30], the CRITIC method [53], and others, can represent the influence of historical data on the weight of evaluation criteria and are extremely objective yet data-dependent. The integrated method [54], which refers to the integrating of subjective and objective weights via additive synthesis, multiplicative synthesis, subjective correction based on objective, range maximization, or other pairing methods, is a more comprehensive strategy, but it is time-consuming.

Many academics utilize subjective methods to assign weights. According to Dragan Pamucar [55], the classic BWM technique misses the potential of numerous evaluation factors having identical meanings in the individual’s preference. He refined the classical BWM approach, minimized the number of comparisons, and demonstrated the revised method’s practicalities through illustration. Focusing on the LBWA method, Sanjib Biswas [56] examined the college and university choice based on location from the perspective of business school candidates, concluding that transportation convenience and commute time are the most important aspects. Alptekin Ulutas [57] applied the FUCOM approach to assess the placement of textile manufacturing facilities in a Turkish province. The FUCOM approach has more dependable standard weight coefficients than the AHP and BWM methods, which lowers the subjective impact and variability of expert choice on the ultimate value of the standard weight. Meanwhile, the FUCOM method’s number of pairwise comparisons is substantially decreased.

In terms of objective weighing methodologies, Bai et al. [58] used the TODIM method to assess the social sustainability of an Iranian manufacturing company’s suppliers. The TODIM method relies on prospect theory’s value function, and it sets the relative superiority function of a solution about other solutions centered on DEs’ psychological behavior, and it opts for solutions based on the level of superiority. Kaviani et al. [59] utilized gray system theory regarding supplier assessment in the energy sector, taking into account DEs’ hesitation caused by a lack of expertise and data. Wang et al. [60] created an adaptive and applicable supplier evaluation and selection framework for the Vietnamese apparel sector founded on the TOPSIS technique. Mukhametzyanov [61] performed a comparative examination of objective approaches for establishing criteria weights in MCDM and found that all objective approaches for assessing criteria weights in MCDM tasks are suspect. The authors of [62] used the DEA method to assess sustainable suppliers in the plastics manufacturing sector concerning their inputs and final products.

Stojanovi I et al. [63] evaluated the logistics efficacy of the GCC countries using an integrated subjective and objective evaluation strategy and chose the United Arab Emirates as a regional logistics center that properly connects the GCC to the global supply chain system. Alossta et al. [63] employed the AHP-RAFIS approach to tackle the site selection issue facing emergency medical centers, and they concluded that the road network is the optimum construction site compared to other sites. Bad et al. [64] adopted the FUCOM-MARCOS approach to assess the Nigerian textile industry’s green innovation aptitude and developed a set of green innovation assessment techniques that may be extended to other industries. Sssessment problems frequently include fuzzy DM; thus, fuzzy theory is commonly employed, usually in conjunction with other approaches like Fuzzy-AHP [65], Fuzzy-TOPSIS [66], and Fuzzy-TODIM [67].

In a nutshell, subjective, objective, and combined subjective and objective weighting approaches each have merits and downsides. Questionnaires, consultations with specialists, and other tasks are often employed in both subjective and combined subjective–objective weighting procedures, which take a long time and have a huge burden. As a result, they are not ideal for quick, short-term assessments. The supplier evaluation in this article is based on short-term dynamic objective data; thus, the objective weighting approach was employed.

The mathematical structure of the devised CRITIC technique is detailed in the subsequent steps.

Step $1^{\prime }$: Decision process initiation

Let $\mathbb{O}=\left\{{\mathbb{O}}_i;i=1\left(1\right)m\right\}$ stand for the collection of alternatives, $\mathbb{C}=\left\{{\mathbb{C}}_j;j=1\left(1\right)n\right\}$ presents the set of possible criteria, and $\mathbb{D}=\left\{{\mathbb{D}}_k;k=1\left(1\right)t\right\}$ denotes the group of t DEs. The weight vector of criteria $\omega ={\left({\omega }_j;j=1\left(1\right)n\right)}^T$ is unknown and the significance degree for the group of DEs is given by the vector $\omega ={\left({\omega }_k;k=1\left(1\right)t\right)}^T$, with ${\omega }_k\ge 0,$ ${\displaystyle \sum _{k=1}^t}\omega k=1.$ The evaluation value of alternative ${\mathbb{O}}_i$ with respect to criteria ${\mathbb{C}}_j$ provided by DE ${\mathbb{D}}_k$ is represented in terms of the $IVp,q^{r}OFN$ ${ß}_{ij}^k=\left(\left[{\underline{\Omega }}_{ij}^k,{\overline{\Omega }}_{ij}^k\right],\left[{\underline{\mho }}_{ij}^k,{\overline{\mho }}_{ij}^k\right]\right)$ to build the $IVp,q^{r}OF$ decision matrix (DMX) ${X^{⏜}}^k={\left({ß}_{ij}^k\right)}_{m\times n};$ $i=1\left(1\right)m$ and $j=1\left(1\right)n$, shown as follows:

$${X^{⏜}}^k=\begin{array}{cccccc}& {\mathbb{C}}_1& \cdots & {\mathbb{C}}_j& \cdots & {\mathbb{C}}_n\\ {\mathbb{O}}_1& \left(\left[{\underline{\Omega }}_{11}^k,{\overline{\Omega }}_{11}^k\right],\left[{\underline{\mho }}_{11}^k,{\overline{\mho }}_{11}^k\right]\right)& \cdots & \left(\left[{\underline{\Omega }}_{1j}^k,{\overline{\Omega }}_{1j}^k\right],\left[{\underline{\mho }}_{1j}^k,{\overline{\mho }}_{1j}^k\right]\right)& \cdots & \left(\left[{\underline{\Omega }}_{1n}^k,{\overline{\Omega }}_{1n}^k\right],\left[{\underline{\mho }}_{1n}^k,{\overline{\mho }}_{1n}^k\right]\right)\\ ⋮& ⋮& \ddots & ⋮& \ddots & ⋮\\ {\mathbb{O}}_i& \left(\left[{\underline{\Omega }}_{i1}^k,{\overline{\Omega }}_{i1}^k\right],\left[{\underline{\mho }}_{i1}^k,{\overline{\mho }}_{i1}^k\right]\right)& \cdots & \left(\left[{\underline{\Omega }}_{ij}^k,{\overline{\Omega }}_{ij}^k\right],\left[{\underline{\mho }}_{ij}^k,{\overline{\mho }}_{ij}^k\right]\right)& \cdots & \left(\left[{\underline{\Omega }}_{in}^k,{\overline{\Omega }}_{in}^k\right],\left[{\underline{\mho }}_{in}^k,{\overline{\mho }}_{in}^k\right]\right)\\ ⋮& ⋮& \ddots & ⋮& \ddots & ⋮\\ {\mathbb{O}}_m& \left(\left[{\underline{\Omega }}_{m1}^k,{\overline{\Omega }}_{m1}^k\right],\left[{\underline{\mho }}_{m1}^k,{\overline{\mho }}_{m1}^k\right]\right)& \cdots & \left(\left[{\underline{\Omega }}_{mj}^k,{\overline{\Omega }}_{mj}^k\right],\left[{\underline{\mho }}_{mj}^k,{\overline{\mho }}_{mj}^k\right]\right)& \cdots & \left(\left[{\underline{\Omega }}_{mn}^k,{\overline{\Omega }}_{mn}^k\right],\left[{\underline{\mho }}_{mn}^k,{\overline{\mho }}_{mn}^k\right]\right)\end{array},$$

where $k=1\left(1\right)t$.

Step $2^{\prime }$: Normalization

Most MCGDM problems have two sorts of criteria: cost type (CT) and benefit type (BT). To avoid unintended consequences from distinct types, we must transform different criteria into the same type. In general, the CT is transformed to the BT by employing Equation (32). For the sake of simplicity, the generated result is still written as ${X^{⏜}}^k={\left({ß}_{ij}\right)}_{m\times n}.$

$${ß}_{ij}=\left\{\begin{array}{cc}\hfill {ß}_{ij},\hfill & \hfill if{\mathbb{C}}_j\mathrm{is}\mathrm{BT}\hfill \\ \hfill {ß}_{ij}^c,\hfill & \hfill if{\mathbb{C}}_j\mathrm{is}\mathrm{CT},\hfill \end{array}\right.$$

(32)

where ${ß}_{ij}^c$ denotes the complement of ${ß}_{ij}$.

Step $3^{\prime }$: Combined judgment matrix

The collective DMX X is acquired by aggregating the individual DMXs ${X^{⏜}}^k;k=1\left(1\right)t$ in light of the DEs’ weight based on the $IVp,q^{r}OFWA$ operator

$$\begin{array}{c}\hfill {ß}_{ij}=\left(\left[{\underline{\Omega }}_{ij},{\overline{\Omega }}_{ij}\right],\left[{\underline{\mho }}_{ij},{\overline{\mho }}_{ij}\right]\right)=IVp,q^{r}OFWA\left({ß}_{ij}^1,{ß}_{ij}^2,\dots ,{ß}_{ij}^t\right)=\hfill \\ \hfill \left(\left[\sqrt[p^{*}]{1-\prod _{k=1}^t{\left(1-{\underline{\Omega }}_{ij}^{p^{*}}\right)}^{{\omega }_k}},\sqrt[p^{*}]{1-\prod _{k=1}^t{\left(1-{\overline{\Omega }}_{ij}^{p^{*}}\right)}^{{\omega }_k}}\right],\left[\prod _{k=1}^t{\underline{\mho }}_{ij}^{{\omega }_k},\prod _{k=1}^t{\overline{\mho }}_{ij}^{{\omega }_k}\right]\right),\end{array}$$

(33)

or based on the $IVp,q^{r}OFWG$ operator

$$\begin{array}{c}\hfill {ß}_{ij}=\left(\left[{\underline{\Omega }}_{ij},{\overline{\Omega }}_{ij}\right],\left[{\underline{\mho }}_{ij},{\overline{\mho }}_{ij}\right]\right)=IVp,q^{r}OFWA\left({ß}_{ij}^1,{ß}_{ij}^2,\dots ,{ß}_{ij}^t\right)=\hfill \\ \hfill \left(\left[\prod _{k=1}^t{\underline{\Omega }}_{ij}^{{\omega }_k},\prod _{k=1}^t{\overline{\Omega }}_{ij}^{{\omega }_k}\right],\left[\sqrt[q^{*}]{1-\prod _{k=1}^t{\left(1-{\underline{\mho }}_{ij}^{q^{*}}\right)}^{{\omega }_k}},\sqrt[q^{*}]{1-\prod _{k=1}^t{\left(1-{\overline{\mho }}_{ij}^{q^{*}}\right)}^{{\omega }_k}}\right]\right).\end{array}$$

(34)

Step $4^{\prime }$: Score matrix

The score matrix $\mathfrak{S}={\left(S^{⏜}\left({ß}_{ij}\right)\right)}_{m\times n}$ is determined, where $S^{⏜}\left({ß}_{ij}\right)$ can be derived using Equation (35):

$$S^{⏜}\left({ß}_{ij}\right)=\frac{1}{2}\left(\left({{\underline{\Omega }}_{ij}}^{p^{*}}-{{\underline{\mho }}_{ij}}^{q^{*}}\right)\left(1+\sqrt[\ell ]{1-\left({\underline{\Omega }}_{ij}^{p^{*}}+{\underline{\mho }}_{ij}^{q^{*}}\right)}\right)+\left({\overline{\Omega }}_{ij}^{p^{*}}-{\overline{\mho }}_{ij}^{q^{*}}\right)\left(1+\sqrt[\ell ]{1-\left({\overline{\Omega }}_{ij}^{p^{*}}+{\overline{\mho }}_{ij}^{q^{*}}\right)}\right)\right).$$

(35)

Step $5^{\prime }$: Normalized score matrix

The score values of the score matrix $\mathfrak{S}={\left(S^{⏜}\left({ß}_{ij}\right)\right)}_{m\times n}$ are normalized through Equation (36)

$$\ddot{S}\left({ß}_{ij}\right)=\frac{S^{⏜}\left({ß}_{ij}\right)-\underset{i}{min}{S}^{⏜}\left({ß}_{ij}\right)}{\underset{i}{max}{S}^{⏜}\left({ß}_{ij}\right)-\underset{i}{min}{S}^{⏜}\left({ß}_{ij}\right)}.$$

(36)

Step $6^{\prime }$: Correlation coefficient

The correlation coefficient of criteria ${\mathbb{C}}_j$ to criteria ${\mathbb{C}}_r$ is computed with Equation (37)

$${\varsigma }_{jr}=\frac{{\displaystyle \sum _{i=1}^m}\left(\ddot{S}\left({ß}_{ij}\right)-\overline{\ddot{S}\left({ß}_j\right)}\right)\left(\ddot{S}\left({ß}_{ir}\right)-\overline{\ddot{S}\left({ß}_r\right)}\right)}{\sqrt{{\displaystyle \sum _{i=1}^m}{\left(\ddot{S}\left({ß}_{ij}\right)-\overline{\ddot{S}\left({ß}_j\right)}\right)}^2·{\displaystyle \sum _{i=1}^m}{\left(\ddot{S}\left({ß}_{ir}\right)-\overline{\ddot{S}\left({ß}_r\right)}\right)}^2}},$$

(37)

where $\overline{\ddot{S}\left({ß}_j\right)}=\frac{1}{m}·{\displaystyle \sum _{i=1}^m}\ddot{S}\left({ß}_{ij}\right)$ and $\overline{\ddot{S}\left({ß}_r\right)}=\frac{1}{m}·{\displaystyle \sum _{i=1}^m}\ddot{S}\left({ß}_{ir}\right)$.

Step $7^{\prime }$: Standard deviation

The standard deviation for each criterion is determined according to Equation (38)

$${\mathfrak{F}}_j=\sqrt{\frac{1}{m-1}\sum _{i=1}^m{\left(\ddot{S}\left({ß}_{ij}\right)-\overline{\ddot{S}\left({ß}_j\right)}\right)}^2},j=1\left(1\right)n.$$

(38)

Step $8^{\prime }$: Information content

The information content of each criterion is derived by Equation (39)

$${\mathcal{I}}_j={\mathfrak{F}}_j\sum _{r=1}^n\left(1-{\varsigma }_{jr}\right),j=1\left(1\right)n.$$

(39)

Step $9^{\prime }$: Weight values

The weight of each criterion is derived through Equation (40)

$${\omega }_j=\frac{{\mathcal{I}}_j}{{\displaystyle \sum _{j=1}^n}{\mathcal{I}}_j},j=1\left(1\right)n.$$

(40)

5.2. Interval-Valued p,q-Rung Orthopair Fuzzy Exp-TODIM with Unknown Weight

Steps 1–3: Same as outlined in the CRITIC approach.

Step 4: Criteria weights

Ascertain the weight vector of criteria via the proposed $IVp,q^{r}OF$ CRITIC approach outlined in subpart Section 5.1.

Step 5: Relative weights

Using Equation (41), calculate the relative weight ${\sout{w}}_j$ of the criteria ${\mathbb{C}}_j$ to the reference point ${\mathbb{C}}_t$

$${{\displaystyle \sout{w}}}_j={\omega }_j/{\omega }_t,j,t=1\left(1\right)n$$

(41)

where ${\omega }_t=max\left\{{\omega }_j|j=1\left(1\right)n\right\}$.

Step 6: Dominance of alternatives

Compute the dominance of alternative ${\mathbb{O}}_i$ over other alternative ${\mathbb{O}}_{ß}$ in light of Equation (42)

$${ð}_i=\sum _{ß=1}^mð\left({\mathbb{O}}_i,{\mathbb{O}}_{ß}\right)=\sum _{ß=1}^m\sum _{j=1}^n{\Phi }_j\left({\mathbb{O}}_i,{\mathbb{O}}_{ß}\right)$$

(42)

$ð\left({\mathbb{O}}_i,{\mathbb{O}}_{ß}\right)={\displaystyle \sum _{j=1}^n}{\Phi }_j\left({\mathbb{O}}_i,{\mathbb{O}}_{ß}\right)$; the preference index ${\Phi }_j\left({\mathbb{O}}_i,{\mathbb{O}}_{ß}\right)$ can be found according to Equation (43)

$${\Phi }_j\left({\mathbb{O}}_i,{\mathbb{O}}_{ß}\right)=\left\{\begin{array}{cc}\frac{{\sout{w}}_j\left(1-{10}^{-\rho d\left({ß}_{ij},{ß}_{ßj}\right)}\right)}{{\sum }_{j=1}^n{\sout{w}}_j},& \hfill \mathrm{if} {ß}_{ij}>{ß}_{ßj}\hfill \\ 0,& \hfill \mathrm{if} {ß}_{ij}={ß}_{ßj}\hfill \\ \frac{-1}{\delta }\frac{{\displaystyle \sum _{j=1}^n}{\sout{w}}_j\left(1-{10}^{-\rho d\left({ß}_{ij},{ß}_{ßj}\right)}\right)}{{\sout{w}}_j},& \hfill \mathrm{if} {ß}_{ij}<{ß}_{ßj},\hfill \end{array}\right.$$

(43)

where $\rho$ and δ are both parameters. The parameter $\rho$ represents the significance of the decision depending on the agent’s perception. Normally, the amplification parameter δ is used to alter the gain and loss for different responses.

Step 7: Overall dominance value

Calculate the overall dominance value for each alternative ${\mathbb{O}}_i$ by Formula (44)

$$ϝ\left({\mathbb{O}}_i\right)=\frac{{ð}_i-\underset{i}{min}{ð}_i}{\underset{i}{max}{ð}_i-\underset{i}{min}{ð}_i};i=1\left(1\right)m.$$

(44)

Step 8: Ranking

Evaluate the best possible alternative using $ϝ\left({\mathbb{O}}_i\right)$ values. The greater the value of $ϝ\left({\mathbb{O}}_i\right)$, the better the alternative ${\mathbb{O}}_i$.

6. Application and Sensitivity Analysis

6.1. Illustrative Example

In this part, the suggested approach is presented utilizing a practical MCGDM problem of green supplier selection (GSS) with $IVp,q^{r}OF$ input.

Problem Background

Supply chain management (SCM) entails acquiring raw materials for manufacturing and delivering completed items to clients. It is an extensive procedure that demands comprehensive effectiveness to meet consumer needs and company objectives [68]. With rising competition and environmental risks, the value of green suppliers cannot be overstated. Green supply chain management (GSCM) incorporates ecological considerations into all supply chain activities and has gained worldwide recognition because of environmental regulations and the sustainable development goals (SDGs) [69,70].

Supplier selection is an important aspect of SCM, and GSS is the initial move toward acknowledging and dealing with environmental hazards in business operations. GSS entails incorporating ecological considerations into traditional supplier selection by taking into account criteria like green products, pollution management, carbon policy, and environmental protection [71]. GSS is a challenging and complicated assignment that can be classified as an MCGDM task. Several studies have established DM algorithms to tackle GSS challenges in several fields [72,73,74,75]. The selection of green suppliers is critical to achieving SDGs for businesses by boosting profit and the quality of products. As a result, their choice should be carefully considered in order to balance corporate earnings with SDGs.

To put the proposed approach into practice, a real-world problem of picking the best green provider for delivering fruits and vegetables in a retail chain is analyzed in the setting of $IVp,q^{r}OFSs$.

Example 8.

Customers in agro-based food sectors are becoming more environmentally sensitive, preferring to purchase products from firms that have set up themselves as environmentally conscientious or have constructed a green image for themselves. Organizations or suppliers of food goods are striving for the top rank amid ever-increasing rivalries. In this case, merchants are not only curious about choices, but they also face the daunting chore of selecting the finest provider from a long list. As a result, retailers that supply these food goods must create an approach for evaluating qualities and selecting the best supplier(s). A strategic and strong partnership between the provider and the retailer or procurer is required. The GSS DM challenge entails selecting the best provider based on a variety of evaluation criteria. It is necessary to discuss such selection criteria in terms of $IVp,q^{r}OFNs$. Assume that a well-established retail chain specializing in fast-moving consumer goods (FMCGs) needs to select the best green provider for fruits and vegetables from a list of five suppliers represented by ${\mathbb{O}}_i;i=1\left(1\right)5$. An advisory board of three DEs is formed and symbolized by ${\mathbb{D}}_k;k=1\left(1\right)3$, whose weight vector $={\left(0.3,0.4,0.3\right)}^T$, in order to evaluate the service providers. These suppliers will be assessed based on a variety of critical characteristics, including traditional features and ecological traits that adhere to ecologically friendly methods. These types of decisions are extremely challenging because there is a trade-off between both quantity and quality. Through DE feedback, research analyses, and questionnaires, a survey approach was utilized to identify the critical qualities for assessing and choosing the best green supplier. A set of the six most important criteria was determined and is summarized below:

1. 

Price $\left({\mathbb{C}}_1\right)$;

2. 

Quality $\left({\mathbb{C}}_2\right)$;

3. 

Transportation cost $\left({\mathbb{C}}_3\right)$;

4. 

Delivery $\left({\mathbb{C}}_4\right)$;

5. 

Green competencies $\left({\mathbb{C}}_5\right)$;

6. 

Use of eco-friendly technology $\left({\mathbb{C}}_6\right)$.

It is worth mentioning that criteria ${\mathbb{C}}_1$ and ${\mathbb{C}}_3$ are CT, while the rest are BT.

To obtain the evaluation data of the DEs corresponding to the alternatives, a survey was carried out with respect to the accessible suppliers, and the DEs were requested to offer their rating values corresponding to the available alternatives ${\mathbb{O}}_i;i=1\left(1\right)5$ considering the established criteria ${\mathbb{C}}_j;j=1\left(1\right)6$. After examining the information regarding the available suppliers, the DEs provided the following DMXs ${X^{⏜}}^k={\left({ß}_{ij}^k\right)}_{5\times 6};$ $k=1\left(1\right)3$, as shown in

Table 2. Green supplier ratings provided by three DEs using the evaluation criteria.

		${\mathbb{C}}_1$	${\mathbb{C}}_2$	${\mathbb{C}}_3$	${\mathbb{C}}_4$	${\mathbb{C}}_5$	${\mathbb{C}}_6$
${\mathbb{D}}_1$	${\mathbb{O}}_1$	$\left(\left[0.30,0.60\right],\left[0.25,0.30\right]\right)$	$\left(\left[0.16,0.26\right],\left[0.15,0.20\right]\right)$	$\left(\left[0.25,0.30\right],\left[0.35,0.40\right]\right)$	$\left(\left[0.15,0.20\right],\left[0.10,0.20\right]\right)$	$\left(\left[0.60,0.70\right],\left[0.40,0.44\right]\right)$	$\left(\left[0.66,0.76\right],\left[0.56,0.62\right]\right)$
	${\mathbb{O}}_2$	$\left(\left[0.40,0.60\right],\left[0.20,0.42\right]\right)$	$\left(\left[0.40,0.50\right],\left[0.28,0.36\right]\right)$	$\left(\left[0.50,0.60\right],\left[0.14,0.18\right]\right)$	$\left(\left[0.65,0.75\right],\left[0.45,0.50\right]\right)$	$\left(\left[0.46,0.52\right],\left[0.30,0.35\right]\right)$	$\left(\left[0.45,0.55\right],\left[0.40,0.45\right]\right)$
	${\mathbb{O}}_3$	$\left(\left[0.50,0.70\right],\left[0.20,0.35\right]\right)$	$\left(\left[0.66,0.76\right],\left[0.18,0.25\right]\right)$	$\left(\left[0.35,0.50\right],\left[0.25,0.30\right]\right)$	$\left(\left[0.66,0.70\right],\left[0.24,0.28\right]\right)$	$\left(\left[0.45,0.55\right],\left[0.42,0.48\right]\right)$	$\left(\left[0.70,0.75\right],\left[0.20,0.30\right]\right)$
	${\mathbb{O}}_4$	$\left(\left[0.45,0.65\right],\left[0.30,0.45\right]\right)$	$\left(\left[0.20,0.60\right],\left[0.45,0.56\right]\right)$	$\left(\left[0.45,0.55\right],\left[0.28,0.38\right]\right)$	$\left(\left[0.62,0.67\right],\left[0.50,0.60\right]\right)$	$\left(\left[0.35,0.40\right],\left[0.30,0.40\right]\right)$	$\left(\left[0.33,0.43\right],\left[0.10,0.15\right]\right)$
	${\mathbb{O}}_5$	$\left(\left[0.10,0.30\right],\left[0.45,0.50\right]\right)$	$\left(\left[0.30,0.60\right],\left[0.40,0.50\right]\right)$	$\left(\left[0.65,0.70\right],\left[0.46,0.52\right]\right)$	$\left(\left[0.44,0.48\right],\left[0.40,0.45\right]\right)$	$\left(\left[0.70,0.75\right],\left[0.20,0.30\right]\right)$	$\left(\left[0.36,0.40\right],\left[0.12,0.22\right]\right)$
${\mathbb{D}}_2$	${\mathbb{O}}_1$	$\left(\left[0.32,0.60\right],\left[0.20,0.50\right]\right)$	$\left(\left[0.18,0.26\right],\left[0.12,0.20\right]\right)$	$\left(\left[0.28,0.30\right],\left[0.32,0.40\right]\right)$	$\left(\left[0.35,0.45\right],\left[0.08,0.18\right]\right)$	$\left(\left[0.62,0.70\right],\left[0.38,0.44\right]\right)$	$\left(\left[0.68,0.78\right],\left[0.54,0.64\right]\right)$
	${\mathbb{O}}_2$	$\left(\left[0.42,0.60\right],\left[0.18,0.42\right]\right)$	$\left(\left[0.42,0.50\right],\left[0.26,0.36\right]\right)$	$\left(\left[0.52,0.60\right],\left[0.15,0.20\right]\right)$	$\left(\left[0.68,0.75\right],\left[0.42,0.50\right]\right)$	$\left(\left[0.48,0.52\right],\left[0.28,0.38\right]\right)$	$\left(\left[0.48,0.55\right],\left[0.35,0.45\right]\right)$
	${\mathbb{O}}_3$	$\left(\left[0.52,0.70\right],\left[0.18,0.35\right]\right)$	$\left(\left[0.80,0.85\right],\left[0.55,0.60\right]\right)$	$\left(\left[0.45,0.52\right],\left[0.22,0.30\right]\right)$	$\left(\left[0.68,0.70\right],\left[0.22,0.28\right]\right)$	$\left(\left[0.48,0.55\right],\left[0.40,0.48\right]\right)$	$\left(\left[0.72,0.76\right],\left[0.18,0.30\right]\right)$
	${\mathbb{O}}_4$	$\left(\left[0.48,0.65\right],\left[0.28,0.45\right]\right)$	$\left(\left[0.20,0.55\right],\left[0.42,0.56\right]\right)$	$\left(\left[0.48,0.55\right],\left[0.25,0.40\right]\right)$	$\left(\left[0.64,0.68\right],\left[0.48,0.60\right]\right)$	$\left(\left[0.32,0.40\right],\left[0.30,0.40\right]\right)$	$\left(\left[0.35,0.45\right],\left[0.12,0.22\right]\right)$
	${\mathbb{O}}_5$	$\left(\left[0.15,0.30\right],\left[0.40,0.50\right]\right)$	$\left(\left[0.35,0.60\right],\left[0.45,0.50\right]\right)$	$\left(\left[0.62,0.70\right],\left[0.45,0.52\right]\right)$	$\left(\left[0.45,0.50\right],\left[0.40,0.45\right]\right)$	$\left(\left[0.65,0.70\right],\left[0.30,0.35\right]\right)$	$\left(\left[0.40,0.45\right],\left[0.15,0.25\right]\right)$
${\mathbb{D}}_3$	${\mathbb{O}}_1$	$\left(\left[0.25,0.40\right],\left[0.25,0.30\right]\right)$	$\left(\left[0.18,0.26\right],\left[0.12,0.20\right]\right)$	$\left(\left[0.20,0.30\right],\left[0.30,0.35\right]\right)$	$\left(\left[0.30,0.40\right],\left[0.15,0.20\right]\right)$	$\left(\left[0.66,0.70\right],\left[0.40,0.44\right]\right)$	$\left(\left[0.60,0.75\right],\left[0.55,0.60\right]\right)$
	${\mathbb{O}}_2$	$\left(\left[0.38,0.45\right],\left[0.25,0.30\right]\right)$	$\left(\left[0.40,0.50\right],\left[0.30,0.35\right]\right)$	$\left(\left[0.45,0.55\right],\left[0.15,0.20\right]\right)$	$\left(\left[0.46,0.52\right],\left[0.45,0.50\right]\right)$	$\left(\left[0.65,0.75\right],\left[0.30,0.35\right]\right)$	$\left(\left[0.50,0.55\right],\left[0.40,0.45\right]\right)$
	${\mathbb{O}}_3$	$\left(\left[0.40,0.46\right],\left[0.30,0.35\right]\right)$	$\left(\left[0.65,0.75\right],\left[0.20,0.25\right]\right)$	$\left(\left[0.40,0.50\right],\left[0.30,0.35\right]\right)$	$\left(\left[0.65,0.70\right],\left[0.25,0.30\right]\right)$	$\left(\left[0.50,0.55\right],\left[0.45,0.50\right]\right)$	$\left(\left[0.40,0.55\right],\left[0.25,0.35\right]\right)$
	${\mathbb{O}}_4$	$\left(\left[0.30,0.50\right],\left[0.20,0.25\right]\right)$	$\left(\left[0.35,0.45\right],\left[0.30,0.40\right]\right)$	$\left(\left[0.45,0.55\right],\left[0.28,0.38\right]\right)$	$\left(\left[0.60,0.75\right],\left[0.60,0.70\right]\right)$	$\left(\left[0.35,0.40\right],\left[0.35,0.40\right]\right)$	$\left(\left[0.30,0.40\right],\left[0.10,0.20\right]\right)$
	${\mathbb{O}}_5$	$\left(\left[0.25,0.30\right],\left[0.55,0.60\right]\right)$	$\left(\left[0.20,0.35\right],\left[0.45,0.50\right]\right)$	$\left(\left[0.65,0.70\right],\left[0.46,0.52\right]\right)$	$\left(\left[0.40,0.45\right],\left[0.45,0.55\right]\right)$	$\left(\left[0.60,0.70\right],\left[0.25,0.30\right]\right)$	$\left(\left[0.35,0.40\right],\left[0.30,0.40\right]\right)$

, accordingly.

Step 1: The data provided by three DEs ${\mathbb{D}}_k;k=1,2,3$ are listed in Table 2.

Step 2: ${\mathbb{C}}_1$ and ${\mathbb{C}}_3$ are cost type. Therefore, normalization of the original DMXs is necessary. Based on Equation (32), the normalized DMXs are constructed as tabulated in

Table 3. Normalized ratings provided by three DEs using the evaluation criteria.

		${\mathbb{C}}_1$	${\mathbb{C}}_2$	${\mathbb{C}}_3$	${\mathbb{C}}_4$	${\mathbb{C}}_5$	${\mathbb{C}}_6$
${\mathbb{D}}_1$	${\mathbb{O}}_1$	$\left(\left[0.25,0.30\right],\left[0.30,0.60\right]\right)$	$\left(\left[0.16,0.26\right],\left[0.15,0.20\right]\right)$	$\left(\left[0.35,0.40\right],\left[0.25,0.30\right]\right)$	$\left(\left[0.15,0.20\right],\left[0.10,0.20\right]\right)$	$\left(\left[0.60,0.70\right],\left[0.40,0.44\right]\right)$	$\left(\left[0.66,0.76\right],\left[0.56,0.62\right]\right)$
	${\mathbb{O}}_2$	$\left(\left[0.20,0.42\right],\left[0.40,0.60\right]\right)$	$\left(\left[0.40,0.50\right],\left[0.28,0.36\right]\right)$	$\left(\left[0.14,0.18\right],\left[0.50,0.60\right]\right)$	$\left(\left[0.65,0.75\right],\left[0.45,0.50\right]\right)$	$\left(\left[0.46,0.52\right],\left[0.30,0.35\right]\right)$	$\left(\left[0.45,0.55\right],\left[0.40,0.45\right]\right)$
	${\mathbb{O}}_3$	$\left(\left[0.20,0.35\right],\left[0.50,0.70\right]\right)$	$\left(\left[0.66,0.76\right],\left[0.18,0.25\right]\right)$	$\left(\left[0.25,0.30\right],\left[0.35,0.50\right]\right)$	$\left(\left[0.66,0.70\right],\left[0.24,0.28\right]\right)$	$\left(\left[0.45,0.55\right],\left[0.42,0.48\right]\right)$	$\left(\left[0.70,0.75\right],\left[0.20,0.30\right]\right)$
	${\mathbb{O}}_4$	$\left(\left[0.30,0.45\right],\left[0.45,0.65\right]\right)$	$\left(\left[0.20,0.60\right],\left[0.45,0.56\right]\right)$	$\left(\left[0.28,0.38\right],\left[0.45,0.55\right]\right)$	$\left(\left[0.62,0.67\right],\left[0.50,0.60\right]\right)$	$\left(\left[0.35,0.40\right],\left[0.30,0.40\right]\right)$	$\left(\left[0.33,0.43\right],\left[0.10,0.15\right]\right)$
	${\mathbb{O}}_5$	$\left(\left[0.45,0.50\right],\left[0.10,0.30\right]\right)$	$\left(\left[0.30,0.60\right],\left[0.40,0.50\right]\right)$	$\left(\left[0.46,0.52\right],\left[0.65,0.70\right]\right)$	$\left(\left[0.44,0.48\right],\left[0.40,0.45\right]\right)$	$\left(\left[0.70,0.75\right],\left[0.20,0.30\right]\right)$	$\left(\left[0.36,0.40\right],\left[0.12,0.22\right]\right)$
${\mathbb{D}}_2$	${\mathbb{O}}_1$	$\left(\left[0.20,0.50\right],\left[0.32,0.60\right]\right)$	$\left(\left[0.18,0.26\right],\left[0.12,0.20\right]\right)$	$\left(\left[0.32,0.40\right],\left[0.28,0.30\right]\right)$	$\left(\left[0.35,0.45\right],\left[0.08,0.18\right]\right)$	$\left(\left[0.62,0.70\right],\left[0.38,0.44\right]\right)$	$\left(\left[0.68,0.78\right],\left[0.54,0.64\right]\right)$
	${\mathbb{O}}_2$	$\left(\left[0.18,0.42\right],\left[0.42,0.60\right]\right)$	$\left(\left[0.42,0.50\right],\left[0.26,0.36\right]\right)$	$\left(\left[0.15,0.20\right],\left[0.52,0.60\right]\right)$	$\left(\left[0.68,0.75\right],\left[0.42,0.50\right]\right)$	$\left(\left[0.48,0.52\right],\left[0.28,0.38\right]\right)$	$\left(\left[0.48,0.55\right],\left[0.35,0.45\right]\right)$
	${\mathbb{O}}_3$	$\left(\left[0.18,0.35\right],\left[0.52,0.70\right]\right)$	$\left(\left[0.80,0.85\right],\left[0.55,0.60\right]\right)$	$\left(\left[0.22,0.30\right],\left[0.45,0.52\right]\right)$	$\left(\left[0.68,0.70\right],\left[0.22,0.28\right]\right)$	$\left(\left[0.48,0.55\right],\left[0.40,0.48\right]\right)$	$\left(\left[0.72,0.76\right],\left[0.18,0.30\right]\right)$
	${\mathbb{O}}_4$	$\left(\left[0.28,0.45\right],\left[0.48,0.65\right]\right)$	$\left(\left[0.20,0.55\right],\left[0.42,0.56\right]\right)$	$\left(\left[0.25,0.40\right],\left[0.48,0.55\right]\right)$	$\left(\left[0.64,0.68\right],\left[0.48,0.60\right]\right)$	$\left(\left[0.32,0.40\right],\left[0.30,0.40\right]\right)$	$\left(\left[0.35,0.45\right],\left[0.12,0.22\right]\right)$
	${\mathbb{O}}_5$	$\left(\left[0.40,0.50\right],\left[0.15,0.30\right]\right)$	$\left(\left[0.35,0.60\right],\left[0.45,0.50\right]\right)$	$\left(\left[0.45,0.52\right],\left[0.62,0.70\right]\right)$	$\left(\left[0.45,0.50\right],\left[0.40,0.45\right]\right)$	$\left(\left[0.65,0.70\right],\left[0.30,0.35\right]\right)$	$\left(\left[0.40,0.45\right],\left[0.15,0.25\right]\right)$
${\mathbb{D}}_3$	${\mathbb{O}}_1$	$\left(\left[0.25,0.30\right],\left[0.25,0.40\right]\right)$	$\left(\left[0.18,0.26\right],\left[0.12,0.20\right]\right)$	$\left(\left[0.30,0.35\right],\left[0.20,0.30\right]\right)$	$\left(\left[0.30,0.40\right],\left[0.15,0.20\right]\right)$	$\left(\left[0.66,0.70\right],\left[0.40,0.44\right]\right)$	$\left(\left[0.60,0.75\right],\left[0.55,0.60\right]\right)$
	${\mathbb{O}}_2$	$\left(\left[0.25,0.30\right],\left[0.38,0.45\right]\right)$	$\left(\left[0.40,0.50\right],\left[0.30,0.35\right]\right)$	$\left(\left[0.15,0.20\right],\left[0.45,0.55\right]\right)$	$\left(\left[0.46,0.52\right],\left[0.45,0.50\right]\right)$	$\left(\left[0.65,0.75\right],\left[0.30,0.35\right]\right)$	$\left(\left[0.50,0.55\right],\left[0.40,0.45\right]\right)$
	${\mathbb{O}}_3$	$\left(\left[0.30,0.35\right],\left[0.40,0.46\right]\right)$	$\left(\left[0.65,0.75\right],\left[0.20,0.25\right]\right)$	$\left(\left[0.30,0.35\right],\left[0.40,0.50\right]\right)$	$\left(\left[0.65,0.70\right],\left[0.25,0.30\right]\right)$	$\left(\left[0.50,0.55\right],\left[0.45,0.50\right]\right)$	$\left(\left[0.40,0.55\right],\left[0.25,0.35\right]\right)$
	${\mathbb{O}}_4$	$\left(\left[0.20,0.25\right],\left[0.30,0.50\right]\right)$	$\left(\left[0.35,0.45\right],\left[0.30,0.40\right]\right)$	$\left(\left[0.28,0.38\right],\left[0.45,0.55\right]\right)$	$\left(\left[0.60,0.75\right],\left[0.60,0.70\right]\right)$	$\left(\left[0.35,0.40\right],\left[0.35,0.40\right]\right)$	$\left(\left[0.30,0.40\right],\left[0.10,0.20\right]\right)$
	${\mathbb{O}}_5$	$\left(\left[0.55,0.60\right],\left[0.25,0.30\right]\right)$	$\left(\left[0.20,0.35\right],\left[0.45,0.50\right]\right)$	$\left(\left[0.46,0.52\right],\left[0.65,0.70\right]\right)$	$\left(\left[0.40,0.45\right],\left[0.45,0.55\right]\right)$	$\left(\left[0.60,0.70\right],\left[0.25,0.30\right]\right)$	$\left(\left[0.35,0.40\right],\left[0.30,0.40\right]\right)$

.

Step 3: Using Equation (33), the normalized DMXs in Table 3 are aggregated as given in

Table 4. Aggregated $IVp,q^{r}OF$ DMX.

	${\mathbb{C}}_1$	${\mathbb{C}}_2$	${\mathbb{C}}_3$	${\mathbb{C}}_4$	${\mathbb{C}}_5$	${\mathbb{C}}_6$
${\mathbb{O}}_1$	$\left(\left[0.4800,0.6230\right],\left[0.2915,0.5313\right]\right)$	$\left(\left[0.4172,0.5099\right],\left[0.1283,0.2000\right]\right)$	$\left(\left[0.5686,0.6208\right],\left[0.2447,0.3000\right]\right)$	$\left(\left[0.5289,0.6069\right],\left[0.1033,0.1917\right]\right)$	$\left(\left[0.7917,0.8367\right],\left[0.3919,0.4400\right]\right)$	$\left(\left[0.8072,0.8748\right],\left[0.5489,0.6218\right]\right)$
${\mathbb{O}}_2$	$\left(\left[0.4556,0.6216\right],\left[0.4017,0.5504\right]\right)$	$\left(\left[0.6388,0.7071\right],\left[0.2775,0.3570\right]\right)$	$\left(\left[0.3834,0.4405\right],\left[0.4921,0.5845\right]\right)$	$\left(\left[0.7845,0.8342\right],\left[0.4378,0.5000\right]\right)$	$\left(\left[0.7301,0.7780\right],\left[0.2918,0.3617\right]\right)$	$\left(\left[0.6909,0.7416\right],\left[0.3792,0.4500\right]\right)$
${\mathbb{O}}_3$	$\left(\left[0.4731,0.5916\right],\left[0.4750,0.6172\right]\right)$	$\left(\left[0.8501,0.8937\right],\left[0.2904,0.3548\right]\right)$	$\left(\left[0.5037,0.5616\right],\left[0.4028,0.5079\right]\right)$	$\left(\left[0.8156,0.8367\right],\left[0.2346,0.2859\right]\right)$	$\left(\left[0.6909,0.7416\right],\left[0.4205,0.4859\right]\right)$	$\left(\left[0.8004,0.8406\right],\left[0.2050,0.3142\right]\right)$
${\mathbb{O}}_4$	$\left(\left[0.5130,0.6296\right],\left[0.4089,0.6008\right]\right)$	$\left(\left[0.4983,0.7339\right],\left[0.3876,0.5062\right]\right)$	$\left(\left[0.5178,0.6230\right],\left[0.4618,0.5500\right]\right)$	$\left(\left[0.7889,0.8367\right],\left[0.5196,0.6284\right]\right)$	$\left(\left[0.5815,0.6325\right],\left[0.3142,0.4000\right]\right)$	$\left(\left[0.5739,0.6553\right],\left[0.1076,0.1906\right]\right)$
${\mathbb{O}}_5$	$\left(\left[0.6810,0.7296\right],\left[0.1548,0.3000\right]\right)$	$\left(\left[0.5410,0.7330\right],\left[0.4344,0.5000\right]\right)$	$\left(\left[0.6753,0.7211\right],\left[0.6378,0.7000\right]\right)$	$\left(\left[0.6576,0.6924\right],\left[0.4144,0.4779\right]\right)$	$\left(\left[0.8076,0.8461\right],\left[0.2515,0.3191\right]\right)$	$\left(\left[0.6111,0.6485\right],\left[0.1727,0.2770\right]\right)$

.

Step 4: In this step, the procedure of the proposed CRITIC technique is adopted to ascertain the criteria weights:

Steps $1^{\prime }$–$3^{\prime }$: These steps have already been completed.

Step $4^{\prime }$: Based on the novel score function, Equation (35), the score matrix $\mathfrak{S}={\left(S^{⏜}\left({ß}_{ij}\right)\right)}_{5\times 6}$ is determined as follows:

$$\mathfrak{S}=\left(\begin{array}{cccccc}0.4547& 0.5436& 0.5540& 0.6136& 0.5779& 0.4916\\ 0.4165& 0.5678& 0.3321& 0.5442& 0.6031& 0.5325\\ 0.3799& 0.6651& 0.4252& 0.6835& 0.5107& 0.6808\\ 0.4169& 0.4666& 0.4167& 0.4860& 0.5122& 0.6397\\ 0.6401& 0.4663& 0.3891& 0.4957& 0.6669& 0.6009\end{array}\right)$$

Step $5^{\prime }$: With the use of Equation (36), the normalized form of the developed score matrix $\mathfrak{S}$ is derived as given below:

$$\mathfrak{N}=\left(\begin{array}{cccccc}0.0000& 0.5595& 0.6249& 1.000& 0.7753& 0.2322\\ 0.3114& 0.8697& 0.0000& 0.7827& 1.000& 0.7395\\ 0.0000& 0.9394& 0.1492& 1.000& 0.4308& 0.9911\\ 0.0009& 0.2238& 0.0000& 0.3108& 0.4283& 1.000\\ 0.9035& 0.2779& 0.0000& 0.3837& 1.000& 0.7624\end{array}\right)$$

Steps $6^{\prime }$–$8^{\prime }$: In accordance with Equation (37)–(39), the correlation coefficient, standard deviation, and information content of criteria are computed and specified in

Table 5. Correlation coefficient, standard deviation, and information content.

	${\mathbb{C}}_1$	${\mathbb{C}}_2$	${\mathbb{C}}_3$	${\mathbb{C}}_4$	${\mathbb{C}}_5$	${\mathbb{C}}_6$	${\mathfrak{F}}_{\mathit{j}}$	${\mathcal{I}}_{\mathit{j}}$
${\mathbb{C}}_1$	1.000	−0.3406	−0.4425	−0.4897	0.7362	0.02895	0.3929	2.164
${\mathbb{C}}_2$	−0.3406	1.000	0.1277	0.8271	−0.01179	0.003127	0.3285	1.444
${\mathbb{C}}_3$	−0.4425	0.1277	1.000	0.6581	−0.04489	−0.8409	0.2706	1.500
${\mathbb{C}}_4$	−0.4897	0.8271	0.6581	1.000	−0.05771	−0.4486	0.3311	1.494
${\mathbb{C}}_5$	0.7362	−0.01179	−0.04489	−0.05771	1.000	−0.4773	0.2865	1.391
${\mathbb{C}}_6$	0.02895	0.003127	−0.8409	−0.4486	0.3019	1.000	0.3118	1.857

.

Step $9^{\prime }$: In the light of Equation (40), the weight vector of criteria is obtained as follows:

${\omega }_1=0.2197$, ${\omega }_2=0.1466$, ${\omega }_3=0.1523$, ${\omega }_4=0.1517$, ${\omega }_5=0.1412$, ${\omega }_6=0.1885$.

Step 5: In accordance with Equation (41), the relative weight of each criteria is calculated as follows:

${\sout{w}}_1=1.000$, ${\sout{w}}_2=0.6673$, ${\sout{w}}_3=0.6932$, ${\sout{w}}_4=0.6905$, ${\sout{w}}_5=0.6427$, ${\sout{w}}_6=0.8580$.

Step 6: With Equation (43), the values of ${\Phi }_j\left({\mathbb{O}}_i,{\mathbb{O}}_{ß}\right)$ are determined (taking $\rho =\delta =1$). The generated outcomes are summarized in

Table 6. Preference index of each alternative over each criteria.

	Criteria	${\mathbb{O}}_1$	${\mathbb{O}}_2$	${\mathbb{O}}_3$	${\mathbb{O}}_4$	${\mathbb{O}}_5$	${\mathbb{O}}_6$
Dominance
${\Phi }_j\left({\mathbb{O}}_1,{\mathbb{O}}_1\right)$		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
${\Phi }_j\left({\mathbb{O}}_1,{\mathbb{O}}_2\right)$		0.01391	−3.119	0.05257	0.01730	−1.843	−2.794
${\Phi }_j\left({\mathbb{O}}_1,{\mathbb{O}}_3\right)$		0.03854	−4.999	0.02759	−3.797	0.04119	−2.093
${\Phi }_j\left({\mathbb{O}}_1,{\mathbb{O}}_4\right)$		0.03517	0.06702	0.03489	0.1067	0.07354	−3.731
${\Phi }_j\left({\mathbb{O}}_1,{\mathbb{O}}_5\right)$		−1.605	0.07252	0.09359	0.05987	−0.7480	−3.632
${\Phi }_j\left({\mathbb{O}}_2,{\mathbb{O}}_1\right)$		−0.2881	0.06703	−2.267	−4.154	0.03674	0.09927
${\Phi }_j\left({\mathbb{O}}_2,{\mathbb{O}}_2\right)$		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
${\Phi }_j\left({\mathbb{O}}_2,{\mathbb{O}}_3\right)$		0.02988	−3.466	−1.517	−1.188	0.01866	−1.634
${\Phi }_j\left({\mathbb{O}}_2,{\mathbb{O}}_4\right)$		−0.5841	0.04072	−1.998	0.02988	0.05211	−1.981
${\Phi }_j\left({\mathbb{O}}_2,{\mathbb{O}}_5\right)$		−1.686	0.03439	−4.285	0.05904	−1.646	−1.770
${\Phi }_j\left({\mathbb{O}}_3,{\mathbb{O}}_1\right)$		−0.7985	0.1074	−1.190	0.08739	−2.066	0.07437
${\Phi }_j\left({\mathbb{O}}_3,{\mathbb{O}}_2\right)$		−0.6192	0.07450	0.03519	0.02734	−0.9358	0.05806
${\Phi }_j\left({\mathbb{O}}_3,{\mathbb{O}}_3\right)$		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
${\Phi }_j\left({\mathbb{O}}_3,{\mathbb{O}}_4\right)$		−0.4279	0.08366	0.02498	0.05057	−2.526	0.09616
${\Phi }_j\left({\mathbb{O}}_3,{\mathbb{O}}_5\right)$		−1.834	0.08047	0.09257	0.06169	−2.298	0.09030
${\Phi }_j\left({\mathbb{O}}_4,{\mathbb{O}}_1\right)$		−0.7286	−3.118	−1.504	−4.634	−3.689	0.1326
${\Phi }_j\left({\mathbb{O}}_4,{\mathbb{O}}_2\right)$		0.05781	−1.894	0.04635	−1.299	−2.614	0.07038
${\Phi }_j\left({\mathbb{O}}_4,{\mathbb{O}}_3\right)$		0.02066	−3.893	−1.077	−2.197	0.05037	−2.706
${\Phi }_j\left({\mathbb{O}}_4,{\mathbb{O}}_4\right)$		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
${\Phi }_j\left({\mathbb{O}}_4,{\mathbb{O}}_5\right)$		−1.649	0.01200	0.08085	−3.358	−3.654	0.01305
${\Phi }_j\left({\mathbb{O}}_5,{\mathbb{O}}_1\right)$		0.07746	−3.374	−4.035	−2.602	0.01491	0.1291
${\Phi }_j\left({\mathbb{O}}_5,{\mathbb{O}}_2\right)$		0.08140	−1.600	0.0994	−2.566	0.03282	0.06289
${\Phi }_j\left({\mathbb{O}}_5,{\mathbb{O}}_3\right)$		0.08853	−3.744	−3.991	−2.681	0.04582	−2.541
${\Phi }_j\left({\mathbb{O}}_5,{\mathbb{O}}_4\right)$		0.07957	−0.5582	−3.486	0.07728	0.07285	−0.3674
${\Phi }_j\left({\mathbb{O}}_5,{\mathbb{O}}_5\right)$		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

. Next, the dominance value of each alternative is calculated utilizing these derived values and Equation (42) as follows:

${ð}_1=-27.62660$, ${ð}_2=-27.99648$, ${ð}_3=-11.65075$, ${ð}_4=-37.53053$, ${ð}_5=-30.68357$.

Step 7: Using Equation (44), the overall dominance values of alternatives are computed as shown below.

$ϝ\left({\mathbb{O}}_1\right)=0.3827$, $ϝ\left({\mathbb{O}}_2\right)=0.3684$, $ϝ\left({\mathbb{O}}_3\right)=1.000$, $ϝ\left({\mathbb{O}}_4\right)=0.0000$, $ϝ\left({\mathbb{O}}_5\right)=0.2646$.

Step 8: The ranked list of suppliers is identified as ${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$. The graphical illustration of the ranking is shown in Figure 1.

6.2. Sensitive Analysis

Once the results were derived through MCGDM techniques, a pertinent question emerged regarding the potential impact of slight variations in the input parameter values on the algorithm’s outcomes. Hence, it is imperative to assess the resilience of the outcomes in the formulated framework and evaluate the susceptibility of the derived results to variations in the input variables for MCGDM approaches. Within the context of this research, the subsequent section focuses on the presentation of a sensitivity analysis.

6.2.1. Influence Analysis of Parameters δ and $\rho$

In Equation (43), the final dominance of the alternatives is influenced by two parameters: δ and $\rho$. The parameter $\delta >0$ represents the loss-aversion coefficient, while $\rho \in {\mathbb{N}}^{+}$ signifies the significance of the decision based on the agent’s perception. Therefore, the sensitivity analysis can be divided into two cases:

$\rho \left(\rho \in {\mathbb{N}}^{+},\rho \in [0,5]\right)$$\left(\delta >1,\delta \in [1,30]\right)$

Case 1: The parameter $\rho$ is fixed $\left(\rho =01\right)$, and the values of the attention factor are varied. Six scenarios $\left(\delta =01,02,05,10,20,30\right)$ are generated in order to analyze the variations in the rankings of green suppliers. The outcomes of these simulations are depicted in

Table 7. Sensitivity analysis with respect to δ and $\rho$.

Parameter Value	Overall Dominance Value					Ranking
	ϝ $\left({\mathbb{O}}_{\mathbf{1}}\right)$	ϝ $\left({\mathbb{O}}_{\mathbf{2}}\right)$	ϝ $\left({\mathbb{O}}_{\mathbf{3}}\right)$	ϝ $\left({\mathbb{O}}_{\mathbf{4}}\right)$	ϝ $\left({\mathbb{O}}_{\mathbf{5}}\right)$
$\delta =01$	0.3827	0.3684	1.000	0.0000	0.2646	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$
$\delta =02$	0.4036	0.3783	1.000	0.0000	0.2872	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$
$\delta =05$	0.3878	0.3366	1.000	0.0000	0.2973	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$
$\delta =10$	0.3931	0.3035	1.000	0.0000	0.3314	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_5>{\mathbb{O}}_2>{\mathbb{O}}_4$
$\delta =20$	0.4734	0.3425	1.000	0.0000	0.4582	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_5>{\mathbb{O}}_2>{\mathbb{O}}_4$
$\delta =30$	0.5657	0.4577	1.000	0.0000	0.5383	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_5>{\mathbb{O}}_2>{\mathbb{O}}_4$
$\rho =01$	0.3827	0.3684	1.000	0.0000	0.2646	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$
$\rho =02$	0.4191	0.2903	1.000	0.0000	0.1026	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$
$\rho =03$	0.4980	0.2813	1.000	0.0000	0.1607	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$
$\rho =05$	0.6051	0.2646	1.000	0.0000	0.1819	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$
$\rho =10$	0.7307	0.2423	1.000	0.0000	0.2095	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$
$\rho =15$	0.7775	0.2329	1.000	0.0000	0.2275	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$

and Figure 2.

Table 7 demonstrates that regardless of variations in δ, the green supplier that consistently exhibits the highest overall dominance value is ${\mathbb{O}}_3$. Additionally, it is worth noting that there is a slight alteration in the ranking when $\delta \ge 10$; specifically, the positions of ${\mathbb{O}}_2$ and ${\mathbb{O}}_5$ are interchanged. It is plausible that there will be a discernible alteration in the level of loss aversion exhibited by the DEs when $\delta \ge 10$, because Wang et al. [76] proposed that the attenuation loss factor is estimated as approximately $2.25$. This analysis suggests that the parameter minimally influences the overall ranking outcomes.

Case 2: In this case, the parameter is kept fixed $\left(\delta =01\right)$, and the overall dominance values of alternatives $ϝ\left({\mathbb{O}}_i\right);i=1\left(1\right)5$ are computed for the various values of $\rho$, i.e., for $\rho =01,02,03,05,10,$ and 15. The obtained results are summarized in Table 7.

Based on an analysis of Figure 3, it is evident that the influence of the parameter $\rho$ on the overall dominance of each alternative is tiny, and there is no alteration in the ranking of green suppliers. Therefore, the developed TODIM strategy exhibits strong stability in relation to the parameter $\rho$. As mentioned before, the parameter $\rho$ serves as a metric for assessing the significance of a decision as perceived by the agent. A higher value of $\rho$ signifies an increased level of sensitivity towards the outcome.

6.2.2. Influence Analysis of Parameters p and q

This section is devoted to carrying out a sensitivity analysis regarding the parameters p and q. A variation in their respective values may cause a shift in the outcomes of the alternatives’ ranking. The present analysis is divided into the following two cases:

Case 1: In order to examine the impact of different values of parameter p on the outcomes of the previous example, we assign the parameter p values of 02, 03, 05, 08, 12, 18, and 25 in our recommended methodology (taking $q=3$). This enables us to determine the final placement of the suppliers. The possible outcomes are expressed in

Table 8. Sensitivity analysis with respect to p and q.

Parameter Value	Overall Dominance Value					Ranking
	ϝ $\left({\mathbb{O}}_{\mathbf{1}}\right)$	ϝ $\left({\mathbb{O}}_{\mathbf{2}}\right)$	ϝ $\left({\mathbb{O}}_{\mathbf{3}}\right)$	ϝ $\left({\mathbb{O}}_{\mathbf{4}}\right)$	ϝ $\left({\mathbb{O}}_{\mathbf{5}}\right)$
$p=02$	0.3827	0.3684	1.000	0.0000	0.2646	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$
$p=03$	0.3140	0.3580	1.000	0.0000	0.1397	${\mathbb{O}}_3>{\mathbb{O}}_2>{\mathbb{O}}_1>{\mathbb{O}}_5>{\mathbb{O}}_4$
$p=05$	0.2959	0.3577	1.000	0.0000	0.1696	${\mathbb{O}}_3>{\mathbb{O}}_2>{\mathbb{O}}_1>{\mathbb{O}}_5>{\mathbb{O}}_4$
$p=08$	0.2724	0.3336	1.000	0.0000	0.1949	${\mathbb{O}}_3>{\mathbb{O}}_2>{\mathbb{O}}_1>{\mathbb{O}}_5>{\mathbb{O}}_4$
$p=12$	0.2670	0.2805	1.000	0.0000	0.1986	${\mathbb{O}}_3>{\mathbb{O}}_2>{\mathbb{O}}_1>{\mathbb{O}}_5>{\mathbb{O}}_4$
$p=18$	0.3295	0.2234	1.000	0.0000	0.1367	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$
$p=25$	0.4749	0.2452	1.000	0.0000	0.1014	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$
$q=03$	0.3827	0.3684	1.000	0.0000	0.2646	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$
$q=05$	0.2856	0.2070	1.000	0.0000	0.09189	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$
$q=08$	0.2666	0.1431	1.000	0.0000	0.1089	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$
$q=12$	0.2609	0.1234	1.000	0.0000	0.1331	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_5>{\mathbb{O}}_2>{\mathbb{O}}_4$
$q=18$	0.2598	0.1216	1.000	0.0000	0.1430	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_5>{\mathbb{O}}_2>{\mathbb{O}}_4$
$q=25$	0.2811	0.1214	1.000	0.0000	0.1444	${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_5>{\mathbb{O}}_2>{\mathbb{O}}_4$

and Figure 4. From Table 8, it is noticeable that varied values of p lead to slightly varied preferable supplier orderings; however, the best and worst suppliers stay the same. For $p=02,18$, and 25, the suppliers are ranked as ${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$, whereas in the reset range ranking, ${\mathbb{O}}_3>{\mathbb{O}}_2>{\mathbb{O}}_1>{\mathbb{O}}_5>{\mathbb{O}}_4$ is achieved.

Case 2: This time, we alter the value of the parameter q from 03 to 25, keeping the value of $p=02$. The results derived under various values of q are tabulated in Table 8 and Figure 5. From Table 8, it is evident that for $q=03,05$, and 08, we obtain the ranking of suppliers as ${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_4$. From Table 8, one can further observe that for $q\ge 12$, the alternatives ${\mathbb{O}}_2$ and ${\mathbb{O}}_5$ swap positions, i.e., the ordering is found as ${\mathbb{O}}_3>{\mathbb{O}}_1>{\mathbb{O}}_5>{\mathbb{O}}_2>{\mathbb{O}}_4$. Hence, altering the value of q leads to one of the aforesaid two scenarios.

7. Comparative Study

The relevance of the created MCGDM approach cannot be fully conveyed by the numerical example described above. In this part, we evaluate the effectiveness of the supplied MCGDM approach to that of several related earlier MCGDM approaches. This comparison is made in order to showcase the effectiveness of the proposed method. The following methodologies have been chosen for analysis: (I). Gao et al.’s VIKOR method [77], (II). Surkan’s CODAS method [78], (III). Jeevaraj’s TOPSIS method [35], and (IV). Zhao et al.’s TODIM method [79]. The four aforementioned techniques are used to solve the problem of selecting green suppliers, and the details of each technique are as follows:

7.1. Comparison with the $IV{q}^{r}OF$ VIKOR Method

To solve the issue of selecting green suppliers, we apply Gao et al.’s VIKOR approach [77] in accordance with the $IV{q}^{r}OF$ weighted averaging ($IV{q}^{r}OF$WA) operator.

First, we used the $IV{q}^{r}OF$WA operator, i.e., Equation (45) (fixing q = 3), to aggregate the $IV{q}^{r}OF$ data of the corresponding DEs. By doing so, the aggregated result matrix was obtained as listed in

Table 9. Aggregated $IV{q}^{r}OF$ DMX.

	${\mathbb{C}}_1$	${\mathbb{C}}_2$	${\mathbb{C}}_3$	${\mathbb{C}}_4$	${\mathbb{C}}_5$	${\mathbb{C}}_6$
${\mathbb{O}}_1$	$\left(\left[0.2326,0.4071\right],\left[0.2915,0.5313\right]\right)$	$\left(\left[0.1745,0.2600\right],\left[0.1283,0.2000\right]\right)$	$\left(\left[0.3242,0.3864\right],\left[0.2447,0.3000\right]\right)$	$\left(\left[0.2978,0.3886\right],\left[0.1033,0.1917\right]\right)$	$\left(\left[0.6274,0.7000\right],\left[0.3919,0.4400\right]\right)$	$\left(\left[0.6526,0.7655\right],\left[0.5489,0.6218\right]\right)$
${\mathbb{O}}_2$	$\left(\left[0.2113,0.3919\right],\left[0.4017,0.5504\right]\right)$	$\left(\left[0.4083,0.5000\right],\left[0.2775,0.3570\right]\right)$	$\left(\left[0.1471,0.1944\right],\left[0.4921,0.5845\right]\right)$	$\left(\left[0.6238,0.7040\right],\left[0.4378,0.5000\right]\right)$	$\left(\left[0.5425,0.6188\right],\left[0.2918,0.3617\right]\right)$	$\left(\left[0.4779,0.5500\right],\left[0.3792,0.4500\right]\right)$
${\mathbb{O}}_3$	$\left(\left[0.2344,0.3500\right],\left[0.4750,0.6172\right]\right)$	$\left(\left[0.7270,0.8003\right],\left[0.2904,0.3548\right]\right)$	$\left(\left[0.2575,0.3168\right],\left[0.4028,0.5079\right]\right)$	$\left(\left[0.6654,0.7000\right],\left[0.2346,0.2859\right]\right)$	$\left(\left[0.4779,0.5500\right],\left[0.4205,0.4859\right]\right)$	$\left(\left[0.6566,0.7130\right],\left[0.2050,0.3142\right]\right)$
${\mathbb{O}}_4$	$\left(\left[0.2683,0.4104\right],\left[0.4089,0.6008\right]\right)$	$\left(\left[0.2649,0.5430\right],\left[0.3876,0.5062\right]\right)$	$\left(\left[0.2688,0.3883\right],\left[0.4618,0.5500\right]\right)$	$\left(\left[0.6227,0.7012\right],\left[0.5196,0.6284\right]\right)$	$\left(\left[0.3387,0.4000\right],\left[0.3142,0.4000\right]\right)$	$\left(\left[0.3304,0.4301\right],\left[0.1076,0.1906\right]\right)$
${\mathbb{O}}_5$	$\left(\left[0.4701,0.5353\right],\left[0.1548,0.3000\right]\right)$	$\left(\left[0.3028,0.5514\right],\left[0.4344,0.5000\right]\right)$	$\left(\left[0.4561,0.5200\right],\left[0.6378,0.7000\right]\right)$	$\left(\left[0.4332,0.4800\right],\left[0.4144,0.4779\right]\right)$	$\left(\left[0.6536,0.7164\right],\left[0.2515,0.3191\right]\right)$	$\left(\left[0.3745,0.4216\right],\left[0.1727,0.2770\right]\right)$

.

$$IV{q}^{r}OFWA\left({\mathfrak{N}}_1,{\mathfrak{N}}_2,\dots ,{\mathfrak{N}}_n\right)=\left(\left[\sqrt[q]{1-\prod _{j=1}^n{\left(1-{\underline{\Omega }}_j^q\right)}^{{\omega }_j}},\sqrt[q]{1-\prod _{j=1}^n{\left(1-{\overline{\Omega }}_j^q\right)}^{{\omega }_j}}\right],\left[\prod _{j=1}^n{\underline{\mho }}_j^{{\omega }_j},\prod _{j=1}^n{\overline{\mho }}_j^{{\omega }_j}\right]\right).$$

(45)

$${\eta }^{+}={\left[{\eta }_{sj}^{+}\right]}_{1\times n}=\left\{\begin{array}{cc}\hfill \left(\left[\underset{i}{max}\left\{{\underline{\Omega }}_{ij}\right\},\underset{i}{max}\left\{{\overline{\Omega }}_{ij}\right\}\right],\left[\underset{i}{min}\left\{{\underline{\mho }}_{ij}\right\},\underset{i}{min}\left\{{\overline{\mho }}_{ij}\right\}\right]\right),\hfill & \hfill \mathrm{if} {\mathbb{C}}_j\mathrm{is}\mathrm{BT}\hfill \\ \hfill \left(\left[\underset{i}{min}\left\{{\underline{\Omega }}_{ij}\right\},\underset{i}{min}\left\{{\overline{\Omega }}_{ij}\right\}\right],\left[\underset{i}{max}\left\{{\underline{\mho }}_{ij}\right\},\underset{i}{max}\left\{{\overline{\mho }}_{ij}\right\}\right]\right),\hfill & \hfill \mathrm{if} {\mathbb{C}}_j\mathrm{is}\mathrm{CT},\hfill \end{array}\right.$$

(46)

$${\eta }^{-}={\left[{\eta }_{sj}^{-}\right]}_{1\times n}=\left\{\begin{array}{cc}\hfill \left(\left[\underset{i}{min}\left\{{\underline{\Omega }}_{ij}\right\},\underset{i}{min}\left\{{\overline{\Omega }}_{ij}\right\}\right],\left[\underset{i}{max}\left\{{\underline{\mho }}_{ij}\right\},\underset{i}{max}\left\{{\overline{\mho }}_{ij}\right\}\right]\right),\hfill & \hfill \mathrm{if} {\mathbb{C}}_j\mathrm{is}\mathrm{BT}\hfill \\ \hfill \left(\left[\underset{i}{max}\left\{{\underline{\Omega }}_{ij}\right\},\underset{i}{max}\left\{{\overline{\Omega }}_{ij}\right\}\right],\left[\underset{i}{min}\left\{{\underline{\mho }}_{ij}\right\},\underset{i}{min}\left\{{\overline{\mho }}_{ij}\right\}\right]\right),\hfill & \hfill \mathrm{if} {\mathbb{C}}_j\mathrm{is}\mathrm{CT},\hfill \end{array}\right.$$

(47)

Secondly, the positive ideal solution (PIS) and negative ideal solution (NIS) were determined by the following Equations (46) and (47), respectively. The results are shown as follows:

$${\eta }^{+}=\left\{\begin{array}{c}\left(\left[0.2113,0.3500\right],\left[0.4750,0.6172\right]\right),\left(\left[0.7270,0.8003\right],\left[0.1283,0.2000\right]\right),\\ \left(\left[0.1471,0.1944\right],\left[0.6378,0.7000\right]\right),\left(\left[0.6654,0.7040\right],\left[0.1033,0.1917\right]\right),\\ \left(\left[0.6536,0.7164\right],\left[0.2515,0.3191\right]\right),\left(\left[0.6566,0.7655\right],\left[0.1076,0.1906\right]\right)\end{array}\right\},$$

$${\eta }^{-}=\left\{\begin{array}{c}\left(\left[0.4701,0.5353\right],\left[0.1548,0.3000\right]\right),\left(\left[0.1745,0.2600\right],\left[0.4344,0.5062\right]\right),\\ \left(\left[0.4561,0.5200\right],\left[0.2447,0.30000\right]\right),\left(\left[0.2978,0.3886\right],\left[0.5196,0.6284\right]\right),\\ \left(\left[0.3387,0.4000\right],\left[0.4205,0.4859\right]\right),\left(\left[0.3304,0.4216\right],\left[0.5489,0.6218\right]\right)\end{array}\right\}.$$

$${Ĩ}_i=\sum _{j=1}^n{\omega }_j\frac{d\left({\eta }_{sj}^{+},{\mathfrak{N}}_{ij}\right)}{d\left({\eta }_{sj}^{+},{\eta }_{sj}^{-}\right)};i=1\left(1\right)m,j=1\left(1\right)n,$$

(48)

$${Ī}_i=\underset{j}{max}\left\{{\omega }_j\frac{d\left({\eta }_{sj}^{+},{\mathfrak{N}}_{ij}\right)}{d\left({\eta }_{sj}^{+},{\eta }_{sj}^{-}\right)}\right\};i=1\left(1\right)m,j=1\left(1\right)n,$$

(49)

where “d” represents the distance measure given by $d\left({\mathfrak{N}}_1,{\mathfrak{N}}_2\right)=\frac{1}{4}(\left|{\underline{\Omega }}_1^q-{\underline{\Omega }}_2^q\right|+\left|{\overline{\Omega }}_1^q-{\overline{\Omega }}_2^q\right|+\left|{\underline{\mho }}_1^q-{\underline{\mho }}_2^q\right|+\left|{\overline{\mho }}_1^q-{\overline{\mho }}_2^q\right|).$

According to Equations (48) and (49), the mean ${Ĩ}_i$ and worst ${Ī}_i$ scores of alternatives ${\mathbb{O}}_i$ are calculated using the criteria weights generated by the proposed CRITIC approach. The findings are listed as follows:

$${Ĩ}_1=0.5232,{Ĩ}_2=0.4204,{Ĩ}_3=0.2285,{Ĩ}_4=0.5468,{Ĩ}_5=0.6017,$$

$${Ī}_1=0.1245,{Ī}_2=0.1099,{Ī}_3=0.1056,{Ī}_4=0.1221,{Ī}_5=0.2197.$$

$${\Theta }_i=\alpha \left(\frac{{Ĩ}_i-{Ĩ}_i^{-}}{{Ĩ}_i^{+}-{Ĩ}_i^{-}}\right)+(1-\alpha )\left(\frac{{Ī}_i-{Ī}_i^{-}}{{Ī}_i^{+}-{Ī}_i^{-}}\right),$$

(50)

where ${Ĩ}_i^{-}=\underset{i}{min}\left\{{Ĩ}_i\right\}$, ${Ĩ}_i^{+}=\underset{i}{max}\left\{{Ĩ}_i\right\}$, where ${Ī}_i^{-}=\underset{i}{min}\left\{{Ī}_i\right\}$, ${Ī}_i^{+}=\underset{i}{max}\left\{{Ī}_i\right\}$, and $\alpha$ is a parameter that represents the coefficient of DM strategies.

Finally, the results of ${\Theta }_i$ via Equation (50) (fixing $\alpha =0.5$) are acquired as given below.

${\Theta }_1=0.4777,{\Theta }_2=0.2759,{\Theta }_3=0.0000,{\Theta }_4=0.4988,{\Theta }_5=1.000.$

The smaller the value of ${\Theta }_i$, the better the alternative is. Thus, the ranking of suppliers is ${\mathbb{O}}_3>{\mathbb{O}}_2>{\mathbb{O}}_1>{\mathbb{O}}_4>{\mathbb{O}}_5$.

7.2. Comparison with the $IV{q}^{r}OF$ CODAS Method

We compared the proposed approach’s results to the work presented in [78]. The steps of Surkan’s CODAS mechanism are applied to the considered MCGDM problem, as detailed below.

First, similar to the VIKOR approach, individual DMXs are combined using the $IV{q}^{r}OF$WA operator, Equation (45), and tabulated in Table 9.

Following Equation (32), the normalized form of the $IV{q}^{r}OF$ aggregated DMX (Table 9) is acquired as provided in

Table 10. Normalized $IV{q}^{r}OF$ DMX.

	${\mathbb{C}}_1$	${\mathbb{C}}_2$	${\mathbb{C}}_3$	${\mathbb{C}}_4$	${\mathbb{C}}_5$	${\mathbb{C}}_6$
${\mathbb{O}}_1$	$\left(\left[0.2915,0.5313\right],\left[0.2326,0.4071\right]\right)$	$\left(\left[0.1745,0.2600\right],\left[0.1283,0.2000\right]\right)$	$\left(\left[0.2447,0.3000\right],\left[0.3242,0.3864\right]\right)$	$\left(\left[0.2978,0.3886\right],\left[0.1033,0.1917\right]\right)$	$\left(\left[0.6274,0.7000\right],\left[0.3919,0.4400\right]\right)$	$\left(\left[0.6526,0.7655\right],\left[0.5489,0.6218\right]\right)$
${\mathbb{O}}_2$	$\left(\left[0.4017,0.5504\right],\left[0.2113,0.3919\right]\right)$	$\left(\left[0.4083,0.5000\right],\left[0.2775,0.3570\right]\right)$	$\left(\left[0.4921,0.5845\right],\left[0.1471,0.1944\right]\right)$	$\left(\left[0.6238,0.7040\right],\left[0.4378,0.5000\right]\right)$	$\left(\left[0.5425,0.6188\right],\left[0.2918,0.3617\right]\right)$	$\left(\left[0.4779,0.5500\right],\left[0.3792,0.4500\right]\right)$
${\mathbb{O}}_3$	$\left(\left[0.4750,0.6172\right],\left[0.2344,0.3500\right]\right)$	$\left(\left[0.7270,0.8003\right],\left[0.2904,0.3548\right]\right)$	$\left(\left[0.4028,0.5079\right],\left[0.2575,0.3168\right]\right)$	$\left(\left[0.6654,0.7000\right],\left[0.2346,0.2859\right]\right)$	$\left(\left[0.4779,0.5500\right],\left[0.4205,0.4859\right]\right)$	$\left(\left[0.6566,0.7130\right],\left[0.2050,0.3142\right]\right)$
${\mathbb{O}}_4$	$\left(\left[0.4089,0.6008\right],\left[0.2683,0.4104\right]\right)$	$\left(\left[0.2649,0.5430\right],\left[0.3876,0.5062\right]\right)$	$\left(\left[0.4618,0.5500\right],\left[0.2688,0.3883\right]\right)$	$\left(\left[0.6227,0.7012\right],\left[0.5196,0.6284\right]\right)$	$\left(\left[0.3387,0.4000\right],\left[0.3142,0.4000\right]\right)$	$\left(\left[0.3304,0.4301\right],\left[0.1076,0.1906\right]\right)$
${\mathbb{O}}_5$	$\left(\left[0.1548,0.3000\right],\left[0.4701,0.5353\right]\right)$	$\left(\left[0.3028,0.5514\right],\left[0.4344,0.5000\right]\right)$	$\left(\left[0.6378,0.7000\right],\left[0.4561,0.5200\right]\right)$	$\left(\left[0.4332,0.4800\right],\left[0.4144,0.4779\right]\right)$	$\left(\left[0.6536,0.7164\right],\left[0.2515,0.3191\right]\right)$	$\left(\left[0.3745,0.4216\right],\left[0.1727,0.2770\right]\right)$

.

$$\AE ={\left[{\ae }_{ij}\right]}_{m\times n}={\left[{\omega }_j\otimes {\mathfrak{N}}_{ij}\right]}_{m\times n}.$$

(51)

According to Equation (51), the weighted DMX is obtained (see

Table 11. Weighted normalized $IV{q}^{r}OF$ DMX.

	${\mathbb{C}}_1$	${\mathbb{C}}_2$	${\mathbb{C}}_3$	${\mathbb{C}}_4$	${\mathbb{C}}_5$	${\mathbb{C}}_6$
${\mathbb{O}}_1$	$\left(\left[0.1765,0.3273\right],\left[0.7258,0.8208\right]\right)$	$\left(\left[0.09208,0.1374\right],\left[0.7401,0.7898\right]\right)$	$\left(\left[0.1309,0.1608\right],\left[0.8424,0.8652\right]\right)$	$\left(\left[0.1594,0.2090\right],\left[0.7087,0.7783\right]\right)$	$\left(\left[0.3399,0.3862\right],\left[0.8761,0.8905\right]\right)$	$\left(\left[0.3905,0.4735\right],\left[0.8931,0.9143\right]\right)$
${\mathbb{O}}_2$	$\left(\left[0.2445,0.3399\right],\left[0.7107,0.8140\right]\right)$	$\left(\left[0.2174,0.2686\right],\left[0.8287,0.8598\right]\right)$	$\left(\left[0.2675,0.3219\right],\left[0.7468,0.7792\right]\right)$	$\left(\left[0.3457,0.3980\right],\left[0.8822,0.9002\right]\right)$	$\left(\left[0.2895,0.3346\right],\left[0.8404,0.8662\right]\right)$	$\left(\left[0.2783,0.3231\right],\left[0.8329,0.8603\right]\right)$
${\mathbb{O}}_3$	$\left(\left[0.2908,0.3853\right],\left[0.7271,0.7940\right]\right)$	$\left(\left[0.4094,0.4641\right],\left[0.8342,0.8591\right]\right)$	$\left(\left[0.2172,0.2766\right],\left[0.8133,0.8394\right]\right)$	$\left(\left[0.3722,0.3952\right],\left[0.8026,0.8270\right]\right)$	$\left(\left[0.2530,0.2938\right],\left[0.8849,0.9031\right]\right)$	$\left(\left[0.3932,0.4333\right],\left[0.7418,0.8039\right]\right)$
${\mathbb{O}}_4$	$\left(\left[0.2490,0.3739\right],\left[0.7490,0.8223\right]\right)$	$\left(\left[0.1401,0.2934\right],\left[0.8703,0.9050\right]\right)$	$\left(\left[0.2502,0.3012\right],\left[0.8187,0.8658\right]\right)$	$\left(\left[0.3450,0.3960\right],\left[0.9055,0.9319\right]\right)$	$\left(\left[0.1774,0.2103\right],\left[0.8492,0.8786\right]\right)$	$\left(\left[0.1904,0.2494\right],\left[0.6569,0.7316\right]\right)$
${\mathbb{O}}_5$	$\left(\left[0.09345,0.1817\right],\left[0.8472,0.8717\right]\right)$	$\left(\left[0.1603,0.2983\right],\left[0.8849,0.9034\right]\right)$	$\left(\left[0.3549,0.3957\right],\left[0.8873,0.9052\right]\right)$	$\left(\left[0.2338,0.2602\right],\left[0.8749,0.8940\right]\right)$	$\left(\left[0.3562,0.3972\right],\left[0.8229,0.8510\right]\right)$	$\left(\left[0.2163,0.2443\right],\left[0.7182,0.7851\right]\right)$

) using the criteria weights generated by the proposed CRITIC approach.

$${\P }^{-}={\left[{\P }_{sj}^{-}\right]}_{1\times n}=\left(\left[\underset{i}{min}\left\{{\underline{\dot{\Omega }}}_{ij}\right\},\underset{i}{min}\left\{{\overline{\dot{\Omega }}}_{ij}\right\}\right],\left[\underset{i}{max}\left\{{\underline{\dot{\mho }}}_{ij}\right\},\underset{i}{max}\left\{{\overline{\dot{\mho }}}_{ij}\right\}\right]\right),$$

(52)

Next, the NIS is determined by the following Equation (52), and the derived results are shown as follows:

$${\P }^{-}=\left\{\begin{array}{c}\left(\left[0.09345,0.1817\right],\left[0.8472,0.8717\right]\right),\left(\left[0.09208,0.1374\right],\left[0.8849,0.9050\right]\right),\\ \left(\left[0.1309,0.1608\right],\left[0.8873,0.9052\right]\right),\left(\left[0.1594,0.2090\right],\left[0.9055,0.9319\right]\right),\\ \left(\left[0.1774,0.2103\right],\left[0.8849,0.9031\right]\right),\left(\left[0.1904,0.2443\right],\left[0.8931,0.9143\right]\right)\end{array}\right\}.$$

$$RA={\left[{\mathfrak{p}}_{it}\right]}_{m\times m},$$

(53)

where ${\mathfrak{p}}_{it}=\left(E{D}_i-E{D}_t\right)+{⏜}^{}\delta \left(E{D}_i-E{D}_t\right)\ast \left(H{D}_i-H{D}_t\right)$ such that ${⏜}^{}\delta \left(x\right)=\left\{\begin{array}{cc}\hfill 1,\hfill & \hfill \mathrm{if} \left|x\right|\ge {⏜}^{}\delta \hfill \\ \hfill 0,\hfill & \hfill \mathrm{otherwise}.\hfill \end{array}\right.$ and ${⏜}^{}\delta$ represents the threshold value, which can be determined by DEs. Further, $ED$ and $HD$ represent the Euclidean and Hamming distances, respectively [78].

Based on Equation (53), a relative assessment (RA) matrix was built (taking ${⏜}^{}\delta$ = 0.05) and shown in

Table 12. RA matrix for the five suppliers.

	${\mathbb{O}}_1$	${\mathbb{O}}_2$	${\mathbb{O}}_3$	${\mathbb{O}}_4$	${\mathbb{O}}_5$
${\mathbb{O}}_1$	0.0000	−0.0395	0.7666	0.0061	0.0801
${\mathbb{O}}_2$	0.0395	0.0000	0.8061	0.0456	1.1196
${\mathbb{O}}_3$	1.2334	0.3039	0.0000	1.2395	1.3135
${\mathbb{O}}_4$	−0.0061	−0.0456	0.7605	0.0000	0.0740
${\mathbb{O}}_5$	−0.0801	0.8804	0.6865	−0.0740	0.0000

.

According to the formula $AS={\displaystyle \sum _{i=1}^m}{\mathfrak{p}}_{it}$, the assessment score (AS) for each alternative was obtained as follows: $A{S}_1=0.8133$, $A{S}_2=2.0108$, $A{S}_3=4.0903$, $A{S}_4=0.7828$, $A{S}_5=1.4128$.

At last, the available alternates are placed in descending order of $AS$ value: ${\mathbb{O}}_3>{\mathbb{O}}_2>{\mathbb{O}}_5>{\mathbb{O}}_1>{\mathbb{O}}_4$.

7.3. Comparison with the IVFF-TOPSIS Method

In this section, we implement the specific procedure of the IVFF-TOPSIS approach [35] to select the most qualified green supplier.

Like the VIKOR method, in the first stage, the $IV{q}^{r}OF$WA operator was utilized to combine the individual DMXs, and the results are presented in Table 9.

According to Equation (54), the weighted DMX is obtained (see

Table 13. Weighted $IV{q}^{r}OF$ DMX.

	${\mathbb{C}}_1$	${\mathbb{C}}_2$	${\mathbb{C}}_3$	${\mathbb{C}}_4$	${\mathbb{C}}_5$	${\mathbb{C}}_6$
${\mathbb{O}}_1$	$\left(\left[0.1406,0.2479\right],\left[0.7627,0.8703\right]\right)$	$\left(\left[0.09208,0.1374\right],\left[0.7401,0.7898\right]\right)$	$\left(\left[0.1740,0.2081\right],\left[0.8070,0.8325\right]\right)$	$\left(\left[0.1594,0.2090\right],\left[0.7087,0.7783\right]\right)$	$\left(\left[0.3399,0.3862\right],\left[0.8761,0.8905\right]\right)$	$\left(\left[0.3905,0.4735\right],\left[0.8931,0.9143\right]\right)$
${\mathbb{O}}_2$	$\left(\left[0.1277,0.2384\right],\left[0.8184,0.8771\right]\right)$	$\left(\left[0.2174,0.2686\right],\left[0.8287,0.8598\right]\right)$	$\left(\left[0.07859,0.1039\right],\left[0.8976,0.9215\right]\right)$	$\left(\left[0.3457,0.3980\right],\left[0.8822,0.9002\right]\right)$	$\left(\left[0.2895,0.3346\right],\left[0.8404,0.8662\right]\right)$	$\left(\left[0.2783,0.3231\right],\left[0.8329,0.8603\right]\right)$
${\mathbb{O}}_3$	$\left(\left[0.1417,0.2124\right],\left[0.8491,0.8994\right]\right)$	$\left(\left[0.4094,0.4641\right],\left[0.8342,0.8591\right]\right)$	$\left(\left[0.1378,0.1700\right],\left[0.8707,0.9020\right]\right)$	$\left(\left[0.3722,0.3952\right],\left[0.8026,0.8270\right]\right)$	$\left(\left[0.2530,0.2938\right],\left[0.8849,0.9031\right]\right)$	$\left(\left[0.3932,0.4333\right],\left[0.7418,0.8039\right]\right)$
${\mathbb{O}}_4$	$\left(\left[0.1623,0.2499\right],\left[0.8216,0.8941\right]\right)$	$\left(\left[0.1401,0.2934\right],\left[0.8703,0.9050\right]\right)$	$\left(\left[0.1439,0.2091\right],\left[0.8890,0.9130\right]\right)$	$\left(\left[0.3450,0.3960\right],\left[0.9055,0.9319\right]\right)$	$\left(\left[0.1774,0.2103\right],\left[0.8492,0.8786\right]\right)$	$\left(\left[0.1904,0.2494\right],\left[0.6569,0.7316\right]\right)$
${\mathbb{O}}_5$	$\left(\left[0.2877,0.3300\right],\left[0.6637,0.7676\right]\right)$	$\left(\left[0.1603,0.2983\right],\left[0.8849,0.9034\right]\right)$	$\left(\left[0.2470,0.2836\right],\left[0.9338,0.9471\right]\right)$	$\left(\left[0.2338,0.2602\right],\left[0.8749,0.8940\right]\right)$	$\left(\left[0.3562,0.3972\right],\left[0.8229,0.8510\right]\right)$	$\left(\left[0.2163,0.2443\right],\left[0.7182,0.7851\right]\right)$

) using the criteria weights generated by the proposed CRITIC approach.

$$\AE ={\left[{\ae }_{ij}\right]}_{m\times n}={\left[{\omega }_j\otimes {\mathfrak{N}}_{ij}\right]}_{m\times n}.$$

(54)

$${\ddot{\P }}^{+}={\left[{\ddot{\P }}_{sj}^{+}\right]}_{1\times n}=\left(\left[\underset{i}{max}\left\{{\underline{\ddot{\Omega }}}_{ij}\right\},\underset{i}{max}\left\{{\overline{\ddot{\Omega }}}_{ij}\right\}\right],\left[\underset{i}{min}\left\{{\underline{\ddot{\mho }}}_{ij}\right\},\underset{i}{min}\left\{{\overline{\ddot{\mho }}}_{ij}\right\}\right]\right),$$

(55)

$${\ddot{\P }}^{-}={\left[{\ddot{\P }}_{sj}^{-}\right]}_{1\times n}=\left(\left[\underset{i}{min}\left\{{\underline{\ddot{\Omega }}}_{ij}\right\},\underset{i}{min}\left\{{\overline{\ddot{\Omega }}}_{ij}\right\}\right],\left[\underset{i}{max}\left\{{\underline{\ddot{\mho }}}_{ij}\right\},\underset{i}{max}\left\{{\overline{\ddot{\mho }}}_{ij}\right\}\right]\right).$$

(56)

In this step, PIS ${\ddot{\P }}^{+}$ and NIS ${\ddot{\P }}^{-}$ are determined according to Equations (55) and (56), respectively. The derived results are recorded as follows:

$${\ddot{\P }}^{+}=\left\{\begin{array}{c}\left(\left[0.2877,0.3300\right],\left[0.6637,0.7676\right]\right),\left(\left[0.4094,0.4641\right],\left[0.7401,0.7898\right]\right),\\ \left(\left[0.2470,0.2836\right],\left[0.8070,0.8325\right]\right),\left(\left[0.3722,0.3980\right],\left[0.7087,0.7783\right]\right),\\ \left(\left[0.3562,0.3972\right],\left[0.8229,0.8510\right]\right),\left(\left[0.3932,0.4735\right],\left[0.6569,0.7316\right]\right),\end{array}\right\},$$

$${\ddot{\P }}^{-}=\left\{\begin{array}{c}\left(\left[0.1406,0.2124\right],\left[0.8491,0.8994\right]\right),\left(\left[0.09208,0.1374\right],\left[0.8849,0.9050\right]\right),\\ \left(\left[0.07859,0.1039\right],\left[0.9338,0.9471\right]\right),\left(\left[0.1594,0.2090\right],\left[0.9055,0.9319\right]\right),\\ \left(\left[0.1774,0.2103\right],\left[0.8849,0.9031\right]\right),\left(\left[0.1904,0.2443\right],\left[0.8931,0.9143\right]\right),\end{array}\right\}.$$

With the equations $d_i\left({\mathbb{O}}_i,{\ddot{\P }}^{+}\right)={\displaystyle \sum _{j=1}^n}{d}_{GE}\left({\ae }_{ij},{\ddot{\P }}_{sj}^{+}\right)$ and $d_i\left({\mathbb{O}}_i,{\ddot{\P }}^{-}\right)={\displaystyle \sum _{j=1}^n}{d}_{GE}\left({\ae }_{ij},{\ddot{\P }}_{sj}^{-}\right)$ (here $d_{GE}$ denotes the generalized Euclidean distance measure given in the literature [35]), the distance between each ${\mathbb{O}}_i$ and PIS and NIS, respectively, is determined. The obtained results are shown as follows: $d_1\left({\mathbb{O}}_1,{\ddot{\P }}^{+}\right)=0.6701$, $d_2\left({\mathbb{O}}_2,{\ddot{\P }}^{+}\right)=0.9234$, $d_3\left({\mathbb{O}}_3,{\ddot{\P }}^{+}\right)=0.7790$, $d_4\left({\mathbb{O}}_4,{\ddot{\P }}^{+}\right)=0.9313$, $d_5\left({\mathbb{O}}_5,{\ddot{\P }}^{+}\right)=0.7087$.

$d_1\left({\mathbb{O}}_1,{\ddot{\P }}^{-}\right)=0.9376$, $d_2\left({\mathbb{O}}_2,{\ddot{\P }}^{-}\right)=0.4215$, $d_3\left({\mathbb{O}}_3,{\ddot{\P }}^{-}\right)=0.5838$, $d_4\left({\mathbb{O}}_4,{\ddot{\P }}^{-}\right)=0.5594$, $d_5\left({\mathbb{O}}_5,{\ddot{\P }}^{-}\right)=0.6754$.

$$C{C}_i\left({\mathbb{O}}_i\right)=\frac{d_i\left({\mathbb{O}}_i,{\ddot{\P }}^{-}\right)}{d_i\left({\mathbb{O}}_i,{\ddot{\P }}^{+}\right)+d_i\left({\mathbb{O}}_i,{\ddot{\P }}^{-}\right)};i=1\left(1\right)m.$$

(57)

In view of Equation (57), the closeness coefficient $C{C}_i\left({\mathbb{O}}_i\right)$ of each alternative is obtained, i.e., $C{C}_1\left({\mathbb{O}}_1\right)=0.5832,$ $C{C}_2\left({\mathbb{O}}_2\right)=0.3134,$ $C{C}_3\left({\mathbb{O}}_3\right)=0.4284,$ $C{C}_4\left({\mathbb{O}}_4\right)=0.3753,$ $C{C}_5\left({\mathbb{O}}_5\right)=0.4880$.

Keeping in mind that the optimal option should have the highest closeness coefficient and vice versa, the suppliers are ordered as follows: ${\mathbb{O}}_1>{\mathbb{O}}_5>{\mathbb{O}}_3>{\mathbb{O}}_4>{\mathbb{O}}_2$.

7.4. Comparison with the IVPyF-TODIM Method

To exemplify the benefits of the provided TODIM approach, we apply the IVPyF-TODIM method [79] to the same example. Interestingly, it is obvious from an examination of Table 2 that IVPyF-TODIM is unable to rank the alternatives since it cannot guarantee that the assessment value meets ${\overline{\Omega }}^2+{\overline{\mho }}^2>1$. As a result, the solution proposed by Zhao et al. fails to address the issue at hand. It only works if the assessment information meets the criterion ${\overline{\Omega }}^2+{\overline{\mho }}^2\le 1$. The provided approach, on the other hand, can manage the assessment data satisfactorily. For instance, if we replace some values of upper bounds with 0.8 or 0.9, by varying the values of p and q, our technique remains valid. As a result, the proposed method has a greater application scope than the prior one. Furthermore, as described in Section 3.3, our created approach employs a novel score function for $IVp,q^{r}OFNs$, the outcomes of which are more reasonable than those of the present ones. Due to its novel scoring function and parameters p and q, this new approach based on $IVp,q^{r}OFNs$ is more powerful and efficient than previous methods.

7.5. Results and Discussion

Based on the data in

Table 14. Ranking results of previous approaches and the developed approach.

Methods
	IVqr OF VIKOR [77]		IVqr OF CODAS [78]		IVFF-TOPSIS [35]		IVPyF-TODIM [79]		Developed Approach
Suppliers	Θi	Rank	ASi	Rank	CCi	Rank	ϝi	Rank	ϝi	Rank
${\mathbb{O}}_1$	0.4777	3	0.8133	4	0.5832	1	—	—	0.3827	2
${\mathbb{O}}_2$	0.2759	2	2.0108	2	0.3134	5	—	—	0.3684	3
${\mathbb{O}}_3$	0.0000	1	4.0903	1	0.4284	3	—	—	1.000	1
${\mathbb{O}}_4$	0.4988	4	0.7828	5	0.3753	4	—	—	0.0000	5
${\mathbb{O}}_5$	1.0000	5	1.4128	3	0.4880	2	—	—	0.2646	4

, we offer the following analysis and commentary:

➊

Both the VIKOR and TODIM techniques share the commonality of considering the DE’s subjective opinion. The presented method, however, is superior to the VIKOR method in its ability to assess the DE’s risk tolerance. The risk attitude is reflected through the attenuation parameter, a representation that is consistent with the psychological behavior of DEs and also reasonably reflects the logic of whether the losses are mitigated or increased. Table 14 displays the ranking results obtained by the $IV{q}^{r}OF$-VIKOR method, which are partially discordant with those from the suggested approach, particularly in regards to the positions of ${\mathbb{O}}_2$, ${\mathbb{O}}_3$, and ${\mathbb{O}}_4$. Two primary reasons cause this difference: Firstly, the prior method [77] mainly responds to MCGDM problems with $IV{q}^{r}OF$ data and fails to handle data requiring different term levels. Secondly, the method in Reference [77] is based on distance measures that ignore the degree of indeterminacy aspect while calculating the mean ${Ĩ}_i$ and worst ${Ī}_i$ scores of alternatives. Thus, it does not use decision data in the full sense.

➋

For the outputs of the $IV{q}^{r}OF$ CODAS [78] and presented TODIM methods, although the best and worst suppliers are the same, the remaining ranking results have slight differences. This is inevitable since the proposed parameterized (p and q) approach provides advantages in dealing with experts’ psychological behaviors. In addition, the distance measure accounted for in $IV{q}^{r}OF$ CODAS does not consider the indeterminacy degrees, which causes information loss and distortion. Consideration of degrees of uncertainty in the measures has been shown to improve the validity and reliability of evaluations in a number of works [39]. Therefore, these degrees of objects should be taken into account during the decision-making process. Furthermore, the created mechanism’s criterion weights are collected using the CRITIC technique, which incorporates a statistical viewpoint into criteria weighting by employing correlation coefficients and standard deviation terms. The entropy method [78] lacks this capability, which also impacts alternative rankings. It is worth mentioning that in the conducted comparison, we employed the weight vectors that the proposed CRITIC has obtained because the entropy formulation (Equation (21) of reference [78]) is only applicable for q-rung orthopair fuzzy data.

➌

The ranking results obtained by the IVFF-TOPSIS [79] method differ significantly from those obtained by the proposed approach, which can be explained by two main aspects: (a) The TOPSIS technique obtains the aggregated matrix using $IV{q}^{r}OF$ AOs, whereas the suggested method obtains the same matrix via $IVp,q^{r}OF$ AOs. Different parameter values, that is, “3” and “2”, were utilized in these two AOs, which have caused the fusion results displayed in Table 4 and Table 9 to be different from each other. (b) It is evident from the mathematical formulas given in [35] for determining the PIS ${\ddot{\P }}^{+}$ and NIS ${\ddot{\P }}^{-}$, i.e., Equations (55) and (56), that these are all sorts of fixed criteria, whether they are BT or CT. Furthermore, there is no normalization step in Jeevaraj’s TOPSIS algorithm. Thus, for certain MCGDM situations involving CT criterion, notably the considered one, ignoring the normalization phase leads to wrong decisions.

➍

Zhao et al.’s [79] approach is basically an extension of the TODIM method in a IVPyF environment and is incapable of handling the aforesaid application.

In light of this analysis, the following are the key merits of the proposed model:

(1).

The model proposed in this study characterizes fuzzy data on a broader scale, and it allows the sum, square sum, and cubic sum of the upper bound MG and thte upper bound NMG to exceed one. The proposed method aims to offer a more enhanced tool for the computational evaluation of uncertainty because it allows for assessments with varying term levels for the upper bound MG and upper bound NMG, i.e., $0\le {\left(\overline{\Omega }\right)}^p+{\left(\overline{\mho }\right)}^q\le 1$; $p,q\ge 1.$ Therefore, its scope of applicability is broader than $IV{q}^{r}OF$ DM methods [35,77,78,79]. The utilization of these two parameters (p,q), in contrast to $IV{q}^{r}OFS$s, which only employ a single parameter (q), yields more accurate and precise selection values in decision outcomes when implemented within the MCGDM framework.

(2).

The ranking order given in earlier methods is determined based on score functions that have several flaws, as outlined in Examples 1–7. On the other hand, in the present research, the ranking order of the $IVp,q^{r}OFNs$ is determined according to the new score specified in Equation (19), which overcomes the drawbacks associated with the available score functions.

(3).

One notable advantage of the proposed TODIM model is its ability to capture the psychological behaviors exhibited by DEs accurately. In contrast, the TOPSIS and CODAS models do not incorporate this particular mechanism.

(4).

The distance measures employed in the VIKOR and CODAS methods are defined on a specific subclass of the $IVp,q^{r}OFS$, and are blind to the degree of indeterminacy. On the other hand, in the present study, we designed the distance measure in the $IVp,q^{r}OF$ setting, incorporating a consideration of the degree of indeterminacy to prevent severe losses of information.

(5).

The proposed $IVp,q^{r}OFS$ TODIM method does not require pre-provision of criteria weights. Instead, it constructs a model to objectively determine the weights, which is a more rational approach. In contrast, the other three methods [35,77,79] rely on decision experts to provide weight information beforehand, which is subjective and can occasionally yield unreasonable outcomes [80].

Alongside these merits, there are also certain demerits associated with this study:

(1).

The introduced AOs in this study are unable to handle the interrelationship among the aggregated arguments during the aggregation procedure.

(2).

The weights of DEs should be predefined in the developed methodology, which makes achieving objectivity and rationality challenging.

(3).

In the practical implementation of this study, it is imperative to incorporate additional criteria to determine the optimal green supplier in the real world accurately.

8. Conclusions and Future Work

In this paper, we put forward a new FS known as a $IVp,q^{r}OFS$, along with its fundamental tenets. The proposal has the advantage of providing a broader measure for the mathematical study of uncertainty, and it also has the enticing feature of allowing the handling of situations requiring assessments with varied term levels for the upper bounds of the MG and NMG. Earlier methods, such as the score function for rating the given numbers from IVIFSs, IVPyFSs, IVFFSs, or IVq-ROFSs, have been observed to miss the objective. The paper has identified the deficiencies of these score functions in Table 1. To conquer these limitations, a refined score function for pairs of $IVp,q^{r}OFNs$ matching the specified objects is built. The superiority and applications of the provided score function are also supported by Examples 1–7 cited in Table 1. To aggregate the $IVp,q^{r}OFS$ data, some AOs have been researched, which make the aggregation process more flexible due to the involvement of extra parameters. Later, the enviable features relating to each operator were explored. Meanwhile, for the case that criteria weights are completely unknown, the CRITIC process has been structured under the background of a $IVp,q^{r}OFS$ to retrieve the weight information objectively. Following this, an advanced MCGDM approach called $IVp,q^{r}OFS$-TODIM is introduced to figure out the ordering of various alternatives, and is capable of dealing with the DEs’ psychological measures and uncertain data. Finally, the deployed framework is used in a case study of green supplier selection, and its effectiveness is proven by a detailed sensitivity analysis and comparisons with earlier methods. The results indicated that the created approach is more accurate and rational in analyzing and ranking alternatives than other approaches.

Future research efforts will aim to address the caveats of this study (as outlined in Section 7.5). In addition, the developed MCGDM mechanism will be expanded to “covering-based q-rung orthopair fuzzy rough sets [81]”, “circular-IFSs [82]”, “(a, b)-fuzzy soft sets [83]”, and “disc Pythagorean fuzzy sets [84]”. Next, we will present a number of novel AOs based on $IVp,q^{r}OF$ information and discuss their features and potential uses.