1 Introduction

This article studies new models of covering via multigranulation fuzzy rough sets by introducing the notions of family fuzzy \(\beta \)-minimal and \(\beta \)-maximal descriptions.

1.1 Background

Pawlak (1982) invented rough set theory as a strategy that scholars can use to address the conflicts and complexity in information systems as well as data processing.

Therefore, in recent years, several improvements have been distinguished, in which the partitioning part has been replaced with different structures, namely binary relations (Herawan et al. 2010) and (Jensen and Shen 2004), neighborhood systems (Atef et al. 2020) and (Yao 1998), Boolean algebras (Liu and Zhu 2008) and (Wu and Zhang 2002), and covering techniques (Bonikowski et al. 1998) and (Pomykala 1987, 1988).

Many academics have worked on covering rough sets. Two couple mixed approximation operators were created by Pomykala (1987, 1988) and then analyzed through the concept of neighborhood by Yao (1998). Couso and Dubois (2011) explored new models of insufficient input, and Bonikowski et al. (1998) discovered a CRS framework using minimum description. Zhu (2007); Zhu and Wang (2003, 2007) and Zhu and Wang (2012) defined different covering methods and examined their relations. Additionally, there were various templates presented by Tsang et al. (2008) and Xu and Zhang (2007) and at the same time (Liu and Sai 2009) provided us with the differences among Zhu’s schemes and structures given by Xu and Zhang (2007). Ma (2012) generated new CRS models via a complementary neighborhood, while (Yao and Yao 2012) identified four neighborhoods and derived six covering models from them. To widen the range of the study, multigranulation rough sets (MGRS) were demonstrated by Qian et al. (2010) using multiple binary relations. The MGRS was generated with variable precision by Xu et al. (2011). Yang et al. (2011) introduced MGRS according to a fuzzy relation. Furthermore, Liu et al. (2014) observed multigranulation covering rough sets (MGCRS), and (Lin et al. 2013) showed a covering through multigranulation rough set (CMGRS). Further information regarding the (MGRS) and its various uses can be obtained from numerous sources. For instance, Yang et al. (2012), Huang et al. (2014), Sun and Ma (2015), Xue et al. (2023), Zhan and Xu (2020) and Chen and Zhu (2023).

RS techniques are incompetent at working real-valued features successfully because RS fundamentally handles qualitative data (Jensen and Shen 2004). So, Zadeh (1975) introduced fuzzy set theory (FS) to treat this problem effectively using a membership function. After that, by combining RS and FS, the views of rough fuzzy sets (RFS) and fuzzy rough sets (FRS) appear by Dubois and Prade (1990). Many investigators have worked on this definition and tried to use distinct samples to develop FRS. A well-thought-out structure is designed utilizing fuzzy relations and fuzzy sets rather than crisp ones.

De Cock et al. (2024) developed FRS through R-foresets of every object in the universe in relation to a fuzzy relation. Li et al. (2008) and Feng et al. (2012) initiated the definition of a fuzzy covering (FC). Then, Ma (2016) worked to determine the barriers of FC and process them by offering a fuzzy \(\beta \) covering (\(\hbox {F}_{\beta }\)CRS). She introduced it by substituting 1 with an undetermined variable \(\beta \) that takes values between 0 and 1. This indicates that if \(\beta =1\), FC is a particular instance from Ma\('\)s manner. After that several scientists extended Ma\('\)s method in different ways such as Yang and Hu (2017, 2019) proposed new kinds of \(\hbox {F}_{\beta }\)CRS and examined their properties utilizing the complementary fuzzy \(\beta \)-neighborhood, fuzzy \(\beta \)-minimum and \(\beta \)-maximum descriptions, D’eer et al. (2017, 2016) debates neighborhoods through CRS and fuzzy neighborhoods via FC, Jiang et al. (2020) and Atef and Azzam (2021) established covering fuzzy rough sets through variable precision, Garg and Atef (2022) created new approaches to a covering using Pythagorean fuzzy sets and q-ROF sets; new extensions via two different global groups are demonstrated by Yang (2022) and Yang and Atef (2023), Atef et al. (2023) examined the topological properties of covering soft rough and used them to solve MGDM problems and the latest models are explored through fermatean and interval-valued fermatean fuzzy sets by Qi et al. (2024). Multigranulation in fuzzy variations of rough sets has been linked to three-way multi-attribute group decision-making Zhang et al. (2021a, b, 2022) recently. Thus, Zhan et al. (2023) presented a survey for three-way behavioral decision-making with a hesitant fuzzy environment. As well as, Zhan et al. (2024) presented a hybrid multivariate time series prediction framework based on fuzzy C-means clustering coupled with feature selection.

1.2 Motivation and contribution of this article

Numerous propositions have been put out to investigate the domain of fuzzification of MGRS. The meaning of multigranulation fuzzy rough sets (MGFRS) was proposed by Xu et al. (2011, 2014) through the utilization of an equivalence relation, drawing inspiration from the ideas put out by Dubois and Prade (1990). Here, let us further emphasize this research to be able to proceed with a larger volume of data, Liu and Wang (2011) and Liu and Miao (2011) discovered coverage fuzzy rough methods through MGRS, Liu and Pedrycz (2016) put forward new covering models by MGFRSs, Zhan et al. (2019) proposed the covering based multigranulation (IT) fuzzy rough sets with practical tests, Ma et al. (2020) discussed certain kinds of covering samples in the view of multigranulation (IT) fuzzy rough sets, Zhan et al. (2020) demonstrated \(\hbox {C}_{{MG}}\)FRS models via the collection of fuzzy \(\beta \)-neighborhoods and Atef and El-Atik (2021) constructed four types of approximations according to the notion of the collection of fuzzy complementary \(\beta \)-neighborhoods, a novel model of fuzzy \(\beta \)-covering through MGRS was presented and applied for attribute picking to reduce the impact of noisy data by Zhang and Dai (2022) explored new types of FRS models in \(\hbox {F}_{\beta }\)CAS to eliminate some of the noise data, Ye et al. (2021) proposed the \((\mathscr {I},\mathscr {T})\)-FRS templates to address MCDM problems, Huang and Li (2022) suggested a new scale of discrimination using fuzzy \(\beta \)-neighborhood and evaluated its effectiveness through the attribute reduction algorithm, Atef and Liu (2024) discovered the grey \(\beta \) coverings and examined their properties. Consequently, Zhang et al. (2021a, b) and Ma et al. (2021) defined the fuzzy \(\alpha \) covering with reflexivity to solve the classical concept issues. Based on this idea, Huang and Li (2024a) presented fuzzy \(\gamma \) neighborhoods with reflexivity and utilized them to investigate a noise-tolerant variable precision discrimination index (VPDI), Huang et al. (2022) established fuzzy \(\beta \) covering based multigranulation rough sets, Huang et al. (2024) improved noise-tolerant ability through introduced multigranulation variable precision via fuzzy \(\beta \) covering and Huang and Li (2024b) proposed fuzzy \(\beta \) covering via multigranulation rough fuzzy sets and used it in a feature selection process.

The study conducted by Sun et al. (2017) focused on examining the procedure of three-way group decision-making utilizing a multigranulation fuzzy decision approach. Group Decision Making (GDM) has been widely recognized as a highly successful approach for addressing intricate challenges encountered in real-world contexts. The utilization of multi-criteria in (GDM) was investigated by Hwang and Lin (1987) in their study. Multi-criteria group decision-making (MCGDM) pertains to the process of evaluating and prioritizing many feasible alternatives based on competing and interactive criteria. Numerous scholars are actively involved in this field through the utilization of MCGDM. For instance, notable studies conducted by Chen et al. (2012), Hong and Choi (2000), and Zhang et al. (2014) exemplify this trend. In their study, Mardani et al. (2015) conducted a comprehensive evaluation of Multiple Criteria Group Decision Making (MCGDM) methods that are grounded in FS. The review specifically focused on the period between 1994 and 2014. Numerous scholars have proposed novel procedures and strategies pertaining to decision-making methods such as Zhu et al. (2023a) constructed MADM with probabilistic linguistic three-way decision method according to the regret theory (RT), Han and Zhan (2023) discovered MAGDM problems in the environment of probabilistic linguistic term sets (PLTSs), Zhu et al. (2023b) investigated three-way consensus model based on RT over the PLTSs to fix MAGDM, Han et al. (2024a) introduced a three-way group consensus methodology grounded in probabilistic linguistic preference relations (PLPRs), Han et al. (2024b) and Ma et al. (2024) use PLPRs to solve complex problems, Xiao et al. (2023) proposed a new method, namely, prospect theory through intuitionistic fuzzy numbers within generalized three way decision framework (PT-IF-G3WD), Guo et al. (2023) established a large scale group decision making through fuzzy preference relations (FPRs) and Xiao et al. (2024) defined novel way to solve real problems under multi-scale decision information systems. Subsequently, Sivaprakasam and Angamuthu (2023) presented the concept of a generalized Z-fuzzy soft-covering-based rough matrices to address MAGDM issues, Pamucar et al. (2022) explored a novel way in the modification of the CRiteria Importance Through Intercriteria Correlation (CRITIC) sample using fuzzy rough numbers, Shen (2024) demonstrated a rough set based bipolar method, Kakati et al. (2024) constructed fermatean fuzzy archimedean heronian mean-based method to fix transport issues, Donbosco and Ganesan (2022) discovered the energy of rough neutrosophic matrix to treat the MCDM problem and Zhang et al. (2024) defined a multi-scale neighborhood for feature selection. Upon reviewing the extant literature, it becomes evident that there is a dearth of research pertaining to the exploration of \(\hbox {C}_{{MG}}\)FRSs, with a specific focus on their applications in the domain of multi-attribute decision-making. Therefore, to widen the range of studying of the given approaches, we present novel \(\hbox {C}_{{MG}}\)FRS models utilizing the definition of the family of fuzzy \(\beta \)-minimum and \(\beta \)-maximum descriptions. According to the literature reviewed above, it is evident that there are restricted studies on \(\hbox {C}_{{MG}}\)FRSs. So, we have the following list that outlines the main works:

  1. 1.

    We introduce the meaning of the family of fuzzy \(\beta \)-minimum and maximum descriptions over the universe.

  2. 2.

    According to these definitions, we investigate two new models of \(\hbox {CO(P)}_{MG}\)FRS (optimistic and pessimistic) and studied their pr.

  3. 3.

    Four samples of \(\hbox {CVP}_{{MG}}\)FRSs are established and their relevant characteristics are informed.

  4. 4.

    Relations and comparisons between these models and the existing models (i.e., Zhan et al. 2019, 2020; Atef and El-Atik 2021 and Ma et al. 2021) are clarified.

  5. 5.

    Numerical examples are presented to show their efficacy in supporting physicians’ decision-making through practical issues in MAGDM and MCGDM

The major goal of this article can be explained in the next Fig. 1.

Fig. 1
figure 1

The major goal of this article

Where white boxes describe existing work and blue boxes shows our research direction in this article.

1.3 Structure of the present paper

The outline of this article is the following. Section 2 gives the principle information about the previous work. Sections 3 and 4 present two new \(\hbox {CO(P)}_{MG}\hbox {FRS}\) methods using the notions of fuzzy \(\beta \)-minimal and maximal descriptions. We propose distinct \(\hbox {CVP}_{MG}\hbox {FRS}\) models and discuss their properties. Section 6 establishes relations among these covering models. Section 7 presents procedures for fixing MAGDM and MCGDM issues that take advantage of our theoretical framework along with some algorithms that facilitate the use of these methods. The main findings and future vision are explained in Sect. 8.

2 Preliminaries

Throughout this section, several fundamental notions will be mentioned for covering models and their generalizations.

Definition 1

(Zhu 2007) Let \(\mathscr {F}\) be a collection of subsets of the common set \(\Omega \) and \(\Omega =\bigcup _{\mathbb {F} \in \mathscr {F}}\mathbb {F}\). Then \(\mathscr {F}\) is a covering of \(\Omega \). The \((\Omega ,\mathscr {F})\) is said to be a (\(\textbf{CAS}\)).

Definition 2

(Feng et al. 2012) Let \((\Omega ,\mathscr {F})\) be a \(\textbf{CAS}\) and \(\mathscr {F}=\{\mathbb {F}_{1}, \mathbb {F}_{2},\ldots , \mathbb {F}_{n}\}\), where \(\mathbb {F}_{j}\in \mathcal {F}(\Omega )(j=1, 2, \ldots , n)\). If \((\bigcup _{i=1}^{n} \mathbb {F}_{j})(a) =1\), \(\forall a \in \Omega \), then \((\Omega ,\mathscr {F})\) is said to be a \(\textbf{FCAS}\).

After that (Ma 2016) developed \(\textbf{FCAS}\) using the parameter \(\beta \) that accepts values (0, 1] as \((\bigcup _{i=1}^{m} \mathbb {F}_{j})(a) \ge \beta \). Thus, \((\Omega ,\mathscr {F})\) is a \({\textbf {F}}_{\beta }\)CAS.

Definition 3

(Ma 2016; Yang and Hu 2017, 2019) Let \((\Omega ,\mathscr {F})\) be a \({\textbf {F}}_{\beta }\)CAS. For all \(a \in \Omega \), we have the following neighborhoods.

  1. 1.

    \(\mathbb {N}^{\beta }_{a}= \cap \{\mathbb {F}_{j} \in \mathscr {F}:\mathbb {F}_{j} \ge \beta \}\).

  2. 2.

    \(\mathbb {M}^{\beta }_{a}(y)= \mathbb {N}^{\beta }_{y}(a)\).

  3. 3.

    \(\mathbb {M}d^{\beta }_{a}=\{\mathbb {F}\in \mathscr {F}: (\mathbb {F}(a)\ge \beta ) \wedge (\forall \mathbb {D}\in \mathscr {F} \wedge \mathbb {D}(a) \ge \beta \wedge \mathbb {D} \subseteq \mathbb {F} \rightarrow \mathbb {D}=\mathbb {F})\}\).

  4. 4.

    \(\mathbb {M}D^{\beta }_{a}= \{\mathbb {F}\in \mathscr {F}: (\mathbb {F}(a)\ge \beta ) \wedge (\forall \mathbb {D}\in \mathscr {F} \wedge \mathbb {D}(a) \ge \beta \wedge \mathbb {D} \supseteq \mathbb {F} \rightarrow \mathbb {D}=\mathbb {F})\}\).

Zhan et al. (2020) gave the \(\hbox {C}_{{MG}}\)FRSs utilizing the concepts of CRS, MGFRS and \({\textbf {F}}_{\beta }\)CAS.

Let \((\Omega ,{\kappa })\) be a \({\textbf {F}}_{\beta }\)CAS and \({\kappa }= \{\mathbb {F}_{1}, \mathbb {F}_{2},\ldots ,\mathbb {F}_{n}\}\) be n fuzzy \(\beta \) coverings of \(\Omega \), where \(\mathbb {F}_{j}= \{\mathbb {F}_{j1}, \mathbb {F}_{j2},\ldots ,\mathbb {F}_{jr}\}\), for all \(j=1,2,\ldots ,n\). Then, we can call \((\Omega ,{\kappa })\) by \({\textbf {MGF}}_{\beta }\)CAS for short.

Definition 4

(Zhan et al. 2020) Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For any \(\mathscr {G} \in {\mathcal {F}}(\Omega ),\) define the i-OLA, i-OUA, i-PLA, and the i-PUA, respectively, as follows:

$$\begin{aligned}{} & {} {_{-i}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{j=1}^{n} \bigwedge \limits _{y\in \Omega }\bigg \{(1-\mathbb {N}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {G}(y)\bigg \},\\{} & {} {_{+i}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigwedge \limits _{j=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\mathbb {N}_{\mathbb {F}_{j}(a)}^{\beta }(y)\wedge \mathscr {G}(y)\bigg \},\\{} & {} {_{-i}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigwedge \limits _{j=1}^{n} \bigwedge \limits _{y\in \Omega }\bigg \{(1-\mathbb {N}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {G}(y)\bigg \},\hbox { and}\\{} & {} {_{+i}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{j=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\mathbb {N}_{\mathbb {F}_{j}(a)}^{\beta }(y)\wedge \mathscr {G}(y)\bigg \},\hbox { for all }a \in \Omega . \end{aligned}$$

Atef and El-Atik (2021) extended the studies of Zhan et al. (2020) using the family of fuzzy complementary \(\beta \) neighborhoods and examined their properties.

Definition 5

(Atef and El-Atik 2021) Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For any \(\mathscr {G} \in {\mathcal {F}}(\Omega ),\) define the ii(resp. iii and iv)-OLA, ii(resp. iii and iv)-OUA, ii(resp. iii and iv)-PLA, and the ii(resp. iii and iv)-PUA, respectively, as follows:

$$\begin{aligned}{} & {} {_{-ii}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{j=1}^{n} \bigwedge \limits _{y\in \Omega }\bigg \{(1-\mathbb {M}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {G}(y)\bigg \},\\{} & {} {_{+ii}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(x)= \bigwedge \limits _{j=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\mathbb {M}_{\mathbb {F}_{j}(x)}^{\beta }(y)\wedge \mathscr {G}(y)\bigg \},\\{} & {} {_{-ii}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigwedge \limits _{j=1}^{n} \bigwedge \limits _{y\in \Omega }\bigg \{(1-\mathbb {M}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {G}(y)\bigg \},\\{} & {} {_{+ii}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{j=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\mathbb {M}_{\mathbb {F}_{j}(a)}^{\beta }(y)\wedge \mathscr {G}(y)\bigg \}.\\{} & {} {_{-iii}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{j=1}^{n} \bigwedge \limits _{y\in \Omega }\bigg \{(1-\mathbb {N}_{\mathbb {F}_{j}(a)}^{\beta }(y)) \vee (1-\mathbb {M}_{\mathbb {F}_{j}(a)}^{\beta }(y)) \vee \mathscr {G}(y)\bigg \},\\{} & {} {_{+iii}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(x)= \bigwedge \limits _{j=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\mathbb {N}_{\mathbb {F}_{j}(a)}^{\beta }(y)\wedge \mathbb {M}_{\mathbb {F}_{j}(x)}^{\beta }(y)\wedge \mathscr {G}(y)\bigg \},\\{} & {} {_{-iii}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigwedge \limits _{j=1}^{n} \bigwedge \limits _{y\in \Omega }\bigg \{(1-\mathbb {N}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee (1-\mathbb {M}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {G}(y)\bigg \},\\{} & {} {_{+iii}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{j=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\mathbb {N}_{\mathbb {F}_{j}(a)}^{\beta }(y) \wedge \mathbb {M}_{\mathbb {F}_{j}(a)}^{\beta }(y)\wedge \mathscr {G}(y)\bigg \}. \end{aligned}$$

and

$$\begin{aligned}{} & {} {_{-iv}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{j=1}^{n} \bigwedge \limits _{y\in \Omega }\bigg \{(1-\mathbb {N}_{\mathbb {F}_{j}(a)}^{\beta }(y)) \wedge (1-\mathbb {M}_{\mathbb {F}_{j}(a)}^{\beta }(y)) \vee \mathscr {G}(y)\bigg \},\\{} & {} {_{+iv}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(x)= \bigwedge \limits _{j=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\mathbb {N}_{\mathbb {F}_{j}(a)}^{\beta }(y)\vee \mathbb {M}_{\mathbb {F}_{j}(x)}^{\beta }(y)\wedge \mathscr {G}(y)\bigg \},\\{} & {} {_{-iv}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigwedge \limits _{j=1}^{n} \bigwedge \limits _{y\in \Omega }\bigg \{(1-\mathbb {N}_{\mathbb {F}_{j}(a)}^{\beta }(y))\wedge (1-\mathbb {M}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {G}(y)\bigg \},\\{} & {} {_{+iv}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{j=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\mathbb {N}_{\mathbb {F}_{j}(a)}^{\beta }(y) \vee \mathbb {M}_{\mathbb {F}_{j}(a)}^{\beta }(y)\wedge \mathscr {G}(y)\bigg \}, \end{aligned}$$

for all \(a \in \Omega \).

3 The first \(\hbox {CO(P)}_{MG}\)FRS model via the fuzzy \(\beta \)-minimal description

This section presents the \(\hbox {CO}_{{MG}}\)FRS and \(\hbox {CP}_{{MG}}\)FRS models, which are based on a set of fuzzy \(\beta \)-minimal descriptions and examine some of their characteristics.

Definition 6

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For all \(a \in \Omega \),

$$\begin{aligned} \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }= \{\mathbb {F}_{ji}\in \mathbb {F}_{j}: (\mathbb {F}_{ji}(a)\ge \beta ) \wedge (\forall \mathbb {F}_{jr}\in \mathbb {F}_{j} \wedge \mathbb {F}_{jr}(a) \ge \beta \wedge \mathbb {F}_{jr} \subseteq \mathbb {F}_{ji} \rightarrow \mathbb {F}_{jr}=\mathbb {F}_{ji})\} \end{aligned}$$

is called the family of fuzzy \(\beta \)-minimal descriptions of a for any \(j=1,\ldots ,n\) and \(i,r=1,\ldots ,m\).

Example 1

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS and \(\kappa = \{\mathbb {F}_{1}, \mathbb {F}_{2}\}\) be a family of \(\hbox {F}_{\beta }\)Cs of \(\Omega \), where \(\Omega =\{a_1,a_2,a_3,a_4,a_5,a_6\}\), \(\beta =0.5\) and \(\mathbb {F}_{1}=\) \(\{\mathbb {F}_{11}, \mathbb {F}_{12},\ldots ,\mathbb {F}_{15}\}\) and \(\mathbb {F}_{2}= \{\mathbb {F}_{21}, \mathbb {F}_{22},\ldots ,\mathbb {F}_{25}\}\) as shown in Tables 1 and 2.

Table 1 Data for \(\mathbb {F}_{1}\)
Table 2 Data for \(\mathbb {F}_{2}\)

The fuzzy beta-minimal description \(\tilde{\mathbb {M}d}_{\mathbb {F}_{j}}^{\beta }(a)\) \((j=1,2)\) can subsequently be computed. See Tables 3 and 4.

Table 3 Data for \(\mathbb {N}_{\kappa }^{0.5}(a)\), \(\tilde{\mathbb {M}d}_{\kappa }^{0.5}(a)\), for \(\mathbb {F}_{1}\)
Table 4 Data for \(\mathbb {N}_{\kappa }^{0.5}(a)\), \(\tilde{\mathbb {M}d}_{\kappa }^{0.5}(a)\), for \(\mathbb {F}_{2}\)

3.1 \(\hbox {CO}_{{MG}}\)FRSs

Definition 7

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For every \(\mathscr {G} \in {\mathcal {F}}(\Omega ),\) the 1-OLA and 1-OUA are defined as follows:

$$\begin{aligned}{} & {} {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {G}(y)\bigg \},\hbox { and}\\{} & {} _{+1}{\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigwedge \limits _{i=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y)\wedge \mathscr {G}(y)\bigg \},\hbox { for all }a \in \Omega , \end{aligned}$$

If \({_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\ne {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\), then \(({_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}), {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}))\) is called a 1-\(\hbox {CO}_{{MG}}\)FRS, otherwise it is definable.

Example 2

Consider Example 1, and when \(\mathscr {G}= \frac{0.6}{a_1}+ \frac{0.3}{a_2}+\) \(\frac{0.7}{a_3}+ \frac{0.1}{a_4}+ \frac{0.5}{a_5}+ \frac{0.8}{a_6}\). Then we compute the \(\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y)\) for \(\kappa = \{\mathbb {F}_{1},\mathbb {F}_{2}\}\) as next in Tables 5 and 6.

Table 5 Data for \(\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{1}(a)}^{0.5}\)
Table 6 Data for \(\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{2}(a)}^{0.5}\)

So,

$$\begin{aligned}{} & {} {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})= \frac{0.4}{a_1}+\frac{0.2}{a_2}+\frac{0.4}{a_3}+\frac{0.2}{a_4}+ \frac{0.1}{a_5}+\frac{0.4}{a_6},\hbox { and}\\{} & {} {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})= \frac{0.7}{a_1}+\frac{0.5}{a_2}+\frac{0.7}{a_3}+ \frac{0.7}{a_4}+\frac{0.7}{a_5}+\frac{0.7}{a_6}. \end{aligned}$$

Theorem 1

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For any \(\mathscr {G} \in {\mathcal {F}}(\Omega ),\) the below items for 1-LAO and 1-UAO hold.

  1. (1)

    \({_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\bigg ({_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}^{c}) \bigg )^c\) and \({_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\bigg ({_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}^{c}) \bigg )^c\).

  2. (2)

    \({_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\Omega )=\Omega \) and \({_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\emptyset )=\emptyset \).

  3. (3)

    If \(\mathscr {G} \subseteq \mathscr {Y}\), then \({_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\) \(\subseteq {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\).

  4. (4)

    \({_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cap \mathscr {Y})={_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cap {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cap \mathscr {Y})\subseteq {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cap {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\).

  5. (5)

    \({_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cup \mathscr {Y}) \supseteq {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cup {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cup \mathscr {Y})={_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cup {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\).

  6. (6)

    If \(1-\mathbb {N}_{\mathbb {F}_{j}(x)}^{\beta }\le \mathscr {G}(a) \le \mathbb {N}_{\mathbb {F}_{j}(a)}^{\beta }\), \(\forall a \in \Omega \), then \({_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\).

Proof

  1. (1)
    $$\begin{aligned} ({_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}^{c}))^c(a)&=\bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y)\wedge \mathscr {G}^{c}(y)\bigg \}^{c}\\&=\bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y)\wedge (1-\mathscr {G}(y))\bigg \}^{c}\\&=\bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {G}(y)\bigg \}\\&={_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a). \end{aligned}$$
  2. (2)

    Since \(\Omega (a)= 1\), for any \(a \in \Omega \),

    \({_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\Omega )(a)=\) \(\bigwedge \limits _{i=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{(1-\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \Omega (a)\bigg \}=1=\Omega (a).\)

  3. (3)

    Let \(\mathscr {G}, \mathscr {Y} \in \mathcal {F}(\Omega )\) such that \(\mathscr {G} \subseteq \mathscr {Y}\) and \(a \in \Omega \). Then

    $$\begin{aligned} {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)&=\bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1- \vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {G}(y)\bigg \}\\&\le \bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {Y}(y)\bigg \}\\&={_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})(a). \end{aligned}$$
  4. (4)

    When \(a \in \Omega \), then

    $$\begin{aligned} {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cap \mathscr {Y})(a)&=\bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee (\mathscr {G}\cap \mathscr {Y})(y)\bigg \}\\&=\bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {G}(y)\bigg \}\\&\cap \bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {Y}(y)\bigg \}\\&={_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)\cap {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})(a). \end{aligned}$$
  5. (5)

    Similar to (4).

  6. (6)

    If \(1-{\mathbb {N}}_{\mathbb {F}_{j}(a)}^{\beta }\le \mathscr {G}(a) \le {\mathbb {N}}_{\mathbb {F}_{j}(a)}^{\beta }\), \(\forall a \in \Omega \), then

    $$\begin{aligned} \mathscr {G}(a)&=\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(a) \wedge \mathscr {G}(a)\le \bigvee \limits _{y\in \Omega }\bigg (\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y) \wedge \mathscr {G}(y)\bigg )\\&\implies \mathscr {G}(a)\le \bigwedge \limits _{i=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y)\wedge \mathscr {G}(y)\bigg \}={_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)\\&\implies \bigwedge \limits _{y\in \Omega }\bigg \{(1-\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {G}(y)\bigg \}\le (1-\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(a))\vee \mathscr {G}(a)\le \mathscr {G}(a) \\&\implies \bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {G}(y)\bigg \}\le \mathscr {G}(a)\\&\implies {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)\le \mathscr {G}(a)\\&\implies {_{-1}\mathscr {R}}^{o}_{\sum _{i=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq \mathscr {G} \subseteq {_{+1}\mathscr {R}}^{o}_{\sum _{i=1}^{n}\mathbb {F}_{j}}(\mathscr {G}). \end{aligned}$$

\(\square \)

Definition 8

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS and for any \(a,y \in \Omega \). A MGFMD \(^{1}\mathscr {D}^{\beta }_{\mathbb {F}_{j}(a_i)}(a,y)\), where \(a_i \in \Omega \) and \(i=1,\ldots ,m\) is shown by

$$\begin{aligned} ^{1}\mathscr {D}^{\beta }_{\mathbb {F}_{j}(a_i)}(a,y)= \frac{\sum _{j=1}^{n}\bigg (\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a_i)}^{\beta }(a)\bigwedge \vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a_i)}^{\beta }(y)\bigg )}{\sum _{j=1}^{n}\bigg (\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a_i)}^{\beta }(a) \bigvee \vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a_i)}^{\beta }(y)\bigg )}. \end{aligned}$$

Example 3

(Continued from Examples 1 and 2) If \(\beta =0.5\), then we have the following in \(\mathbb {F}_{1}\):

  1. (1)

    \(^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_1,a_1)=1, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_1,a_2)=0.47, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_1,a_3)=0.7, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_1,a_4)=0.64, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_1,a_5)=0.66,\)

    \(^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_1,a_6)=0.82\).

  2. (2)

    \(^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_2,a_1)=0.47, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_2,a_2)=1, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_2,a_3)=0.4, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_2,a_4)=0.49, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_2,a_5)=0.66,\)

    \(^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_2,a_6)=0.42\).

  3. (3)

    \(^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_3,a_1)=0.7, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_3,a_2)=0.4, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_3,a_3)=1, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_3,a_4)=0.78, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_3,a_5)=0.66,\)

    \(^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_3,a_6)=0.86\).

  4. (4)

    \(^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_4,a_1)=0.64, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_4,a_2)=0.49, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_4,a_3)=0.78, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_4,a_4)=1, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_4,a_5)=0.74, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_4,a_6)=0.73\).

  5. (5)

    \(^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_5,a_1)=0.66, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_5,a_2)=0.66, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_5,a_3)=0.66, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_5,a_4)=0.74, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_5,a_5)=1, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_5,a_6)=0.62\).

  6. (6)

    \(^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_6,a_1)=0.82, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_6,a_2)=0.42, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_6,a_3)=0.86, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_6,a_4)=0.73, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_6,a_5)=0.62, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_6,a_6)=1\).

And in \(\mathbb {F}_{2}\):

  1. (i)

    \(^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_1,a_1)=1, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_1,a_2)=0.49, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_1,a_3)=0.87, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_1,a_4)=0.85, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_1,a_5)=0.7, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_1,a_6)=0.88\).

  2. (ii)

    \(^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_2,a_1)=0.49, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_2,a_2)=1, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_2,a_3)=0.49, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_2,a_4)=0.54, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_2,a_5)=0.63, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_2,a_6)=0.54\).

  3. (iii)

    \(^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_3,a_1)=0.87, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_3,a_2)=0.49, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_3,a_3)=1, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_3,a_4)=0.78, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_3,a_5)=0.7, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_3,a_6)=0.88\).

  4. (iv)

    \(^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_4,a_1)=0.85, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_4,a_2)=0.54, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_4,a_3)=0.78, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_4,a_4)=1, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_4,a_5)=0.76, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_4,a_6)=0.75\).

  5. (v)

    \(^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_5,a_1)=0.7, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_5,a_2)=0.63, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_5,a_3)=0.7, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_5,a_4)=0.76, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_5,a_5)=1, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_5,a_6)=0.74\).

  6. (vi)

    \(^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_6,a_1)=0.88, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_6,a_2)=0.54, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_6,a_3)=0.88, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_6,a_4)=0.75, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_6,a_5)=0.74, ^{1}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_6,a_6)=1\).

Definition 9

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS and \(^{1}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}\) be a MGFMD of \(\Omega \). For every \(\mathscr {G} \in {\mathcal {F}}(\Omega )\) and \(\alpha \in [0,1)\), the 1-O\(\alpha \)LA and the 1-O\(\alpha \)UA are given by

$$\begin{aligned}{} & {} {_{-1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{\mathscr {G}(y): ^{1}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}(a,y)> \alpha \bigg \}\hbox { and }\\{} & {} {_{+1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigwedge \limits _{i=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\mathscr {G}(y): ^{1}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}(a,y)> \alpha \bigg \},\hbox { for every }a \in \Omega . \end{aligned}$$

Example 4

(Continued from Example 3) When \(\alpha =0.65\),

$$\begin{aligned}{} & {} {_{-1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})= \frac{0.5}{a_1}+\frac{0.3}{a_2}+\frac{0.1}{a_3}+\frac{0.1}{a_4}+ \frac{0.1}{a_5}+\frac{0.1}{a_6},\hbox { and } \\{} & {} {_{+1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})= \frac{0.8}{a_1}+\frac{0.3}{a_2}+\frac{0.8}{a_3}+ \frac{0.8}{a_4}+\frac{0.7}{a_5}+\frac{0.8}{a_6}. \end{aligned}$$

Theorem 2

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For every \(\mathscr {G} \in {\mathcal {F}}(\Omega )\), the below items for 1-O\(\alpha \)LA and 1-O\(\alpha \)UA hold.

  1. (1)

    \({_{-1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\bigg ({_{+1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}^c)\bigg )^c\) and \({_{+1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\bigg ({_{-1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}^c)\bigg )^c\).

  2. (3)

    \({_{-1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\Omega )=\) \(\Omega \) and \({_{+1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\emptyset )=\) \(\emptyset \)

  3. (4)

    If \(\mathscr {G} \subseteq \mathscr {Y}\), then \({_{-1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\) \(\subseteq {_{-1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\) \(\subseteq {_{+1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\).

  4. (5)

    \({_{-1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cap \mathscr {Y})=\) \({_{-1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cap {_{-1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cap \mathscr {Y})\subseteq \) \({_{+1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cap {_{+1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\)

  5. (6)

    \({_{-1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cup \mathscr {Y})\supseteq {_{-1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cup {_{-1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cup \mathscr {Y})= {_{+1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cup {_{+1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\).

  6. (7)

    If \(1-\mathbb {N}_{\mathbb {F}_{j}}^{\beta }(a)\le \mathscr {G}(a) \le \mathbb {N}_{\mathbb {F}_{j}}^{\beta }(a)\), \(\forall a \in \Omega \), then \({_{-1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}) \subseteq \mathscr {G} \subseteq {_{+1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\).

Proof

Similarly, by Theorem 1. \(\square \)

Now, we give another MGFRC form by \(^{1}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}\).

Definition 10

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS and \(^{1}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}\) be a MGFMD of \(\Omega \). For every \(\mathscr {G} \in {\mathcal {F}}(\Omega )\), the 1-O\(\mathscr {D}\)LA and 1-O\(\mathscr {D}\)UA are given by

$$\begin{aligned}{} & {} {_{-1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-^{1}\mathscr {D}_{\mathbb {F}_{j}}^{\beta }(a,y))\vee \mathscr {G}(y) \bigg \}\hbox { and } \\{} & {} {_{+1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigwedge \limits _{i=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\ ^{1}\mathscr {D}_{\mathbb {F}_{j}}^{\beta }(a,y)\wedge \mathscr {G}(y) \bigg \},\hbox { for every }a \in \Omega . \end{aligned}$$

Example 5

(Continued from Example 3) We have,

$$\begin{aligned}{} & {} {_{-1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})= \frac{0.36}{a_1}+\frac{0.3}{a_2}+\frac{0.22}{a_3}+ \frac{0.1}{a_4}+\frac{0.26}{a_5}+\frac{0.27}{a_6},\hbox { and} \\{} & {} {_{+1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})= \frac{0.8}{a_1}+\frac{0.5}{a_2}+\frac{0.8}{a_3}+ \frac{0.73}{a_4}+\frac{0.66}{a_5}+\frac{0.8}{a_6}. \end{aligned}$$

Theorem 3

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For every \(\mathscr {G} \in {\mathcal {F}}(\Omega )\), the below items for 1-O\(\mathscr {D}\)LA and 1-O\(\mathscr {D}\)UA hold.

  1. (1)

    \({_{-1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\bigg ({_{+1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}^c)\bigg )^c\) and \({_{+1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\bigg ({_{-1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}^c)\bigg )^c\).

  2. (2)

    \({_{-1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\Omega )=\) \(\Omega \) and \({_{+1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\emptyset )=\) \(\emptyset \)

  3. (3)

    If \(\mathscr {G} \subseteq \mathscr {Y}\), then \({_{-1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\) \(\subseteq {_{-1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\) \(\subseteq {_{+1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\).

  4. (4)

    \({_{-1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cap \mathscr {Y})=\) \({_{-1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cap {_{-1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cap \mathscr {Y})\subseteq \) \({_{+1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cap {_{+1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\)

  5. (5)

    \({_{-1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cup \mathscr {Y})\supseteq {_{-1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cup {_{-1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cup \mathscr {Y})= {_{+1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cup {_{+1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\).

  6. (6)

    If \(1-\mathbb {N}_{\mathbb {F}_{j}}^{\beta }(a)\le \mathscr {G}(a) \le \mathbb {N}_{\mathbb {F}_{j}}^{\beta }(a)\), \(\forall a \in \Omega \), then \({_{-1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}) \subseteq \mathscr {G} \subseteq {_{+1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\).

Proof

The proof is obvious by Theorem 1. \(\square \)

3.2 \(\hbox {CP}_{{MG}}\)FRSs

Definition 11

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For any \(\mathscr {G} \in {\mathcal {F}}(\Omega ),\) the 1-PLA and the 1-PUA are introduced respectively as next.

$$\begin{aligned}{} & {} {_{-1}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigwedge \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {G}(y)\bigg \}, \\{} & {} _{+1}{\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{i=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }(y)\wedge \mathscr {G}(y)\bigg \},\hbox { for all }a \in \Omega . \end{aligned}$$

If \({_{-1}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\ne {_{+1}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\), then \(\mathscr {G}\) is called a 1-\(\hbox {CP}_{{MG}}\)FRS, otherwise it is definable.

Definition 12

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS and \(^{1}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}\) be an MGFMD of \(\Omega \). For any \(\mathscr {G} \in {\mathcal {F}}(\Omega )\) and \(\alpha \in [0,1)\), the 1-P\(\alpha \)LA and the 1-P\(\alpha \)UA are given by

$$\begin{aligned}{} & {} {_{-1}\mathscr {R}}^{p,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigwedge \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{\mathscr {G}(y): ^{1}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}(a,y)> \alpha \bigg \},\hbox { and } \\{} & {} {_{+1}\mathscr {R}}^{p,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{i=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\mathscr {G}(y): ^{1}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}(a,y)> \alpha \bigg \},\hbox { for any }a \in \Omega . \end{aligned}$$

Definition 13

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS and \(^{1}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}\) be an MGFMD of \(\Omega \). For every \(\mathscr {G} \in {\mathcal {F}}(\Omega )\), the 1-P\(\mathscr {D}\)LA and 1-P\(\mathscr {D}\)UA, where \(a_i \in \Omega \) and \(i=1,..,m\) are given by

$$\begin{aligned}{} & {} {_{-1}\mathscr {R}}^{p,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}(a_i)}(\mathscr {G})(a)= \bigwedge \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\ ^{1}\mathscr {D}_{\mathbb {F}_{j}(a_i)}^{\beta }(a,y))\vee \mathscr {G}(y) \bigg \}\hbox { and } \\{} & {} {_{+1}\mathscr {R}}^{p,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}(a_i)}(\mathscr {G})(a)= \bigvee \limits _{i=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\ ^{1}\mathscr {D}_{\mathbb {F}_{j}(a_i)}^{\beta }(y)\wedge \mathscr {G}(y) \bigg \},\hbox { for every }a \in \Omega . \end{aligned}$$

Example 6

(Continued from Examples 1 and 2) Let \(\mathscr {G}=\frac{0.6}{a_1}+\frac{0.3}{a_2}+\frac{0.7}{a_3}+\) \(\frac{0.1}{a_4}+\frac{0.5}{a_5}+\frac{0.8}{a_6}\) and \(\alpha =0.65\). Then we have

  1. (1)

    \({_{-1}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\frac{0.4}{a_1}+\frac{0.2}{a_2}+\frac{0.3}{a_3}+\frac{0.1}{a_4}+\) \(\frac{0.1}{a_5}+\frac{0.1}{a_6}\) and

    \({_{+1}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\frac{0.7}{a_1}+\frac{0.5}{a_2}+\frac{0.7}{a_3}+\frac{0.7}{a_4}+\) \(\frac{0.7}{a_5}+\frac{0.7}{a_6}\).

  2. (2)

    \({_{-1}\mathscr {R}}^{p,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\frac{0.5}{a_1}+\frac{0.3}{a_2}+\frac{0.1}{a_3}+\frac{0.1}{a_4}+\) \(\frac{0.1}{a_5}+\frac{0.1}{a_6},\) and

    \({_{+1}\mathscr {R}}^{p,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\frac{0.8}{a_1}+\frac{0.5}{a_2}+\frac{0.8}{a_3}+\frac{0.8}{a_4}+\) \(\frac{0.8}{a_5}+\frac{0.8}{a_6}.\)

  3. (3)

    \({_{-1}\mathscr {R}}^{p,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\frac{0.15}{a_1}+\frac{0.3}{a_2}+\frac{0.22}{a_3}+\) \(\frac{0.1}{a_4}+\frac{0.24}{a_5}+\frac{0.25}{a_6},\) and

    \({_{+1}\mathscr {R}}^{p,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\frac{0.8}{a_1}+\frac{0.52}{a_2}+\frac{0.8}{a_3}+\frac{0.75}{a_4}+\) \(\frac{0.74}{a_5}+\frac{0.8}{a_6}.\)

4 The second \(\hbox {CO(P)}_{MG}\)FRS model via the fuzzy \(\beta \)-maximal description

This section aims to construct \(\hbox {CO}_{{MG}}\)FRS and \(\hbox {CP}_{{MG}}\)FRS models by using a family of a fuzzy \(\beta \)-maximal description, and their relations are investigated.

Definition 14

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For every \(a \in \Omega \),

$$\begin{aligned} \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a)}^{\beta }=\{\mathbb {F}_{ji}\in \mathbb {F}_{j}: (\mathbb {F}_{ji}(a)\ge \beta ) \wedge (\forall \mathbb {F}_{jr}\in \mathbb {F}_{j} \wedge \mathbb {F}_{jr}(a) \ge \beta \wedge \mathbb {F}_{jr} \supseteq \mathbb {F}_{ji} \rightarrow \mathbb {F}_{jr}=\mathbb {F}_{ji})\} \end{aligned}$$

is called the family of fuzzy \(\beta \)-maximal description of a for every \(j=1,\ldots ,n\) and \(j,r=1,\ldots ,m\).

Example 7

(Continued from Example 2) Then we can calculate the \(\tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a_i)}^{\beta }\) \((i=1,2,3,4,5,6)\) for \(\kappa \). See Tables 7 and 8.

Table 7 Table for \(\tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a_i)}^{\beta }\) for \(\mathbb {F}_{1}\)
Table 8 Table for \(\tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a_i)}^{\beta }\) for \(\mathbb {F}_{2}\)

4.1 \(\hbox {CO}_{{MG}}\)FRSs

Definition 15

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For every \(a \in \Omega \), the 2-OLA and the 2-OUA are defined as follows.

$$\begin{aligned}{} & {} {_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {G}(y)\bigg \},\hbox { and }\\{} & {} _{+2}{\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigwedge \limits _{i=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a)}^{\beta }(y)\wedge \mathscr {G}(y)\bigg \},\hbox { for every }a \in \Omega , \end{aligned}$$

If \({_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\ne {_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\), then \(({_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}), {_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}))\) is called a 2-\(\hbox {CO}_{{MG}}\)FRS, otherwise it is definable.

Example 8

(Continued from Example 7) The below results hold, if \(\mathscr {G}= \frac{0.6}{a_1}+ \frac{0.3}{a_2}+\) \(\frac{0.7}{a_3}+ \frac{0.1}{a_4}+ \frac{0.5}{a_5}+ \frac{0.8}{a_6}\). Then we calculate the \(\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a)}^{\beta }\) for \(\kappa = \{\mathbb {F}_{1}, \mathbb {F}_{2}\}\) as below in Tables 9 and 10.

Table 9 Data for \(\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{1}(a_i)}^{0.5}\)
Table 10 Data for \(\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{2}(a_i)}^{0.5}\)

So,

$$\begin{aligned}{} & {} {_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})= \frac{0.5}{a_1}+\frac{0.4}{a_2}+\frac{0.4}{a_3}+\frac{0.5}{a_4}+ \frac{0.2}{a_5}+\frac{0.5}{a_6},\hbox { and }\\{} & {} {_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})= \frac{0.6}{a_1}+\frac{0.3}{a_2}+\frac{0.7}{a_3}+ \frac{0.3}{a_4}+\frac{0.5}{a_5}+\frac{0.6}{a_6}. \end{aligned}$$

Theorem 4

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For any \(\mathscr {G}\in \mathscr {F}(\Omega )\), the below properties hold for 2-OLA and 2-OUA.

  1. (1)

    \({_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\bigg ({_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}^c) \bigg )^c\) and \({_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\bigg ({_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}^c) \bigg )^c\).

  2. (2)

    \({_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\Omega )=\Omega \) and \({_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\emptyset )=\emptyset \).

  3. (3)

    If \(\mathscr {G} \subseteq \mathscr {Y}\), then \({_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\).

  4. (4)

    \({_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cap \mathscr {Y})={_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cap {_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cup \mathscr {Y})={_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cup {_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\).

  5. (5)

    \({_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cup \mathscr {Y}) \supseteq {_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cup {_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cap \mathscr {Y}) \subseteq {_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cap {_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\).

  6. (6)

    If \(1-\mathbb {N}_{\mathbb {F}_{j}}^{\beta }(a)\le \mathscr {G}(a) \le \mathbb {N}_{\mathbb {F}_{j}}^{\beta }(a)\), \(\forall a \in \Omega \), then \({_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\).

Proof

  1. (1)
    $$\begin{aligned} ({_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}^c))^c(a)&=\bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}}^{\beta }(y)\wedge \mathscr {G}^{c}(y)\bigg \}^{c}\\ {}&=\bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}}^{\beta }(y)\wedge (1-\mathscr {G}(y))\bigg \}^{c}\\ {}&=\bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}}^{\beta }(y))\vee \mathscr {G}(y)\bigg \}\\ {}&={_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a). \end{aligned}$$
  2. (2)

    Since \(\Omega (a)= 1\), \(\forall a \in \Omega \),

    \({_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\Omega )=\) \(\bigwedge \limits _{i=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{(1-\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}}^{\beta }(y))\vee \Omega (a)\bigg \}=1=\Omega (a).\)

  3. (3)

    Let \(\mathscr {G}, \mathscr {Y} \in \mathcal {F}(\Omega )\) such that \(\mathscr {G} \subseteq \mathscr {Y}\) and \(a \in \Omega \). Then

    $$\begin{aligned} {_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)&=\bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}}^{\beta }(y))\vee \mathscr {G}(y)\bigg \}\\&\le \bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}}^{\beta }(y))\vee \mathscr {Y}(y)\bigg \}\\&={_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})(a). \end{aligned}$$
  4. (4)

    If \(a \in \Omega \), then we have

    $$\begin{aligned} {_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cap \mathscr {Y})(a)&=\bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}}^{\beta }(y))\vee (\mathscr {G}\cap \mathscr {Y})(y)\bigg \}\\&=\bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}}^{\beta }(y))\vee \mathscr {G}(y)\bigg \}\\&\cap \bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}}^{\beta }(y))\vee \mathscr {Y}(y)\bigg \}\\&={_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)\cap {_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})(a). \end{aligned}$$
  5. (5)

    Similar to (4).

  6. (6)

    If \(1-\mathbb {N}_{\mathbb {F}_{j}}^{\beta }(a)\le \mathscr {G}(a) \le \mathbb {N}_{\mathbb {F}_{j}}^{\beta }(a)\), for all \(a \in \Omega \), then

    $$\begin{aligned} \mathscr {G}(a)&=\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}}^{\beta }(a) \wedge \mathscr {G}(a)\le \bigvee \limits _{y\in \Omega }\bigg (\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}}^{\beta }(y) \wedge \mathscr {G}(y)\bigg )\\&\implies \mathscr {G}(a)\le \bigwedge \limits _{i=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}}^{\beta }(y)\wedge \mathscr {G}(y)\bigg \}={_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)\\&\implies \bigwedge \limits _{y\in \Omega }\bigg \{(1-\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}}^{\beta }(y))\vee \mathscr {G}(y)\bigg \}\le (1-\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}}^{\beta }(a))\vee \mathscr {G}(a)\le \mathscr {G}(a) \\&\implies \bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}}^{\beta }(y))\vee \mathscr {G}(y)\bigg \}\le \mathscr {G}(a)\\&\implies {_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)\le \mathscr {G}(a)\\&\implies {_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq \mathscr {G} \subseteq {_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}). \end{aligned}$$

\(\square \)

Definition 16

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS and \(a,y \in \Omega \). A MGFMD \(^{2}\mathscr {D}^{\beta }_{\mathbb {F}_{j}(a_i)}(a,y)\), where \(a_i \in \Omega \) and \(i=1,\ldots ,m\) is shown by

$$\begin{aligned} ^{2}\mathscr {D}^{\beta }_{\mathbb {F}_{j}(a_i)}(a,y)= \frac{\sum _{j=1}^{n}\bigg (\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a_i)}^{\beta }(a)\bigwedge \wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a_i)}^{\beta }(y)\bigg )}{\sum _{j=1}^{n}\bigg (\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a_i)}^{\beta }(a) \bigvee \wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a_i)}^{\beta }(y)\bigg )}. \end{aligned}$$

Example 9

(Continued from Example 7) If \(\beta =0.5\), then the below outcomes hold for \(\mathbb {F}_{1}\):

  1. (1)

    \(^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_1,a_1)=1, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_1,a_2)=0.37, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_1,a_3)=0.54, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_1,a_4)=0.61, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_1,a_5)=0.8, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_1,a_6)=0.85\).

  2. (2)

    \(^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_2,a_1)=0.37, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_2,a_2)=1, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_2,a_3)=0.5, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_2,a_4)=0.34, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_2,a_5)=0.41,\)

    \(^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_2,a_6)=0.32\).

  3. (3)

    \(^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_3,a_1)=0.54, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_3,a_2)=0.5, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_3,a_3)=1, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_3,a_4)=0.58, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_3,a_5)=0.63,\)

    \(^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_3,a_6)=0.57\).

  4. (4)

    \(^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_4,a_1)=0.61, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_4,a_2)=0.34, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_4,a_3)=0.58, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_4,a_4)=1, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_4,a_5)=0.64, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_4,a_6)=0.63\).

  5. (5)

    \(^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_5,a_1)=0.8, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_5,a_2)=0.41, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_5,a_3)=0.63, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_5,a_4)=0.64, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_5,a_5)=1, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_5,a_6)=0.69\).

  6. (6)

    \(^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_6,a_1)=0.85, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_6,a_2)=0.32, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_6,a_3)=0.57, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_6,a_4)=0.63, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_6,a_5)=0.69, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{1}}(a_6,a_6)=1\).

And for \(\mathbb {F}_{2}\):

  1. (i)

    \(^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_1,a_1)=1, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_1,a_2)=0.34, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_1,a_3)=0.81, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_1,a_4)=0.71, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_1,a_5)=0.81, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_1,a_6)=0.86\).

  2. (ii)

    \(^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_2,a_1)=0.34, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_2,a_2)=1, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_2,a_3)=0.29, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_2,a_4)=0.29, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_2,a_5)=0.36, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_2,a_6)=0.28\).

  3. (iii)

    \(^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_3,a_1)=0.81, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_3,a_2)=0.29, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_3,a_3)=1, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_3,a_4)=0.74, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_3,a_5)=0.67, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_3,a_6)=0.71\).

  4. (iv)

    \(^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_4,a_1)=0.71, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_4,a_2)=0.29, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_4,a_3)=0.74, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_4,a_4)=1, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_4,a_5)=0.69, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_4,a_6)=0.77\).

  5. (v)

    \(^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_5,a_1)=0.81, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_5,a_2)=0.36, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_5,a_3)=0.67, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_5,a_4)=0.69, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_5,a_5)=1, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_5,a_6)=0.7\).

  6. (vi)

    \(^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_6,a_1)=0.86, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_6,a_2)=0.28, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_6,a_3)=0.71, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_6,a_4)=0.77, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_6,a_5)=0.7, ^{2}\mathscr {D}^{0.5}_{\mathbb {F}_{2}}(a_6,a_6)=1\).

Definition 17

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS and \(^{2}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}\) be an MGFMD of \(\Omega \). For any \(\mathscr {G} \in {\mathcal {F}}(\Omega )\) and \(\alpha \in [0,1)\), the 2-O\(\alpha \)LA and the 2-O\(\alpha \)UA are given by

$$\begin{aligned}{} & {} {_{-2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{\mathscr {G}(y): ^{2}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}(a,y)> \alpha \bigg \}\hbox { and } \\{} & {} {_{+2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigwedge \limits _{i=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\mathscr {G}(y): ^{2}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}(a,y)> \alpha \bigg \},\hbox { for every }a \in \Omega . \end{aligned}$$

Example 10

(Continued from Example 9) If \(\alpha =0.65\), then we have

$$\begin{aligned}{} & {} {_{-2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})= \frac{0.5}{a_1}+\frac{0.3}{a_2}+\frac{0.7}{a_3}+\frac{0.1}{a_4}+ \frac{0.5}{a_5}+\frac{0.5}{a_6},\hbox { and } \\{} & {} {_{+2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})= \frac{0.8}{a_1}+\frac{0.3}{a_2}+\frac{0.7}{a_3}+ \frac{0.1}{a_4}+\frac{0.8}{a_5}+\frac{0.8}{a_6}. \end{aligned}$$

Theorem 5

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For any \(\mathscr {G} \in {\mathcal {F}}(\Omega )\), the next properties hold for 2-O\(\alpha \)LA and 2-O\(\alpha \)UA.

  1. (1)

    \({_{-2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\bigg ({_{+2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}^c)\bigg )^c\) and \({_{+2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\bigg ({_{-2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}^c)\bigg )^c\).

  2. (2)

    \({_{-2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\Omega )=\) \(\Omega \) and \({_{+2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\emptyset )=\) \(\emptyset \)

  3. (3)

    If \(\mathscr {G} \subseteq \mathscr {Y}\), then \({_{-2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\) \(\subseteq {_{-2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\) \(\subseteq {_{+2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\).

  4. (4)

    \({_{-2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cap \mathscr {Y})=\) \({_{-2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cap {_{-2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cap \mathscr {Y})\subseteq \) \({_{+2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cap {_{+2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\)

  5. (5)

    \({_{-2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cup \mathscr {Y})\supseteq {_{-2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cup {_{-2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cup \mathscr {Y})= {_{+2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cup {_{+2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\).

  6. (6)

    If \(1-\mathbb {N}_{\mathbb {F}_{j}}^{\beta }(a)\le \mathscr {G}(a) \le \mathbb {N}_{\mathbb {F}_{j}}^{\beta }(a)\), \(\forall a \in \Omega \), then \({_{-2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}) \subseteq \mathscr {G} \subseteq {_{+2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\).

Proof

Similarly, follows from Theorem 4. \(\square \)

Now, we give another MGFRC form by \(^{2}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}\).

Definition 18

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS and \(^{2}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}\) be an MGFMD of \(\Omega \). For any \(\mathscr {G} \in {\mathcal {F}}(\Omega )\), the 2-O\(\mathscr {D}\)LA and 2-O\(\mathscr {D}\)UA are given by

$$\begin{aligned}{} & {} {_{-2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-^{2}\mathscr {D}_{\mathbb {F}_{j}}^{\beta }(y))\vee \mathscr {G}(y) \bigg \}\hbox { and } \\{} & {} {_{+2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigwedge \limits _{i=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\ ^{2}\mathscr {D}_{\mathbb {F}_{j}}^{\beta }(y)\wedge \mathscr {G}(y) \bigg \},\hbox { for every }a \in \Omega . \end{aligned}$$

Example 11

(Continued from Examples 1 and 9) We have,

$$\begin{aligned}{} & {} {_{-2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})= \frac{0.39}{a_1}+\frac{0.3}{a_2}+\frac{0.42}{a_3}+ \frac{0.1}{a_4}+\frac{0.36}{a_5}+\frac{0.37}{a_6},\hbox { and } \\{} & {} {_{+2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})= \frac{0.8}{a_1}+\frac{0.26}{a_2}+\frac{0.7}{a_3}+ \frac{0.63}{a_4}+\frac{0.69}{a_5}+\frac{0.8}{a_6}. \end{aligned}$$

Theorem 6

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For any \(\mathscr {G} \in {\mathcal {F}}(\Omega )\), the below properties hold for 2-O\(\mathscr {D}\)LA and 2-O\(\mathscr {D}\)UA.

  1. (1)

    \({_{-2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\bigg ({_{+2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}^c)\bigg )^c\) and \({_{+2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\bigg ({_{-2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}^c)\bigg )^c\).

  2. (2)

    \({_{-2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\Omega )=\) \(\Omega \) and \({_{+2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\emptyset )=\) \(\emptyset \)

  3. (3)

    If \(\mathscr {G} \subseteq \mathscr {Y}\), then \({_{-2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\) \(\subseteq {_{-2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\) \(\subseteq {_{+2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\).

  4. (4)

    \({_{-2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cap \mathscr {Y})=\) \({_{-2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cap {_{-2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cap \mathscr {Y})\subseteq \) \({_{+2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cap {_{+2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\)

  5. (5)

    \({_{-2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cup \mathscr {Y})\supseteq {_{-2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cup {_{-2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\) and \({_{+2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}\cup \mathscr {Y})= {_{+2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\cup {_{+2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {Y})\).

  6. (6)

    If \(1-\mathbb {N}_{\mathbb {F}_{j}}^{\beta }(a)\le \mathscr {G}(a) \le \mathbb {N}_{\mathbb {F}_{j}}^{\beta }(a)\), \(\forall a \in \Omega \), then \({_{-2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}) \subseteq \mathscr {G} \subseteq {_{+2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\).

Proof

The evidence is comparable to Theorem 4. \(\square \)

4.2 \(\hbox {CP}_{{MG}}\)FRSs

Definition 19

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For any \(\mathscr {G} \in {\mathcal {F}}(\Omega ),\) the 2-PLA and the 2-PUA are established as indicated below.

$$\begin{aligned}{} & {} {_{-2}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigwedge \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a)}^{\beta }(y))\vee \mathscr {G}(y)\bigg \},\hbox { and}\\{} & {} _{+2}{\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{i=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a)}^{\beta }(y)\wedge \mathscr {G}(y)\bigg \},\hbox { for all }a \in \Omega . \end{aligned}$$

If \({_{-2}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\ne {_{+2}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\), then \(({_{-2}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}), {_{+2}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}))\) is called a 2-\(\hbox {CP}_{{MG}}\)FRS, otherwise it is definable.

Definition 20

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS and \(^{2}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}\) be a MGFMD of \(\Omega \). For any \(\mathscr {G} \in {\mathcal {F}}(\Omega )\) and \(\alpha \in [0,1)\), the 2-P\(\alpha \)LA and 2-P\(\alpha \)UA are given by

$$\begin{aligned}{} & {} {_{-2}\mathscr {R}}^{p,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigwedge \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{\mathscr {G}(y): ^{2}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}(a,y)\ge \alpha \bigg \},\hbox { and } \\{} & {} {_{+2}\mathscr {R}}^{p,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{i=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\mathscr {G}(y): ^{2}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}(a,y)\ge \alpha \bigg \},\hbox { for every }a \in \Omega . \end{aligned}$$

Definition 21

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS and \(^{2}\mathscr {D}^{\beta }_{\mathbb {F}_{j}}\) be a MGFMD of \(\Omega \). For every \(\mathscr {G} \in {\mathcal {F}}(\Omega )\), the 2-P\(\mathscr {D}\)LA and 2-P\(\mathscr {D}\)UA are given by

$$\begin{aligned}{} & {} {_{-2}\mathscr {R}}^{p,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigwedge \limits _{i=1}^{n}\bigwedge \limits _{y\in \Omega }\bigg \{(1-\ ^{2}\mathscr {D}_{\mathbb {F}_{j}}^{\beta }(y))\vee \mathscr {G}(y) \bigg \}\hbox { and} \\{} & {} {_{+2}\mathscr {R}}^{p,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})(a)= \bigvee \limits _{i=1}^{n}\bigvee \limits _{y\in \Omega }\bigg \{\ ^{2}\mathscr {D}_{\mathbb {F}_{j}}^{\beta }(y)\wedge \mathscr {G}(y) \bigg \},\hbox { for every }a \in \Omega . \end{aligned}$$

Example 12

(Continued from Examples 7 and 8) Let \(\mathscr {G}=\frac{0.6}{a_1}+\frac{0.3}{a_2}+\frac{0.7}{a_3}+\) \(\frac{0.1}{a_4}+\frac{0.5}{a_5}+\frac{0.8}{a_6}\) and \(\alpha =0.65\). Then we have

  1. (1)

    \({_{-2}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\frac{0.4}{a_1}+\frac{0.4}{a_2}+\frac{0.1}{a_3}+\frac{0.4}{a_4}+\) \(\frac{0.2}{a_5}+\frac{0.4}{a_6}\) and

    \({_{+2}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\frac{0.7}{a_1}+\frac{0.3}{a_2}+\frac{0.7}{a_3}+\frac{0.4}{a_4}+\) \(\frac{0.5}{a_5}+\frac{0.7}{a_6}\).

  2. (2)

    \({_{-2}\mathscr {R}}^{p,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\frac{0.1}{a_1}+\frac{0.3}{a_2}+\frac{0.1}{a_3}+\frac{0.1}{a_4}+\) \(\frac{0.1}{a_5}+\frac{0.1}{a_6},\) and

    \({_{+2}\mathscr {R}}^{p,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\frac{0.8}{a_1}+\frac{0.3}{a_2}+\frac{0.8}{a_3}+\frac{0.8}{a_4}+\) \(\frac{0.8}{a_5}+\frac{0.8}{a_6}.\)

  3. (3)

    \({_{-2}\mathscr {R}}^{p,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\frac{0.29}{a_1}+\frac{0.3}{a_2}+\frac{0.26}{a_3}+\) \(\frac{0.1}{a_4}+\frac{0.31}{a_5}+\frac{0.23}{a_6},\) and

    \({_{+2}\mathscr {R}}^{p,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})=\) \(\frac{0.8}{a_1}+\frac{0.41}{a_2}+\frac{0.71}{a_3}+\frac{0.77}{a_4}+\) \(\frac{0.7}{a_5}+\frac{0.8}{a_6}.\)

5 Four kinds of \(\hbox {CVP}_{{MG}}\)FRS models

In this section, we investigate four novel forms of \(\hbox {CVP}_{{MG}}\)FRSs samples and their relevant characteristics are obvious from the above sections, so we introduce the definitions only.

Definition 22

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For every \(\mathscr {G} \in {\mathcal {F}}(\Omega )\), and a variable precision attribute \(\gamma \in [0,1]\).

  1. 1.

    The 1-VPLA and 1-VPUA are given as next

    $$\begin{aligned} _{-1}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})(a)= \bigvee \limits _{j=1}^{n}\bigg (\bigwedge \limits _{\mathscr {G}(y)\le \gamma }((1-\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta })\vee \gamma )\wedge \bigwedge \limits _{\mathscr {G}(y)> \gamma }(1-\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta })\vee \mathscr {G}(y)\bigg ), \end{aligned}$$

    and

    $$\begin{aligned} _{+1}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})(a)= \bigwedge \limits _{j=1}^{n}\bigg (\bigvee \limits _{\mathscr {G}(y)\le \gamma }(\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }\wedge (1-\gamma ))\vee \bigvee \limits _{\mathscr {G}(y)> \gamma }(\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }\wedge \mathscr {G}(y))\bigg ). \end{aligned}$$
  2. 2.

    The 2-VPLA and 2-VPUA are defined as next

    $$\begin{aligned} _{-2}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})(a)= \bigwedge \limits _{j=1}^{n}\bigg (\bigwedge \limits _{\mathscr {G}(y)\le \gamma }((1-\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta })\vee \gamma )\wedge \bigwedge \limits _{\mathscr {G}(y)> \gamma }(1-\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta })\vee \mathscr {G}(y)\bigg ), \end{aligned}$$

    and

    $$\begin{aligned} _{+2}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})(a)= \bigvee \limits _{j=1}^{n}\bigg (\bigvee \limits _{\mathscr {G}(y)\le \gamma }(\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }\wedge (1-\gamma ))\vee \bigvee \limits _{\mathscr {G}(y)> \gamma }(\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}(a)}^{\beta }\wedge \mathscr {G}(y))\bigg ). \end{aligned}$$
  3. 3.

    The 3-VPLA and 3-VPUA are established as next

    $$\begin{aligned} _{-3}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})(a)= \bigvee \limits _{j=1}^{n}\bigg (\bigwedge \limits _{\mathscr {G}(y)\le \gamma }((1-\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a)}^{\beta })\vee \gamma )\wedge \bigwedge \limits _{\mathscr {G}(y)> \gamma }(1-\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a)}^{\beta })\vee \mathscr {G}(y)\bigg ), \end{aligned}$$

    and

    $$\begin{aligned} _{+3}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})(a)= \bigwedge \limits _{j=1}^{n}\bigg (\bigvee \limits _{\mathscr {G}(y)\le \gamma }(\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a)}^{\beta }\wedge (1-\gamma ))\vee \bigvee \limits _{\mathscr {G}(y)> \gamma }(\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a)}^{\beta }\wedge \mathscr {G}(y))\bigg ). \end{aligned}$$
  4. 4.

    The 4-VPLA and 4-VPUA are shown as next

    $$\begin{aligned} _{-4}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})(a)= \bigwedge \limits _{j=1}^{n}\bigg (\bigwedge \limits _{\mathscr {G}(y)\le \gamma }((1-\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a)}^{\beta })\vee \gamma )\wedge \bigwedge \limits _{\mathscr {G}(y)> \gamma }(1-\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a)}^{\beta })\vee \mathscr {G}(y)\bigg ), \end{aligned}$$

    and

    $$\begin{aligned} _{+4}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})(a)=\bigvee \limits _{j=1}^{n}\bigg (\bigvee \limits _{\mathscr {G}(y)\le \gamma }(\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a)}^{\beta }\wedge (1-\gamma ))\vee \bigvee \limits _{\mathscr {G}(y)> \gamma }(\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}(a)}^{\beta }\wedge \mathscr {G}(y))\bigg ). \end{aligned}$$

    For all \(a \in \Omega \) and if \(_{-H}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})\ne _{+H}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})\), then \((\ _{-H}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G}),\ _{+H}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G}))\) is called a H-\(\hbox {VP}_{{MG}}\)FRS, otherwise it is definable, where \(H \in \{1,2,3,4\}\).

Example 13

(Continued from Examples 1 and 7). If \(\gamma =0.5\), then we have,

  1. (1)
    $$\begin{aligned}{} & {} _{-1}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})=\frac{0.5}{a_1}+ \frac{0.5}{a_2}+ \frac{0.5}{a_3}+ \frac{0.5}{a_4}+ \frac{0.5}{a_5}+ \frac{0.5}{a_6},\hbox { and } \\{} & {} _{+1}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})=\frac{0.5}{a_1}+ \frac{0.5}{a_2}+ \frac{0.5}{a_3}+ \frac{0.5}{a_4}+ \frac{0.5}{a_5}+ \frac{0.5}{a_6}. \end{aligned}$$
  2. (2)
    $$\begin{aligned}{} & {} _{-2}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})=\frac{0.5}{a_1}+ \frac{0.5}{a_2}+ \frac{0.5}{a_3}+ \frac{0.5}{a_4}+ \frac{0.5}{a_5}+ \frac{0.5}{a_6},\hbox { and } \\{} & {} _{+2}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})=\frac{0.5}{a_1}+ \frac{0.5}{a_2}+ \frac{0.5}{a_3}+ \frac{0.5}{a_4}+ \frac{0.5}{a_5}+ \frac{0.5}{a_6}. \end{aligned}$$
  3. (3)
    $$\begin{aligned}{} & {} _{-3}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})=\frac{0.5}{a_1}+ \frac{0.5}{a_2}+ \frac{0.5}{a_3}+ \frac{0.5}{a_4}+ \frac{0.5}{a_5}+ \frac{0.5}{a_6},\hbox { and } \\{} & {} _{+3}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})=\frac{0.5}{a_1}+ \frac{0.2}{a_2}+ \frac{0.5}{a_3}+ \frac{0.3}{a_4}+ \frac{0.5}{a_5}+ \frac{0.5}{a_6}. \end{aligned}$$
  4. (4)
    $$\begin{aligned}{} & {} _{-4}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})=\frac{0.5}{a_1}+ \frac{0.5}{a_2}+ \frac{0.5}{a_3}+ \frac{0.5}{a_4}+ \frac{0.5}{a_5}+ \frac{0.5}{a_6},\hbox { and } \\{} & {} _{+4}\mathscr {R}_{\sum _{j=1}^{n}\mathbb {F}_{j}}^\gamma (\mathscr {G})=\frac{0.5}{a_1}+ \frac{0.3}{a_2}+ \frac{0.5}{a_3}+ \frac{0.4}{a_4}+ \frac{0.5}{a_5}+ \frac{0.5}{a_6}. \end{aligned}$$

6 Connections between the \(\hbox {CO}_{{MG}}\)FRS and \(\hbox {CP}_{{MG}}\)FRS models

Now, the relationships among presented models are given. From the above study, we have the next outcomes.

Proposition 1

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For any \(\mathscr {G} \in {\mathcal {F}}(\Omega )\), we have

  1. (1)

    \({_{-1}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}).\)

  2. (2)

    \({_{-2}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}).\)

  3. (3)

    \({_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{+1}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}).\)

  4. (4)

    \({_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{+2}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}).\)

Proof

Follows from Definitions 7, 11, 15 and 19. \(\square \)

Proposition 2

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For any \(\mathscr {G} \in {\mathcal {F}}(\Omega )\), if \(1-\mathbb {N}_{\mathbb {F}_{j}}^{\beta }(a)\le \mathscr {G}(a) \le \mathbb {N}_{\mathbb {F}_{j}}^{\beta }(a)\), \(\forall a \in \Omega \),

  1. (1)

    \({_{-1}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq \mathscr {G} \subseteq {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{+1}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}).\)

  2. (2)

    \({_{-2}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq \mathscr {G} \subseteq {_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{+2}\mathscr {R}}^{p}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}).\)

Proof

Directly from Theorems 1 and 4. \(\square \)

Proposition 3

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For every \(\mathscr {G} \in {\mathcal {F}}(\Omega )\),

  1. (1)

    \({_{-1}\mathscr {R}}^{p,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{-1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq \mathscr {G} \subseteq {_{+1}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{+1}\mathscr {R}}^{p,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}).\)

  2. (2)

    \({_{-2}\mathscr {R}}^{p,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{-2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq \mathscr {G}\subseteq {_{+2}\mathscr {R}}^{o,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{+2}\mathscr {R}}^{p,\alpha }_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}).\)

Proof

Directly from Theorems 2 and 5. \(\square \)

Proposition 4

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For every \(\mathscr {G} \in {\mathcal {F}}(\Omega )\),

  1. (1)

    \({_{-1}\mathscr {R}}^{p,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{-1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq \mathscr {G} \subseteq {_{+1}\mathscr {R}}^{o,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{+1}\mathscr {R}}^{p,^{1}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}).\)

  2. (2)

    \({_{-2}\mathscr {R}}^{p,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}) \subseteq {_{-2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}) \subseteq \mathscr {G} \subseteq {_{+2}\mathscr {R}}^{o,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{+2}\mathscr {R}}^{p,^{2}\mathscr {D}}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}).\)

Proof

Follows from Theorems 3 and 6. \(\square \)

Proposition 5

Let \((\Omega ,{\kappa })\) be a \({\textbf {MGF}}_{\beta }\)CAS. For any \(\mathscr {G} \in {\mathcal {F}}(\Omega )\), we have

  1. (1)

    \({_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{-2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}).\)

  2. (2)

    \({_{+2}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\subseteq {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}).\)

Proof

Its obvious. \(\square \)

The correlations can be depicted in Figs. 2 and 3.

Fig. 2
figure 2

The relationships between 1-\(\hbox {CO}_{{MG}}\)FRLA and 1-\(\hbox {CP}_{{MG}}\)FRLA models as in Sect. 3

Fig. 3
figure 3

The relationships between 2-\(\hbox {CO}_{{MG}}\)FRLA and 2-\(\hbox {CP}_{{MG}}\)FRLA models as in Sect. 4

7 Approaches to MCGDM and MAGDM based on the presented methods

In this section, using the concepts presented in Sects. 3 and 4, we apply the mentioned methods to decide on real-life problems. Next, we construct novel approaches to MCGDM and MAGDM based on \(\hbox {C}_{{MG}}\)FRS. Further, we demonstrate the main characterization of the MCGDM problem in Sect. 7.1 and MAGDM problem in Sect. 7.2.

7.1 MCGDM problem

7.1.1 Description and process

Let \(\Omega =\{a_r: r=1, 2, \ldots , n\}\) be the group of n patients, \(S=\{y_i: i= 1, 2, \ldots , m\}\) be m significant indicators and manifestations (fever, cough, sore throat, headache, and a chill) of \(\mathscr {A}\) sickness, and \(\kappa =\{\mathbb {F}_{j}(a_r)\}\) specify the effectiveness value of \(a_r\) for \(y_i\) signs, for some \(\beta \in (0,1]\), \((\Omega ,\kappa )\) is a \({\textbf {MGF}}_{\beta }\)CAS.

We provide a decision-making algorithm according to the proposed approaches to reach the outcome by the following phases.

Step 1: The on-call physician \(\mathcal {P}\) calls n specialists \(\mathcal {E}_i (i = 1, 2,\ldots , n)\) to estimate each sick \(a_k (k = 1, 2,\ldots , l)\) and every decision maker \(\mathcal {E}_i\) deemed valid each patient \(a_k \in \Omega \) has a clear sign impact denote by \(\mathbb {F}_{ij}(a_k)\).

Step 2: Experts evaluates each \(a_k \in \Omega \) with respect to their symptoms by the following formula \(\{y_i:\mathbb {F}_{ij}(a)\ge \beta \}\).

Step 3: According to the decision maker (physician \(\mathcal {D}\)), consider a FS \(\mathscr {G}\) represents the potential of the drugs in \(\Omega \) to treat the \(\mathcal {A}\) sickness. This assists the decision maker in determining whether or not those suffering \(a_k \in \Omega \) have the disease \(\mathcal {A}\).

Step 4: Evaluate the OLA and the OUA by the proposed models.

Step 5: Compute the MCGDM of \(\mathscr {G}\) by

$$\begin{aligned} \sum _{i=1}^{n}\Re _i(\mathscr {G})=\psi \ {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})+(1-\psi ){_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}),\ \text {where}\ \psi \in [0,1] \end{aligned}$$

, and hence make a decision.

We provide an algorithm to answer decision-making issues depending on the Definitions 7 and 15 based on these stages. Algorithm 1 summarizes the stages that relate to it.

Algorithm 1. Algorithm for MCGDM approach.

Input: Information systems.

Output: Decision Making.

1: Enter \(\mathscr {G}\), \(\beta \), \(\kappa \) and \(\Omega \).

2: From Definition 3, compute the \(\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{j}}^{\beta }(a_i)\) ( \(\wedge \tilde{\mathbb {M}D}_{\mathbb {F}_{j}}^{\beta }(a_i)\)), \(\forall a \in \Omega , i=1,..,n\).

3: From Step 2, calculate \({_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\) and \({_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\).

4: Enter the value of \(\psi \in [0,1]\).

5: Calculate the MCGDM of \(\mathscr {G}\) by \(\sum _{i=1}^{n}\Re _i(\mathscr {G})\).

6: Obtain the decision.

7.1.2 An illustrative example

The preceding steps are built numerically in the below example.

Example 14

To treat a condition, various types of medications are generally mixed. Let \(\Omega =\{a_j: j=1, 2, \ldots , 6\}\) be the group of 6 illness, \(S=\{y_i: i= 1, 2, \ldots , 5\}\) be 5 the primary signs (e.g., fever, cough, sore, headache, and chill) of \(\mathcal {A}\) ailments, and \(\mathbb {F}_{j}(a_i)\) represent the therapeutic associated with \(a_i\) for \(y_i\) disorders. Let \(\beta =0.6\) be the crucial value. If the physician \(\mathcal {D}\) asks the two specialists \(\mathcal {E}_i (i=1,2)\) to examine patients \(a_j (j=1,2,\ldots ,6)\) as shown in Tables 1 and 2.

The phases of the specified algorithm are carried out next.

Step 1. As shown in Example 1, the physician \(\mathcal {D}\) calculates the probable value \(\mathscr {G}\) of a sickness \(\mathcal {A}\) for each patient.

$$\begin{aligned} \mathscr {G}= \frac{0.6}{a_1}+ \frac{0.3}{a_2}+ \frac{0.7}{a_3}+ \frac{0.1}{a_4}+ \frac{0.5}{a_5}+ \frac{0.8}{a_6} \end{aligned}$$

.

Step 2. Compute \(\vee \tilde{\mathbb {M}d}^{\beta }_{\mathbb {F}_{j}}(a_i)\) as listed on Tables 5 and 6.

Step 3. Compute the 1-OLA and the 1-OUA as the following

$$\begin{aligned}{} & {} {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})= \frac{0.4}{a_1}+\frac{0.3}{a_2}+\frac{0.4}{a_3}+\frac{0.2}{a_4}+ \frac{0.1}{a_5}+\frac{0.4}{a_6},\hbox { and}\\{} & {} {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})= \frac{0.7}{a_1}+\frac{0.5}{a_2}+\frac{0.7}{a_3}+ \frac{0.7}{a_4}+\frac{0.7}{a_5}+\frac{0.7}{a_6}. \end{aligned}$$

Step 4. Based on these approximations, compute the MCGDM of \(\mathscr {G}\) by the following method

$$\begin{aligned} \sum _{i=1}^{n}\Re _i(\mathscr {G})=\psi {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})+(1-\psi ){_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G}). \end{aligned}$$

So, if \(\psi =0.6\), then we have

$$\begin{aligned} \sum _{i=1}^{2}\Re _i(\mathscr {G})&=0.6{_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})+0.4{_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\\&=\frac{0.52}{a_1}+ \frac{0.38}{a_2}+\frac{0.52}{a_3}+ \frac{0.48}{a_4}+ \frac{0.34}{a_5}+ \frac{0.52}{a_6}, \end{aligned}$$

and hence the order of the degree of susceptible to infection is obtained as the next.

\(a_1\approx a_3 \approx a_6\ge a_4 \ge a_2 \ge a_5.\)

From the computation, the doctor \(\mathcal {D}\) determines that the patients \(a_1, a_3\), and \(a_6\) are more vulnerable to contamination for the condition \(\mathcal {A}\).

Remark 1

The value of \(\psi \) represents the decision-makers perspectives on the dangers of deciding difficulties. When the decision-maker prefers risk, the higher the value of \(\psi \). When an administrator is risk-averse, the outcome of \(\psi \) tends to be lower. Consequently, the professional can ascertain the conclusion of \(\psi \) by considering the objective. We also have the next conditions.

  1. 1.

    \(\sum _{i=1}^{n}\Re _i(\mathscr {G})={_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\), when \(\psi =1\).

  2. 2.

    \(\sum _{i=1}^{n}\Re _i(\mathscr {G})={_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\), when \(\psi =0\).

In a word, we face a slew of complex concerns fraught with uncertainty. As a result, the expert can change the result of \(\psi \) to fix specific problems in the actual world.

7.2 MAGDM problem

7.2.1 Description and process

Let \(\Omega =\{a_1,a_2,\ldots ,a_n\}\) be n candidates and \(\mathcal {X}=\{t_1,t_2,\ldots ,t_r\}\) be r decision makers. Assume that for each \(\varPsi _i\) is a vector correspondingly to \(t_i\), with \(\varPsi _i\ge 0\) for \(r=1,\ldots ,i\) and \(\sum _{i=1}^{r}\varPsi _i=1\). Hence, \(\mathbb {F}_{j}= \{\mathbb {F}_{j1}, \mathbb {F}_{j2},\ldots ,\mathbb {F}_{jm}\}\), for any \(j=1,2,\ldots ,n\) is the attribute set. A collection of functions \(\mathcal {H}=\{h_r: \Omega \times \mathbb {F}_{j} \rightarrow [0,1]\}\). Thus, we establish the MAGDM through the mentioned schemes to obtain the suitable decision is such problems \((\Omega ,\mathbb {F}_{j},\mathcal {X},\mathcal {H}).\)

Step 1: Create the decision-making entity with fuzzy information. We have the following formulas using the TOPSIS method.

$$\begin{aligned}{} & {} \overline{\mathscr {G}}_{r}= \left\{ \mathbb {F}_{rj}, \bigvee \limits _{1\le i\le n}\bigg (h_{r}(a_i,\mathbb {F}_{rj})\bigg ): (j=1,\ldots ,m) (r = 1,\ldots ,t)\right\} \\{} & {} \quad =\left\{ (\mathbb {F}_{r1},\bigvee (h_{r1})),\ldots , (\mathbb {F}_{rm},\bigvee (h_{rm})) \right\} . \end{aligned}$$

and

$$\begin{aligned}{} & {} \underline{\mathscr {G}}_{r}= \left\{ \mathbb {F}_{rj}, \bigwedge \limits _{1\le i\le n}\bigg (h_{l}(a_i,\mathbb {F}_{rj})\bigg ): (j=1,\ldots ,m) (l = 1,\ldots ,t)\right\} \\{} & {} \quad =\left\{ (\mathbb {F}_{r1},\bigwedge (h_{r1})),\ldots , (\mathbb {F}_{rm},\bigwedge (h_{rm})) \right\} , \end{aligned}$$

where \(\bigvee \) and \(\bigwedge \) denotes to "max" and "min", respectively.

Step 2: Calculate the distances \(\overline{\mathscr {S}}\) and \(\underline{\mathscr {S}}\) as below:

$$\begin{aligned} \overline{\mathscr {S}}_{r}=\mathscr {S}(\mathbb {F}_{rj}(a_i),\mathbb {F}_{rj}(\underline{\mathscr {G}})) =\sqrt{\frac{1}{m}\sum _{j=1}^{m}(\mathbb {F}_{rj}(a_i)-\bigvee (h_{rj}))^2} \end{aligned}$$

and

$$\begin{aligned} \underline{\mathscr {S}}_{r}=\mathscr {S}(\mathbb {F}_{rj}(a_i),\mathbb {F}_{rj}(\overline{\mathscr {G}})) =\sqrt{\frac{1}{m}\sum _{j=1}^{m}(\mathbb {F}_{rj}(u_i)-\bigwedge (h_{rj}))^{2}}, \end{aligned}$$

where \(\mathscr {S}(\hat{L},\hat{P})=\sqrt{\frac{1}{m}\sum _{j=1}^{m}(\hat{L}(a_i)-\hat{P}(a_i))^{2}}\).

Step 3: Achieve the presented method of approximation using Definition 7, to get the values of the OLA and OUA.

$$\begin{aligned} {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\underline{\mathscr {S}}_{r}),\ {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\overline{\mathscr {S}}_{r}), {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\underline{\mathscr {S}}_{r}) \hbox { and }{_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\overline{\mathscr {S}}_{r}). \end{aligned}$$

Step 4: Calculate the closeness coefficient degree by \(\mathscr {C}_{r}(a_i)=\frac{\underline{\mathcal {V}_{r}}(a_i)}{\underline{\mathcal {V}_{r}}(a_i)+\overline{\mathcal {V}_{r}}(a_i)}.\)

Where \(\underline{\mathcal {V}_{r}}(a_i)=\mathcal {S}_{{p}}\bigg ({_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\underline{\mathscr {S}}_{r}(a_i)),{_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\underline{\mathscr {S}}_{r}(a_i))\bigg ),\) and

\(\overline{\mathcal {V}_{r}}(a_i)=\mathcal {S}_{{p}}\bigg ({_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\overline{\mathscr {S}}_{r}(a_i)),{_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\overline{\mathscr {S}}_{r}(a_i))\bigg )\),

and \(0\le \underline{\mathcal {V}_{r}}(a_i),\overline{\mathcal {V}_{r}}(a_i)\le 1\) with \(\mathcal {S}_{{p}}(a,b)=a+b-a*b.\)

Step 5: Compute the arranging function \(\mathscr {C}(a_i)=\sum _{r=1}^{t}\varpi _{r}\mathscr {C}_{r}(a_i),\) and then rank the candidates.

By following the aforementioned stages, we develop an algorithm to address the problems associated with decision making, based on the suggested template outlined in Algorithm 2.

Algorithm 2. Algorithm for MAGDM approach.

Input: Fuzzy information systems \((\Omega ,\kappa , {\mathcal {X}},{\mathcal {H}})\).

Output: Decision Making.

1: Enter \(\mathscr {G}\), \(\beta \), \(\kappa =\{\mathbb {F}_{j}:j=1,\ldots ,n\}\) and \(\Omega =\{a_i:i=1,\ldots ,m\}\).

2: From Definition 3, compute \(\mathbb {N}^{\beta }_{\mathbb {F}_{j}}(a)\) and \(\mathbb {M}^{\beta }_{\mathbb {F}_{j}}(a)\).

3: From Steps 2, compute \(\tilde{\mathbb {M}d}_{\mathbb {F}_{j}}^{\beta }(a).\)

4: Get \(\overline{\mathscr {G}}_{r}\) and \(\underline{\mathscr {G}}_{r}\).

5: Compute the distances \(\overline{\mathscr {S}}_{r}\) and \(\underline{\mathscr {S}}_{r}\).

6: From Definition 7, compute the 1-OLA \({_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\) and the 1-OUA \({_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\mathscr {G})\).

7: Calculate the least well and greatest decision-making factors \(\overline{\mathcal {V}_{r}}\) and \(\underline{\mathcal {V}_{r}}\).

8: Compute the ordering mapping \(\mathcal {C}_{r}\).

9: Compute the set sorting mapping \(\mathcal {C}\).

10: Obtain the decision.

7.2.2 Numerical example

Now, we give an illustrative example to explain the above steps in Algorithm 2.

Example 15

Let \(\Omega =\{a_1,a_2,\ldots ,a_6\}\) be six engineers and \(\kappa =\{\) steadiness (\(\mathbb {F}_{1}\)), skills (\(\mathbb {F}_{2}\)), personality (\(\mathbb {F}_{3}\)), experience (\(\mathbb {F}_{4}\)), aplomb (\(\mathbb {F}_{5}\)) \(\}\) be the description of the candidates’ attribute set. Assume that three specialists \(t_1, t_2\) and \(t_3\) are called to estimate the alternatives based on their information. Every expert’s ratings are: \(\varpi _1=0.4, \varpi _2 = 0.1\) and \(\varpi _3 = 0.5\).

Step 1: Experts estimate each alternative as summarized in Tables 11, 12 and 13.

Table 11 Table for \(\mathbb {F}_{1}\)
Table 12 Table for \(\mathbb {F}_{2}\)
Table 13 Table for \(\mathbb {F}_{3}\)

Step 2: Based on the significant of these parameters, we have the next outcomes for each decision maker.

$$\begin{aligned} \overline{\mathscr {G}}_{1}= \{(\mathbb {F}_{11},0.82),(\mathbb {F}_{12},0.76),(\mathbb {F}_{13},0.74),(\mathbb {F}_{14},0.78),(\mathbb {F}_{15},0.91)\}, \end{aligned}$$

and

$$\begin{aligned}{} & {} \underline{\mathscr {G}}_{1}=\{(\mathbb {F}_{11},0.28),(\mathbb {F}_{12},0.32),(\mathbb {F}_{13},0.36), (\mathbb {F}_{14},0.45),(\mathbb {F}_{15},0.43)\}. \\{} & {} \overline{\mathscr {G}}_{2}= \{(\mathbb {F}_{21},0.85),(\mathbb {F}_{22},0.77),(\mathbb {F}_{23},0.79), (\mathbb {F}_{24},0.81),(\mathbb {F}_{25},0.71)\}, \end{aligned}$$

and

$$\begin{aligned}{} & {} \underline{\mathscr {G}}_{2}=\{(\mathbb {F}_{21},0.35),(\mathbb {F}_{22},0.34),(\mathbb {F}_{23},0.46), (\mathbb {F}_{24},0.26),(\mathbb {F}_{25},0.43)\}. \\{} & {} \overline{\mathscr {G}}_{3}= \{(\mathbb {F}_{31},0.84),(\mathbb {F}_{32},0.75),(\mathbb {F}_{33},0.74), (\mathbb {F}_{34},0.69),(\mathbb {F}_{35},0.78)\}, \end{aligned}$$

and

$$\begin{aligned} \underline{\mathscr {G}}_{3}=\{(\mathbb {F}_{31},0.37),(\mathbb {F}_{32},0.36),(\mathbb {F}_{33},0.35), (\mathbb {F}_{34},0.42),(\mathbb {F}_{35},0.48)\}. \end{aligned}$$

Step 3: If \(\beta = 0.5\), then \(\mathbb {N}^{\beta }_{\mathbb {F}_{j}}, \mathbb {M}^{\beta }_{\mathbb {F}_{j}},\) and \(\tilde{\mathbb {M}d}_{\mathbb {F}_{j}}^{\beta }\) as put in Tables 14, 15, 16, 17, 18 and 19 below.

Table 14 Table for \(\mathbb {N}_{\mathbb {F}_{1}}^{0.5}(a_i)\)
Table 15 Table for \(\mathbb {N}_{\mathbb {F}_{2}}^{0.5}(a_i)\)
Table 16 Table for \(\mathbb {N}_{\mathbb {F}_{3}}^{0.5}(a_i)\)
Table 17 Table for \(\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{1}}^{0.5}(a_i)\)
Table 18 Table for \(\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{2}}^{0.5}(a_i)\)
Table 19 Table for \(\vee \tilde{\mathbb {M}d}_{\mathbb {F}_{3}}^{0.5}(a_i)\)

Step 4: Determine the distances \(\overline{\mathscr {S}}_{r}\) and \(\underline{\mathscr {S}}_{r}\) as next.

$$\begin{aligned}{} & {} \overline{\mathscr {S}}_{1}=\frac{0.248}{a_1}+ \frac{0.269}{a_2}+\frac{0.315}{a_3}+ \frac{0.189}{a_4}+ \frac{0.261}{a_5}+ \frac{0.306}{a_6},\\{} & {} \quad \underline{\mathscr {D}}_{1}=\frac{0.307}{a_1}+ \frac{0.307}{a_2}+\frac{0.241}{a_3}+ \frac{0.317}{a_4}+ \frac{0.247}{a_5}+ \frac{0.259}{a_6}. \\{} & {} \overline{\mathscr {S}}_{2}=\frac{0.276}{a_1}+ \frac{0.276}{a_2}+\frac{0.293}{a_3}+ \frac{0.146}{a_4}+ \frac{0.261}{a_5}+ \frac{0.228}{a_6},\\{} & {} \quad \underline{\mathscr {D}}_{2}=\frac{0.246}{a_1}+ \frac{0.278}{a_2}+\frac{0.209}{a_3}+ \frac{0.352}{a_4}+ \frac{0.293}{a_5}+ \frac{0.278}{a_6}. \\{} & {} \overline{\mathscr {S}}_{3}=\frac{0.243}{a_1}+ \frac{0.233}{a_2}+ \frac{0.21}{a_3}+ \frac{0.219}{a_4}+ \frac{0.184}{a_5}+ \frac{0.259}{a_6}, \\{} & {} \quad \underline{\mathscr {D}}_{3}=\frac{0.225}{a_1}+ \frac{0.231}{a_2}+ \frac{0.25}{a_3}+ \frac{0.237}{a_4}+ \frac{0.232}{a_5}+ \frac{0.212}{a_6}. \end{aligned}$$

Step 5: Compute the 1-OLA \({_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}\) and the 1-OUA \({_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}\) as follows.

When \(t=t_1\), we have

$$\begin{aligned}{} & {} {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\overline{\mathscr {S}}_{1})=\frac{0.33}{a_1}+ \frac{0.34}{a_2}+\frac{0.33}{a_3}+ \frac{0.32}{a_4}+ \frac{0.34}{a_5}+ \frac{0.34}{a_6}, \\{} & {} \quad {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\overline{\mathscr {S}}_{1})=\frac{0.315}{a_1}+ \frac{0.315}{a_2}+ \frac{0.315}{a_3}+ \frac{0.315}{a_4}+ \frac{0.315}{a_5}+ \frac{0.315}{a_6} \\{} & {} {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\underline{\mathscr {S}}_{1})=\frac{0.33}{a_1}+ \frac{0.34}{a_2}+\frac{0.33}{a_3}+ \frac{0.32}{a_4}+ \frac{0.34}{a_5}+ \frac{0.34}{a_6}, \\{} & {} \quad {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\underline{\mathscr {S}}_{1})=\frac{0.317}{a_1}+ \frac{0.317}{a_2}+ \frac{0.317}{a_3}+ \frac{0.317}{a_4}+ \frac{0.317}{a_5}+ \frac{0.317}{a_6} \end{aligned}$$

When \(t=t_2,\) we have

$$\begin{aligned}{} & {} {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\overline{\mathscr {S}}_{2})=\frac{0.33}{a_1}+ \frac{0.34}{a_2}+\frac{0.33}{a_3}+ \frac{0.32}{a_4}+ \frac{0.34}{a_5}+ \frac{0.34}{a_6},\\{} & {} \quad {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\overline{\mathscr {S}}_{2})=\frac{0.276}{a_1}+ \frac{0.276}{a_2}+ \frac{0.276}{a_3}+ \frac{0.276}{a_4}+ \frac{0.276}{a_5}+ \frac{0.276}{a_6} \\{} & {} {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\underline{\mathscr {S}}_{2})=\frac{0.33}{a_1}+ \frac{0.34}{a_2}+\frac{0.33}{a_3}+ \frac{0.32}{a_4}+ \frac{0.34}{a_5}+ \frac{0.34}{a_6},\\{} & {} \quad {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\underline{\mathscr {S}}_{2})=\frac{0.352}{a_1}+ \frac{0.352}{a_2}+ \frac{0.352}{a_3}+ \frac{0.352}{a_4}+ \frac{0.352}{a_5}+ \frac{0.352}{a_6} \end{aligned}$$

When \(t=t_3,\) we have

$$\begin{aligned}{} & {} {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\overline{\mathscr {S}}_{3})=\frac{0.33}{a_1}+ \frac{0.34}{a_2}+\frac{0.33}{a_3}+ \frac{0.32}{a_4}+ \frac{0.34}{a_5}+ \frac{0.34}{a_6},\\{} & {} \quad {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\overline{\mathscr {S}}_{3})=\frac{0.259}{a_1}+ \frac{0.259}{a_2}+\frac{0.259}{a_3}+ \frac{0.259}{a_4}+ \frac{0.259}{a_5}+ \frac{0.259}{a_6}\\{} & {} {_{-1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\underline{\mathscr {S}}_{3})=\frac{0.33}{a_1}+ \frac{0.34}{a_2}+\frac{0.33}{a_3}+ \frac{0.32}{a_4}+ \frac{0.34}{a_5}+ \frac{0.34}{a_6},\\{} & {} \quad {_{+1}\mathscr {R}}^{o}_{\sum _{j=1}^{n}\mathbb {F}_{j}}(\underline{\mathscr {S}}_{3})=\frac{0.255}{a_1}+ \frac{0.255}{a_2}+\frac{0.255}{a_3}+ \frac{0.255}{a_4}+ \frac{0.255}{a_5}+ \frac{0.255}{a_6} \end{aligned}$$

Step 6: The following are the most destructive and finest decision-making belongings.

$$\begin{aligned}{} & {} \overline{\mathcal {V}_{1}}=\frac{0.54105}{a_1}+ \frac{0.5479}{a_2}+\frac{0.54105}{a_3}+ \frac{0.5342}{a_4}+ \frac{0.5479}{a_5}+ \frac{0.5479}{a_6}, \\{} & {} \underline{\mathcal {V}_{1}}=\frac{0.54239}{a_1}+ \frac{0.54922}{a_2}+\frac{0.54239}{a_3}+ \frac{0.53556}{a_4}+ \frac{0.54922}{a_5}+ \frac{0.54922}{a_6} \\{} & {} \overline{\mathcal {V}_{2}}=\frac{0.51492}{a_1}+ \frac{0.52216}{a_2}+\frac{0.51492}{a_3}+ \frac{0.50768}{a_4}+ \frac{0.52216}{a_5}+ \frac{0.52216}{a_6}, \\{} & {} \underline{\mathcal {V}_{2}}=\frac{0.56584}{a_1}+ \frac{0.57232}{a_2}+\frac{0.56584}{a_3}+ \frac{0.55936}{a_4}+ \frac{0.57232}{a_5}+ \frac{0.57232}{a_6}. \\{} & {} \overline{\mathcal {V}_{3}}=\frac{0.50353}{a_1}+ \frac{0.51094}{a_2}+\frac{0.50353}{a_3}+ \frac{0.49612}{a_4}+ \frac{0.51094}{a_5}+ \frac{0.51094}{a_6}, \\{} & {} \underline{\mathcal {V}_{3}}=\frac{0.50085}{a_1}+ \frac{0.5083}{a_2}+\frac{0.50085}{a_3}+ \frac{0.4934}{a_4}+ \frac{0.5083}{a_5}+ \frac{0.5083}{a_6}. \end{aligned}$$

Then, as seen below, we calculate the nearby factor.

$$\begin{aligned}{} & {} \mathscr {C}_{1}=\frac{0.50062}{a_1}+ \frac{0.5006}{a_2}+\frac{0.50062}{a_3}+ \frac{0.50064}{a_4}+ \frac{0.5006}{a_5}+ \frac{0.5006}{a_6}. \\{} & {} \mathscr {C}_{2}=\frac{0.52356}{a_1}+ \frac{0.52291}{a_2}+\frac{0.52356}{a_3}+ \frac{0.52422}{a_4}+ \frac{0.52291}{a_5}+ \frac{0.52291}{a_6}. \\{} & {} \mathscr {C}_{3}=\frac{0.498666}{a_1}+ \frac{0.498705}{a_2}+\frac{0.498666}{a_3}+ \frac{0.498626}{a_4}+ \frac{0.498705}{a_5}+ \frac{0.498705}{a_6}. \end{aligned}$$

Step 7: SO, we compute \(\mathscr {C}\) as follows.

$$\begin{aligned} \mathscr {C}=\frac{0.501937}{a_1}+ \frac{0.5018835}{a_2}+\frac{0.501937}{a_3}+ \frac{0.501991}{a_4}+ \frac{0.5018835}{a_5}+ \frac{0.5018835}{a_6}, \end{aligned}$$

and thus get the order as

\(a_4\ge a_1 \approx a_3 \ge a_6 \approx a_2 \approx a_5.\)

As a consequence, we employ our methods based MAGDM with a fuzzy information template to carry out the decision-making for the best possible choice. We can attain the ultimate optimal solution of the fourth engineer by considering the previous phases.

7.3 Comparative analysis

This study presents a new approach for MAGDM by utilizing the \(\hbox {C}_{{MG}}\)FRSs with the aid of fuzzy \(\beta \)-minimum and maximum descriptions. This framework presents novel mathematical structures that are seen as an extension of the works by Zhan et al. (2020), Atef and El-Atik (2021), Zhan et al. (2019), and Ma et al. (2020). Here, we provide an explanation of the comparisons between the given methods and the previous one (i.e., Zhan et al. 2020; Atef and El-Atik 2021; Zhan et al. 2019 and Ma et al. 2020) to show the efficiency and the performance of these studies. We divide it into two items, that is, MCGDM and MAGDM, respectively.

  1. 1.

    We explain the differences between MCGDM approach and the last ones (i.e., Zhan et al. 2020 as in Definition 4 and Atef and El-Atik (2021) as in Definition 5) as in Table 20.

    From Table 20, in Zhan et al. (2020) and Atef and El-Atik (2021), the patients \(a_3\) and \(a_6\) are susceptible to infection for the disease \(\mathcal {A}\) and here the patients \(a_3\), \(a_6\) and \(a_1\) are susceptible to infection for the disease \(\mathcal {A}\). There is slight difference in the ranking of the other alternatives. So, our approach is effective and reliable.

  2. 2.

    Here, we clarify the comparisons between MAGDM approach and the previous studies such as the Zhan et al. (2019) scheme and Ma et al. (2020) model. The ordering of the alternatives by these different models and ours are established in Table 21.

    The above Table 21 provides us with the sorting of these different approaches and it is clear that the \(a_4\) engineer is the optimal candidate for all methods (i.e., Zhan et al. 2019 model, Ma et al. (2020) template and our approach). This leads that our presented work is effective.

  3. 3.

    Additionally, the ranking of candidates among several approaches is elucidated in Fig. 4. These methods include our proposed methods, the samples presented by Zhan et al. (2019) and (2020), Atef and El-Atik (2021) and the template proposed by Ma et al. (2020). Based on the visual representation provided, it is evident that choice \(a_3\) exhibits superior qualities compared to the other alternatives under consideration for the first technique and \(a_4\) for the second strategy.

  4. 4.

    Hence, to further explain the differences in the selection of optimal candidates with the difference value of the risk parameter \(\psi \), you can see Figs. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and 15 which shows comparisons among alternatives in various methods (i.e., 1-\(\hbox {CO}_{{MG}}\)FRS, 2-\(\hbox {CO}_{{MG}}\)FRS, i-\(\hbox {CO}_{{MG}}\)FRS, ii-\(\hbox {CO}_{{MG}}\)FRS, iii-\(\hbox {CO}_{{MG}}\)FRS and iv-\(\hbox {CO}_{{MG}}\)FRS) with different outcomes of \(\psi \). Subsequently, Table 22 explains the variations of ordering the global set using the 1-\(\hbox {CO}_{{MG}}\)FRS model, because this is the better one according to maximum the lower approximation and minimum the upper approximation that helps us to decrease the boundary area and increase the accuracy via several results of \(\psi \).

  5. 5.

    As is generally known, aggregation algorithms are mostly used in conventional MAGDM situations. This means that the choices of various decision-makers can be predicted by employing the selected aggregation operators. The fundamental issues of conventional MAGDM difficulties then focus on selecting effective aggregation operators. In this paper, we investigate an alternate technique for MAGDM models using CMGFRS. This article attempts to address the issue. The decision-making concept and procedure of the CMGFRS-based MAGDM model and approach differ from those of the aggregation operator strategy for MAGDM problems.

    The downside of the suggested approach is that it is more sophisticated than traditional MAGDM issues, but we can solve it using computer tools (Matlab and Excel).

Table 20 Table for the ordering outcomes
Table 21 Table for the ordering outcomes
Fig. 4
figure 4

The systematic evaluation of alternatives using different techniques

Table 22 Table for the ordering outcomes with different \(\psi \)
Fig. 5
figure 5

Ordering candidates with \(\psi =0\)

Fig. 6
figure 6

Ordering candidates with \(\psi =0.1\)

Fig. 7
figure 7

Ordering candidates with \(\psi =0.2\)

Fig. 8
figure 8

Ordering candidates with \(\psi =0.3\)

Fig. 9
figure 9

Ordering candidates with \(\psi =0.4\)

Fig. 10
figure 10

Ordering candidates with \(\psi =0.5\)

Fig. 11
figure 11

Ordering candidates with \(\psi =0.6\)

Fig. 12
figure 12

Ordering candidates with \(\psi =0.7\)

Fig. 13
figure 13

Ordering candidates with \(\psi =0.8\)

Fig. 14
figure 14

Ordering candidates with \(\psi =0.9\)

Fig. 15
figure 15

Ordering candidates with \(\psi =1\)

8 Conclusion and future work

In this research, we set the definitions of the family of fuzzy \(\beta \)-minimum and maximum descriptions and then construct novel covering strategies called \({\textbf {MGF}}_{\beta }\)CAS with respect to the (O,P,VP) FRSs. According to these concepts, we extend a new structure for a multigranulation fuzzy measure degree and utilize it to construct two models of optimistic (pessimistic) \(\alpha \) lower and upper approximations. By these approximations, we introduce two methods of optimistic (pessimistic) \(\mathscr {D}\) lower and upper approximations and discuss their features. Relationships between these models are also examined. Then, using MCGDM and MAGDM methodologies, we demonstrate the dependability of our methods. Finally, we provide numerical comparisons between the presented models and the previous research to demonstrate their effectiveness as shown in Tables 20 and 21, and Fig. 4. Consequently, we examine the sensitivity analysis of our presented models through Table 22 and Figs. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and 15 that analyze the outcomes over different values of \(\psi \). Another advantage for this article is to introduce the minimal set of attributes and this will allow us to examine the reduction for this approach in further research. Furthermore, we aim to investigate the algebraic and topological aspects of \(\hbox {C}_{{MG}}\)FRS models.