Covering-based \((\alpha , \beta )\)-multi-granulation bipolar fuzzy rough set model under bipolar fuzzy preference relation with decision-making applications
verfasst von:
Rizwan Gul, Muhammad Shabir, Ahmad N. Al-Kenani
The rough set (RS) and multi-granulation rough set (MGRS) theories have been successfully extended to accommodate preference analysis by substituting the equivalence relation (ER) with the dominance relation (DR). On the other hand, bipolarity refers to the explicit handling of positive and negative aspects of data. In this paper, with the help of bipolar fuzzy preference relation (BFPR) and bipolar fuzzy preference \(\delta \)-covering (BFP\(\delta \)C), we put forward the idea of BFP\(\delta \)C based optimistic multi-granulation bipolar fuzzy rough set (BFP\(\delta \)C-OMG-BFRS) model and BFP\(\delta \)C based pessimistic multi-granulation bipolar fuzzy rough set (BFP\(\delta \)C-PMG-BFRS) model. We examine several significant structural properties of BFP\(\delta \)C-OMG-BFRS and BFP\(\delta \)C-PMG-BFRS models in detail. Moreover, we discuss the relationship between BFP\(\delta \)C-OMG-BFRS and BFP\(\delta \)C-PMG-BFRS models. Eventually, we apply the BFP\(\delta \)C-OMG-BFRS model for solving multi-criteria decision-making (MCDM). Furthermore, we demonstrate the effectiveness and feasibility of our designed approach by solving a numerical example. We further conduct a detailed comparison with certain existing methods. Last but not least, theoretical studies and practical examples reveals that our suggested approach dramatically enriches the MGRS theory and offers a novel strategy for knowledge discovery, which is practical in real-world circumstances.
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Introduction
The real world is full of uncertainty and indeterminacy. We frequently deal with problems that are vague rather than precise. Because of many kinds of uncertainties in these problems, classical techniques are not always effective. In 1965, Zadeh [68] pioneered the idea of the fuzzy set (FS), which opened the doors for researchers to capture the imprecision of the data. FS theory uses the fuzzy membership function (MF), through which we can determine the membership degree (MD) of an item with respect to a set. The higher the MD, the greater the belongingness of that object to the corresponding set. There are many generalizations of the FS including vague set, neutrosophic FS, intuitionistic FS, Pythagorean FS, interval-valued FS etc.
RS theory [41, 42], proposed by Pawlak, has been recognized as an effective mathematical tool for dealing with intelligent systems characterized by uncertainty and imprecision. This comparatively new soft computing tool has attracted extensive interest in recent years, and its effectiveness has been successfully confirmed and applied in many domains, such as pattern recognition, conflict analysis, knowledge discovery, data mining, image processing, medical diagnosis, neural networks, and so on. The key notion in RS theory is equivalence relation (ER), which describes the indiscernibility relation between arbitrary objects. Although RS theory has been applied successfully in multiple disciplines, certain shortcomings may cause to limit the application domain of the RS theory. These shortcomings could be the result of inaccurate information regarding the objects under consideration. Sometimes in incomplete information, such as an ER, is difficult to find. Therefore, under different conditions, the RS model has accomplished several interesting and meaningful generalizations in recent years, which include the RS model based on tolerance relations [52], RS based on neighborhood operators [66], fuzzy RS (FRS) model [7], rough FS (RFS) model [6], dominance-based RSs [11, 12, 17], fuzzy dominance-based RSs [9, 10], dominance-based neighborhood RS [5], variable precision RS [75], Bayesian RS [53], covering-based RSs [69] and grey tolerance RS model [19, 20].
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It is worth noting that every matter has two sides, and bipolarity, as well as fuzziness, is an intrinsic aspect of human cognition. Bipolar reasoning is important in human cognitive processes, according to research in cognitive psychology. Positive and negative effects do not appear to be processed in the same part of the brain. Experts in different domains, such as database querying, DM, and classification, have noticed importance of bipolarity.
Fuzziness and bipolarity are two independent but complementary notions devised to model different aspects of human thinking. The former focuses on linguistic imprecision, while the latter emphasizes the relevance and polarity of data. Two notions have gradually shown high relevance in recent research. Under this background, in 1994, Zhang [72] postulated bipolar FSs (BFSs) as an extension of FSs. According to BFS theory, the MD is enlarged from [0, 1] to \([-1, 1]\). The MD “0” of an object illustrates that the element is irrelevant to the related property, the MD in (0, 1] of an object indicates that the object somewhat satisfies the property, and the MD in of an object \([-1, 0)\) means that the object somewhat fulfills the implicit counter-property. BFS theory has been extensively used to address real-life problems. Many efforts have also been made to hybridize RS theory and BFS theory (see [18, 64, 65]). Gul [14] proposed some bipolar fuzzy aggregation operators and then used them to model a multiple attribute DM (MADM) problem. Later, in the same environment, Wei et al. [57] developed bipolar fuzzy Hamacher aggregation operators in MADM. Wei et al. [56] initiated a multiple attribute DM approach using interval-valued bipolar fuzzy information. Gul and Shabir [15] introduced a new idea of the roughness of a crisp set based on \((\alpha , \beta )\)-indiscernibility of a bipolar fuzzy relation. Ali et al. [3] proposed attribute reductions of bipolar fuzzy relation decision systems. Jun and Park [24], Jun et al. [25], and Lee [30] applied BFSs to BCK/BCI-algebras. Kim et al. [27] found some properties of bases, neighborhoods, and continuities in bipolar fuzzy topological spaces. Malik and Shabir [36] developed a consensus model based on rough bipolar fuzzy approximations.
Pawlak’s RS and most of its extensions depend on on a single relation on a specific universe, which are referred to as single granulation RS models in the context of granular computing. There are certain drawbacks to the single granulation RS in some real-world applications, such as knowledge discovery from data with high dimensions, multi-source data analysis, and distributive information systems [43]. In granular computing, granulating a complex problem into many finer sub-problems can effectively tackle the complexity at a certain level. One vital issue in solving a complicated problem in granular computing is constructing multi-granulation spaces. Constructing multiple granular spaces mainly concentrates on multiple binary relations. To overcome such problems, Qian et al. [45] proposed an MGRS model in 2010 for the first time, in which the set approximations of a target concept are constructed by multiple ERs on a universe under related complex issues. Compared with Pawlak’s RS, MGRS holds dominant position for rule extraction and knowledge discovery (particularly when there is a conflict or inconsistent link between two attribute sets in an information system). In terms of granular computing, the MGRS theory offers a distinctive perspective. It has gained popularity in artificial intelligence and management science and has sparked a wide range of theoretical and practical studies.
A brief review on MGRS
To date, the MGRS theory has progressed rapidly and has drawn a wide variety of research from both theoretical and applied perspectives. For instance, in Qian et al.’s [45] MGRS theory, there are two fundamental models: one is the optimistic MGRS, the other is the pessimistic MGRS [43]. Xu et al. [59] offered two new types of MGRS. Yang et al. [63] came up with the hierarchical structural properties of MGRSs. She and He [50] explored the topological structures of MGRSs. By following Qian et al.’s approach, Yang et al. [62] extended MGRSs into fuzzy environment and initiated the multi-granulation FRSs (MGFRSs). Sun et al. [54] offered a MGFRS model over two universes with application to DM. She et al. [51] introduced a multiple-valued logic approach for MGRS. Kong et al. [28] put forth attribute reducts of multi-granulation information system. Liu et al. [35] studied multi-granulation FRSs based on fuzzy preference relations with applications.
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Zhan and Xu [70] projected two different kinds of coverings based on multi-granulation RFSs. Zhan et al. [71] developed the idea of covering based multi-granulation FRSs and their corresponding applications. Lin et al. [33] pioneered a neighborhood-based MGRS model. Xu et al. [60] proposed the idea of generalized MGRSs. Sun et al. [55] initiated multi-granulation vague RS over two universes with application to DM. Qian et al. [46] established three kinds of multi-granulation decision-theoretic RS models. Feng and Mi [8] studied variable precision multi-granulation fuzzy decision-theoretic RSs in an information system. Li et al. [31] proposed a double-quantitative multi-granulation decision-theoretic RFS model. Zhang et al. [73] developed four types of constructive methods of rough approximation operators from existing RSs and offered the non-dual MGRs and hybrid MGRs. Lin et al. [34] designed a two-grade fusion approach involved in the evidence theory and MGRS theory based on a well-defined distance function among granulation structures and developed three types of covering-based MGRSs. Pan et al. [40] suggested a multi-granulation preference relation RS model for the ordinal system. Mandal and Ranadive [38] offered fuzzy multi-granulation decision-theoretic RSs based on fuzzy preference relation. Zhang et al. [74] developed multi-granulation hesitant FRSs along with DM application. Huang et al. [22] developed an intuitionistic fuzzy MGRS (IFMGRS) and three types of IFMGRSs that are generalizations of three existing intuitionistic FRS models. Liang et al. [32] offered an efficient rough feature selection algorithm for large-scale data sets based on the idea from MGRSs. Ali et al. [2] proposed new types of dominance-based MGRSs with applications in conflict analysis. Hu et al. [21] pioneered dynamic dominance-based MGRSs approaches with evolving ordered data. You et al. [67] studied the relative reduction of neighborhood-covering pessimistic MGRS using evidence theory. Xue et al. [61] established three-way decisions based on multi-granulation support intuitionistic fuzzy probabilistic RSs. Qian et al. [47] introduced multi-granulation sequential three-way decisions based on multiple thresholds. Mandal and Ranadive [37] introduced multi-granulation bipolar-valued fuzzy probabilistic RSs and their corresponding three-way decisions over two universes. Recently, Gul and Shabir [16] projected the concept of \((\alpha , \beta )\)-multi-granulation bipolar fuzzified RS using a finite collection of bipolar fuzzy tolerance relations and discuss its applications to DM. Kang et al. [26] initiated a grey MGRSs model.
Research gap and motivations of this study
Based on the above contents, the research gaps and motivations of this article are summarized as follows:
Preference relation (PR) is a powerful tool to model DM problems, where decision-makers articulate their preference information over alternatives through pairwise comparisons. With various representations of preference information, numerous kinds of PRs have been put forth and investigated, such as the multiplicative PR [48], fuzzy PR (FPR) [4, 39], and BFPRs [17]. Meanwhile, RS theory has drawn great attention in recent decades, among which MGRS is an arresting perspective. It constructs a formal theoretical framework to solve complicated problems in the context of multiple binary relations. However, according to the best of our knowledge, there does not exist any study where the hybridization of MGRS theory and BFPRs have been discussed for acquiring knowledge. Therefore, this article fills this research gap by introducing the ideas of BFP\(\delta \)C-OMG-BFRS and BFP\(\delta \)C-PMG-BFRS models.
BFS is a critical mathematical tool to cope with inconsistent and imprecise data that contain both positive and negative sides in real-life dilemmas. Fuzzy \(\beta \)-covering RS set models are incompetent to capture the bipolarity in the data. Inspiration from the studies, as mentioned earlier, shows that there does not exist any investigation on the appropriate fusion of fuzzy covering with BFRSs under MGRS environment. Consequently, its potential applications to MCDM with bipolar information are also missing. This research gap motivates the current research on the bipolar fuzzy preference covering-based BFRS model under MGRS environment and its applicability to DM.
Aim and contributions of this study
In a nutshell, to extend the theory of MGRS, the main aim of this work is to develop a novel framework of MGRS by incorporating BFPR.
The main contributions of this script is as follows:
(1)
By combining the MGRS theory and BFSs with BFPRs, the idea of BFP\(\delta \)C-OMG-BFRS and BFP\(\delta \)C-PMG-BFRS models are introduced.
(2)
Some axiomatic systems of BFP\(\delta \)C-OMG-BFRS and BFP\(\delta \)C-PMG-BFRS models are carefully analyzed.
(3)
A connection among the BFP\(\delta \)C-BFRS, BFP\(\delta \)C-OMG-BFRS, and BFP\(\delta \)C-PMG-BFRS has been established.
(4)
A comprehensive MCDM technique in the framework of the BFP\(\delta \)C-OMG-BFRS model is introduced, and a practical example also validates the practicality of this technique.
(5)
To demonstrate the advantages of the suggested strategy, a detailed comparison with other existing methods is conducted.
Some fundamental notions related to RS, MGRS, FS, BFS, and BFRSs are mentioned in this section.
RS theory
In the RS theory [41], ER assumes a vital role to deal with uncertainty. The ER divides the universe into classes, which are commonly referred to as information granules. Hence, in RS theory, we need to deal with bunches of objects instead of a single entity.
Definition 2.1
[41] An approximation space (AS) is an object of the form \((\mho , \vartheta )\), where \(\mho \) is a non-void finite universe and \(\vartheta \) is an ER on \(\mho \). Given any subset \({\mathcal {T}}\) of \(\mho \), \({\mathcal {T}}\) may or may not be expressed as a union of some equivalence classes induced by \(\vartheta \). If it is possible to express \({\mathcal {T}}\) as the later, then it is called definable, if not it is a RS. If \({\mathcal {T}}\) is a RS, then it can be approximated by the following two definable sets:
are called the positive, boundary, and negative regions of \({\mathcal {T}} \subseteq \mho \), respectively. The semantics of these regions are as follows:
\(q \in {\mathcal {P}}os_\vartheta ({\mathcal {T}})\) indicates that \({\mathcal {T}}\) certainly contains the element \(q \in \mho \).
\(q \in {\mathcal {B}}nd_\vartheta ({\mathcal {T}})\) shows that \({\mathcal {T}}\) may or may not contain the element \(q \in \mho \).
\(q \in {\mathcal {N}}eg_\vartheta ({\mathcal {T}})\) reveals that \({\mathcal {T}}\) definitely does not contain the element \(q \in \mho \).
Clearly, \({\mathcal {T}}\) is definable if \(\underline{{\mathcal {T}}}_\vartheta = \overline{{\mathcal {T}}}^\vartheta \); equivalently \({\mathcal {B}}nd_\vartheta ({\mathcal {T}}) = \varnothing \). \({\mathcal {T}}\) is a RS if \(\underline{{\mathcal {T}}}_\vartheta \ne \overline{{\mathcal {T}}}^\vartheta \); equivalently \({\mathcal {B}}nd_\vartheta ({\mathcal {T}}) \ne \varnothing \).
Pawlak’s RS theory is based on a single ER. Using a finite collection of ERs, Qian et al. [45] formulated the idea of the MGRS. In Qian et al.’s MGRS theory, two different approaches have been proposed. The first one is the optimistic MGRS (OMGRS) and the second one is the pessimistic MGRS (PMGRS).
OMGRS model
Each ER can induce a partition in the universe, and such a partition can be regarded as a granulation space. Therefore, a family of ERs can induce a family of granulation spaces. In optimistic multi-granulation lower approximation, the term “optimistic" means that in multi-independent granulation spaces, we need only at least one of the granulation spaces to satisfy the inclusion condition between the equivalence class and the approximated target. The upper approximation of optimistic MGRS is defined by the complement of the optimistic multi-granulation lower approximation.
Definition 2.2
[45] Let \(\Theta =\{ \vartheta _1, \vartheta _2, \ldots , \vartheta _n \}\) a be finite collection of n independent ERs over \(\mho \) and \({\mathcal {T}} \subseteq \mho \). The optimistic multi-granulation lower and upper approximations of \({\mathcal {T}} \subseteq \mho \) are, respectively, defined as
where \({\mathcal {T}}^c\) is the complement of set \({\mathcal {T}}\). If \({\underline{\Theta }}_{opt}({\mathcal {T}})\ne {\overline{\Theta }}^{opt}({\mathcal {T}})\), then \({\mathcal {T}}\) is referred as an OMGRS, otherwise it is an optimistic definable. The boundary region of \({\mathcal {T}}\subseteq \mho \) under OMGRS environment is defined as
In the PMGRS, the target is still approximated through a family of ERs. However, the pessimistic case is different from the optimistic case. In pessimistic multi-granulation lower approximation, the term “pessimistic" means that we need all of the granulation spaces to satisfy the inclusion condition between the equivalence class and the approximated target. The upper approximation of PMGRS is still defined by the complement of the pessimistic multi-granulation lower approximation.
Definition 2.3
[43] Let \(\Theta = \{\vartheta _1, \vartheta _2, \ldots , \vartheta _n \}\) be a finite collection of n independent ERs over \(\mho \) and \({\mathcal {T}} \subseteq \mho \). The pessimistic multi-granulation lower and upper approximations of \({\mathcal {T}}\subseteq \mho \) are, respectively, defined as
If \({\underline{\Theta }}_{pes}({\mathcal {T}})\ne {\overline{\Theta }}^{pes}({\mathcal {T}})\), then \({\mathcal {T}}\) is called a PMGRS, otherwise it is a pessimistic definable. The boundary region of \({\mathcal {T}}\subseteq \mho \) under PMGRS context is characterized as
[68] A FS \({\tilde{\digamma }}\) on a non-void finite universe \(\mho \) is defined by a function \({\tilde{\digamma }}: \mho \longrightarrow [0, 1]\), where the value \({\tilde{\digamma }}(q)\) signifies the MD of \(q \in \mho \) in \({\tilde{\digamma }}\).
Definition 2.5
[72] A BFS \(\zeta \) over \(\mho \) is a structure of the form:
where \(\zeta ^P: \mho \longrightarrow [0, 1]\) and \(\zeta ^N: \mho \longrightarrow [-1, 0]\) are called positive MD and negative MD, respectively. The positive MD \(\zeta ^P (q)\) denotes the satisfaction degree of an element q to the property and the negative MD \(\zeta ^N (q)\) denotes the satisfaction degree of q to somewhat implicit counter-property.
Henceforth, we write \({{\mathcal {B}}}{{\mathcal {F}}}(\mho )\) to represent the collection of all BFSs over the universe \(\mho \).
Definition 2.6
[72] Let \(\lambda , \zeta \in \mathcal{B}\mathcal{F}(\mho )\). Then for all \(q\in \mho \), we have
[29] The whole BFS over \(\mho \) is symbolized by \( {\mathfrak {U}}= \big \langle {\mathfrak {U}}^P, {\mathfrak {U}}^N \big \rangle \) and is defined as \({\mathfrak {U}}^P(x) = 1\) and \({\mathfrak {U}}^N(x)= 0\), for all \(x\in \mho \). The null BFS over \(\mho \) is symbolized by \( \Theta = \big \langle \Theta ^P, \Theta ^N \big \rangle \) and is defined as \(\Theta ^P(x) = 0\) and \(\Theta ^N(x)= -1\), for all \(x\in \mho \).
Definition 2.8
[63] A BFR \({\mathcal {B}}\) over \(\mho \) is a BFS over \(\mathcal {\mho \times \mho }\). Therefore, it can be denoted as
where \(\mu ^P_{{\mathcal {B}}}: \mathcal {\mho \times \mho } \longrightarrow [0, 1]\) and \(\mu ^N_{{\mathcal {B}}}: \mathcal {\mho \times \mho } \longrightarrow [-1, 0]\) are mappings.
For a BFR \({\mathcal {B}}\) over \(\mho \), \(\mu ^P_{{\mathcal {B}}}(q, r)\) is the positive MD which shows the satisfaction degree of an object (q, r) to the property corresponding \({\mathcal {B}}\), and its negative MD \(\mu ^N_{{\mathcal {B}}}(q, r)\) represents satisfaction degree to some implicit counter-property associated with \({\mathcal {B}}\).
Definition 2.9
[15] Let \({\mathcal {B}} = \big \langle \mu ^P_{{\mathcal {B}}}(q, r), \mu ^N_{{\mathcal {B}}}(q, r) \big \rangle \) be a BFR over \(\mho = \{x_1, x_2, \ldots , x_n \}\). By taking \(a_{ij} = \mu ^P_{{\mathcal {B}}}(q_i, r_j)\) and \(b_{ij} = \mu ^N_{{\mathcal {B}}}(q_i, r_j)\), \(i = 1, 2,..., n\); \(j = 1, 2,..., n\), the BFR \({\mathcal {B}}\) can be expressed by a pair of matrices given as
where \(a_{ij} \in [0, 1]\) and \(b_{ij} \in [-1, 0]\).
Recently, Gul and Shabir [17] initiated the idea of BFPR which is stated as follows.
Definition 2.10
[17] A BFPR \({\mathfrak {B}}\) is a BFS over \(\mathcal {\mho \times \mho }\), which is described by its positive and negative MFs given as \(\mu ^P_{{\mathfrak {B}}}: \mho \times \mho \longrightarrow [0, 1]\) and \(\mu ^N_{{\mathfrak {B}}}: \mho \times \mho \longrightarrow [-1, 0]\). For \(\mho = \{x_1, x_2, \ldots , x_n \}\), we can express it by an \(n\times n\) matrix as
where \(\langle a_{ij}, b_{ij}\rangle \) denotes the bipolar fuzzy preference degree (BFPD) of alternative \(x_i\) over alternative \(x_j\), \(a_{ij}\in [0, 1]\), \(b_{ij}\in [-1, 0]\). Moreover, \(a_{ij}\) and \(b_{ij}\) satisfy the following conditions, \(a_{ij} + a_{ji} = 1, b_{ij} + b_{ji} = -1, a_{ii} = 0.5\ \text {and}\ b_{ii} = -0.5\ \forall i, j=1, 2, \ldots , n\). Particularly,
\(a_{ij}=0.5, b_{ij}= -0.5\) indicates indifference between alternatives \(x_i\) and \(x_j\);
\(a_{ij}> 0.5, b_{ij} > - 0.5\) demonstrates that alternative \(x_i\) is better than alternative \(x_j\);
\(a_{ij}< 0.5, b_{ij} < - 0.5\) indicates that alternative \(x_j\) is better than alternative \(x_i\);
\(a_{ij} = 1, b_{ij} = 0\) shows that alternative \(x_i\) is absolutely better than alternative \(x_j\);
\(a_{ij} = 0, b_{ij} = -1\) means alternative \(x_j\) is absolutely better than alternative \(x_i\).
Definition 2.11
[17] A BFPR \({\mathfrak {B}} = \Big ( \langle a_{ij}, b_{ij}\rangle \Big )_{n \times n}\) is said to be an additive consistent, if \(\forall i, j, k\in \{1, 2, \cdots , n\}\) the following conditions hold:
(1)
\(a_{ij} = a_{ik} - a_{jk} + 0.5\),
(2)
\(b_{ij} = b_{ik} - b_{jk} + 0.5\).
Definition 2.12
[17] Let \(\mho = \{x_i: i=1, 2, \ldots , n\}\) be a non-empty finite universe of n objects and \({\mathfrak {C}} = \{{\mathfrak {C}}_k: k=1, 2, \ldots , m \}\) be a non-empty finite set of m criteria. Let \(f: \mho \times {\mathfrak {C}} \longrightarrow [0, 1]\) be a fuzzy membership function and \(g: \mho \times {\mathfrak {C}} \longrightarrow [-1, 0]\) is a fuzzy non-membership function. Then, we define the transfer functions to compute BFPD of any two objects \(x_i, x_j \in \mho \) about the criterion \({\mathfrak {C}}_k\) as follows:
For a BFPR on the criteria \({\mathfrak {C}}_k\)\({\mathfrak {B}}_{{\mathfrak {C}}_k}(x_i, x_j) = \Big (\!\big \langle a^{{\mathfrak {C}}_k}_{ij}, b^{{\mathfrak {C}}_k}_{ij} \big \rangle \!\Big )_{n \times n}\), the above transfer functions (10) and (11) satisfy the following for the objects \(x_i, x_j, x_k \in \mho \):
(1)
\(a^{{\mathfrak {C}}_k}_{ii} = 0.5\) and \(b^{{\mathfrak {C}}_k}_{ii} = - 0.5\).
[17] The bipolar fuzzy preference class of an object \(x_i \in \mho \) generated by a BFPR \({\mathfrak {B}}_{{\mathfrak {C}}_k}(x_i, x_j) = \Big ( \big \langle a^{{\mathfrak {C}}_k}_{ij}, b^{{\mathfrak {C}}_k}_{ij} \big \rangle \Big )_{n \times n}\) is defined as
The BFPR \({\mathfrak {B}}_{{\mathfrak {C}}_k}\) form a family of bipolar fuzzy information granules from the universe, which constitute the bipolar fuzzy preference granular structure given as
In [17], Gul and Shabir adopted the transfer functions (10) and (11) to introduce the notion of BFP\(\delta \)-nhd and BP\(\delta \)-nhd in a BFP\(\delta \)C-approximation space (\(BFP\delta C\)-AS) and relative properties are investigated. Eventually, they built a novel BFP\(\delta \)C-BFRS model using BFP\(\delta \)-nhd.
Definition 2.14
[17] Let \(\mho \) be an arbitrary non-void finite universe and \({\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_k}\big )\) be a bipolar fuzzy preference granular structure. Then for each \(\delta = \langle \alpha , \beta \rangle \in (0, 1] \times [-1, 0)\), we call \({\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_k}\big )\) a \(BFP\delta C\) of \(\mho \), if
Moreover, the pair \(\big (\mho , {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_k}\big ) \big )\) is said to be a \(BFP\delta C\)-AS.
Definition 2.15
[17] Let \(\big (\mho , {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_k}\big ) \big )\) be a \(BFP\delta C\)-AS. For each object \(x\in \mho \), we define the BFP\(\delta \)-nghd \(\aleph ^{\delta }_{x}\) of x as follows:
If \(\underline{{BF}}_{{\mathfrak {C}}} (\lambda ) \ne \overline{{BF}}_{{\mathfrak {C}}} (\lambda )\), then \(\lambda \) is called bipolar fuzzy preference \(\delta \)-covering based bipolar fuzzy rough set; otherwise \(\lambda \) is said to be bipolar fuzzy definable.
BFP\(\delta \)C-OMG-BFRS model
In this section, we propose the idea of BFRS model by means of BFP\(\delta \)-nhd in the context of optimistic multi-granulation environment and investigate some axiomatic systems.
Definition 3.1
Let \(\big (\mho , {\widetilde{\Upsilon }} \big )\) be a \(BFP\delta C\)-AS and \({\widetilde{\Upsilon }} = \Big \{ {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_1}\big ), {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_2}\big ), \ldots , {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_k}\big ) \Big \}\) be k\(BFP\delta C\) of \(\mho \) for some \(\delta = \langle \alpha , \beta \rangle \in (0, 1] \times [-1, 0)\), where \({\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_t}\big ) = \Big \{[x_1]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}, [x_2]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}, \ldots , [x_n]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}\Big \}\) for all \(t = 1, 2, \ldots , k\). Assume that \(^{{\mathfrak {C}}_t}\aleph ^{\delta }_{x} = \Big \langle \ ^{{\mathfrak {C}}_t} \aleph ^{\alpha }_{x}, ^{{\mathfrak {C}}_t}\aleph ^{\beta }_{x} \Big \rangle \) is a BFP\(\delta \)-nghd of x in \(\mho \) induced by \({\mathfrak {C}}_t, t = 1, 2, \ldots , k\). Then we define the optimistic multi-granulation BFP\(\delta \)C lower and upper approximations of a BFS \(\lambda = \big \langle \lambda ^P, \lambda ^N \big \rangle \in \mathcal{B}\mathcal{F}(\mho )\) in \(\mho \) with respect to \(\big (\mho , {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_k}\big ) \big )\) as follows:
If \(\underline{{BF}}^o_{\sum _{t=1}^{k}{{\mathfrak {C}}_t}} (\lambda ) \ne \overline{{BF}}^o_{\sum _{t=1}^{k}{{\mathfrak {C}}_t}} (\lambda )\), then \(\lambda \) is called BFP\(\delta \)C-OMG-BFRS; otherwise it is optimistic multi-granulation bipolar fuzzy definable.
Remark 3.2
If \({\mathfrak {C}} = {\mathfrak {C}}_1 = {\mathfrak {C}}_2 = \cdots = {\mathfrak {C}}_t\), then the operators given in Eq. (21) degenerates into a bipolar fuzzy preference \(\delta \)-covering based bipolar fuzzy rough set proposed by Gul and Shabir [17].
Example 3.3
Table 1 depicts a bipolar fuzzy information matrix, where \(\mho = \{x_1, x_2, x_3, x_4, x_5\}\) and \({\mathfrak {C}} = \{{\mathfrak {C}}_1, {\mathfrak {C}}_2\}\).
Table 1
Bipolar fuzzified information matrix
\(\mho / {\mathfrak {C}}\)
\({\mathfrak {C}}_1\)
\({\mathfrak {C}}_2\)
\(x_1\)
(0.5, \(-0.25\))
(0.8, \(-0.7\))
\(x_2\)
(0.2, \(-0.8\))
(0.9, \(-0.4\))
\(x_3\)
(0.33, \(-0.25\))
(0.75, \(-0.4\))
\(x_4\)
(0.65, \(-0.6\))
(0.3, \(-0.75\))
\(x_5\)
(1, \(-0.5\))
(0.4, \(-0.35\))
Based on criteria \({\mathfrak {C}}_1\) and \({\mathfrak {C}}_2\), we can construct the BFPRs of alternative \(x_i\) to the alternative \(x_j (i, j=1, 2, \ldots , 5)\) by using formulas (10) and (11), we obtain:
The bipolar fuzzy preference classes \([x_i]_{{\mathfrak {B}}_{{\mathfrak {C}}_1}}\) and \([x_i]_{{\mathfrak {B}}_{{\mathfrak {C}}_2}}\) for \(i = 1, 2, \ldots , 5\) are given in Tables 2 and 3:
Clearly, we can observe that from Tables 2 and 3\({\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_k}\big ) = \Big \{[x_i]_{{\mathfrak {B}}_{{\mathfrak {C}}_k}}: i=1, 2, \ldots , 5, k=1, 2\Big \}\) is a \(BFP\delta C\) of \(\mho \) for \(\delta = \langle 0.500, -0.500 \rangle \). Now let \(\alpha = 0.500\) and \(\beta = -0.500\). Then according to Definition 2.15, we have
Since \(\underline{{BF}}^o_{{{\mathfrak {C}}_1 + {{\mathfrak {C}}_2}}} (\lambda ) \ne \overline{{BF}}^o_{{{\mathfrak {C}}_1 + {{\mathfrak {C}}_2}}} (\lambda )\), so \(\lambda \) is a BFP\(\delta \)C based OMG-BFRS.
Proposition 3.4
Let \(\big (\mho , {\widetilde{\Upsilon }} \big )\) be a \(BFP\delta C\)-AS and \({\widetilde{\Upsilon }} = \Big \{ {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_1}\big ), {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_2}\big ), \ldots , {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_k}\big ) \Big \}\) be k\(BFP\delta C\) of \(\mho \) for some \(\delta = \langle \alpha , \beta \rangle \in (0, 1] \times [-1, 0)\), where \({\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_t}\big ) = \Big \{[x_1]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}, [x_2]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}, \ldots , [x_n]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}\Big \}\) for all \(t = 1, 2, \ldots , k\). For any \(\lambda = \big \langle \lambda ^P, \lambda ^N \big \rangle \in \mathcal{B}\mathcal{F}(\mho )\), we have
As \(\lambda \subseteq \mu \ \text {so}\ \lambda ^P (x) \le \mu ^P (x)\) and \(\lambda ^N (x) \ge \mu ^N (x)\). To prove \(\underline{{BF}}^o_{\sum _{t=1}^{k}{{\mathfrak {C}}_t}} (\lambda ) \subseteq \underline{{BF}}^o_{\sum _{t=1}^{k}{{\mathfrak {C}}_t}} (\mu )\), we have to show that for \(x\in \mho \)
In order to prove \(\underline{{BF}}^o_{\sum _{t=1}^{k}{{\mathfrak {C}}_t}} (\lambda \cap \mu ) = \underline{{BF}}^o_{\sum _{t=1}^{k}{{\mathfrak {C}}_t}} (\lambda ) \cap \underline{{BF}}^o_{\sum _{t=1}^{k}{{\mathfrak {C}}_t}} (\mu )\), we have to show that
Now, according to part (3), if \(\lambda \subseteq \mu \), then \(\underline{{BF}}^o_{\sum _{t=1}^{k}{{\mathfrak {C}}_t}} (\lambda ) \subseteq \underline{{BF}}^o_{\sum _{t=1}^{k}{{\mathfrak {C}}_t}} (\mu )\);
In this section, we propose the concept of BFRS model based on BFP\(\delta \)-nhd in the context pessimistic multi-granulation environment and investigate some axiomatic systems. Due to the same method, we only provide the idea and omit some similar properties.
Definition 4.1
Let \(\big (\mho , {\widetilde{\Upsilon }} \big )\) be a BFP\(\delta \)C-AS and \({\widetilde{\Upsilon }} = \Big \{ {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_1}\big ), {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_2}\big ), \ldots , {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_k}\big ) \Big \}\) be k\(BFP\delta C\) of \(\mho \) for some \(\delta = \langle \alpha , \beta \rangle \in (0, 1] \times [-1, 0)\), where \({\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_t}\big ) = \Big \{[x_1]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}, [x_2]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}, \ldots , [x_n]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}\Big \}\) for all \(t = 1, 2, \ldots , k\). Assume that \(^{{\mathfrak {C}}_t}\aleph ^{\delta }_{x} = \Big \langle \ ^{{\mathfrak {C}}_t} \aleph ^{\alpha }_{x}, ^{{\mathfrak {C}}_t}\aleph ^{\beta }_{x} \Big \rangle \) is a BFP\(\delta \)-nghd of x in \(\mho \) induced by \({\mathfrak {C}}_t, t = 1, 2, \ldots , k\). Then we define the pessimistic multi-granulation BFP\(\delta \)C bipolar fuzzy preference \(\delta \)-covering lower and upper approximations of a BFS \(\lambda = \big \langle \lambda ^P, \lambda ^N \big \rangle \in \mathcal{B}\mathcal{F}(\mho )\) in \(\mho \) with respect to \(\big (\mho , {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_k}\big ) \big )\) as follows:
If \(\underline{{BF}}^p_{\sum _{t=1}^{k}{{\mathfrak {C}}_t}} (\lambda ) \ne \overline{{BF}}^p_{\sum _{t=1}^{k}{{\mathfrak {C}}_t}} (\lambda )\), then \(\lambda \) is called BFP\(\delta \)C-PMG-BFRS; otherwise it is pessimistic multi-granulation bipolar fuzzy definable.
Remark 4.2
If \({\mathfrak {C}} = {\mathfrak {C}}_1 = {\mathfrak {C}}_2 = \cdots = {\mathfrak {C}}_t\), then the operators given in Eq. (25) degenerates into a bipolar fuzzy preference \(\delta \)-covering based bipolar fuzzy rough set proposed by Gul and Shabir [17].
Example 4.3
(Continued from Example 3.3) Let \(\big (\mho , {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_k}\big ) \big )\) be the \(BFP\delta C\)-AS in Example 3.3. Then, for
Since \(\underline{{BF}}^p_{{{\mathfrak {C}}_1 + {{\mathfrak {C}}_2}}} (\lambda ) \ne \overline{{BF}}^p_{{{\mathfrak {C}}_1 + {{\mathfrak {C}}_2}}} (\lambda )\), so \(\lambda \) is a BFP\(\delta \)C based PMG-BFRS.
Proposition 4.4
Let \(\big (\mho , {\widetilde{\Upsilon }} \big )\) be a \(BFP\delta C\)-AS and \({\widetilde{\Upsilon }} = \Big \{ {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_1}\big ), {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_2}\big ), \ldots , {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_k}\big ) \Big \}\) be k\(BFP\delta C\) of \(\mho \) for some \(\delta = \langle \alpha , \beta \rangle \in (0, 1] \times [-1, 0)\), where \({\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_t}\big ) = \Big \{[x_1]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}, [x_2]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}, \cdots , [x_n]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}\Big \}\) for all \(t = 1, 2, \cdots , k\). For any \(\lambda = \big \langle \lambda ^P, \lambda ^N \big \rangle \in \mathcal{B}\mathcal{F}(\mho )\), we have
Analogous to the proof of Proposition 3.5. \(\square \)
Relationship among the BFP\(\delta \)C-BFRS, BFP\(\delta \)C-OMG-BFRS and BFP\(\delta \)C-PMG-BFRS
In this section, the relationships among the BFP\(\delta \)C-BFRS, BFP\(\delta \)C-OMG-BFRS and BFP\(\delta \)C-PMG-BFRS models will be further explored.
The following proposition gives the relationship of containment among the BFP\(\delta \)C-OMG-BFRS and BFP\(\delta \)C-PMG-BFRS approximations of \(\lambda \in \mathcal{B}\mathcal{F}(\mho )\).
Proposition 5.1
Let \(\big (\mho , {\widetilde{\Upsilon }} \big )\) be a \(BFP\delta C\)-AS and \({\widetilde{\Upsilon }} = \Big \{ {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_1}\big ), {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_2}\big ), \ldots , {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_k}\big ) \Big \}\) be k\(BFP\delta C\) of \(\mho \) for some \(\delta = \langle \alpha , \beta \rangle \in (0, 1] \times [-1, 0)\), where \({\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_t}\big ) = \Big \{[x_1]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}, [x_2]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}, \ldots , [x_n]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}\Big \}\) for all \(t = 1, 2, \ldots , k\). For any \(\lambda = \big \langle \lambda ^P, \lambda ^N \big \rangle \in \mathcal{B}\mathcal{F}(\mho )\), we have
The following proposition gives the relationship between the BFP\(\delta \)C approximations and the optimistic multi-granulation BFP\(\delta \)C approximations of a BFS \(\lambda \in \mathcal{B}\mathcal{F}(\mho )\).
Proposition 5.2
Let \(\big (\mho , {\widetilde{\Upsilon }} \big )\) be a \(BFP\delta C\)-AS and \({\widetilde{\Upsilon }} = \Big \{ {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_1}\big ), {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_2}\big ), \ldots , {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_k}\big ) \Big \}\) be k\(BFP\delta C\) of \(\mho \) for some \(\delta = \langle \alpha , \beta \rangle \in (0, 1] \times [-1, 0)\), where \({\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_t}\big ) = \Big \{[x_1]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}, [x_2]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}, \ldots , [x_n]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}\Big \}\) for all \(t = 1, 2, \ldots , k\). For any \(\lambda = \big \langle \lambda ^P, \lambda ^N \big \rangle \in \mathcal{B}\mathcal{F}(\mho )\), we have
Direct consequence of Definition 2.16 and Definition 3.1. \(\square \)
The following proposition gives the relationship between the BFP\(\delta \)C approximations and the pessimistic multi-granulation BFP\(\delta \)C approximations of a BFS \(\lambda \in \mathcal{B}\mathcal{F}(\mho )\).
Proposition 5.3
Let \(\big (\mho , {\widetilde{\Upsilon }} \big )\) be a \(BFP\delta C\)-AS and \({\widetilde{\Upsilon }} = \Big \{ {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_1}\big ), {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_2}\big ), \ldots , {\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_k}\big ) \Big \}\) be k\(BFP\delta C\) of \(\mho \) for some \(\delta = \langle \alpha , \beta \rangle \in (0, 1] \times [-1, 0)\), where \({\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_t}\big ) = \Big \{[x_1]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}, [x_2]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}, \ldots , [x_n]_{{\mathfrak {B}}_{{\mathfrak {C}}_t}}\Big \}\) for all \(t = 1, 2, \ldots , k\). For any \(\lambda = \big \langle \lambda ^P, \lambda ^N \big \rangle \in \mathcal{B}\mathcal{F}(\mho )\), we have
An application of MCDM methods by BFP\(\delta \)C-OMG-BFRS model
In this section, we develop a novel MCDM technique under the environment of the BFP\(\delta \)C based BFRS model in which attribute (criteria) weight are real numbers and attributes values are bipolar fuzzy numbers. We subsequently apply this approach to solve the problem of smartphone selection.
The problem statement
Assume that \(\mho = \{x_i: i = 1, 2, \ldots , n\}\) be the finite set of n alternatives and \({\mathfrak {C}} = \{{\mathfrak {C}}_k: k=1, 2, \ldots , m \}\) be a finite set of m criteria that are determined by a decision-maker (expert). Let \({\mathcal {W}}= (\varpi _1, \varpi _2, \ldots , \varpi _m )^T\) be the weight vector of all criteria, representing the importance of each criteria, where \(0\le \varpi _j\le 1\) and satisfies the normalized condition \(\sum _{j=1}^m \varpi _j = 1\). Let the BFS \(\lambda = \big \{ \big \langle x, \lambda ^P (x), \lambda ^N (x)\big \rangle : x\in \mho \big \}\) over \(\mho \) be the description af all alternatives by the expert (decision-maker). Let \({\mathcal {E}}\) represent a finite set of the domain for the information functions \(f(x_\ell , {\mathfrak {C}}_k)\) and \(g(x_\ell , {\mathfrak {C}}_k)\), where \(f(x_\ell , {\mathfrak {C}}_k)\in [0, 1] \) stands for the positive membership (or satisfaction degree) of the alternative \(x_\ell \) with respect to \({\mathfrak {C}}_k\) given by the expert and \(g(x_\ell , {\mathfrak {C}}_k)\in [-1, 0]\) stands for the negative membership (or dissatisfaction degree) of the alternative \(x_\ell \) with respect to \({\mathfrak {C}}_k\) given by the expert. The bipolar fuzzy preference class \([x_i]_{{\mathfrak {B}}_{{\mathfrak {C}}_k}}(x_j)\) stands for the efficacy value of alternatives \(x_j\)’s with respect to \({\mathfrak {C}}_k\). For a critical value \(\delta = \langle \alpha , \beta \rangle \in (0, 1] \times [-1, 0)\), let for each alternative \(x_i\in \mho \), there is at least one criteria \({\mathfrak {C}}_k\in {\mathfrak {C}}\) such that the efficacy value of alternatives \(x_j\) for the criteria \({\mathfrak {C}}_k\) is not less than \(\alpha \) and greater than \(\beta \), and \({\mathscr {P}}\big ({\mathfrak {B}}_{{\mathfrak {C}}_k}\big )\) is a \(BFP\delta C\) of \(\mho \). Then the BFP\(\delta \)-nghd \(\aleph ^{\delta }_{x_j}\) of \(x_j\) with respect to \({\mathfrak {C}}_k\) is a BFS given as
which express the minimum of all efficacy values for each alternatives \(x_t\) with respect to \({\mathfrak {C}}_k\).
If \(\lambda = \big \{ \big \langle x_i, \lambda ^P (x_i), \lambda ^N (x_i)\big \rangle : x_i\in \mho \big \} \in \mathcal{B}\mathcal{F}(\mho )\) represent possible membership and non-membership degree of each alternative \(x_i\in \mho \) given by the expert, then the DM for MCDM problem is how to select an optimal alternative among all.
We denote the bipolar fuzzified information system \((\mho , {\mathfrak {C}}, {\mathcal {W}}, {\mathcal {E}})\).
Algorithm for the proposed MCDM using BFP\(\delta \)C-OMG-BFRS model
To select the optimal alternative among the available ones, here we propose a DM algorithm in the context of the BFP\(\delta \)C based BFRS model. The related steps are summarized as follows:
Input: Bipolar fuzzified information system \((\mho , {\mathfrak {C}}, {\mathcal {W}}, {\mathcal {E}})\).
Step 1:
Evaluate \({\mathfrak {B}}_{{\mathfrak {C}}_k}; k =1, 2, \cdots , m\) using transfer functions given in Eqs. (10) and (11).
Step 2:
Calculate \([x_i]_{{\mathfrak {B}}_{{\mathfrak {C}}_k}}\) of \(x_i\) with respect to \({\mathfrak {C}}_k\).
Step 3:
Construct \(^{{\mathfrak {C}}_k}\aleph ^{\delta }_{x_i} = \Big \langle \ ^{{\mathfrak {C}}_k}\aleph ^{\alpha }_{x_i},\ ^{{\mathfrak {C}}_k}\aleph ^{\beta }_{x_i} \Big \rangle \) of \(x_i\) with respect to \({\mathfrak {C}}_k\).
Step 4:
Use the method of bipolar fuzzy TOPSIS to determine a positive ideal solution \({{\mathcal {I}}_k^+}\) and negative ideal solution \({{\mathcal {I}}_k^-}\) as
Rank the alternatives with respect to the values of \(\partial (x_j)\) and make the decision.
Output: A ranking result of all alternatives.
A flowchart depiction of the proposed MCDM problem is given in Fig. 1.
Fig. 1
Summary of the proposed DM algorithm
×
An applied example
According to the problem statement in “The problem statement” section, we give an example for solving the smartphone selection problem to verify the validity of the suggested model.
Example 6.1
Consider a decision-maker confused about selecting a smartphone among five smartphones displayed for sale in a store. Assume that he is focused on the following characteristics (criteria) to acquire the best smartphone: colour, memory, elegancy, and camera zoom of a smartphone. Since it is known that every aspect affects the cost and benefit of the intended smartphone. So, these smartphones denote the alternatives, and the mentioned features represent the criteria in our MCDM problem.
Let \(\mho = \{S_1, S_2, S_3, S_4, S_5\}\) be the collection of concerned smartphones and \({\mathfrak {C}} = \{ {\mathfrak {C}}_1, {\mathfrak {C}}_2, {\mathfrak {C}}_3, {\mathfrak {C}}_4 \}\) be the set of concerned features. The weights of all criteria are given as \({\mathcal {W}}= (0.25, 0.3, 0.25, 0.2)^T\). The ratings of alternatives with respect to criteria are portrayed in Table 6.
Table 6
Bipolar fuzzified information matrix
\(\mho / {\mathfrak {C}}\)
\({\mathfrak {C}}_1\)
\({\mathfrak {C}}_2\)
\({\mathfrak {C}}_3\)
\({\mathfrak {C}}_4\)
\(S_1\)
(0.5, \(-0.25\))
(0.8, \(-0.7\))
(0.3, \(-0.1\))
(0.6, \(-0.6\))
\(S_2\)
(0.2, \(-0.8\))
(0.9, \(-0.4\))
(0.6, \(-0.3\))
(0.55, \(-0.5\))
\(S_3\)
(0.33, \(-0.25\))
(0.75, \(-0.4\))
(0.25, \(-0.7\))
(0.3, \(-0.1\))
\(S_4\)
(0.65, \(-0.6\))
(0.3, \(-0.75\))
(0.8, \(-0.35\))
(0.65, \(-0.7\))
\(S_5\)
(1, \(-0.5\))
(0.4, \(-0.35\))
(0.2, \(-0.6\))
(0.25, \(-0.65\))
For the selection of the most suitable smartphone, the calculation steps are shown as follows:
Step 1:
Based on criteria \({\mathfrak {C}}_1, {\mathfrak {C}}_2, {\mathfrak {C}}_3, {\mathfrak {C}}_4\) and using formulas (10) and (11) to evaluate the BFPRs of alternative \(S_i\) to the alternative \(S_j (i, j=1, 2, \ldots , 5)\), we obtain:
As the weight of each criterion is given as \(\varpi _1 = 0.25, \varpi _2 = 0.3, \varpi _3 = 0.25, \varpi _4 = 0.2\), so in the light of formula (30), the optimal index can be calculated as follows:
From the ranking result, we conclude that \(S_1\) and \(S_4\) are the most suitable smartphones. The graphic representation of the ranking of the smartphones is displayed in Fig. 2.
Fig. 2
Ranking of smartphones
×
Comparative analysis
This section examines the advantages of the designed methodology and compares it to some existing approaches.
Table 15
Characteristics comparison of different methods with proposed method
In summary, the advantages of the proposed MCDM method in the context of the BFP\(\delta \)C based OMG-BFRS model are listed as follows:
(1)
In the study of MCDM problems with bipolar fuzzy information, there are many DM approaches based on BFR. However, not all MADM problems can be characterized by a BFR. For this reason, we set forth the method to solve MCDM problems with bipolar fuzzy information based on BFP\(\delta \)C based BFRSs.
(2)
If we compare our suggested technique with the methods presented in [4‐7, 10, 11, 33, 35, 38, 49], we observe that these techniques are incompetent to capture bipolarity in the DM process which is an indispensable component of human thinking and behavior.
(3)
Fuzzy DM approaches are successfully applied to solve problems with only one-sided information; that is, alternatives are ranked based only on the positive MD of alternatives. By using fuzzy structure to DM, we cannot provide information about the dissatisfaction behavior or degree of alternatives corresponding to different criteria. Thus, we propose BFP\(\delta \)C based OMG-BFRSs model to rank the alternatives.
Comparison with some other techniques
In this subsection, we conduct the comparative analysis from qualitative and quantitative aspects with certain existing techniques to demonstrate the effectiveness and superiority of the established DM methodology, and the specific comparison process is as follows.
Qualitative comparison
We compare the characteristics of the established technique and the introduced approaches in [1, 7, 8, 15, 18, 24, 35, 38, 50, 58, 64] from qualitative aspect, and the comparison results are given in Table 15. We conduct qualitative comparison from five features: membership function (MF), non MF (NMF), multi-granulation environment (MGE), covering and ranking of alternatives to illustrate its superiority. From Table 15, it can be observes that the proposed method has all listed characteristics, but the mentioned methods do not have all of them. Thus, the proposed approach is superior to the DM methods under the FS, RS, MGRS environments in many practical DM problems.
Quantitative comparison
We compare our suggested technique with certain existing DM approaches quantitatively by numerical experiments. According to different DM methods proposed in Wei et al. [57], Malik and Shabir [36], Jana et al. [23], we can evaluate the ranking results, which are displayed in Table 16. Also, the ranking results are plotted graphically in Fig. 3.
Fig. 3
Graphical representation of ranking of alternatives using different methods
×
From 16, we can observe that the ranking order of alternatives of the other schemes is different, mainly due to changes in the DM environment. However, the optimal alternative is the same. This phenomenon is normal in decision-making theory. In general, decision-makers can set different values of the parameters \(\alpha \) and \(\beta \) based on their preferences and actual requirements.
Conclusion and future work
Recent years have witnessed the rising of MGRS, which is a new branch of RS theory. It provides a new theoretical paradigm for solving the complicated problem under the circumstance of multiple binary relations on the universe, and offers a new direction based on multi-granulation analysis for knowledge acquisition and DM. Meanwhile, bipolarity refers to an explicit handling of positive and negative sides of data. A variety of human decisions are based on based on positive and negative or bipolar judgements.
Core content and continuous work of this article are summarized as follows:
(1)
By hybridizing the idea of BFPR with MGRSs and BFS theory, we have initiated BFP\(\delta \)C-OMG-BFRS and BFP\(\delta \)C-PMG-BFRS models. Several significant structural properties of these model have also been carefully investigated.
(2)
Furthermore, we have established the interrelationship among BFP\(\delta \)C-BFRS, BFP\(\delta \)C-OMG-BFRS and BFP\(\delta \)C-PMG-BFRS models.
(3)
To illustrate the application of the designed technique with bipolar fuzzified information, we have established a novel approach for solving MCDM problems using the theory of BFP\(\delta \)C-OMG-BFRS model.
(4)
The DM methodology and an algorithm of our suggested approach have been presented. Furthermore, a practical example has been provided to demonstrate the importance of the suggested approach.
(5)
Finally, a comparative analysis has been made to show the effectiveness of the proposed technique.
We anticipate that our research will shed more light on the foundations of the theories of BFS and MGRSs and lead to more robust mathematical approaches to approximate reasoning in soft computing. At the same time, we acknowledge that there remain several avenues for further theoretical research in this perspective. Future work will generally contain the following recommendations:
(1)
The real-world applications of the designed technique in solving a wider variety of selection problems, including TOSIS, VIKOR, ELECTRE, AHP, COPRAS, PROMETHEE, etc.
(2)
The attribute reduction of BFP\(\delta \)C-OMG-BFRSs and BFP\(\delta \)C-PMG-BFRSs should be analyzed, and comprehensive experimental investigations, and comparisons with current methodologies should also be verified and addressed.
(3)
Further study can be done to establish fruitful algorithms for different kinds of DM problems.
(4)
Another direction is to investigate the topological properties and similarity measures of the proposed BFP\(\delta \)C-OMG-BFRS and BFP\(\delta \)C-PMG-BFRS model to establish a concrete platform for future research.
Declarations
Conflict of interest
The authors declare no conflict of interest.
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