Multi-attribute group decision making based on cubic bipolar fuzzy information using averaging aggregation operators

Abstract

A huge range of human decisions is involved bipolar subjective thoughts. For illustration, effects and side effects are two different aspects of decision analysis. The equilibrium and mutual coexistence of these two aspects are treated as a key for balanced social environment. A verity of bipolar fuzzy decision making with different technique is available for bipolar fuzzy characterizations of the universe of options that depend on a limited number of grades. So, the concept of simple bipolar fuzzy set is insufficient to provide the information about the occurrence of ranking with accuracy because information is limited. In this regard, we use cubic bipolar fuzzy sets (CBFSs) as the generalization of bipolar fuzzy sets. In human decisions, the second important part is ranking of alternatives obtained after evaluation. The motivation behind this research is to develop an appropriate aggregation method which is simple reliable and efficient enough to handle cubic bipolar fuzzy data. We propose aggregation operators, including, cubic bipolar fuzzy weighted averaging operator, cubic bipolar fuzzy ordered weighted averaging operator and cubic bipolar fuzzy hybrid weighted averaging operator for $P$ -order and $R$ -order. We demonstrate simplicity and efficiency of proposed aggregation operators and algorithm by discussing multi-attribute group decision making (MAGDM) problem under cubic bipolar fuzzy information. We also discuss the comparison analysis of the proposed method of aggregation.

Keywords

Cubic bipolar fuzzy set Averaging aggregation operators for CBFSs MAGDM

1 Introduction

In daily life, we encounter with many decision making circumstances, which involve ambiguities and uncertainties due to insufficient knowledge, meager information, incompatible data, inconsistent and rare information. In 1965, Zadeh [52] brought out the idea of fuzzy set theory to overcome such problems. Fuzzy set theory has been successfully used in decision making problems to handle uncertain and ambiguous data from many decades. In fuzzy decision making, Zadeh [53] introduced similarity relations and fuzzy ordering. He also described the rating of each alternative and the weight of each criterion using linguistic terms induced by fuzzy linguistic variables which can be expressed in triangular fuzzy numbers [54, 55]. In short, fuzzy set theory has been auspiciously adapted in fields of computer sciences [5], management sciences [22], engineering [11], medical sciences [38] and physics [28] etc.

After fuzzy set theory, many set theories have been developed, including, interval valued set theory [53], intuitionistic fuzzy set theory [6], bipolar fuzzy set theory [60], neutrosophic set theory [36], soft set theory [32], fuzzy soft set theory [33], pythagorean fuzzy set [50] and m-polar fuzzy set theory [10] etc. All these theories have been developed according to necessity of handling specific type of data and its useability in different suitable domains. For other hybrid theories, terminologies and applications, the readers are referred to [8 , 62].

For data analyzing of many types, bipolarity of knowledge is a vital part to be considered while developing a mathematical framework for most of the situations. Bipolarity indicates the positive and negative aspects of a particular problem. The concept behind the bipolarity is that a huge range of human decisions analysis is involved bipolar subjective thoughts. For illustration, happiness and grief, sweetness and sourness, effects and side effects are two different aspects of decision analysis. The equilibrium and mutual coexistence of these two aspects are treated as a key for balanced social environment. Zhang [58] introduced the extension of fuzzy set with bipolarity, called, bipolar-valued fuzzy sets. Bipolar fuzzy set is suitable for information which involve property as well as its counter property. In 2000, Lee [29] discussed some basic operations of bipolar-valued fuzzy set. Lee [30] proposed a comparison of intuitionistic fuzzy set, interval-valued set and bipolar fuzzy set. Dubois and Prade [12] introduced three major types of bipolar fuzzy set. Zhang [59] established a computational framework for bipolar cognitive modeling and multi attribute decision analysis. In bipolar-valued fuzzy set interval of membership value is [-1, 1]. The bipolar fuzzy set involves positive and negative memberships. The elements with 0 membership indicate that they are not satisfying the specific property, the interval (0, 1] indicates elements satisfying property with different degrees of membership, whereas [-1, 0) shows that elements satisfying implicit counter property. Bipolar fuzzy sets have been also used in many fields and decision making problems which involve bipolar type information [1–3 , 35].

In 2013, Jun et al. [27] introduced the concept of cubic set (extension of interval-valued fuzzy set and fuzzy set) and its operations. They presented idea of internal and external cubic sets and their properties. the concept has been extended with many other theories. Cubic set has been also successfully used in decision making processes due to its efficiency of containing enough information of a particular type of data.

Multi-criteria group decision making provides a unanimous decision on basis of different criterions to seek the most accurate solution of real world problems. Decision making is an integral part of modern management and business. Essentially, rational or sound decision making is a primary goal of the management. In primitive times, decisions were framed without handling the uncertainties in the data, which may lead to inadequate results toward the real-life operating situations. Decision making has a vital role in our daily life problems. We encounter with many difficulties to handle a problem which contains big amount of data. In this regard, different techniques with aggregation operators have been introduced to handle such type of data. Fahmi et al. studied aggregation operators and their application in MCGDM based on triangular cubic hesitant fuzzy and triangular neutrosophic cubic fuzzy set [13, 14]. Gul [23] worked on aggregation operators under environment of bipolar fuzzy set with MCGDM. Beg and Rashid [7] studied group decision making in the context of intuitionistic hesitant fuzzy set. Garg and Arora introduced different aggregation operators and their applications in MCDM [17 –20]. Xia and Xu [47, 48] introduced aggregation operators on different extensions of fuzzy set with MCGDM. Many authors have been auspiciously adopted many capable techniques to solve decision making problems. Ghodsypour and Brien discussed decision making of supplier selection using AHP technique [21]. Hwang and Yoon [25, 26] introduced technique of TOPSIS first in 1988. Gulcin and Cifci studied many efficient techniques of MCDM including DEMATEL, ANP, TOPSIS on fuzzy sets [24]. Liu et al. [31] studied group decision making with multiple-attribute strategic weight manipulation with minimum cost. Peng and Selvachandran discussed decision making methods for pythagorean fuzzy sets. Zhan et al. [56, 57] studied many MCGDM method for domain of rough set. Zhang and Xu [61] introduced TOPSIS for pythagoren fuzzy set with decision making. Zhang et al. [64] studied priority weights and consistency for incomplete hesitant fuzzy preference relations. Zhang and Guo [65] discussed deriving priority weights from intuitionistic multiplicative preference relations under group decision-making settings. Yu et al. studied extended TODIM for multi-criteria group decision making based on unbalanced hesitant fuzzy linguistic term sets [51].

Consensus decision making or consensus reaching process (CRPs) is a type of group decision making process in which group members develop, and agree to support a decision in the best interest of the whole group or common goal. It is used to describe both the decision and the process of reaching a decision. CRPs is thus concerned with the process of finalizing a decision, and the social, economic, legal, environmental and political effects of applying this process. In short it has great importance in group decision making. Zhang et al. [66] reviewed some existed CRPs and introduced a series of CRPs as the comparison object. They also developed new multi-stage optimization-based CRPs. There are many MAGDM problems which need to assign two rank levels only, so as to create a ranking of one subset of alternatives above another subset. This type of MAGDM problem is called a 2-rank MAGDM problem. Zhang et al. [63] investigated the 2-rank MAGDM problem under the multi granular linguistic context, and proposed a 2-rank consensus reaching framework with the minimum adjustments. In multiple attribute decision making (MADM), strategic weight manipulation is understood as a deliberate manipulation of attribute weight setting to achieve a desired ranking of alternatives. Liu et al. [31] studied the strategic weight manipulation in a group decision making (GDM) context with interval attribute weight information. Zhang et al. [68] developed a novel consensus reaching process for MAGDM with hesitant fuzzy linguistic term sets (HFLTSs). Based on the proposed consensus rule and aggregation model, They presented a consensus reaching process for MAGDM with HFLTSs. Information loss is an other issue in multiple attribute group decision making. In real-world decision problems, decision makers usually express their opinions with different preference structures. In order to deal with the heterogeneous preference information in group decision making, Zhang et al. [67] presented an optimization-based consensus model for group decision making with heterogeneous preference structures. They developed the technique to minimize the information loss between decision makers’ heterogeneous preference information and individual preference vectors and solution with a consensus.In MADM decision makers may not be honest, to deal with this issue, Liu et al. [31, 69] presented some useful methods for MADM.

The motivation behind this research is to develop an appropriate aggregation method which is simple reliable and efficient enough to handle cubic bipolar fuzzy data. From latest research surveys of hybrid fuzzy set models, most of the researchers in fuzzy-set-inspired models focused on real numbers between 0 and 1. But in most of the real life problems, we encounter with the negative part of a particular decision. For example, a medicine which is not effective may not has any side effect. So the bipolarity is an important aspect of human decisions. In human decisions, the second important part is ranking and rating of different alternatives obtained after particular evaluation. A verity of bipolar fuzzy decision making with different technique is available in literature. This range of applications of the theories can be used to deal with bipolar vagueness and uncertainty, which introduced the simple bipolar fuzzy characterizations of the universe of options that depend on a limited number of grades. So the concept of simple bipolar fuzzy set is insufficient to provide the information about the occurrence of ratings or grades with accuracy because information is limited, and it is also unable to describe the occurrence of uncertainty and vagueness very well specially, when sensitive cases are involved in decision making problems. Similarly, the concept of interval valued bipolar fuzzy sets(IVBFSs) is also insufficient to provide the information about the opinion of experts depending upon the properties of alternatives. For this purpose, Riaz and Tehrim [42] introduced the novel model with application, called cubic bipolar fuzzy sets as the generalization of bipolar fuzzy sets. This model provides more accuracy and flexibility as compared to previously existing approaches, because it contains more information and it is more comprehensive and reasonable. The model provides complete information about occurrence of ratings, uncertainty and bipolarity. On the other hand, an arithmetic mean is useful when comparing different items finding a single “ figure of merit” for these items when each item has multiple properties that have different numeric ranges. The arithmetic operations are useful in multiple-attribute group decision making because these operations provide an average value of a collection of observational data against each attribute. Yager [49] introduced averaging aggregation operators first in 1988. These operators satisfied some important properties and are useful in many fields including economics, business and finance etc. The motivation of this paper is to introduce a series of aggregation operators on cubic bipolar fuzzy set using arithmetic operations. It is also valuable because we extend the method of aggregation operators by using arithmetic operations under cubic bipolar fuzzy data for MAGDM. We compile the final decision by using these extended operators method because of the complex structure of proposed model. In this paper, firstly, we use the concept of bipolar fuzzy set and cubic set to initiate the notion of cubic bipolar fuzzy set, secondly, we introduce six cubic bipolar fuzzy averaging aggregation operators to aggregate the cubic bipolar fuzzy data then we apply both concepts to a multi-criteria group decision making problem to handle a specific type of data with bipolar imprecision which is inexact. We also check and compare results obtained by these averaging operators.

The proposed aggregation operators on cubic bipolar fuzzy set have following merits.

The presented operators are simple in calculation and efficient to calculate huge amount of data, which is often unable or time-taking to handle in group decision making with multi attributes.

The proposed algorithm with these averaging aggregation operator is suitable to handle business or related MAGDM.

The average based operations are suitable in multiple-attribute group decision making because these operations provide an average value of a collection of observational data against each attribute. The loss of data is less as compare to other techniques.

These averaging aggregation operators on cubic bipolar fuzzy set provide a dual check because of dual perspective that is $P$ -order and $R$ -order.

There is need of good technique for handling bipolar fuzzy information in simple as well as efficient manners. The proposed aggregation operators method is combined with cubic bipolar fuzzy set is more strong but ease.

The paper is organized as follows: In Section 2 we discuss some preliminaries, in Section 3 we introduce averaging aggregation operators on CBFSs under $P$ -Order and $R$ -Order. We also examine capability and reliability of operators in aggregating the cubic bipolar fuzzy data. In Section 4, we see the useability and application of these operators. The Section 5 is devoted for careful analysis and comparison with interval valued bipolar fuzzy set(IVBFS). In Section 6, we discuss merits of proposed aggregation operators and method base comparison with existed models. In Section 7 we finally concluded our study.

2 Preliminaries

In the present section, we review some basic definitions which are useful throughout this research work.

2.1 Fuzzy set, bipolar fuzzy set, interval-valued bipolar fuzzy set and cubic set

Definition 2.1. [52] Consider the universal set V and membership function f : V → [0, 1]. Then V_f is a fuzzy set on V if each element ς ∈ V is associated with degree of membership, which is a real number in [0, 1] and it is denoted by f (ς).

Definition 2.2. [29] A bipolar fuzzy set on V is of the form $K = {(ς, δ_{K}^{P} (ς), δ_{K}^{N} (ς)) : for all ς \in V},$ where $δ_{K}^{P} (ς)$ denotes the positive memberships ranges over [0, 1] and $δ_{K}^{N} (ς)$ denotes the negative memberships ranges over [-1, 0]. A bipolar fuzzy element(BFE) is simply written as $K = (δ_{K}^{N}, δ_{K}^{P})$ .

Definition 2.3. [42] Let V be a non empty universal set and $[J]$ be the collection of all closed subintervals of [-1, 1]. An interval-valued bipolar fuzzy set(IVBFS) is of the form $K = {< ς, {M_{K}^{P} (ς), M_{K}^{N} (ς)} >}$ where,

$M_{K}^{P} (ς) : V \to [0, 1]$ and $M_{K}^{N} (ς) : V \to [- 1, 0]$ are interval-valued positive and negative membership degrees of ς ∈ V. An interval-valued bipolar fuzzy element(IVBFE) can be written as

${M_{K}^{P} (ς), M_{K}^{N} (ς)} = {M^{P}, M^{N}}$ where $δ_{K ℓ}^{P}$ , $δ_{K u}^{P}$ , $δ_{K ℓ}^{N}$ and $δ_{K u}^{N}$ are called upper and lower limits of the IVBFEs $M^{P}$ and $M^{N}$ respectively.

Definition 2.4. [42] Let V be non empty set. A cubic set on V can be defined as

$K = {< ς, \ddot{j} (ς), V_{f} (ς) > ∣ ς \in V}$ where $\ddot{j}$ is interval-valued fuzzy set on V and V_f is a fuzzy set on V.

2.2 Cubic bipolar fuzzy set with operations

Definition 2.5. [42] Consider a universal set V. A cubic bipolar fuzzy set (CBFS) can be defined as $\ddot{K} = {< ς, {M_{\ddot{K}}^{P} (ς), M_{\ddot{K}}^{N} (ς)}, N_{\ddot{K}} (ς) > : ς \in V}$ (1) where $M = {M_{\ddot{K}}^{P} (ς), M_{\ddot{K}}^{N} (ς)}$ is called Interval Valued Bipolar Fuzzy Set (IVBFS) whereas $N = N_{\ddot{K}} (ς)$ is called Bipolar Fuzzy Set (BFS). Consider the Interval $I = [- 1, 1]$ . Suppose that $[I]$ and $[J]$ be the collection of all sub-intervals of [0, 1] and [-1, 0] respectively. Then we obtain the mappings $M_{\ddot{K}}^{P} (ς) \to [I] ∣ M_{\ddot{K}}^{P} (ς) = [δ_{\ddot{K} ℓ}^{P} (ς), δ_{\ddot{K} u}^{P} (ς)]$ and $M_{\ddot{K}}^{N} (ς) \to [J] ∣ M_{\ddot{K}}^{N} (ς) = [δ_{\ddot{K} ℓ}^{N} (ς), δ_{\ddot{K} u}^{N} (ς)]$ similarly we get, $N_{\ddot{K}} (ς) \to I ∣ N_{\ddot{K}} (ς) = [ρ_{\ddot{K}}^{P} (ς), ρ_{\ddot{K}}^{N} (ς)]$ , so (1) becomes $\ddot{K} = < M, N > = {< ς, {[δ_{\ddot{K} ℓ}^{P} (ς), δ_{\ddot{K} u}^{P} (ς)], [δ_{\ddot{K} ℓ}^{N} (ς), δ_{\ddot{K} u}^{N} (ς)]}, {ρ_{\ddot{K}}^{P} (ς), ρ_{\ddot{K}}^{N} (ς)} > : ς \in V}$ . Now the set $\ddot{K}$ is called cubic bipolar fuzzy set. A cubic bipolar fuzzy element(CBFE) can be denoted by $\ddot{K} = {< {[δ_{\ddot{K} ℓ}^{P}, δ_{\ddot{K} u}^{P}], [δ_{\ddot{K} ℓ}^{N}, δ_{\ddot{K} u}^{N}]}, {ρ_{\ddot{K}}^{P}, ρ_{\ddot{K}}^{N}} >}$

Definition 2.6. [42] ( $P$ -Order) Let $\ddot{K_{1}} = < M_{1}, N_{1} >$ and $\ddot{K_{2}} = < M_{2}, N_{2} >$ be two CBFSs on V. Then $\ddot{K_{1}} \subseteq_{P} \ddot{K_{2}}$ , if (i) $[δ_{\ddot{K} ℓ_{1}}^{P} (ς), δ_{\ddot{K} u_{1}}^{P} (ς)] \leq [δ_{\ddot{K} ℓ_{2}}^{P} (ς), δ_{\ddot{K} u_{2}}^{P} (ς)]$ and $[δ_{\ddot{K} ℓ_{1}}^{N} (ς), δ_{\ddot{K} u_{1}}^{N} (ς)] \geq [δ_{\ddot{K} ℓ_{2}}^{N} (ς), δ_{\ddot{K} u_{2}}^{N} (ς)]$ (ii) ${ρ_{\ddot{K_{1}}}^{P} (ς) \leq ρ_{{\ddot{K}}_{2}}^{P} (ς)}$ and ${ρ_{\ddot{K_{1}}}^{N} (ς) \geq ρ_{{\ddot{K}}_{2}}^{N} (ς)}$ .

Definition 2.7. [42] ( $R$ -Order) Let $\ddot{K_{1}} = < M_{1}, N_{1} >$ and $\ddot{K_{2}} = < M_{2}, N_{2} >$ be two CBFSs on V. Then $\ddot{K_{1}} \subseteq_{R} \ddot{K_{2}}$ , if (i) $[δ_{\ddot{K} ℓ_{1}}^{P} (ς), δ_{\ddot{K} u_{1}}^{P} (ς)] \leq [δ_{\ddot{K} ℓ_{2}}^{P} (ς), δ_{\ddot{K} u_{2}}^{P} (ς)]$ and $[δ_{\ddot{K} ℓ_{1}}^{N} (ς), δ_{\ddot{K} u_{1}}^{N} (ς)] \geq [δ_{\ddot{K} ℓ_{2}}^{N} (ς), δ_{\ddot{K} u_{2}}^{N} (ς)]$ (ii) ${ρ_{\ddot{K_{1}}}^{P} (ς) \geq ρ_{{\ddot{K}}_{2}}^{P} (ς)}$ and ${ρ_{\ddot{K_{1}}}^{N} (ς) \leq ρ_{{\ddot{K}}_{2}}^{N} (ς)}$ .

Definition 2.8. [42] The complement of $\ddot{K}$ can be defined as ${\ddot{K}}^{c} = {< ς, {[1 - δ_{\ddot{K} u}^{P} (ς), 1 - δ_{\ddot{K} ℓ}^{P} (ς)], [- 1 - δ_{\ddot{K} u}^{N} (ς), - 1 - δ_{\ddot{K} ℓ}^{N} (ς)]}, {1 - ρ_{\ddot{K}}^{P} (ς), - 1 - ρ_{\ddot{K}}^{N} (ς)} > : ς \in V}$

Definition 2.9. [42] ( $P$ -Union) Let $\ddot{K_{ɛ}} = < M_{ɛ}, N_{ɛ} >$ be a collection of CBFSs on V, then we define $P$ -Union as $\underset{ɛ = 1}{\overset{n}{\cup}} \ddot{K_{ɛ}} = {< \max_{ɛ \in Ω} [δ_{\ddot{K} ℓ_{ɛ}}^{P} (ς), δ_{\ddot{K} u_{ɛ}}^{P} (ς)], \min_{ɛ \in Ω} [δ_{\ddot{K} ℓ_{ɛ}}^{N} (ς), δ_{\ddot{K} u_{ɛ}}^{N} (ς)]}, {\max_{ɛ \in Ω} (ρ_{{\ddot{K}}_{ɛ}}^{P} (ς)), \min_{ɛ \in Ω} (ρ_{{\ddot{K}}_{ɛ}}^{N} (ς)) >}$ . It is denoted by $\cup_{P}$ .

Definition 2.10. [42] ( $P$ -Intersection) Let $\ddot{K_{ɛ}} = < M_{ɛ}, N_{ɛ} >$ be a collection of CBFSs on V, then we define $P$ -Intersection as $\underset{ɛ = 1}{\overset{n}{\cap}} \ddot{K_{ɛ}} = {< \min_{ɛ \in Ω} [δ_{\ddot{K} ℓ_{ɛ}}^{P} (ς), δ_{\ddot{K} u_{ɛ}}^{P} (ς)], \max_{ɛ \in Ω} [δ_{\ddot{K} ℓ_{ɛ}}^{N} (ς), δ_{\ddot{K} u_{ɛ}}^{N} (ς)]}, {\min_{ɛ \in Ω} (ρ_{{\ddot{K}}_{ɛ}}^{P} (ς)), \max_{ɛ \in Ω} (ρ_{{\ddot{K}}_{ɛ}}^{N} (ς)) >}$ . It is denoted by $\cap_{P}$ .

Definition 2.11. [42] ( $R$ -Union) Let $\ddot{K_{ɛ}} = < M_{ɛ}, N_{ɛ} >$ be a collection of CBFSs on V, then we define $R$ -Union as $\underset{ɛ = 1}{\overset{n}{\cup}} \ddot{K_{ɛ}} = {< \max_{ɛ \in Ω} [δ_{\ddot{K} ℓ_{ɛ}}^{P} (ς), δ_{\ddot{K} u_{ɛ}}^{P} (ς)], \min_{ɛ \in Ω} [δ_{\ddot{K} ℓ_{ɛ}}^{N} (ς), δ_{\ddot{K} u_{ɛ}}^{N} (ς)]}, {\min_{ɛ \in Ω} (ρ_{{\ddot{K}}_{ɛ}}^{P} (ς)), \max_{ɛ \in Ω} (ρ_{{\ddot{K}}_{ɛ}}^{N} (ς)) >}$ . It is denoted by $\cup_{R}$ .

Definition 2.12. [42] ( $R$ -Intersection) Let $\ddot{K_{ɛ}} = < M_{ɛ}, N_{ɛ} >$ be a collection of CBFSs on V, then we define $R$ -Intersection as $\underset{ɛ = 1}{\overset{n}{\cap}} \ddot{K_{ɛ}} = {< \min_{ɛ \in Ω} [δ_{\ddot{K} ℓ_{ɛ}}^{P} (ς), δ_{\ddot{K} u_{ɛ}}^{P} (ς)], \max_{ɛ \in Ω} [δ_{\ddot{K} ℓ_{ɛ}}^{N} (ς), δ_{\ddot{K} u_{ɛ}}^{N} (ς)]}, {\max_{ɛ \in Ω} (ρ_{{\ddot{K}}_{ɛ}}^{P} (ς)), \min_{ɛ \in Ω} (ρ_{{\ddot{K}}_{ɛ}}^{N} (ς)) >}$ . It is denoted by $\cap_{R}$ .

Definition 2.13. [42] Let $\ddot{K_{1}} = < M_{1}, N_{1} >$ , $\ddot{K_{2}} =$ $< M_{2}, N_{2} >$ be two CBFSs on V and ϱ > 0 be a real number. Then we define following algebraic operations for

$P$ -Order. $\ddot{K_{1}} \oplus_{P} \ddot{K_{2}} = {〈 {[δ_{\ddot{K} ℓ_{1}}^{P} (ς) + δ_{\ddot{K} ℓ_{2}}^{P} (ς) - (δ_{\ddot{K} ℓ_{1}}^{P} (ς) δ_{\ddot{K} ℓ_{2}}^{P} (ς)), δ_{\ddot{K} u_{1}}^{P} (ς) + δ_{\ddot{K} u_{2}}^{P} (ς) - (δ_{\ddot{K} u_{1}}^{P} (ς) δ_{\ddot{K} u_{2}}^{P} (ς))], [- (δ_{\ddot{K} ℓ_{1}}^{N} (ς) δ_{\ddot{K} ℓ_{2}}^{N} (ς)), - (δ_{\ddot{K} u_{1}}^{N} (ς) δ_{\ddot{K} u_{2}}^{N} (ς))]}, {ρ_{\ddot{K_{1}}}^{P} (ς) + ρ_{{\ddot{K}}_{2}}^{P} (ς) - (ρ_{\ddot{K_{1}}}^{P} (ς) ρ_{{\ddot{K}}_{2}}^{P} (ς)), - (ρ_{\ddot{K_{1}}}^{N} (ς) ρ_{{\ddot{K}}_{2}}^{N} (ς))} 〉}$

$\ddot{K_{1}} \otimes_{P} \ddot{K_{2}} = {〈 {[(δ_{\ddot{K} ℓ_{1}}^{P} (ς) δ_{\ddot{K} ℓ_{2}}^{P} (ς)), (δ_{\ddot{K} u_{1}}^{P} (ς) δ_{\ddot{K} u_{2}}^{P} (ς))], [- (- δ_{\ddot{K} ℓ_{1}}^{N} (ς) - δ_{\ddot{K} ℓ_{2}}^{N} (ς) - (δ_{\ddot{K} ℓ_{1}}^{N} (ς) δ_{\ddot{K} ℓ_{2}}^{N} (ς))), - (- δ_{\ddot{K} u_{1}}^{N} (ς) - δ_{\ddot{K} u_{2}}^{N} (ς) - (δ_{\ddot{K} u_{1}}^{N} (ς) δ_{\ddot{K} u_{2}}^{N} (ς)))]}, {(ρ_{\ddot{K_{1}}}^{P} (ς) ρ_{{\ddot{K}}_{2}}^{P} (ς)), - (- ρ_{\ddot{K_{1}}}^{N} (ς) - ρ_{{\ddot{K}}_{2}}^{N} (ς) - (ρ_{\ddot{K_{1}}}^{N} (ς) ρ_{{\ddot{K}}_{2}}^{N} (ς)))} 〉}$

$\ddot{K_{1}^{ϱ}} = {〈 {[{(δ_{\ddot{K} ℓ_{1}}^{P} (ς))}^{ϱ}, {(δ_{\ddot{K} u_{1}}^{P} (ς))}^{ϱ}], [- (1 - {(1 - (- δ_{\ddot{K} ℓ_{1}}^{N} (ς)))}^{ϱ}), - (1 - {(1 - (- δ_{\ddot{K} u_{1}}^{N} (ς)))}^{ϱ})]}, {{(ρ_{\ddot{K_{1}}}^{P} (ς))}^{ϱ}, - (1 - {(1 - (- ρ_{\ddot{K_{1}}}^{N} (ς)))}^{ϱ})} 〉}$

$ϱ \ddot{K_{1}} = {〈 {[1 - {(1 - δ_{\ddot{K} ℓ_{1}}^{P} (ς))}^{ϱ}, 1 - {(1 - δ_{\ddot{K} u_{1}}^{P} (ς))}^{ϱ}], [- {(- δ_{\ddot{K} ℓ_{1}}^{N} (ς))}^{ϱ}, - {(- δ_{\ddot{K} u_{1}}^{N} (ς))}^{ϱ}]}, {1 - {(1 - ρ_{\ddot{K_{1}}}^{P} (ς))}^{ϱ}, - {(- ρ_{\ddot{K_{1}}}^{N} (ς))}^{ϱ}} 〉}$

Definition 2.14. [42] Let $\ddot{K_{1}} = 〈 M_{1}, N_{1} >$ , $\ddot{K_{2}} = 〈 M_{2}, N_{2} >$ be two CBFSs on V and ϱ > 0 be a real number. Then we define following algebraic operations for $R$ -Order.

$\ddot{K_{1}} \oplus_{R} \ddot{K_{2}} = {〈 {[δ_{\ddot{K} ℓ_{1}}^{P} (ς) + δ_{\ddot{K} ℓ_{2}}^{P} (ς) - (δ_{\ddot{K} ℓ_{1}}^{P} (ς) δ_{\ddot{K} ℓ_{2}}^{P} (ς)), δ_{\ddot{K} u_{1}}^{P} (ς) + δ_{\ddot{K} u_{2}}^{P} (ς) - (δ_{\ddot{K} u_{1}}^{P} (ς) δ_{\ddot{K} u_{2}}^{P} (ς))], [- (δ_{\ddot{K} ℓ_{1}}^{N} (ς) δ_{\ddot{K} ℓ_{2}}^{N} (ς)), - (δ_{\ddot{K} u_{1}}^{N} (ς) δ_{\ddot{K} u_{2}}^{N} (ς))] {(ρ_{\ddot{K_{1}}}^{P} (ς) ρ_{{\ddot{K}}_{2}}^{P} (ς)), - (- ρ_{\ddot{K_{1}}}^{N} (ς) - ρ_{{\ddot{K}}_{2}}^{N} (ς) - (ρ_{\ddot{K_{1}}}^{N} (ς) ρ_{{\ddot{K}}_{2}}^{N} (ς)))} 〉}$

$\ddot{K_{1}} \otimes_{R} \ddot{K_{2}} = {〈 {[(δ_{\ddot{K} ℓ_{1}}^{P} (ς) δ_{\ddot{K} ℓ_{2}}^{P} (ς)), (δ_{\ddot{K} u_{1}}^{P} (ς) δ_{\ddot{K} u_{2}}^{P} (ς))], [- (- δ_{\ddot{K} ℓ_{1}}^{N} (ς) - δ_{\ddot{K} ℓ_{2}}^{N} (ς) - (δ_{\ddot{K} ℓ_{1}}^{N} (ς) δ_{\ddot{K} ℓ_{2}}^{N} (ς))), - (- δ_{\ddot{K} u_{1}}^{N} (ς) - δ_{\ddot{K} u_{2}}^{N} (ς) - (δ_{\ddot{K} u_{1}}^{N} (ς) δ_{\ddot{K} u_{2}}^{N} (ς)))]},}, {ρ_{\ddot{K_{1}}}^{P} (ς) + ρ_{{\ddot{K}}_{2}}^{P} (ς) - (ρ_{\ddot{K_{1}}}^{P} (ς) ρ_{{\ddot{K}}_{2}}^{P} (ς)), - (ρ_{\ddot{K_{1}}}^{N} (ς) ρ_{{\ddot{K}}_{2}}^{N} (ς))} 〉}$

$\ddot{K_{1}^{ϱ}} = {〈 {[{(δ_{\ddot{K} ℓ_{1}}^{P} (ς))}^{ϱ}, {(δ_{\ddot{K} u_{1}}^{P} (ς))}^{ϱ}], [- (1 - {(1 - (- δ_{\ddot{K} ℓ_{1}}^{N} (ς)))}^{ϱ}), - (1 - {(1 - (- δ_{\ddot{K} u_{1}}^{N} (ς)))}^{ϱ})]}, {1 - {(1 - ρ_{\ddot{K_{1}}}^{P} (ς))}^{ϱ}, - {(- ρ_{\ddot{K_{1}}}^{N} (ς))}^{ϱ}} 〉}$

$ϱ \ddot{K_{1}} = {〈 {[1 - {(1 - δ_{\ddot{K} ℓ_{1}}^{P} (ς))}^{ϱ}, 1 - {(1 - δ_{\ddot{K} u_{1}}^{P} (ς))}^{ϱ}], [- {(- δ_{\ddot{K} ℓ_{1}}^{N} (ς))}^{ϱ}, - {(- δ_{\ddot{K} u_{1}}^{N} (ς))}^{ϱ}] {{(ρ_{\ddot{K_{1}}}^{P} (ς))}^{ϱ}, - (1 - {(1 - (- ρ_{\ddot{K_{1}}}^{N} (ς)))}^{ϱ})} 〉}$

3 Averaging aggregation operators for CBFSs

In the present section, firstly we define an improved score function for cubic bipolar fuzzy data for both orders. Secondly, we define a collection of aggregation operators for CBFSs. We define different averaging operators for both $P$ -Order and $R$ -Order of CBFSs. We discuss some certain characteristics of these operators. We also see useability and efficiency of these operators in aggregating the cubic bipolar fuzzy data.

3.1 Score and accuracy functions for CBFEs

In [42], we define score function for $P$ -Order and $R$ -Order. Here, we define an improved score function for a CBFE, which we use to compare CBFE.

Definition 3.1. The score function of a CBFE under $P$ -Order can be defined as $S_{P} (\ddot{K}) = \frac{[δ_{\ddot{K} ℓ}^{P} (ς) + δ_{\ddot{K} u}^{P} (ς)] + [1 + δ_{\ddot{K} ℓ}^{N} (ς) + 1 + δ_{\ddot{K} u}^{N} (ς)] - ρ_{\ddot{K}}^{P} (ς) + 1 + ρ_{\ddot{K}}^{N} (ς)}{6}$ The score function of a CBFE under $R$ -Order can be defined as $S_{R} (\ddot{K}) = \frac{[δ_{\ddot{K} ℓ}^{P} (ς) + δ_{\ddot{K} u}^{P} (ς)] + [1 + δ_{\ddot{K} ℓ}^{N} (ς) + 1 + δ_{\ddot{K} u}^{N} (ς)] + ρ P_{\ddot{K}} (ς) - (1 + ρ N_{\ddot{K}} (ς))}{6}$ where $0 \leq S_{P, R} (\ddot{K}) \leq 1$ . If $S (\ddot{K_{1}}) \leq S (\ddot{K_{2}})$ , then we say $\ddot{K_{1}}$ is weaker than $\ddot{K_{2}}$ and we denote it by $\ddot{K_{1}} ≺ \ddot{K_{2}}$ . If $S (\ddot{K_{1}}) \geq S (\ddot{K_{2}})$ , then we say $\ddot{K_{1}}$ is stronger than $\ddot{K_{2}}$ and we denote it by $\ddot{K_{1}} ≻ \ddot{K_{2}}$ . If $S (\ddot{K_{1}}) = S (\ddot{K_{2}})$ , then we say $\ddot{K_{1}}$ is equal to $\ddot{K_{2}}$ and we denote it by $\ddot{K_{1}} = \ddot{K_{2}}$ , in this condition we use accuracy function which is defined below.

Definition 3.2. The accuracy function of a CBFE can be defined as $A (\ddot{K}) = \frac{[δ_{\ddot{K} ℓ}^{P} (ς) + δ_{\ddot{K} u}^{P} (ς)] - [δ_{\ddot{K} ℓ}^{N} (ς) + δ_{\ddot{K} u}^{N} (ς)] + ρ P_{\ddot{K}} (ς) - ρ N_{\ddot{K}} (ς)}{6}$ where $0 \leq A (\ddot{K}) \leq 1$ . If $A (\ddot{K_{1}}) \leq A (\ddot{K_{2}})$ , then we say $\ddot{K_{1}}$ is weaker than $\ddot{K_{2}}$ and we denote it by $\ddot{K_{1}} ≺ \ddot{K_{2}}$ . If $A (\ddot{K_{1}}) \geq A (\ddot{K_{2}})$ , then we say $\ddot{K_{1}}$ is stronger than $\ddot{K_{2}}$ and we denote it by $\ddot{K_{1}} ≻ \ddot{K_{2}}$ . If $A (\ddot{K_{1}}) = A (\ddot{K_{2}})$ , then we say $\ddot{K_{1}}$ is equal to $\ddot{K_{2}}$ and we denote it by $\ddot{K_{1}} = \ddot{K_{2}}$ .

3.2 $P$ -order cubic bipolar fuzzy weighted average( $P$ -CBFWA) Operator

Suppose that $\ddot{K_{ɛ}} = {< {[δ_{\ddot{K} ℓ_{ɛ}}^{P}, δ_{\ddot{K} u_{ɛ}}^{P}], [δ_{\ddot{K} ℓ_{ɛ}}^{N}, δ_{\ddot{K} u_{ɛ}}^{N}]}, {ρ_{{\ddot{K}}_{ɛ}}^{P}, ρ_{{\ddot{K}}_{ɛ}}^{N}} 〉}$ , for {ɛ = 1, 2, . . . , n}, be the family of CBFEs. Then consider a weighted vector $ϒ = {w_{1}, w_{2}, . . ., w_{3}}^{T}$ , where $\underset{ɛ = 1}{\overset{n}{Σ}} w_{ɛ} = 1$ and $0 \leq w_{ɛ} \leq 1$ then cubic bipolar fuzzy weighted averaging(CBFWA) operator is a mapping $A : {\ddot{K}}^{n} \to \ddot{K}$ which can be computed under $P$ -Order as follows $P - CBFWA (\ddot{K_{1}}, \ddot{K_{2}}, . . ., \ddot{K_{n}}) = {\underset{ɛ = 1}{\overset{n}{\oplus}}}_{P} (w_{ɛ} (\ddot{K_{ɛ}})) = {{[1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - δ_{\ddot{K} ℓ_{ɛ}}^{P})^{w_{ɛ}}, 1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - δ_{\ddot{K} u_{ɛ}}^{P})^{w_{ɛ}}], [- \underset{ɛ = 1}{\overset{n}{Π}} (- δ_{\ddot{K} ℓ_{ɛ}}^{N})^{w_{ɛ}}, - \underset{ɛ = 1}{\overset{n}{Π}} (- δ_{\ddot{K} u_{ɛ}}^{N})^{w_{ɛ}}]}, {1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - ρ_{{\ddot{K}}_{ɛ}}^{P})^{w_{ɛ}}, - \underset{ɛ = 1}{\overset{n}{Π}} (- ρ_{{\ddot{K}}_{ɛ}}^{N})^{w_{ɛ}}}}$

Example 3.3 Let us consider CBFEs $\ddot{K_{1}} = {[0.34, 0.53], [- 0.45, - 0.31], {0.23, - 0.34}}$ , ${\ddot{K}}_{2} = {[0.23, 0.33], [- 0.53, - 0.43], {0.54, - 0.41}}$ , ${\ddot{K}}_{3} = {[0.45, 0.49], [- 0.62, - 0.52], {0.61, - 0.51}}$ , ${\ddot{K}}_{4} = {[0.52, 0.61], [- 0.71, - 0.60], {0.43, - 0.63}}$ with weight vector ϒ = {0.1, 0.2, 0.3, 0.4} ^T. $\underset{ɛ = 1}{\overset{4}{Π}} (1 - δ_{\ddot{K} ℓ_{ɛ}}^{P})^{w_{ɛ}} = (1 - 0.34)^{0.1} \times (1 - 0.23)^{0.2} \times (1 - 0.45)^{0.3} \times (1 - 0.52)^{0.4} = 0.567$ $\underset{ɛ = 1}{\overset{4}{Π}} (1 - δ_{\ddot{K} u_{ɛ}}^{P})^{w_{ɛ}} = (1 - 0.53)^{0.1} \times (1 - 0.33)^{0.2} \times (1 - 0.49)^{0.3} \times (1 - 0.61)^{0.4} = 0.479$ $- \underset{ɛ = 1}{\overset{4}{Π}} (- δ_{\ddot{K} ℓ_{ɛ}}^{N})^{w_{ɛ}} = - ((0.45)^{0.1} \times (0.53)^{0.2} \times (0.62)^{0.3} \times (0.71)^{0.4}) = - 0.614$ $- \underset{ɛ = 1}{\overset{4}{Π}} (- δ_{\ddot{K} u_{ɛ}}^{N})^{w_{ɛ}} = - ((0.31)^{0.1} \times (0.43)^{0.2} \times (0.52)^{0.3} \times (0.60)^{0.4}) = - 0.503$ $\underset{ɛ = 1}{\overset{4}{Π}} (1 - ρ_{{\ddot{K}}_{ɛ}}^{P})^{w_{ɛ}} = (1 - 0.23)^{0.1} \times (1 - 0.54)^{0.2} \times (1 - 0.61)^{0.3} \times (1 - 0.43)^{0.4} = 0.502$ $- \underset{ɛ = 1}{\overset{4}{Π}} (- ρ_{{\ddot{K}}_{ɛ}}^{N})^{w_{ɛ}} = - ((0.34)^{0.1} \times (0.41)^{0.2} \times (0.51)^{0.3} \times (0.63)^{0.4}) = - 0.510$ $P - CBFWA (\ddot{K_{1}}, \ddot{K_{2}}, {\ddot{K}}_{3}, {\ddot{K}}_{4}) = {[0.433, 0.521], [- 0.614, - 0.503],$ {0.498, - 0.510}}

3.3 $P$ -Order cubic bipolar fuzzy ordered weighted averaging( $P$ -CBFOWA) Operator

which can be computed under $P$ -Order as follows $P - CBFOWA (\ddot{K_{1}}, \ddot{K_{2}}, . . ., {\ddot{K}}_{n}) = {\underset{ɛ = 1}{\overset{n}{\oplus}}}_{P} (w_{ɛ} (\ddot{K_{o ɛ}})) = {{[1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - δ_{\ddot{K} ℓ_{o ɛ}}^{P})^{w_{ɛ}}, 1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - δ_{\ddot{K} u_{o ɛ}}^{P})^{w_{ɛ}}], [- \underset{ɛ = 1}{\overset{n}{Π}} (- δ_{\ddot{K} ℓ_{o ɛ}}^{N})^{w_{ɛ}}, - \underset{ɛ = 1}{\overset{n}{Π}} (- δ_{\ddot{K} u_{o ɛ}}^{N})^{w_{ɛ}}]}, {1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - ρ_{{\ddot{K}}_{o ɛ}}^{P})^{w_{ɛ}}, - \underset{ɛ = 1}{\overset{n}{Π}} (- ρ_{{\ddot{K}}_{o ɛ}}^{N})^{w_{ɛ}}}}$ In this operator we first arrange $\ddot{K_{ɛ}}$ in an order by using score function then we associate weights with them.

3.4 $P$ -order cubic bipolar fuzzy hybrid weighted averaging( $P$ -CBFHWA) operator

Let $\ddot{K_{ɛ}} = {< {[δ_{\ddot{K} ℓ_{ɛ}}^{P}, δ_{\ddot{K} u_{ɛ}}^{P}], [δ_{\ddot{K} ℓ_{ɛ}}^{N}, δ_{\ddot{K} u_{ɛ}}^{N}]}, {ρ_{{\ddot{K}}_{ɛ}}^{P}, ρ_{{\ddot{K}}_{ɛ}}^{N}} >}$ , for {ɛ = 1, 2, . . . , n} be a family of CBFEs. Then consider a weighted vector $ϒ = {w_{1}, w_{2}, . . ., w_{3}}^{T}$ , where $\underset{ɛ = 1}{\overset{n}{Σ}} w_{ɛ} = 1$ and $0 \leq w_{ɛ} \leq 1$ . Let o_ɛ be an alteration which shows the greatest $\ddot{K_{ɛ}^{'}} = m w_{ɛ} \ddot{K_{ɛ}}$ . where m is number of weight vector in ϒ, then cubic bipolar fuzzy hybrid weighted averaging(CBFHWA) operator is a mapping $A : {\ddot{K^{'}}}^{n} \to \ddot{K^{'}}$ which can be computed under $P$ -Order as follows $P - CBFHWA (\ddot{K_{1}}, \ddot{K_{2}}, . . ., {\ddot{K}}_{n}) = {\underset{ɛ = 1}{\overset{n}{\oplus}}}_{P} (w_{ɛ} (\ddot{K_{o ɛ}^{'}})) = {{[1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - δ_{\ddot{K^{'}} ℓ_{o ɛ}}^{P})^{w_{ɛ}}, 1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - δ_{\ddot{K^{'}} u_{o ɛ}}^{P})^{w_{ɛ}}], [- \underset{ɛ = 1}{\overset{n}{Π}} (- δ_{\ddot{K^{'}} ℓ_{o ɛ}}^{N})^{w_{ɛ}}, - \underset{ɛ = 1}{\overset{n}{Π}} (- δ_{\ddot{K^{'}} u_{o ɛ}}^{N})^{w_{ɛ}}]}, {1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - ρ_{{\ddot{K^{'}}}_{o ɛ}}^{P})^{w_{ɛ}}, - \underset{ɛ = 1}{\overset{n}{Π}} (- ρ_{{\ddot{K^{'}}}_{o ɛ}}^{N})^{w_{ɛ}}}}$

In this operator we first assign weights to parameters and rearrange the $\ddot{K_{ɛ}^{'}}$ in an order by using score function then we associate weights with them.

Example 3.4. Let us consider CBFEs

$\ddot{K_{1}} = {[0.80, 0.90], [- 0.32, - 0.24], {0.24, - 0.32}}$ ,

${\ddot{K}}_{2} = {[0.23, 0.33], [- 0.53, - 0.43], {0.54, - 0.41}}$ ,

${\ddot{K}}_{3} = {[0.45, 0.49], [- 0.62, - 0.52], {0.61, - 0.51}}$ ,

${\ddot{K}}_{4} = {[0.52, 0.61], [- 0.71, - 0.60], {0.43, - 0.63}}$ with weight vector ϒ = {0.1, 0.3, 0.3, 0.2} ^T. Since

$\ddot{K_{ɛ}^{'}} = m w_{ɛ} \ddot{K_{ɛ}}$ and $m = 4$ . Thus we get, $\ddot{K_{1}^{'}} = {{[1 - (1 - 0.80)^{0.4}, 1 - (1 - 0.90)^{0.4}], [- (0.32)^{0.4}, - (0.24)^{0.4}]}, {1 - (1 - 0.24)^{0.4}, - (0.32)^{0.4}}}$ $\ddot{K_{2}^{'}} = {{[1 - (1 - 0.23)^{1.2}, 1 - (1 - 0.33)^{1.2}], [- (0.53)^{1.2}, - (0.43)^{1.2}]}, {1 - (1 - 0.54)^{1.2}, - (0.41)^{1.2}}}$ $\ddot{K_{3}^{'}} = {{[1 - (1 - 0.45)^{1.2}, 1 - (1 - 0.49)^{1.2}], [- (0.62)^{1.2}, - (0.52)^{1.2}]}, {1 - (1 - 0.61)^{1.2}, - (0.51)^{1.2}}}$ $\ddot{K_{4}^{'}} = {{[1 - (1 - 0.52)^{0.8}, 1 - (1 - 0.61)^{0.8}], [- (0.71)^{0.8}, - (0.60)^{0.8}]}, {1 - (1 - 0.43)^{0.8}, - (0.63)^{0.8}}}$ this implies that

${\ddot{K^{'}}}_{1} = {[0.474, 0.601], [- 0.633, - 0.565], {0.103, - 0.633}}$

${\ddot{K^{'}}}_{2} = {[0.269, 0.381], [- 0.466, - 0.363], {0.606, - 0.343}}$

${\ddot{K^{'}}}_{3} = {[0.511, 0.554], [- 0.563, - 0.456], {0.676, - 0.445}}$

${\ddot{K^{'}}}_{4} = {[0.444, 0.529], [- 0.760, - 0.664], {0.362, - 0.690}}$

now we compute the score function

$S ({\ddot{K^{'}}}_{1}) = - 0.108$ , $S ({\ddot{K^{'}}}_{2}) = 0.014$ , $S ({\ddot{K^{'}}}_{3}) = 0.046$ and $S ({\ddot{K^{'}}}_{4}) = - 0.129$ , which shows that $S ({\ddot{K^{'}}}_{3}) \geq S ({\ddot{K^{'}}}_{2}) \geq S ({\ddot{K^{'}}}_{1}) \geq S ({\ddot{K^{'}}}_{4})$ $\Rightarrow {\ddot{K^{'}}}_{3} \geq {\ddot{K^{'}}}_{2} \geq {\ddot{K^{'}}}_{1} \geq {\ddot{K^{'}}}_{4}$ we rearrange the CBFEs

${\ddot{K^{'}}}_{o 1} = {[0.511, 0.554], [- 0.563, - 0.456], {0.676, - 0.445}}$

${\ddot{K^{'}}}_{o 2} = {[0.269, 0.381], [- 0.466, - 0.363], {0.606, - 0.343}}$

${\ddot{K^{'}}}_{o 3} = {[0.474, 0.601], [- 0.633, - 0.565], {0.103, - 0.633}}$

${\ddot{K^{'}}}_{o 4} = {[0.444, 0.529], [- 0.760, - 0.664], {0.362, - 0.690}}$ Again we associate the weight vector to the alteration. Let ϒ = {0.2, 0.2, 0.2, 0.4} $\underset{ɛ = 1}{\overset{4}{Π}} (1 - δ_{\ddot{K^{'}} ℓ_{ɛ}}^{P})^{w_{ɛ}} = (1 - 0.474)^{0.2} \times (1 - 0.511)^{0.2} \times (1 - 0.269)^{0.2} \times (1 - 0.444)^{0.4} = 0.566$ $\underset{ɛ = 1}{\overset{4}{Π}} (1 - δ_{\ddot{K^{'}} u_{ɛ}}^{P})^{w_{ɛ}} = (1 - 0.601)^{0.2} \times (1 - 0.554)^{0.2} \times (1 - 0.381)^{0.2} \times (1 - 0.529)^{0.4} = 0.476$ $- \underset{ɛ = 1}{\overset{4}{Π}} (- δ_{\ddot{K^{'}} ℓ_{ɛ}}^{N})^{w_{ɛ}} = - ((0.633)^{0.2} \times (0.563)^{0.2} \times (0.466)^{0.2} \times (0.760)^{0.4}) = - 0.625$ $- \underset{ɛ = 1}{\overset{4}{Π}} (- δ_{\ddot{K^{'}} u_{ɛ}}^{N})^{w_{ɛ}} = - ((0.565)^{0.2} \times (0.456)^{0.2} \times (0.363)^{0.2} \times (0.664)^{0.4}) = - 0.528$ $\underset{ɛ = 1}{\overset{4}{Π}} (1 - ρ_{{\ddot{K^{'}}}_{ɛ}}^{P})^{w_{ɛ}} = (1 - 0.103)^{0.2} \times (1 - 0.676)^{0.2} \times (1 - 0.606)^{0.2} \times (1 - 0.362)^{0.4} = 0.541$ $- \underset{ɛ = 1}{\overset{4}{Π}} (- ρ_{{\ddot{K^{'}}}_{ɛ}}^{N})^{w_{ɛ}} = - ((0.633)^{0.2} \times (0.445)^{0.2} \times (0.343)^{0.2} \times (0.690)^{0.4}) = - 0.540$ $P - CBFHWA ({\ddot{K^{'}}}_{o 1}, \ddot{K_{o 2}^{'}}, {\ddot{K^{'}}}_{o 3}, {\ddot{K^{'}}}_{o 4}) = {[0.434, 0.524], [- 0.625, - 0.528], {0.459, - 0.540}}$

Now we define all averaging operators under $R$ -Order, which share similar properties as in $P$ -Order.

3.5 $R$ -order cubic bipolar fuzzy weighted averaging( $R$ -CBFWA) Operator

Suppose that $\ddot{K_{ɛ}} = {< {[δ_{\ddot{K} ℓ_{ɛ}}^{P}, δ_{\ddot{K} u_{ɛ}}^{P}], [δ_{\ddot{K} ℓ_{ɛ}}^{N},$ $δ_{\ddot{K} u_{ɛ}}^{N}]}, {ρ_{{\ddot{K}}_{ɛ}}^{P}, ρ_{{\ddot{K}}_{ɛ}}^{N}} >}$ , for {ɛ = 1, 2, . . . , n}, be a family of CBFEs. Then consider a weight vector $ϒ = {w_{1}, w_{2}, . . ., w_{3}}^{T}$ , where $\underset{ɛ = 1}{\overset{n}{Σ}} w_{ɛ} = 1$ and $0 \leq w_{ɛ} \leq 1$ then cubic bipolar fuzzy averaging(CBFWA) is a mapping $A : {\ddot{K}}^{n} \to \ddot{K}$ which can be computed under $R$ -Order as follows $R - CBFWA (\ddot{K_{1}}, \ddot{K_{2}}, . . ., {\ddot{K}}_{n}) = {\underset{ɛ = 1}{\overset{n}{\oplus}}}_{R} (w_{ɛ} (\ddot{K_{ɛ}})) = {{[1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - δ_{\ddot{K} ℓ_{ɛ}}^{P})^{w_{ɛ}}, 1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - δ_{\ddot{K} u_{ɛ}}^{P})^{w_{ɛ}}], [- \underset{ɛ = 1}{\overset{n}{Π}} (- δ_{\ddot{K} ℓ_{ɛ}}^{N})^{w_{ɛ}}, - \underset{ɛ = 1}{\overset{n}{Π}} (- δ_{\ddot{K} u_{ɛ}}^{N})^{w_{ɛ}}]}, {\underset{ɛ = 1}{\overset{n}{Π}} (ρ_{{\ddot{K}}_{ɛ}}^{P})^{w_{ɛ}}, - (1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - (- ρ_{{\ddot{K}}_{ɛ}}^{N}))^{w_{ɛ}})}}$

3.6 $R$ -order cubic bipolar fuzzy ordered weighted averaging( $R$ -CBFOWA) Operator

Suppose that $\ddot{K_{ɛ}} = {< {[δ_{\ddot{K} ℓ_{ɛ}}^{P}, δ_{\ddot{K} u_{ɛ}}^{P}], [δ_{\ddot{K} ℓ_{ɛ}}^{N}, δ_{\ddot{K} u_{ɛ}}^{N}]}, {ρ_{{\ddot{K}}_{ɛ}}^{P}, ρ_{{\ddot{K}}_{ɛ}}^{N}} >}$ , for {ɛ = 1, 2, . . . , n}, be a family of CBFEs. Then consider a weighted vector $ϒ = {w_{1}, w_{2}, . . ., w_{3}}^{T}$ , where $\underset{ɛ = 1}{\overset{n}{Σ}} w_{ɛ} = 1$ and $0 \leq w_{ɛ} \leq 1$ . Let o_ɛ be an alteration which shows the greatest $\ddot{K_{ɛ}}$ then cubic bipolar fuzzy ordered weighted averaging(CBFOWA) operator is a mapping $A : {\ddot{K}}^{n} \to \ddot{K}$ which can be computed under $R$ -Order as follows $R - CBFOWA (\ddot{K_{1}}, \ddot{K_{2}}, . . ., {\ddot{K}}_{n}) = {\underset{ɛ = 1}{\overset{n}{\oplus}}}_{R} (w_{ɛ} (\ddot{K_{o ɛ}})) = {{[1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - δ_{\ddot{K} ℓ_{o ɛ}}^{P})^{w_{ɛ}}, 1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - δ_{\ddot{K} u_{o ɛ}}^{P})^{w_{ɛ}}], [- \underset{ɛ = 1}{\overset{n}{Π}} (- δ_{\ddot{K} ℓ_{o ɛ}}^{N})^{w_{ɛ}}, - \underset{ɛ = 1}{\overset{n}{Π}} (- δ_{\ddot{K} u_{o ɛ}}^{N})^{w_{ɛ}}]}, {\underset{ɛ = 1}{\overset{n}{Π}} (ρ_{{\ddot{K}}_{o ɛ}}^{P})^{w_{ɛ}}, - (1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - (- ρ_{{\ddot{K}}_{o ɛ}}^{N}))^{w_{ɛ}})}}$

3.7 $R$ -Order cubic bipolar fuzzy hybrid weighted averaging( $R$ -CBFHWA) operator

Let $\ddot{K_{ɛ}} = {< {[δ_{\ddot{K} ℓ_{ɛ}}^{P}, δ_{\ddot{K} u_{ɛ}}^{P}], [δ_{\ddot{K} ℓ_{ɛ}}^{N}, δ_{\ddot{K} u_{ɛ}}^{N}]}, {ρ_{{\ddot{K}}_{ɛ}}^{P}, ρ_{{\ddot{K}}_{ɛ}}^{N}} >}$ , for {ɛ = 1, 2, . . . , n}, be a family of CBFEs. Then consider a weight vector $ϒ = {w_{1}, w_{2}, . . ., w_{3}}^{T}$ , where $\underset{ɛ = 1}{\overset{n}{Σ}} w_{ɛ} = 1$ and $0 \leq w_{ɛ} \leq 1$ . Let o_ɛ be an alteration which shows the greatest $\ddot{K_{ɛ}^{'}} = m w_{ɛ} \ddot{K_{ɛ}}$ . where m is number of weight vector in ϒ, then cubic bipolar fuzzy hybrid weighted averaging(CBFHWA) operator is a mapping $A : {\ddot{K^{'}}}^{n} \to \ddot{K^{'}}$ which can be computed under $R$ -Order as follows $R - CBFHWA (\ddot{K_{1}}, \ddot{K_{2}}, . . ., {\ddot{K}}_{n}) = {\underset{ɛ = 1}{\overset{n}{\oplus}}}_{R} (w_{ɛ} (\ddot{K_{o ɛ}^{'}})) = {{[1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - δ_{\ddot{K^{'}} ℓ_{o ɛ}}^{P})^{w_{ɛ}}, 1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - δ_{\ddot{K^{'}} u_{o ɛ}}^{P})^{w_{ɛ}}], [- \underset{ɛ = 1}{\overset{n}{Π}} (- δ_{\ddot{K^{'}} ℓ_{o ɛ}}^{N})^{w_{ɛ}}, - \underset{ɛ = 1}{\overset{n}{Π}} (- δ_{\ddot{K^{'}} u_{o ɛ}}^{N})^{w_{ɛ}}]}, {\underset{ɛ = 1}{\overset{n}{Π}} (ρ_{{\ddot{K^{'}}}_{o ɛ}}^{P})^{w_{ɛ}}, - (1 - \underset{ɛ = 1}{\overset{n}{Π}} (1 - (- ρ_{{\ddot{K^{'}}}_{o ɛ}}^{N}))^{w_{ɛ}})}}$

Theorem 3.5. (Properties) Let $\ddot{K_{ɛ}} = {< {[δ_{\ddot{K} ℓ_{ɛ}}^{P}, δ_{\ddot{K} u_{ɛ}}^{P}], [δ_{\ddot{K} ℓ_{ɛ}}^{N}, δ_{\ddot{K} u_{ɛ}}^{N}]}, {ρ_{{\ddot{K}}_{ɛ}}^{P}, ρ_{{\ddot{K}}_{ɛ}}^{N}} >}$ be a family of CBFEs and Φ be any operator from above defined operators including, $P$ -CBFWA, $P$ -CBFOWA, $P$ -CBFHWA $R$ -CBFWA, $R$ -CBFOWA and $R$ -CBFHWA, then Φ satisfies the following properties under operations of both orders.

(Idempotent) For $\ddot{K_{ɛ}} = \ddot{K}$ , we have $Φ (\ddot{K_{1}}, \ddot{K_{2}}, . . ., {\ddot{K}}_{n}) = \ddot{K}$

(Bounded) $\cap_{P} Φ (\ddot{K_{1}}, \ddot{K_{2}}, . . ., {\ddot{K}}_{n}) \leq Φ (\ddot{K_{1}}, \ddot{K_{2}}, . . ., {\ddot{K}}_{n}) \leq \cup_{P} Φ (\ddot{K_{1}}, \ddot{K_{2}}, . . ., {\ddot{K}}_{n})$

(Commutative) If $({\ddot{K}}_{1}^{'}, {\ddot{K_{2}}}^{'}, . . ., {\ddot{K}}_{n}^{'})$ be different alteration of $(\ddot{K_{1}}, \ddot{K_{2}}, . . ., {\ddot{K}}_{n})$ , then $Φ (\ddot{K_{1}}, \ddot{K_{2}}, . . ., {\ddot{K}}_{n}) = Φ ({\ddot{K}}_{1}^{'}, {\ddot{K_{2}}}^{'}, . . ., {\ddot{K}}_{n}^{'})$

(Monotonic) Let $\ddot{K_{ɛ}}$ and $\ddot{K_{ɛ}^{'}}$ be two families of CBFEs then for $\ddot{K_{ɛ}} \leq \ddot{K_{ɛ}^{'}}$ , we get $Φ (\ddot{K_{1}}, \ddot{K_{2}}, . . ., {\ddot{K}}_{n}) \leq Φ ({\ddot{K}}_{1}^{'}, {\ddot{K_{2}}}^{'}, . . ., {\ddot{K}}_{n}^{'})$

Proof. The proofs are straightforward. □

Lemma 3.6 Let $\ddot{K_{ɛ}} = {< {[δ_{\ddot{K} ℓ_{ɛ}}^{P}, δ_{\ddot{K} u_{ɛ}}^{P}], [δ_{\ddot{K} ℓ_{ɛ}}^{N},$ $δ_{\ddot{K} u_{ɛ}}^{N}]}, {ρ_{{\ddot{K}}_{ɛ}}^{P}, ρ_{{\ddot{K}}_{ɛ}}^{N}} >}$ , for {ɛ = 1, 2, . . . , n}, be a family of CBFEs. Then their aggregating value calculated by Φ operator is a CBFE.

4 CBFSs application to multi-criteria group decision making using averaging aggregation operators

The multi-attribute group decision making (also called conjoint decision making with a collection of alternatives) is a process in which a panel cooperatively take a decision from a collection of different alternatives. The choice can not be further attributable to a single member of the panel. This is so because in such processes the whole panel and related social aspects contribute in final choice. The final results obtained by groups are clearly strong and correct as compare to obtained by a single person. This type of decision making is more useful when the problems involve more uncertainties and ambiguities. More precisely, a collective decision making has extreme tendency to manifest an inclination towards discussing shared material. The role of bipolarity is also important in decision making process, which involve uncertain bipolar type information. Here we use a collection of averaging aggregation operators on cubic bipolar fuzzy set to tackle a multi-attribute group decision making problem.

4.1 Proposed technique

In this subsection, we introduce the technique of cubic bipolar fuzzy geometric aggregation operators to solve a multi-criteria group decision making problem under the environment of cubic bipolar fuzzy data. In this terminology the following notions are included.

Step1 : Let $X = {ς_{1}, ς_{2}, . . ., ς_{m}}$ be the set of alternatives $Y = {{\dot{y}}_{1}, {\dot{y}}_{2}, . . ., {\dot{y}}_{k}}$ be the set of criterions or attribute and and $Z = {{\dot{z}}_{1}, {\dot{z}}_{2}, . . ., {\dot{z}}_{n}}$ be the set of experts and $\dot{ϒ} = {\dot{w_{1}}, \dot{w_{2}}, . . ., \dot{w_{n}}}^{T}$ be the weight vector associated with the set of experts under the condition that $0 \leq w \leq 1$ and $\underset{ɛ = 1}{\overset{n}{Σ}} w_{ɛ} = 1$ .

Step2 : We further assume that $M = [b_{ı ȷ}]_{m \times k}$ , for ı = {1, 2, . . . m} and ȷ = {1, 2, . . . k}, be decision matrices provided by decision makers, where each $b_{ı ȷ} = {{[δ_{ℓ_{ı ȷ}}^{P}, δ_{u_{ı ȷ}}^{P}], [δ_{ℓ_{ı ȷ}}^{N}, δ_{u_{ı ȷ}}^{N}]}, {ρ_{ℓ_{ı ȷ}}^{P}, ρ_{u_{ı ȷ}}^{N}}}$ is CBFE. These CBFEs are specified by the set of alternatives with respect to the set of criterions. In decision terms, we mostly consider two main criterions including, benefit and cost. In multi-criteria decision making, the greatest value of benefit attribute and lower value of cost attribute leads us to success.

Step3 : The value of loss attribute case can be converted into value of benefit attribute by normalizing $M_{ɛ} = [b_{ı ȷ}^{ɛ}]_{n \times m \times k}$ which we can obtain by ${M^{'}}_{ɛ} = [{b^{'}}_{ı ȷ}^{ɛ}]_{n \times m \times k}$ where, ${b^{'}}_{ı ȷ}^{ɛ} = {\begin{matrix} (b_{ı ȷ}^{ɛ}), & for value of benefit attribute \\ (b_{ı ȷ}^{ɛ})^{c}, & for value of cost attribute \end{matrix}$ and $\begin{matrix} (b_{ı ȷ}^{ɛ})^{c} = & {\begin{matrix} {\begin{matrix} [\begin{matrix} 1 - δ_{u_{ı ȷ}}^{ɛ P}, \\ 1 - δ_{ℓ_{ı ȷ}}^{ɛ P} \end{matrix}], [\begin{matrix} - 1 - δ_{u_{ı ȷ}}^{ɛ N}, \\ - 1 - δ_{ℓ_{ı ȷ}}^{ɛ N} \end{matrix}] \end{matrix}}, {\begin{matrix} 1 - ρ_{ı ȷ}^{ɛ P}, \\ - 1 - ρ_{ı ȷ}^{ɛ N} \end{matrix}} \end{matrix}} \end{matrix}$

Step4 : Apply proposed averaging aggregation operator to ${M^{'}}_{ɛ} = [{b^{'}}_{ı ȷ}^{ɛ}]_{n \times m \times k}$ until we obtain aggregating column matrix

Step5 : Calculate $S_{P} (\ddot{K}) = \frac{[δ_{\ddot{K} ℓ}^{P} (ς) + δ_{\ddot{K} u}^{P} (ς)] + [1 + δ_{\ddot{K} ℓ}^{N} (ς) + 1 + δ_{\ddot{K} u}^{N} (ς)] - ρ_{\ddot{K}}^{P} (ς) + 1 + ρ_{\ddot{K}}^{N} (ς)}{6}$ or $S_{R} (\ddot{K}) = \frac{[δ_{\ddot{K} ℓ}^{P} (ς) + δ_{\ddot{K} u}^{P} (ς)] + [1 + δ_{\ddot{K} ℓ}^{N} (ς) + 1 + δ_{\ddot{K} u}^{N} (ς)] + ρ P \ddot{K} (ς) - (1 + ρ N \ddot{K} (ς))}{6}$

Step6 : Rank the alternatives.

The effectiveness of technique can be seen by the application given below.

4.2 Computative example of averaging aggregation operators in multi-attribute group decision making problem

Suppose that four business partners want to initiate a new business. They have three preferences as alternatives including, ς₁ = land business, ς₂ = construction business and ς₃ = transport business before them. Obviously, they want a business with more profit and lesser cost and loss. Three business experts ${{\dot{z}}_{1}, {\dot{z}}_{2}, {\dot{z}}_{3}}$ analyze all related factors and conditions to examine all three terms. The experts consider three parameters or attributes including, $\dot{y_{1}} =$ reasonable profit, ${\dot{y}}_{2} =$ business profile and ${\dot{y}}_{3} =$ risk factors to examine all three alternatives. The experts associate positive membership degree with benefits of a particular alternative whereas negative membership degrees associate with flaws of that alternative with respect to each attribute. The experts $\dot{z_{ɛ}}$ construct three decision matrices(see Table 1), for alternatives ς_ı with respect to attributes ${\dot{y}}_{ȷ}$ , where ɛ = 1, 2, 3, ı=1, 2, 3 and ȷ=1, 2, 3.

Since $\dot{y_{1}}$ and ${\dot{y}}_{2}$ are attributes of benefits whereas ${\dot{y}}_{3}$ is only cost attribute so after normalizing we obtain the matrices(see Table 2).

Table 1
$M_{ɛ} = [b_{ı ȷ}^{ɛ}]_{n \times m \times k}$ collection of three decision matrices

$X$ vs $Y$ $\dot{y_{1}}$ ${\dot{y}}_{2}$ ${\dot{y}}_{3}$

ς ₁ {, [-0.29, - 0.18] , {0.22, - 0.13}} {[0.14, 0.24] , [-0.25, - 0.15] , {0.26, - 0.30}} {[0.81, 0.90] , [-0.87, - 0.71] , {0.87, - 0.90}}

ς ₂ {[0.43, 0.54] , [-0.53, - 0.40] , {0.51, - 0.53}} {[0.45, 0.54] , [-0.43, - 0.40] , {0.54, - 0.56}} {[0.60, 0.61] , [-0.54, - 0.46] , {0.54, - 0.61}}

ς ₃ {[0.81, 0.92] , [-0.87, - 0.78] , {0.76, - 0.89}} {[0.76, 0.82] , [-0.84, - 0.79] , {0.83, - 0.71}} {[0.24, 0.27] , [-0.29, - 0.19] , {0.22, - 0.33}}

ς ₁ {[0.12, 0.23] , [-0.25, - 0.20] , {0.16, - 0.19}} {[0.10, 0.29] , [-0.27, - 0.11] , {0.25, - 0.18}} {[0.83, 0.87] , [-0.76, - 0.70] , {0.85, - 0.76}}

ς ₂ {[0.54, 0.58] , [-0.42, - 0.35] , {0.28, - 0.40}} {[0.54, 0.58] , [-0.43, - 0.40] , {0.58, - 0.53}} {[0.52, 0.59] , [-0.65, - 0.60] , {0.56, - 0.59}}

ς ₃ {[0.75, 0.79] , [-0.83, - 0.78] , {0.76, - 0.80}} {[0.82, 0.93] , [-0.87, - 0.72] , {0.73, - 0.74}} {[0.17, 0.27] , [-0.30, - 0.25] , {0.21, - 0.35}}

ς ₁ {[0.20, 0.27] , [-0.30, - 0.23] , {0.26, - 0.18}} {[0.24, 0.32] , [-0.26, - 0.15] , {0.28, - 0.16}} {[0.71, 0.78] , [-0.82, - 0.80] , {0.26, - 0.32}}

ς ₂ {[0.46, 0.49] , [-0.56, - 0.46] , {0.57, - 0.65}} {[0.34, 0.44] , [-0.54, - 0.36] , {0.54, - 0.32}} {[0.52, 0.54] , [-0.65, - 0.60] , {0.57, - 0.63}}

ς ₃ {[0.76, 0.80] , [-0.82, - 0.72] , {0.72, - 0.89}} {[0.77, 0.88] , [-0.82, - 0.73] , {0.71, - 0.79}} {[0.23, 0.27] , [-0.30, - 0.27] , {0.18, - 0.22}}

$X$ vs $Y$	$\dot{y_{1}}$	${\dot{y}}_{2}$	${\dot{y}}_{3}$
ς ₁	{, [-0.29, - 0.18] , {0.22, - 0.13}}	{[0.14, 0.24] , [-0.25, - 0.15] , {0.26, - 0.30}}	{[0.81, 0.90] , [-0.87, - 0.71] , {0.87, - 0.90}}
ς ₂	{[0.43, 0.54] , [-0.53, - 0.40] , {0.51, - 0.53}}	{[0.45, 0.54] , [-0.43, - 0.40] , {0.54, - 0.56}}	{[0.60, 0.61] , [-0.54, - 0.46] , {0.54, - 0.61}}
ς ₃	{[0.81, 0.92] , [-0.87, - 0.78] , {0.76, - 0.89}}	{[0.76, 0.82] , [-0.84, - 0.79] , {0.83, - 0.71}}	{[0.24, 0.27] , [-0.29, - 0.19] , {0.22, - 0.33}}
ς ₁	{[0.12, 0.23] , [-0.25, - 0.20] , {0.16, - 0.19}}	{[0.10, 0.29] , [-0.27, - 0.11] , {0.25, - 0.18}}	{[0.83, 0.87] , [-0.76, - 0.70] , {0.85, - 0.76}}
ς ₂	{[0.54, 0.58] , [-0.42, - 0.35] , {0.28, - 0.40}}	{[0.54, 0.58] , [-0.43, - 0.40] , {0.58, - 0.53}}	{[0.52, 0.59] , [-0.65, - 0.60] , {0.56, - 0.59}}
ς ₃	{[0.75, 0.79] , [-0.83, - 0.78] , {0.76, - 0.80}}	{[0.82, 0.93] , [-0.87, - 0.72] , {0.73, - 0.74}}	{[0.17, 0.27] , [-0.30, - 0.25] , {0.21, - 0.35}}
ς ₁	{[0.20, 0.27] , [-0.30, - 0.23] , {0.26, - 0.18}}	{[0.24, 0.32] , [-0.26, - 0.15] , {0.28, - 0.16}}	{[0.71, 0.78] , [-0.82, - 0.80] , {0.26, - 0.32}}
ς ₂	{[0.46, 0.49] , [-0.56, - 0.46] , {0.57, - 0.65}}	{[0.34, 0.44] , [-0.54, - 0.36] , {0.54, - 0.32}}	{[0.52, 0.54] , [-0.65, - 0.60] , {0.57, - 0.63}}
ς ₃	{[0.76, 0.80] , [-0.82, - 0.72] , {0.72, - 0.89}}	{[0.77, 0.88] , [-0.82, - 0.73] , {0.71, - 0.79}}	{[0.23, 0.27] , [-0.30, - 0.27] , {0.18, - 0.22}}

Table 2

${M^{'}}_{ɛ} = [{b^{'}}_{ı ȷ}^{ɛ}]_{n \times m \times k}$ are matrices after normalizing the decision matrices

$X$ vs $Y$	$\dot{y_{1}}$	${\dot{y}}_{2}$	${\dot{y}}_{3}$
ς ₁	{, [-0.29, - 0.18] , {0.22, - 0.13}}	{[0.14, 0.24] , [-0.25, - 0.15] , {0.26, - 0.30}}	{[0.10, 0.19] , [-0.29, - 0.13] , {0.13, - 0.10}}
ς ₂	{[0.43, 0.54] , [-0.53, - 0.40] , {0.51, - 0.53}}	{[0.45, 0.54] , [-0.43, - 0.40] , {0.54, - 0.56}}	{[0.39, 0.40] , [-0.54, - 0.46] , {0.46, - 0.39}}
ς ₃	{[0.81, 0.92] , [-0.87, - 0.78] , {0.76, - 0.89}}	{[0.76, 0.82] , [-0.84, - 0.79] , {0.83, - 0.71}}	{[0.73, 0.76] , [-0.81, - 0.71] , {0.78, - 0.67}}
ς ₁	{[0.12, 0.23] , [-0.25, - 0.20] , {0.16, - 0.19}}	{[0.10, 0.29] , [-0.27, - 0.11] , {0.25, - 0.18}}	{[0.13, 0.17] , [-0.30, - 0.24] , {0.15, - 0.24}}
ς ₂	{[0.54, 0.58] , [-0.42, - 0.35] , {0.28, - 0.40}}	{[0.54, 0.58] , [-0.43, - 0.40] , {0.58, - 0.53}}	{[0.41, 0.48] , [-0.40, - 0.35] , {0.44, - 0.41}}
ς ₃	{[0.75, 0.79] , [-0.83, - 0.78] , {0.76, - 0.80}}	{[0.82, 0.93] , [-0.87, - 0.72] , {0.73, - 0.74}}	{[0.73, 0.83] , [-0.75, - 0.70] , {0.79, - 0.65}}
ς ₁	{[0.20, 0.27] , [-0.30, - 0.23] , {0.26, - 0.18}}	{[0.24, 0.32] , [-0.26, - 0.15] , {0.28, - 0.16}}	{[0.22, 0.29] , [-0.20, - 0.18] , {0.74, - 0.68}}
ς ₂	{[0.46, 0.49] , [-0.56, - 0.46] , {0.57, - 0.65}}	{[0.34, 0.44] , [-0.54, - 0.36] , {0.54, - 0.32}}	{[0.46, 0.48] , [-0.40, - 0.35] , {0.43, - 0.37}}
ς ₃	{[0.76, 0.80] , [-0.82, - 0.72] , {0.72, - 0.89}}	{[0.77, 0.88] , [-0.82, - 0.73] , {0.71, - 0.79}}	{[0.73, 0.77] , [-0.73, - 0.70] , {0.82, - 0.78}}

Now we apply $P$ -CBFWA operator with weight vector ϒ = {0.5, 0.3, 0.2} ^T on normalized matrices to find their aggregating matrix(see Table 3), for ɛ = 1, 2, 3 and ı=1, 2, 3 $P - CBFWA ({M^{'}}_{ɛ}) = {{[1 - \underset{ȷ = 1}{\overset{3}{Π}} (1 - δ_{ℓ_{ı ȷ}}^{ɛ P})^{w_{ȷ}}, 1 - \underset{ȷ = 1}{\overset{3}{Π}} (1 - δ_{u_{ı ȷ}}^{ɛ P})^{w_{ȷ}}], [- \underset{ȷ = 1}{\overset{3}{Π}} (- δ_{ℓ_{ı ȷ}}^{ɛ N})^{w_{ȷ}}, - \underset{ȷ = 1}{\overset{3}{Π}} (- δ_{u_{ı ȷ}}^{ɛ N})^{w_{ȷ}}]}, {1 - \underset{ȷ = 1}{\overset{3}{Π}} (1 - ρ_{ı ȷ}^{ɛ P})^{w_{ȷ}}, - \underset{ȷ = 1}{\overset{3}{Π}} (- ρ_{ı ȷ}^{ɛ N})^{w_{ȷ}}}}$

Table 3

$M^{″} = [{b^{″}}_{ı}^{ɛ}]_{n \times m}$ is matrix after applying $P$ -CBFWA operator on normalized matrices

$X$ vs $Z$	${\dot{z}}_{1}$	${\dot{z}}_{2}$	${\dot{z}}_{3}$
ς ₁	{[0.15, 0.23] , [-0.28, - 0.16] , {0.22, - 0.16}}	{[0.12, 0.24] , [-0.27, - 0.17] , {0.19, - 0.20}}	{[0.22, 0.29] , [-0.27, - 0.19] , {0.40, - 0.23}}
ς ₂	{, [-0.50, - 0.41] , {0.51, - 0.51}}	{[0.52, 0.56] , [-0.42, - 0.41] , {0.42, - 0.44}}	{[0.43, 0.47] , [-0.52, - 0.40] , {0.54, - 0.47}}
ς ₃	{[0.78, 0.87] , [-0.85, - 0.76] , {0.79, - 0.79}}	{[0.77, 0.86] , [-0.82, - 0.75] , {0.76, - 0.75}}	{[0.76, 0.82] , [-0.80, - 0.72] , {0.74, - 0.84}}

we further aggregate ${M^{″}}_{ɛ}$ by using again $P$ -CBFWA operator with weight vector ϒ′ = {0.2, 0.3, 0.5} ^T, thus for ı=1, 2, 3 we have $P - CBFWA ({M^{″}}_{ɛ}) = {{[1 - \underset{ɛ = 1}{\overset{3}{Π}} (1 - δ_{ℓ_{ı}}^{ɛ P})^{w_{ɛ}}, 1 - \underset{ɛ = 1}{\overset{3}{Π}} (1 - δ_{u_{ı}}^{ɛ P})^{w_{ɛ}}], [- \underset{ɛ = 1}{\overset{3}{Π}} (- δ_{ℓ_{ı}}^{ɛ N})^{w_{ɛ}}, - \underset{ɛ = 1}{\overset{3}{Π}} (- δ_{u_{ı}}^{ɛ N})^{w_{ɛ}}]}, {1 - \underset{ɛ = 1}{\overset{3}{Π}} (1 - ρ_{ı}^{ɛ P})^{w_{ɛ}}, - \underset{ɛ = 1}{\overset{3}{Π}} (- ρ_{ı}^{ɛ N})^{w_{ɛ}}}}$ . We obtain the averaging aggregating column matrix $M ″' = [b ″' ı]_{m}$ $\begin{matrix} M ″' = & [\begin{matrix} ς_{1} = {[0.177, 0.263], [- 0.271, - 0.177], {0.308, - 0.205}} \\ ς_{2} = {[0.458, 0.506], [- 0.483, - 0.404], {0.500, - 0.468}} \\ ς_{3} = {[0.767, 0.843], [- 0.815, - 0.736], {0.756, - 0.802}} \end{matrix}] \end{matrix}$ (2) now calculate score function for ranking of alternatives. $S_{P} (b ″' ı) = \frac{[δ_{ℓ_{ı}}^{P} + δ_{u_{ı}}^{P}] + [1 - δ_{ℓ_{ı}}^{N} + 1 - δ_{u_{ı}}^{N}] - ρ_{ı}^{P} + 1 - ρ_{ı}^{N}}{6}$ we get $S_{P} (ς_{1}) = 0.413$ , $S_{P} (ς_{2}) = 0.351$ and $S_{P} (ς_{3}) = 0.250$ . Thus we obtain the ranking $S (ς_{1}) ≻ S_{P} (ς_{2}) ≻ S_{P} (ς_{3})$ this implies that

ς₁ ≻ ς₂ ≻ ς₃. The best alternative is ς₁. Now we apply $R$ -CBFWA operator with weight vector ϒ = {0.5, 0.3, 0.2} ^T on normalized matrices to find their aggregating matrix for ɛ = 1, 2, 3 and ı=1, 2, 3 $R - CBFWA ({M^{'}}_{ɛ}) = {{[1 - \underset{ȷ = 1}{\overset{3}{Π}} (1 - δ_{ℓ_{ı ȷ}}^{ɛ P})^{w_{ȷ}}, 1 - \underset{ȷ = 1}{\overset{3}{Π}} (1 - δ_{u_{ı ȷ}}^{ɛ P})^{w_{ȷ}}], [- \underset{ȷ = 1}{\overset{3}{Π}} (- δ_{ℓ_{ı ȷ}}^{ɛ N})^{w_{ȷ}}, - \underset{ȷ = 1}{\overset{3}{Π}} (- δ_{u_{ı ȷ}}^{ɛ N})^{w_{ȷ}}]}, {\underset{ȷ = 1}{\overset{3}{Π}} (ρ_{ı ȷ}^{ɛ P})^{w_{ȷ}}, - (1 - \underset{ȷ = 1}{\overset{3}{Π}} (1 - (- ρ_{ı ȷ}^{ɛ N}))^{w_{ȷ}})}}$ Similarly, we compute the aggregating matrix (see Table 4).

Fig.1

Flow chart of the whole technique

Table 4

$M^{″} = [{b^{″}}_{ı}^{ɛ}]_{n \times m}$ is matrix after applying $R$ -CBFWA operator on normalized matrices

$X$ vs $Z$	${\dot{z}}_{1}$	${\dot{z}}_{2}$	${\dot{z}}_{3}$
ς ₁	{[0.15, 0.23] , [-0.28, - 0.16] , {0.21, - 0.18}}	{[0.12, 0.24] , [-0.27, - 0.17] , {0.18, - 0.20}}	{[0.22, 0.29] , [-0.27, - 0.19] , {0.33, - 0.32}}
ς ₂	{, [-0.50, - 0.41] , {0.51, - 0.51}}	{[0.52, 0.56] , [-0.42, - 0.41] , {0.38, - 0.44}}	{[0.43, 0.47] , [-0.52, - 0.40] , {0.53, - 0.52}}
ς ₃	{[0.78, 0.87] , [-0.85, - 0.76] , {0.78, - 0.82}}	{[0.77, 0.86] , [-0.82, - 0.75] , {0.76, - 0.76}}	{[0.76, 0.82] , [-0.80, - 0.72] , {0.73, - 0.85}}

We further aggregate ${M^{″}}_{ɛ}$ by using again $R$ -CBFWA operator with weight vector ϒ′ = {0.2, 0.3, 0.5} ^T, thus for ı=1, 2, 3 we have $R - CBFWA ({M^{″}}_{ɛ}) = {{[1 - \underset{ɛ = 1}{\overset{3}{Π}} (1 - δ_{ℓ_{ı}}^{ɛ P})^{w_{ɛ}}, 1 - \underset{ɛ = 1}{\overset{3}{Π}} (1 - δ_{u_{ı}}^{ɛ P})^{w_{ɛ}}], [- \underset{ɛ = 1}{\overset{3}{Π}} (- δ_{ℓ_{ı}}^{ɛ N})^{w_{ɛ}}, - \underset{ɛ = 1}{\overset{3}{Π}} (- δ_{u_{ı}}^{ɛ N})^{w_{ɛ}}]}, {\underset{ɛ = 1}{\overset{3}{Π}} (ρ_{ı}^{ɛ P})^{w_{ɛ}}, - (1 - \underset{ɛ = 1}{\overset{3}{Π}} (1 - (- ρ_{ı}^{ɛ N}))^{w_{ɛ}})}}$ we obtain the averaging aggregating column matrix $M ″' = [b ″' ı]_{m}$ $\begin{matrix} M ″' = & [\begin{matrix} ς_{1} = {[0.177, 0.263], [- 0.271, - 0.177], {0.251, - 0.258}} \\ ς_{2} = {[0.458, 0.506], [- 0.483, - 0.404], {0.475, - 0.495}} \\ ς_{3} = {[0.767, 0.843], [- 0.815, - 0.736], {0.748, - 0.820}} \end{matrix}] \end{matrix}$ now calculate score function for ranking of alternatives. $S_{R} (b ″' ı) = \frac{[δ_{ℓ_{ı}}^{P} + δ_{u_{ı}}^{P}] + [1 - δ_{ℓ_{ı}}^{N} + 1 - δ_{u_{ı}}^{N}] + ρ_{ı}^{P} - (1 - ρ_{ı}^{N})}{6}$ we get $S_{R} (ς_{1}) = 0.250$ , $S_{R} (ς_{2}) = 0.341$ and $S_{R} (ς_{3}) = 0.437$ . Thus we obtain the ranking $S_{R} (ς_{3}) ≻ S_{R} (ς_{2}) ≻ S_{R} (ς_{1})$ this implies that

ς₃ ≻ ς₂ ≻ ς₁. The best optimal choice is ς₃. The rankings of alternatives by both operators can be seen in Figs. 3 and 4.

Fig.2

Algorithm A. for application of $P$ -CBFWA and $R$ -CBFWA operators

Fig.3

Ranking of alternatives using $P$ -CBFWA

Fig.4

Ranking of alternatives using $R$ -CBFWA

Fig.5

Algorithm B. for application of $P$ -CBFOWA and $R$ -CBFOWA operators

We apply $P$ -CBFOWA operator to examine the best alternative. First we ordered each alternative in all normalized matrices. We calculate score function to obtain the matrices with ordered alternatives(see Table 5). Consider $S_{P} ({b^{'}}_{ı ȷ}^{ɛ}) = \frac{[δ_{ℓ_{ı}}^{P} + δ_{u_{ı}}^{P}] + [1 - δ_{ℓ_{ı}}^{N} + 1 - δ_{u_{ı}}^{N}] - ρ_{ı}^{P} + (1 - ρ_{ı}^{N})}{6}$ $S_{P} ({b^{'}}_{11}^{1}) = 0.43$ , $S_{P} ({b^{'}}_{12}^{1}) = 0.40$ , $S_{P} ({b^{'}}_{13}^{1}) = 0.44$ , $S_{P} ({b^{'}}_{21}^{1}) = 0.33$ , $S_{P} ({b^{'}}_{22}^{1}) = 0.68$ , $S_{P} ({b^{'}}_{23}^{1}) = 0.32$ , $S_{P} ({b^{'}}_{31}^{1}) = 0.238$ , $S_{P} ({b^{'}}_{32}^{1}) = 0.235$ , $S_{P} ({b^{'}}_{33}^{1}) = 0.253$ , $S_{P} ({b^{'}}_{11}^{2}) = 0.42$ , $S_{P} ({b^{'}}_{12}^{2}) = 0.43$ , $S_{P} ({b^{'}}_{13}^{2}) = 0.39$ , $S_{P} ({b^{'}}_{21}^{2}) = 0.44$ , $S_{P} ({b^{'}}_{22}^{2}) = 0.36$ , $S_{P} ({b^{'}}_{23}^{2}) = 0.38$ , $S_{P} ({b^{'}}_{31}^{2}) = 0.22$ , $S_{P} ({b^{'}}_{32}^{2}) = 0.28$ , $S_{P} ({b^{'}}_{33}^{2}) = 0.27$ , $S_{P} ({b^{'}}_{11}^{3}) = 0.41$ , $S_{P} ({b^{'}}_{12}^{3}) = 0.45$ , $S_{P} ({b^{'}}_{13}^{3}) = 0.28$ , $S_{P} ({b^{'}}_{21}^{3}) = 0.28$ , $S_{P} ({b^{'}}_{22}^{3}) = 0.33$ , $S_{P} ({b^{'}}_{23}^{3}) = 0.39$ , $S_{P} ({b^{'}}_{31}^{3}) = 0.23$ , $S_{P} ({b^{'}}_{32}^{3}) = 0.26$ and $S_{P} ({b^{'}}_{33}^{3}) = 0.24$ . $S_{P} ({b^{'}}_{13}^{1}) ≻ S_{P} ({b^{'}}_{11}^{1}) ≻ S_{P} ({b^{'}}_{12}^{1}) \Rightarrow {b^{'}}_{13}^{1} ≻ {b^{'}}_{11}^{1} ≻ {b^{'}}_{12}^{1}, S_{P} ({b^{'}}_{22}^{1}) ≻ S_{P} ({b^{'}}_{21}^{1}) ≻ S_{P} ({b^{'}}_{23}^{1}) \Rightarrow {b^{'}}_{22}^{1} ≻ {b^{'}}_{21}^{1} ≻ {b^{'}}_{23}^{1}$ , $S_{P} ({b^{'}}_{33}^{1}) ≻ S_{P} ({b^{'}}_{31}^{1}) ≻ S_{P} ({b^{'}}_{32}^{1}) \Rightarrow {b^{'}}_{33}^{1} ≻ {b^{'}}_{31}^{1} ≻ {b^{'}}_{32}^{1}$ , $S_{P} ({b^{'}}_{12}^{2}) ≻ S_{P} ({b^{'}}_{11}^{2}) ≻ S_{P} ({b^{'}}_{13}^{2}) \Rightarrow {b^{'}}_{12}^{2} ≻ {b^{'}}_{11}^{2} ≻ {b^{'}}_{13}^{2}, S_{P} ({b^{'}}_{21}^{2}) ≻ S_{P} ({b^{'}}_{23}^{2}) ≻ S_{P} ({b^{'}}_{22}^{2}) \Rightarrow {b^{'}}_{21}^{2} ≻ {b^{'}}_{23}^{2} ≻ {b^{'}}_{22}^{2}$ , $S_{P} ({b^{'}}_{32}^{2}) ≻ S_{P} ({b^{'}}_{33}^{2}) ≻ S_{P} ({b^{'}}_{31}^{2}) \Rightarrow {b^{'}}_{32}^{2} ≻ {b^{'}}_{33}^{2} ≻ {b^{'}}_{31}^{2}$ , $S_{P} ({b^{'}}_{12}^{3}) ≻ S_{P} ({b^{'}}_{11}^{3}) ≻ S_{P} ({b^{'}}_{13}^{3}) \Rightarrow {b^{'}}_{12}^{3} ≻ {b^{'}}_{11}^{3} ≻ {b^{'}}_{13}^{3}, S_{P} ({b^{'}}_{23}^{3}) ≻ S_{P} ({b^{'}}_{22}^{3}) ≻ S_{P} ({b^{'}}_{21}^{3}) \Rightarrow {b^{'}}_{23}^{3} ≻ {b^{'}}_{22}^{3} ≻ {b^{'}}_{21}^{3}$ , $S_{P} ({b^{'}}_{32}^{3}) ≻ S_{P} ({b^{'}}_{33}^{3}) ≻ S_{P} ({b^{'}}_{31}^{3}) \Rightarrow {b^{'}}_{32}^{3} ≻ {b^{'}}_{33}^{3} ≻ {b^{'}}_{31}^{3}$

Table 5

${M^{″}}_{ɛ} = [{b^{″}}_{ı ȷ}^{ɛ}]_{n \times m \times k}$ are matrices with ordered alternatives

$X$ vs $Y$	$\dot{y_{1}}$	${\dot{y}}_{2}$	${\dot{y}}_{3}$
ς ₁	{, [-0.29, - 0.13] , {0.13, - 0.10}}	{[0.17, 0.23] , [-0.29, - 0.18] , {0.22, - 0.13}}	{[0.14, 0.24] , [-0.25, - 0.15] , {0.26, - 0.30}}
ς ₂	{[0.45, 0.54] , [-0.43, - 0.40] , {0.54, - 0.56}}	{[0.43, 0.54] , [-0.53, - 0.40] , {0.51, - 0.53}}	{[0.39, 0.40] , [-0.54, - 0.46] , {0.46, - 0.39}}
ς ₃	{[0.73, 0.76] , [-0.81, - 0.71] , {0.78, - 0.67}}	{[0.81, 0.92] , [-0.87, - 0.78] , {0.76, - 0.89}}	{[0.76, 0.82] , [-0.84, - 0.79] , {0.83, - 0.71}}
ς ₁	{[0.10, 0.29] , [-0.27, - 0.11] , {0.25, - 0.18}}	{[0.12, 0.23] , [-0.25, - 0.20] , {0.16, - 0.19}}	{[0.13, 0.17] , [-0.30, - 0.24] , {0.15, - 0.24}}
ς ₂	{[0.54, 0.58] , [-0.42, - 0.35] , {0.28, - 0.40}}	{[0.41, 0.48] , [-0.40, - 0.35] , {0.44, - 0.41}}	{[0.54, 0.58] , [-0.43, - 0.40] , {0.58, - 0.53}}
ς ₃	{[0.82, 0.93] , [-0.87, - 0.72] , {0.73, - 0.74}}	{[0.73, 0.83] , [-0.75, - 0.70] , {0.79, - 0.65}}	{[0.75, 0.79] , [-0.83, - 0.78] , {0.76, - 0.80}}
ς ₁	{[0.24, 0.32] , [-0.26, - 0.15] , {0.28, - 0.16}}	{[0.20, 0.27] , [-0.30, - 0.23] , {0.26, - 0.18}}	{[0.22, 0.29] , [-0.20, - 0.18] , {0.74, - 0.68}}
ς ₂	{[0.46, 0.48] , [-0.40, - 0.35] , {0.43, - 0.37}}	{[0.34, 0.44] , [-0.54, - 0.36] , {0.54, - 0.32}}	{[0.46, 0.49] , [-0.56, - 0.46] , {0.57, - 0.65}}
ς ₃	{[0.77, 0.88] , [-0.82, - 0.73] , {0.71, - 0.79}}	{[0.73, 0.77] , [-0.73, - 0.70] , {0.82, - 0.78}}	{[0.76, 0.80] , [-0.82, - 0.72] , {0.72, - 0.89}}

now we use $P$ -CBFOWA operator with weight vector ϒ = {0.4, 0.3, 0.3} ^T on ordered alternatives matrices to find an aggregating matrix for ɛ = 1, 2, 3 and ı=1, 2, 3 $P - CBFOWA ({M^{″}}_{ɛ}) = {{[1 - \underset{ȷ = 1}{\overset{3}{Π}} (1 - δ_{ℓ_{ı ȷ}}^{ɛ P})^{w_{ȷ}}, 1 - \underset{ȷ = 1}{\overset{3}{Π}} (1 - δ_{u_{ı ȷ}}^{ɛ P})^{w_{ȷ}}], [- \underset{ȷ = 1}{\overset{3}{Π}} (- δ_{ℓ_{ı ȷ}}^{ɛ N})^{w_{ȷ}}, - \underset{ȷ = 1}{\overset{3}{Π}} (- δ_{u_{ı ȷ}}^{ɛ N})^{w_{ȷ}}]}, {1 - \underset{ȷ = 1}{\overset{3}{Π}} (1 - ρ_{ı ȷ}^{ɛ P})^{w_{ȷ}}, - \underset{ȷ = 1}{\overset{3}{Π}} (- ρ_{ı ȷ}^{ɛ N})^{w_{ȷ}}}}$ after computing $P$ -CBFOWA operator we obtain aggregated matrix $M ″' ɛ = [{b ″'}_{ı}^{ɛ}]_{n \times m}$ (see Table 6).

Table 6

$M ″' = [{b ″'}_{ı}^{ɛ}]_{n \times m}$ is matrix after applying $P$ -CBFOWA operator on ordered alternatives matrices

$X$ vs $Z$	${\dot{z}}_{1}$	${\dot{z}}_{2}$	${\dot{z}}_{3}$
ς ₁	{[0.13, 0.22] , [-0.28, - 0.15] , {0.20, - 0.15}}	{[0.12, 0.24] , [-0.27, - 0.17] , {0.19, - 0.20}}	{[0.22, 0.30] , [-0.25, - 0.18] , {0.47, - 0.26}}
ς ₂	{, [-0.49, - 0.42] , {0.50, - 0.50}}	{[0.50, 0.55] , [-0.42, - 0.36] , {0.43, - 0.44}}	{[0.43, 0.47] , [-0.48, - 0.38] , {0.50, - 0.42}}
ς ₃	{[0.76, 0.84] , [-0.84, - 0.75] , {0.79, - 0.74}}	{[0.78, 0.87] , [-0.82, - 0.73] , {0.76, - 0.73}}	{[0.76, 0.83] , [-0.79, - 0.72] , {0.75, - 0.82}}

Now aggregate $M ″' ɛ$ by using again $P$ -CBFWA operator with weight vector ϒ′ = {0.3, 0.3, 0.4} ^T, thus for ı=1, 2, 3 we have $P - CBFWA (M ″' ɛ) = {{[1 - \underset{ɛ = 1}{\overset{3}{Π}} (1 - δ_{ℓ_{ı}}^{ɛ P})^{w_{ɛ}}, 1 - \underset{ɛ = 1}{\overset{3}{Π}} (1 - δ_{u_{ı}}^{ɛ P})^{w_{ɛ}}], [- \underset{ɛ = 1}{\overset{3}{Π}} (- δ_{ℓ_{ı}}^{ɛ N})^{w_{ɛ}}, - \underset{ɛ = 1}{\overset{3}{Π}} (- δ_{u_{ı}}^{ɛ N})^{w_{ɛ}}]}, {1 - \underset{ɛ = 1}{\overset{3}{Π}} (1 - ρ_{ı}^{ɛ P})^{w_{ɛ}}, - (\underset{ɛ = 1}{\overset{3}{Π}} (- ρ_{ı}^{ɛ N})^{w_{ɛ}})}}$ thus we get the averaging aggregating column matrix $M ″ ″ = [b ″ ″ ı]_{m}$ $\begin{matrix} M ″ ″ = & [\begin{matrix} ς_{1} = {[0.160, 0.260], [- 0.260, - 0.170], {0.320, - 0.200}} \\ ς_{2} = {[0.450, 0.500], [- 0.460, - 0.390], {0.480, - 0.450}} \\ ς_{3} = {[0.770, 0.850], [- 0.810, - 0.730], {0.770, - 0.770}} \end{matrix}] \end{matrix}$ now calculate score function for ranking of alternatives. $S_{P} ({b^{'}}_{ı}) = \frac{[δ_{ℓ_{ı}}^{P} + δ_{u_{ı}}^{P}] + [1 - δ_{ℓ_{ı}}^{N} + 1 - δ_{u_{ı}}^{N}] - ρ_{ı}^{P} + (1 - ρ_{ı}^{N})}{6}$ we get $S_{P} (ς_{1}) = 0.411$ , $S_{P} (ς_{2}) = 0.361$ and $S_{P} (ς_{3}) = 0.256$ . Thus we obtain the ranking $S_{P} (ς_{1}) ≻ S_{P} (ς_{2}) ≻ S_{P} (ς_{3})$ this implies that ς₁ ≻ ς₂ ≻ ς₃. The best optimal choice is ς₁. We apply $R$ -CBFOWA operator to chose the best alternatives. Firstly, we calculate the ordering of the alternatives by computing score function of $R$ -Order. Consider $S_{R} ({b^{'}}_{ı ȷ}^{ɛ}) = \frac{[δ_{ℓ_{ı}}^{P} + δ_{u_{ı}}^{P}] + [1 - δ_{ℓ_{ı}}^{N} + 1 - δ_{u_{ı}}^{N}] + ρ_{ı}^{P} - (1 - ρ_{ı}^{N})}{6}$ we obtain collection of matrices with ordered alternatives (see Table 7).

Table 7

${M^{″}}_{ɛ} = [{b^{″}}_{ı ȷ}^{ɛ}]_{n \times m \times k}$ are matrices with ordered alternatives

$X$ vs $Y$	$\dot{y_{1}}$	${\dot{y}}_{2}$	${\dot{y}}_{3}$
ς ₁	{, [-0.25, - 0.15] , {0.26, - 0.30}}	{[0.17, 0.23] , [-0.29, - 0.18] , {0.22, - 0.13}}	{[0.10, 0.19] , [-0.29, - 0.13] , {0.13, - 0.10}}
ς ₂	{[0.45, 0.54] , [-0.43, - 0.40] , {0.54, - 0.56}}	{[0.43, 0.54] , [-0.53, - 0.40] , {0.51, - 0.53}}	{[0.39, 0.40] , [-0.54, - 0.46] , {0.46, - 0.39}}
ς ₃	{[0.81, 0.92] , [-0.87, - 0.78] , {0.76, - 0.89}}	{[0.76, 0.82] , [-0.84, - 0.79] , {0.83, - 0.71}}	{[0.73, 0.76] , [-0.81, - 0.71] , {0.78, - 0.67}}
ς ₁	{[0.10, 0.29] , [-0.27, - 0.11] , {0.25, - 0.18}}	{[0.12, 0.23] , [-0.25, - 0.20] , {0.16, - 0.19}}	{[0.13, 0.17] , [-0.30, - 0.24] , {0.15, - 0.24}}
ς ₂	{[0.41, 0.48] , [-0.40, - 0.35] , {0.44, - 0.41}}	{[0.54, 0.58] , [-0.42, - 0.35] , {0.28, - 0.40}}	{[0.54, 0.58] , [-0.43, - 0.40] , {0.58, - 0.53}}
ς ₃	{[0.75, 0.79] , [-0.83, - 0.78] , {0.76, - 0.80}}	{[0.82, 0.93] , [-0.87, - 0.72] , {0.73, - 0.74}}	{[0.73, 0.83] , [-0.75, - 0.70] , {0.79, - 0.65}}
ς ₁	{[0.24, 0.32] , [-0.26, - 0.15] , {0.28, - 0.16}}	{[0.22, 0.29] , [-0.20, - 0.18] , {0.74, - 0.68}}	{[0.20, 0.27] , [-0.30, - 0.23] , {0.26, - 0.18}}
ς ₂	{[0.46, 0.49] , [-0.56, - 0.46] , {0.57, - 0.65}}	{[0.46, 0.48] , [-0.40, - 0.35] , {0.43, - 0.37}}	{[0.34, 0.44] , [-0.54, - 0.36] , {0.54, - 0.32}}
ς ₃	{[0.73, 0.77] , [-0.73, - 0.70] , {0.82, - 0.78}}	{[0.76, 0.80] , [-0.82, - 0.72] , {0.72, - 0.89}}	{[0.77, 0.88] , [-0.82, - 0.73] , {0.71, - 0.79}}

After applying $R$ -CBFOWA operator with weight vector ϒ = {0.4, 0.3, 0.3} ^T on ordered alternatives matrices we obtain an aggregated matrix, $M ″' ɛ = [{b ″'}_{ı}^{ɛ}]_{n \times m}$ , for ɛ = 1, 2, 3 and ı=1, 2, 3 (see Table 8).

Table 8

$M ″' = [{b ″'}_{ı}^{ɛ}]_{n \times m}$ is matrix after applying $R$ -CBFOWA operator on ordered alternatives matrices

$X$ vs $Z$	${\dot{z}}_{1}$	${\dot{z}}_{2}$	${\dot{z}}_{3}$
ς ₁	{[0.14, 0.22] , [-0.27, - 0.15] , {0.20, - 0.19}}	{[0.12, 0.24] , [-0.27, - 0.17] , {0.19, - 0.64}}	{[0.22, 0.30] , [-0.25, - 0.18] , {0.37, - 0.38}}
ς ₂	{, [-0.49, - 0.42] , {0.51, - 0.50}}	{[0.49, 0.54] , [-0.41, - 0.36] , {0.42, - 0.45}}	{[0.43, 0.47] , [-0.50, - 0.38] , {0.52, - 0.49}}
ς ₃	{[0.78, 0.86] , [-0.84, - 0.75] , {0.79, - 0.80}}	{[0.77, 0.86] , [-0.82, - 0.74] , {0.76, - 0.74}}	{[0.75, 0.82] , [-0.78, - 0.71] , {0.76, - 0.82}}

Now we get, aggregate $M ″' ɛ$ by using again $R$ -CBFWA operator with weight vector ϒ′ = {0.3, 0.3, 0.4} ^T, thus for ı=1, 2, 3 we obtain averaging aggregating column matrix $M ″ ″ = [b ″ ″ ı]_{m}$ $\begin{matrix} M ″ ″ = & [\begin{matrix} ς_{1} = {[0.160, 0.260], [- 0.260, - 0.170], {0.250, - 0.430}} \\ ς_{2} = {[0.450, 0.500], [- 0.470, - 0.390], {0.480, - 0.480}} \\ ς_{3} = {[0.770, 0.850], [- 0.810, - 0.730], {0.770, - 0.790}} \end{matrix}] \end{matrix}$

Now calculate score function for ranking of alternatives. After computing score function, we get $S_{R} (ς_{1}) = 0.280$ , $S_{R} (ς_{2}) = 0.341$ and $S_{R} (ς_{3}) = 0.440$ . Thus we obtain the ranking $S_{R} (ς_{3}) ≻ S_{R} (ς_{2}) ≻ S_{R} (ς_{1})$ this implies that ς₃ ≻ ς₂ ≻ ς₁. The best optimal choice is ς₃. The rankings of alternatives by both operators can be seen in Figs. 6 and 7.

Fig.6

Ranking of alternatives using $P$ -CBFOWA

Fig.7

Ranking of alternatives using $R$ -CBFOWA

Fig.8

Algorithm C. for application of $P$ -CBFHWA and $R$ -CBFHWA operators

We apply now $P$ -CBFHWA operator on normalized decision matrices. Firstly, we assume weighted vector ϒ = {0.2, 0.6, 0.2} ^T, and compute the entries ${b^{″}}_{ı ȷ}^{ɛ} = n w_{ȷ} {b^{'}}^{ɛ} ı ȷ$ , where $n$ is number of weight vector in ϒ.

${b^{″}}_{11}^{1} = {{[1 - (1 - 0.17)^{0.6}, 1 - (1 - 0.23)^{0.6}], [- (0.29)^{0.6}, - (0.18)^{0.6}]}, {1 - (1 - 0.22)^{0.6}, - (0.13)^{0.6}}}$ ${b^{″}}_{12}^{1} = {{[1 - (1 - 0.14)^{0.6}, 1 - (1 - 0.24)^{0.6}], [- (0.25)^{0.6}, - (0.15)^{0.6}]}, {1 - (1 - 0.26)^{0.6}, - (0.30)^{0.6}}}$ ${b^{″}}_{13}^{1} = {{[1 - (1 - 0.10)^{0.6}, 1 - (1 - 0.19)^{0.6}], [- (0.29)^{0.6}, - (0.13)^{0.6}]}, {1 - (1 - 0.13)^{0.6}, - (0.10)^{0.6}}}$ ${b^{″}}_{21}^{1} = {{[1 - (1 - 0.43)^{0.6}, 1 - (1 - 0.54)^{0.6}], [- (0.53)^{0.6}, - (0.40)^{0.6}]}, {1 - (1 - 0.51)^{0.6}, - (0.53)^{0.6}}}$ ${b^{″}}_{22}^{1} = {{[1 - (1 - 0.45)^{0.6}, 1 - (1 - 0.54)^{0.6}], [- (0.43)^{0.6}, - (0.40)^{0.6}]}, {1 - (1 - 0.54)^{0.6}, - (0.56)^{0.6}}}$ ${b^{″}}_{23}^{1} = {{[1 - (1 - 0.39)^{0.6}, 1 - (1 - 0.40)^{0.6}], [- (0.54)^{0.6}, - (0.46)^{0.6}]}, {1 - (1 - 0.46)^{0.6}, - (0.39)^{0.6}}}$ ${b^{″}}_{31}^{1} = {{[1 - (1 - 0.81)^{0.6}, 1 - (1 - 0.92)^{0.6}], [- (0.87)^{0.6}, - (0.78)^{0.6}]}, {1 - (1 - 0.76)^{0.6}, - (0.89)^{0.6}}}$ ${b^{″}}_{32}^{1} = {{[1 - (1 - 0.76)^{0.6}, 1 - (1 - 0.82)^{0.6}], [- (0.84)^{0.6}, - (0.79)^{0.6}]}, {1 - (1 - 0.83)^{0.6}, - (0.71)^{0.6}}}$ ${b^{″}}_{33}^{1} = {{[1 - (1 - 0.73)^{0.6}, 1 - (1 - 0.76)^{0.6}], [- (0.81)^{0.6}, - (0.71)^{0.6}]}, {1 - (1 - 0.78)^{0.6}, - (0.67)^{0.6}}}$ similarly, we calculate all entries and obtain weighted matrices (see Table 9).

Table 9

${M^{″}}_{ɛ} = [{b^{″}}_{ı ȷ}^{ɛ}]_{n \times m \times k}$ are weighted matrices

$X$ vs $Y$	$\dot{y_{1}}$	${\dot{y}}_{2}$	${\dot{y}}_{3}$
ς ₁	{, [-0.48, - 0.36] , {0.14, - 0.29}}	{[0.87, 0.15] , [-0.44, - 0.32] , {0.17, - 0.49}}	{[0.61, 0.12] , [-0.48, - 0.29] , {0.80, - 0.25}}
ς ₂	{[0.29, 0.37] , [-0.68, - 0.58] , {0.35, - 0.68}}	{[0.30, 0.37] , [-0.60, - 0.58] , {0.37, - 0.71}}	{[0.26, 0.26] , [-0.69, - 0.63] , {0.31, - 0.57}}
ς ₃	{[0.63, 0.78] , [-0.92, - 0.86] , {0.58, - 0.93}}	{[0.58, 0.64] , [-0.90, - 0.87] , {0.65, - 0.81}}	{[0.54, 0.58] , [-0.88, - 0.81] , {0.60, - 0.79}}
ς ₁	{[0.21, 0.38] , [-0.08, - 0.05] , {0.27, - 0.05}}	{[0.17, 0.46] , [-0.09, - 0.02] , {0.40, - 0.05}}	{[0.22, 0.28] , [-0.11, - 0.08] , {0.25, - 0.08}}
ς ₂	{[0.75, 0.79] , [-0.21, - 0.15] , {0.45, - 0.19}}	{[0.75, 0.79] , [-0.22, - 0.19] , {0.79, - 0.32}}	{[0.61, 0.69] , [-0.19, - 0.15] , {0.65, - 0.20}}
ς ₃	{[0.92, 0.94] , [-0.71, - 0.64] , {0.92, - 0.67}}	{[0.95, 0.99] , [-0.78, - 0.55] , {0.91, - 0.58}}	{[0.91, 0.96] , [-0.60, - 0.53] , {0.94, - 0.46}}
ς ₁	{[0.13, 0.17] , [-0.49, - 0.41] , {0.17, - 0.36}}	{[0.15, 0.21] , [-0.45, - 0.32] , {0.18, - 0.33}}	{[0.14, 0.19] , [-0.38, - 0.36] , {0.55, - 0.79}}
ς ₂	{[0.31, 0.33] , [-0.71, - 0.63] , {0.40, - 0.77}}	{[0.22, 0.29] , [-0.69, - 0.54] , {0.37, - 0.50}}	{[0.31, 0.32] , [-0.58, - 0.53] , {0.29, - 0.55}}
ς ₃	{[0.58, 0.62] , [-0.89, - 0.82] , {0.53, - 0.93}}	{[0.59, 0.72] , [-0.89, - 0.83] , {0.52, - 0.87}}	{[0.54, 0.59] , [-0.83, - 0.81] , {0.64, - 0.86}}

Now calculate score function and get the matrices with ordered alternatives (see Table 10).

Table 10

$M ″' ɛ = [{b ″'}_{ı ȷ}^{ɛ}]_{n \times m \times k}$ are matrices with ordered alternatives

$X$ vs $Y$	$\dot{y_{1}}$	${\dot{y}}_{2}$	${\dot{y}}_{3}$
ς ₁	{, [-0.44, - 0.32] , {0.17, - 0.49}}	{[0.11, 0.15] , [-0.48, - 0.36] , {0.14, - 0.29}}	{[0.61, 0.12] , [-0.48, - 0.29] , {0.80, - 0.25}}
ς ₂	{[0.30, 0.37] , [-0.60, - 0.58] , {0.37, - 0.71}}	{[0.29, 0.37] , [-0.68, - 0.58] , {0.35, - 0.68}}	{[0.26, 0.26] , [-0.69, - 0.63] , {0.31, - 0.57}}
ς ₃	{[0.63, 0.78] , [-0.92, - 0.86] , {0.58, - 0.93}}	{[0.54, 0.58] , [-0.88, - 0.81] , {0.60, - 0.79}}	{[0.58, 0.64] , [-0.90, - 0.87] , {0.65, - 0.81}}
ς ₁	{[0.21, 0.38] , [-0.08, - 0.05] , {0.27, - 0.05}}	{[0.17, 0.46] , [-0.09, - 0.02] , {0.40, - 0.05}}	{[0.22, 0.28] , [-0.11, - 0.08] , {0.25, - 0.08}}
ς ₂	{[0.75, 0.79] , [-0.21, - 0.15] , {0.45, - 0.19}}	{[0.61, 0.69] , [-0.19, - 0.15] , {0.65, - 0.20}}	{[0.75, 0.79] , [-0.22, - 0.19] , {0.79, - 0.32}}
ς ₃	{[0.91, 0.96] , [-0.60, - 0.53] , {0.94, - 0.46}}	{[0.95, 0.99] , [-0.78, - 0.55] , {0.91, - 0.58}}	{[0.92, 0.94] , [-0.71, - 0.64] , {0.92, - 0.67}}
ς ₁	{[0.15, 0.21] , [-0.45, - 0.32] , {0.18, - 0.33}}	{[0.13, 0.17] , [-0.49, - 0.41] , {0.17, - 0.36}}	{[0.14, 0.19] , [-0.38, - 0.36] , {0.55, - 0.79}}
ς ₂	{[0.31, 0.32] , [-0.58, - 0.53] , {0.29, - 0.55}}	{[0.22, 0.29] , [-0.69, - 0.54] , {0.37, - 0.50}}	{[0.31, 0.33] , [-0.71, - 0.63] , {0.40, - 0.77}}
ς ₃	{[0.59, 0.72] , [-0.89, - 0.83] , {0.52, - 0.87}}	{[0.58, 0.62] , [-0.89, - 0.82] , {0.53, - 0.93}}	{[0.54, 0.59] , [-0.83, - 0.81] , {0.64, - 0.86}}

We apply $P$ -CBFHWA operator with weight vector ϒ = {0.6, 0.2, 0.2} to compute aggregating matrix $M ″ ″ ɛ = [{b ″ ″}_{ı}^{ɛ}]_{n \times m}$ , for ɛ = 1, 2, 3 and ı=1, 2, 3 (see Table 11).

Table 11

$M ″ ″ = [{b ″ ″}_{ı}^{ɛ}]_{n \times m}$ is matrix after applying $P$ -CBFHWA operator on ordered alternatives matrices

$X$ vs $Z$	${\dot{z}}_{1}$	${\dot{z}}_{2}$	${\dot{z}}_{3}$
ς ₁	{[0.76, 0.14] , [-0.46, - 0.32] , {0.37, - 0.39}}	{[0.20, 0.38] , [-0.87, - 0.46] , {0.29, - 0.55}}	{[0.14, 0.20] , [-0.44, - 0.34] , {0.27, - 0.40}}
ς ₂	{, [-0.63, - 0.59] , {0.35, - 0.67}}	{[0.73, 0.77] , [-0.21, - 0.16] , {0.59, - 0.21}}	{[0.29, 0.32] , [-0.63, - 0.55] , {0.33, - 0.58}}
ς ₃	{[0.60, 0.72] , [-0.91, - 0.85] , {0.60, - 0.88}}	{[0.92, 0.97] , [-0.65, - 0.55] , {0.93, - 0.52}}	{[0.58, 0.68] , [-0.88, - 0.82] , {0.55, - 0.88}}

We get, aggregate $M ″ ″' ɛ$ by using again $P$ -CBFWA operator with weight vector ϒ′ = {0.8, 0.1, 0.1} ^T, thus for ı=1, 2, 3 we obtain averaging aggregating column matrix $M ″ ″' = [b ″ ″' ı] m$ $\begin{matrix} M ″ ″' = & [\begin{matrix} ς_{1} = {[0.692, 0.173], [- 0.488, - 0.310], {0.352, - 0.404}} \\ ς_{2} = {[0.355, 0.411], [- 0.564, - 0.514], {0.377, - 0.588}} \\ ς_{3} = {[0.657, 0.773], [- 0.876, - 0.810], {0.659, - 0.834}} \end{matrix}] \end{matrix}$ now calculate score function for ranking of alternatives. After computing score function, we get $S_{P} (ς_{1}) = 0.385$ , $S_{P} (ς_{2}) = 0.287$ and $S_{P} (ς_{3}) = 0.208$ . Thus we obtain the ranking $S_{P} (ς_{1}) ≻ S_{P} (ς_{2}) ≻ S_{P} (ς_{1})$ this implies that ς₁ ≻ ς₂ ≻ ς₃. The best optimal choice is ς₁. We apply $R$ -CBFHWA operator to examine best alternatives. consider the collection of weighted matrices (see Table 12).

Table 12

${M^{″}}_{ɛ} = [{b^{″}}_{ı ȷ}^{ɛ}]_{n \times m \times k}$ are weighted matrices

$X$ vs $Y$	$\dot{y_{1}}$	${\dot{y}}_{2}$	${\dot{y}}_{3}$
ς ₁	{, [-0.48, - 0.36] , {0.14, - 0.29}}	{[0.87, 0.15] , [-0.44, - 0.32] , {0.17, - 0.49}}	{[0.61, 0.12] , [-0.48, - 0.29] , {0.80, - 0.25}}
ς ₂	{[0.29, 0.37] , [-0.68, - 0.58] , {0.35, - 0.68}}	{[0.30, 0.37] , [-0.60, - 0.58] , {0.37, - 0.71}}	{[0.26, 0.26] , [-0.69, - 0.63] , {0.31, - 0.57}}
ς ₃	{[0.63, 0.78] , [-0.92, - 0.86] , {0.58, - 0.93}}	{[0.58, 0.64] , [-0.90, - 0.87] , {0.65, - 0.81}}	{[0.54, 0.58] , [-0.88, - 0.81] , {0.60, - 0.79}}
ς ₁	{[0.21, 0.38] , [-0.08, - 0.05] , {0.27, - 0.05}}	{[0.17, 0.46] , [-0.09, - 0.02] , {0.40, - 0.05}}	{[0.22, 0.28] , [-0.11, - 0.08] , {0.25, - 0.08}}
ς ₂	{[0.75, 0.79] , [-0.21, - 0.15] , {0.45, - 0.19}}	{[0.75, 0.79] , [-0.22, - 0.19] , {0.79, - 0.32}}	{[0.61, 0.69] , [-0.19, - 0.15] , {0.65, - 0.20}}
ς ₃	{[0.92, 0.94] , [-0.71, - 0.64] , {0.92, - 0.67}}	{[0.95, 0.99] , [-0.78, - 0.55] , {0.91, - 0.58}}	{[0.91, 0.96] , [-0.60, - 0.53] , {0.94, - 0.46}}
ς ₁	{[0.13, 0.17] , [-0.49, - 0.41] , {0.17, - 0.36}}	{[0.15, 0.21] , [-0.45, - 0.32] , {0.18, - 0.33}}	{[0.14, 0.19] , [-0.38, - 0.36] , {0.55, - 0.79}}
ς ₂	{[0.31, 0.33] , [-0.71, - 0.63] , {0.40, - 0.77}}	{[0.22, 0.29] , [-0.69, - 0.54] , {0.37, - 0.50}}	{[0.31, 0.32] , [-0.58, - 0.53] , {0.29, - 0.55}}
ς ₃	{[0.58, 0.62] , [-0.89, - 0.82] , {0.53, - 0.93}}	{[0.59, 0.72] , [-0.89, - 0.83] , {0.52, - 0.87}}	{[0.54, 0.59] , [-0.83, - 0.81] , {0.64, - 0.86}}

We get the collection of matrices with ordered alternatives by calculating score function (see Table 13). We compute $R$ -CBFHWA operator with weight vector ϒ = {0.6, 0.2, 0.2} to aggregate matrices $M ″ ″ ɛ = [{b ″ ″}_{ı}^{ɛ}]_{n \times m}$ , for ɛ = 1, 2, 3 and ı=1, 2, 3 (see Table 14).

now we obtain, aggregated matrix ${M ″ ″'}_{ɛ}$ by using again $R$ -CBFWA operator with weight vector ϒ′ = {0.8, 0.1, 0.1} ^T, thus for ı=1, 2, 3 we obtain averaging aggregating column matrix $M ″ ″' = [b ″ ″' ı] m$ $\begin{matrix} M ″ ″' = & [\begin{matrix} ς_{1} = {[0.692, 0.173], [- 0.488, - 0.310], {0.396, - 0.338}} \\ ς_{2} = {[0.355, 0.411], [- 0.564, - 0.514], {0.376, - 0.653}} \\ ς_{3} = {[0.657, 0.773], [- 0.876, - 0.810], {0.620, - 0.876}} \end{matrix}] \end{matrix}$

Table 13

$M ″' ɛ = [{b ″'}_{ı ȷ}^{ɛ}] n \times m \times k$ are matrices with ordered alternatives

$X$ vs $Y$	$\dot{y_{1}}$	${\dot{y}}_{2}$	${\dot{y}}_{3}$
ς ₁	{, [-0.48, - 0.29] , {0.80, - 0.25}}	{[0.87, 0.15] , [-0.44, - 0.32] , {0.17, - 0.49}}	{[0.11, 0.15] , [-0.48, - 0.36] , {0.14, - 0.29}}
ς ₂	{[0.30, 0.37] , [-0.60, - 0.58] , {0.37, - 0.71}}	{[0.29, 0.37] , [-0.68, - 0.58] , {0.35, - 0.68}}	{[0.26, 0.26] , [-0.69, - 0.63] , {0.31, - 0.57}}
ς ₃	{[0.63, 0.78] , [-0.92, - 0.86] , {0.58, - 0.93}}	{[0.58, 0.64] , [-0.90, - 0.87] , {0.65, - 0.81}}	{[0.54, 0.58] , [-0.88, - 0.81] , {0.60, - 0.79}}
ς ₁	{[0.17, 0.46] , [-0.09, - 0.02] , {0.40, - 0.05}}	{[0.21, 0.38] , [-0.08, - 0.05] , {0.27, - 0.05}}	{[0.22, 0.28] , [-0.11, - 0.08] , {0.25, - 0.08}}
ς ₂	{[0.75, 0.79] , [-0.22, - 0.19] , {0.79, - 0.32}}	{[0.75, 0.79] , [-0.21, - 0.15] , {0.45, - 0.19}}	{[0.61, 0.69] , [-0.19, - 0.15] , {0.65, - 0.20}}
ς ₃	{[0.95, 0.99] , [-0.78, - 0.55] , {0.91, - 0.58}}	{[0.91, 0.96] , [-0.60, - 0.53] , {0.94, - 0.46}}	{[0.92, 0.94] , [-0.71, - 0.64] , {0.92, - 0.67}}
ς ₁	{[0.14, 0.19] , [-0.38, - 0.36] , {0.55, - 0.79}}	{[0.15, 0.21] , [-0.45, - 0.32] , {0.18, - 0.33}}	{[0.13, 0.17] , [-0.49, - 0.41] , {0.17, - 0.36}}
ς ₂	{[0.31, 0.33] , [-0.71, - 0.63] , {0.40, - 0.77}}	{[0.31, 0.32] , [-0.58, - 0.53] , {0.29, - 0.55}}	{[0.22, 0.29] , [-0.69, - 0.54] , {0.37, - 0.50}}
ς ₃	{[0.58, 0.62] , [-0.89, - 0.82] , {0.53, - 0.93}}	{[0.54, 0.59] , [-0.83, - 0.81] , {0.64, - 0.86}}	{[0.59, 0.72] , [-0.89, - 0.83] , {0.52, - 0.87}}

Table 14

$M ″ ″ = [{b ″ ″}_{ı}^{ɛ}]_{n \times m}$ is matrix after applying $R$ -CBFHWA operator on ordered alternatives matrices

$X$ vs $Z$	${\dot{z}}_{1}$	${\dot{z}}_{2}$	${\dot{z}}_{3}$
ς ₁	{[0.76, 0.14] , [-0.46, - 0.32] , {0.41, - 0.31}}	{[0.20, 0.38] , [-0.87, - 0.46] , {0.34, - 0.05}}	{[0.14, 0.20] , [-0.44, - 0.34] , {0.35, - 0.67}}
ς ₂	{, [-0.63, - 0.59] , {0.35, - 0.68}}	{[0.73, 0.77] , [-0.21, - 0.16] , {0.68, - 0.27}}	{[0.29, 0.32] , [-0.63, - 0.55] , {0.37, - 0.69}}
ς ₃	{[0.60, 0.72] , [-0.91, - 0.85] , {0.60, - 0.89}}	{[0.92, 0.97] , [-0.65, - 0.55] , {0.92, - 0.58}}	{[0.58, 0.68] , [-0.88, - 0.82] , {0.55, - 0.91}}

Table 15

Method base comparison with existed method

Techniques	Simplicity	Dual Check	Generalized
Wei et al. 2017	×	×	✓
Proposed Method	✓	✓	✓

Now calculate score function for ranking of alternatives. $S_{R} (b ″ ″ ı) = \frac{[δ_{ℓ_{ı}}^{P} + δ_{u_{ı}}^{P}] + [1 - δ_{ℓ_{ı}}^{n} + 1 - δ_{u_{ı}}^{N}] + ρ_{ı}^{P} - (1 - ρ_{ı}^{N})}{6}$ After computing score function, we get $S_{R} (ς_{1}) = 0.300$ , $S_{R} (ς_{2}) = 0.286$ and $S_{R} (ς_{3}) = 0.373$ . Thus we obtain the ranking $S_{R} (ς_{3}) ≻ S_{R} (ς_{2}) ≻ S_{R} (ς_{1})$ this implies that ς₃ ≻ ς₂ ≻ ς₁. The best optimal choice is ς₃. The rankings of alternatives can be seen in Figs. 9 and 10. A complete ranking of all operators can be seen in Figs. 11 and 12.

Fig.9

Ranking of alternatives using $P$ -CBFHWA

Fig.10

Ranking of alternatives using $R$ -CBFHWA

Fig.11

Ranking of alternatives of all operators

Fig.12

Ranking of alternatives of all operators

5 Comparison Analysis

In this section, firstly we compare our domain with interval valued bipolar fuzzy(IVBF) domain. Secondly, we differentiate between proposed operators. We also make an analysis between all proposed operators and point out the best one. Thirdly, we compare our aggregation operators with existed operators proposed in [46].

5.1 IVBFS vs CBFS

Firstly, we see the difference of particular information in both domains. The IVBF domain contains less information as compare to CBF domain. In CBF domain we have triplet of bipolar fuzzy information with two types of operations, including, $P$ -Order and $R$ -Order which help us to examine the information with double point of view. On the other hand IVBF domain restrict us at usual fuzzy methodology. For example, if we consider IVBF domain and apply same problem, we get final aggregated values by IVBFWA, IVBFOWA and IVBFHWA operators (which can be obtained by definition 2.7) respectively. We have, $\begin{matrix} M ″' = & [\begin{matrix} ς_{1} = {[0.177, 0.263], [- 0.271, - 0.177]} \\ ς_{2} = {[0.458, 0.506], [- 0.483, - 0.404]} \\ ς_{3} = {[0.767, 0.843], [- 0.815, - 0.736]} \end{matrix}] \end{matrix}$ $\begin{matrix} M ″ ″ = & [\begin{matrix} ς_{1} = {[0.160, 0.260], [- 0.260, - 0.170]} \\ ς_{2} = {[0.450, 0.500], [- 0.460, - 0.390]} \\ ς_{3} = {[0.770, 0.850], [- 0.810, - 0.730]} \end{matrix}] \end{matrix}$ $\begin{matrix} M ″ ″' = & [\begin{matrix} ς_{1} = {[0.692, 0.173], [- 0.488, - 0.310]} \\ ς_{2} = {[0.355, 0.411], [- 0.564, - 0.514]} \\ ς_{3} = {[0.657, 0.773], [- 0.876, - 0.810]} \end{matrix}] \end{matrix}$

Define the score function for IVBF domain as $S (b_{ı}) = \frac{[δ_{ℓ_{ı}}^{P} + δ_{u_{ı}}^{P}] + [δ_{ℓ_{ı}}^{N} + δ_{u_{ı}}^{N}]}{4}$ we get the rankings ς₂ ≻ ς₃ ≻ ς₁ by IVBFWA operator, ς₂ ≻ ς₃ ≻ ς₁ by IVBFOWA operator and ς₁ ≻ ς₃ ≻ ς₂ by IVBFHWA operator. We can see the difference in final rankings. The IVBF domain gives preference to the alternative which is at low position by applying six CBF operators. So, CBF domain gives more proper knowledge of a particular problem as compare to IVBF domain. The CBF averaging operators contain efficiency of handling data in more generalized form as compare to IVBF averaging operators.

5.2 Analysis of proposed aggregation operators

In this subsection, we differentiate among proposed operators. We develop three operators cubic bipolar fuzzy averaging operator, cubic bipolar fuzzy ordered weighted averaging operator and cubic bipolar fuzzy hybrid weighted averaging operator with dual order due to cubic set. In $P$ -CBFWA and $R$ -CBFWA we just assign weights to each attribute and calculate average of CBF data using arithmetic operations. Whereas, in $P$ -CBFOWA and $R$ -CBFOWA we use score function and obtain an ordering of attributes then repeat process of CBFWA. In $P$ -CBFHWA, and $P$ -CBFHWA operators we first obtain weighted matrices then calculate ordered matrices and repeat process of CBFWA operators for both order. In real decision making CBFHWA operator is more accurate because it tackle information with more sensitivity. Although, in presented example three operators give the same results according to their chosen order but it may not be same in other cases. After applying all these operators to a numerical example we see that $P$ -CBFWA, $P$ -CBFOWA and $P$ -CBFHWA provide the ranking, ς₁ ≻ ς₂ ≻ ς₃ and $R$ -CBFWA, $R$ -CBFOWA and $R$ -CBFHWA give the ranking as ς₃ ≻ ς₂ ≻ ς₁. This shows that mutual understanding of decision makers is necessary to obtain best results in group decision making. In both order ς₂ is second choice whereas ς₁ and ς₃ are alternate choices. The ς₁ is best for $P$ -order and ς₃ is $R$ -order.

5.3 Comparison with existed aggregation operator

Our third focus is methodology, the proposed method has three futures including, simplicity, double order, and generalized.

i . The existed operators are simple as compare to proposed in [46]. The proposed algorithm is also simple and effective for business MAGDM because of its normalizing property whereas algorithms presented in [46] is more complicated for MAGDM. Calculations are feasible and as a result aggregated values are easy to calculate.

ii . The presented approach tackle a particular problem in two perspectives i.e. $P$ -Order and $R$ -Order of proposed operator. This concept is a dual check which provides a more reliable analysis of a particular data for a healthy and beneficial decision. The approaches in [46] does not contain such property.

iii . The proposed method is more suitable to handle group decision making whereas operators in [46] is time consuming in case of big data. So presented technique unambiguously solves numerous incompatible criteria or points of comparison in decision making in the terms such as government, business, medicine, physics, information technology etc.

6 Conclusion

For data analyzing of many types, bipolarity of knowledge is a vital part to be considered while developing a mathematical framework for most of the situations. Bipolarity indicates the positive and negative aspects of a particular problem. The concept behind the bipolarity is that a huge range of human decisions analysis is involved bipolar subjective thoughts. A verity of bipolar fuzzy decision making with different technique is available in literature. But the concept of simple bipolar fuzzy set is insufficient to provide the information about the occurrence of ratings or grades with accuracy because information is limited, and it is also unable to describe the occurrence of uncertainty and vagueness very well specially, when sensitive cases are involved in decision making problems. For this purpose, we considered cubic bipolar fuzzy sets as the generalization of bipolar fuzzy sets. This model provides more accuracy and flexibility as compared to previously existing approaches, because it contains more information and it is more comprehensive and reasonable. In human decisions, the second important part is ranking and rating of different alternatives obtained after particular evaluation. we developed an appropriate aggregation method which is simple reliable and efficient enough to handle cubic bipolar fuzzy data. In this study, we discussed a multi-attribute group decision making problem under cubic bipolar fuzzy environment. We deduce some averaging aggregate operators, including, $P$ -CBFWA, $R$ -CBFWA, $P$ -CBFOWA, $R$ -CBFOWA, $P$ -CBFHWA and $R$ -CBFHWA to aggregate cubic bipolar fuzzy data. The technique is feasible and efficient enough to huge amount of data. The ranking of each alternative by using different operator can be clearly seen in following figures. The comparison and analysis is also presented in this research. In future work, we shall study different techniques and methods including, TOPSIS, AHP, ELECTRE and VIKOR under cubic bipolar fuzzy information by discussing multi-attributes group decision making problems.

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Multi-attribute group decision making based on cubic bipolar fuzzy information using averaging aggregation operators

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Fuzzy set, bipolar fuzzy set, interval-valued bipolar fuzzy set and cubic set

2.2 Cubic bipolar fuzzy set with operations

3.1 Score and accuracy functions for CBFEs

3.2 P -order cubic bipolar fuzzy weighted average( P -CBFWA) Operator

3.3 P -Order cubic bipolar fuzzy ordered weighted averaging( P -CBFOWA) Operator

3.4 P -order cubic bipolar fuzzy hybrid weighted averaging( P -CBFHWA) operator

3.5 R -order cubic bipolar fuzzy weighted averaging( R -CBFWA) Operator

3.6 R -order cubic bipolar fuzzy ordered weighted averaging( R -CBFOWA) Operator

3.7 R -Order cubic bipolar fuzzy hybrid weighted averaging( R -CBFHWA) operator

4 CBFSs application to multi-criteria group decision making using averaging aggregation operators

4.1 Proposed technique

4.2 Computative example of averaging aggregation operators in multi-attribute group decision making problem

5.1 IVBFS vs CBFS

5.2 Analysis of proposed aggregation operators

5.3 Comparison with existed aggregation operator

6 Conclusion

References

3.2 $P$ -order cubic bipolar fuzzy weighted average( $P$ -CBFWA) Operator

3.3 $P$ -Order cubic bipolar fuzzy ordered weighted averaging( $P$ -CBFOWA) Operator

3.4 $P$ -order cubic bipolar fuzzy hybrid weighted averaging( $P$ -CBFHWA) operator

3.5 $R$ -order cubic bipolar fuzzy weighted averaging( $R$ -CBFWA) Operator

3.6 $R$ -order cubic bipolar fuzzy ordered weighted averaging( $R$ -CBFOWA) Operator

3.7 $R$ -Order cubic bipolar fuzzy hybrid weighted averaging( $R$ -CBFHWA) operator