Cubic bipolar fuzzy Dombi averaging aggregation operators with application to multi-criteria decision-making

Abstract

A cubic bipolar fuzzy set (CBFS) is a new approach in computational intelligence and decision-making under uncertainty. This model is the generalization of bipolar fuzzy sets to deal with two-sided contrasting features which can describe the information with a bipolar fuzzy number and an interval-valued bipolar fuzzy number simultaneously. In this paper, the Dombi’s operations are analyzed for information aggregation of cubic bipolar fuzzy numbers (CBFNs). The Dombi’s operations carry the advantage of more pliability and reliability due to the existence of their operational parameters. Owing to the pliable nature of Dombi’s operators, this research work introduces new aggregation operators named as cubic bipolar fuzzy Dombi weighted averaging (CBFDWA) operator and cubic bipolar fuzzy Dombi ordered weighted averaging (CBFDOWA) operator with $ℙ$ -order and $ℝ$ -order, respectively. Additionally, this paper presents some significant characteristics of suggested operators including, idempotency, boundedness and monotonicity. Moreover, a robust multi-criteria decision making (MCDM) technique is developed by using $ℙ$ -CBFDWA and $ℝ$ -CBFDWA operators. Based on the suggested operators a practical application is demonstrated towards MCDM under uncertainty. The comparison analysis of suggested Dombi’s operators with existing operators is also given to discuss the rationality, efficiency and applicability of these operators.

Keywords

Cubic bipolar fuzzy sets Dombi’s operations cubic bipolar fuzzy Dombi weighted averaging operator cubic bipolar fuzzy Dombi ordered weighted averaging operator ℙ-order and ℝ-order operations multi-criteria decision making

1 Introduction

Decision making is an important phenomenon to attain an optimal alternative among the feasible alternatives. However, this process involves uncertain and vague information due to incomplete data and inherent human judgements. Due to these reasons, classical approaches are unable to identify the optimal alternative under uncertainty. To resolve such type of real-life problems, Zadeh [1] presented the notion of fuzzy set (FS) which has been substantially applied to handle various real-life problems. Later on, Zadeh [2] gave the abstraction of interval-valued fuzzy set (IVFS) which is superior to fuzzy set. Owing to the fact that FS theory deals vagueness with the membership function only, Atanassov [3, 4] extended Zadeh’s FS to the idea of intuitionistic fuzzy set (IFS) which has both membership function (MF) and non-membership function (NMF) with the constraint that sum of the membership degree (MD) μ and non-membership degree (NMD) ν must be less than or equal to one i.e., μ + ν ≤ 1.

Xu and Yager [5], for simplicity, called (μ, ν) an intuitionistic fuzzy number (IFN). The constraint of IFN often fails in decision analysis when decision makers (DMs) assign (μ, ν) to an alternative against a criteria such that μ + ν > 1. To deal with this concern and to handle the ambiguity in the decision analysis, Yager [6, 7] introduced Pythagorean fuzzy set (PFS) with the modified condition that μ² + ν² ≤ 1, in this way, a PFS is a superior than IFS. The only difference between PFS and IFS is the said conditions associated with MD and NMD. Zhang and Xu [8], for simplicity, gave the idea of Pythagorean fuzzy number (PFN) and introduced an extension of TOPSIS “a technique for ordering preference through the ideal solution” towards PFSs for MCDM problem. Yager [9] proposed the idea of q-rung orthopair fuzzy set (q-ROFS) as an efficient tool to deal with vagueness of the (MCDM) problems. Yager [9] further proposed the idea of q-rung orthopair fuzzy number (q-ROFN). The main condition for q-ROFN (μ, ν) is that μ^q + ν^q ≤ 1 which provide a larger space to choose MD and NMD, in this way, a q-ROFN is superior than both PFN and IFN. These models have been successfully used by many researchers in the last decades. All of these models have been developed on the basis of the need to address uncertainty in real-life problems.

In many real-world situations, it has been observed that there is a counter property corresponding to each property in the data analysis of alternatives/objects. In this view, Zhang [10, 11] originated the concept of bipolar fuzzy sets (BFSs). The BFS theory amalgamates both polarity and fuzziness into a unified model. The bipolar-valued fuzzy set(BVFS) contains two constituents, a positive membership grade whose value is taken from interval [0,1] and a negative membership grade that belongs to the interval [-1,0]. Lee [12, 13] discussed the comparison of IVFSs, IFSs, and BVFSs. Alcantud et al. [14] proposed the notion of dual extended hesitant fuzzy sets. Wei et al. [15] proposed the idea of interval-valued bipolar fuzzy set (IVBFS) and discussed the MCDM under IVBF information and their application to emerging technology commercialization evaluation.

In the real situations, the membership grade in some fuzzy problems cannot be described completely by using an exact value or an interval-value. Therefore, Jun et al. [16] inaugurated the abstraction of cubic set (CS) (hybrid set of IVFS and FS) and defined some of its operations. Cubic sets have the ability to describe vagueness by using an exact value and an interval-value simultaneously. Jun et al. [16] defined internal cubic sets (ICSs) and external cubic sets (ECSs). They also defined the notions of $ℙ$ -union, $ℙ$ -intersection, $ℝ$ -union and $ℝ$ -intersection of cubic sets and proved some related results.

Since, bipolarity is an important ingredient in human decisions, some researchers have worked on different decision making techniques under bipolar fuzzy information [17 –23]. The study of aggregation operators has been of great interest to researchers as they have the ability to aggregate the individual input data into one single input data. For more information, the readers are referred to [24 –33].

Dombi [34] introduced a general family of fuzzy operators and fuzziness measures induced by fuzzy operators. They introduced the notions of score function, accuracy function, and operational rules of IFNs. Liu et al. [35] introduced MADM based on some intuitionistic fuzzy Dombi Bonferroni mean operators. Liu et al. [36] developed multiple attribute group decision making based on q-rung orthopair fuzzy Heronian mean operators. Liu et al. [37] presented q-rung orthopair fuzzy multiple attribute group decision-making method based on normalized bidirectional projection model and generalized knowledge-based entropy measure. Liu and Wang [38] proposed q-ROF aggregation operators and applied them in multiple-attribute decision making (MADM). Jana et al. [39] extended Dombi operations to bipolar fuzzy sets. Akram et al. [40] and Khan et al. [41] proposed Pythagorean Dombi fuzzy aggregation operators. Chen and Ye [42] solved a MCDM problem utilizing Dombi aggregation operators under the single-valued neutrosophic environment. Riaz and Hashmi [43] introduced the idea of Pythagorean m-polar fuzzy Dombi’s aggregation operators for the censuses process. Shi and Ye [44] introduced cubic neutrosophic Dombi aggregation operators and applied them in MCDM.

However, sometimes, the simple BFS does not provide adequate information about the occurrence of grades because the information is meagre and it also fails to address the uncertainty properly. Similarly, interval-valued bipolar fuzzy set, although a better version than bipolar fuzzy set, has some loopholes as, sometimes, it is insufficient to provide expert opinion information depending on the features of alternatives. Recently, Riaz and Tehrim [45 –47] initiated a novel model named as cubic bipolar fuzzy set (CBFS) which is a hybrid set of BFS and IVBFS. This model gives more precision and pliability as compared to the existing models because it accommodates bipolar and interval-valued bipolar fuzzy information simultaneously. So, this model provides maximum details about the occurrence of ratings, inexactness and bipolarity. They proposed some aggregation operators like cubic bipolar fuzzy weighted geometric (CBFWG) aggregation operators, cubic bipolar fuzzy averaging (CBFA) aggregation operators, ordered weighted and hybrid weighted averaging operators under $ℙ$ ( $ℝ$ )-order and applied these operators in some multi-criteria group decision making (MCGDM) problems.

A bipolar fuzzy set is a strong model to express various contrasting features of real-life for instance happiness and grief of human thoughts, effects and side effects of drugs, hopeful and hopeless in faith, sweetness and sourness of things. The existence and harmony of these two features are considered as a key for the stability of a social system. A broad variety of human decision-making is based on two-sided or bipolar judgmental reasoning with a positive and a negative side. The literature includes a number of bipolar fuzzy decisions with various approaches.

The main objectives and advantages of the manuscript are listed as follows.

The main objective of this paper is to extend Dombi’s operations to cubic bipolar fuzzy numbers (CBFNs) and to investigate properties these numbers. A CBFN is superior that a bipolar fuzzy number as it carries more comprehensive and reasonable information. CBFNs is the extension of bipolar fuzzy numbers to deal with two-sided contrasting features which can describe the information with a bipolar fuzzy number and an interval-valued bipolar fuzzy number simultaneously.

m A secondary objective of this paper is to introduce some fundamental operations on CBFNs, their key properties, and related significant results. Suggested operations are very helpful to strengthen CBFS theory.

Since Dombi’s aggregation operators for cubic bipolar fuzzy numbers (CBFNs) have not been established so far, motivated by the above discussion, this paper presents novel averaging aggregation operators under cubic bipolar fuzzy information, including, cubic bipolar fuzzy Dombi weighted averaging (CBFDWA) and cubic bipolar fuzzy Dombi ordered weighted averaging (CBFDOWA) operators under both $ℙ$ -order and $ℝ$ -order.

An algorithm for new MCDM technique is developed based on proposed Dombi’s aggregation operators using cubic bipolar fuzzy information. Proposed technique is also demonstrated by a numerical illustration. The comparison analysis of the final ranking computed by proposed technique with some existing MCDM techniques is given to justify the rationality, feasibility, and reliability of proposed technique.

The remainder of the article is structured as follows. In Section 2, we review some basic concepts for better understanding of cubic bipolar fuzzy numbers (CBFNs). In Section 3, Dombi operations are defined on CBFNs under both $ℙ$ -order and $ℝ$ -order. Section 4 introduces the Dombi’s aggregation operators named as $ℙ$ -order cubic bipolar fuzzy Dombi weighted averaging ( $ℙ$ -CBFDWA) and $ℙ$ -order cubic bipolar fuzzy Dombi ordered weighted averaging ( $ℙ$ -CBFDOWA). Section 5 introduces the Dombi’s aggregation operators named as $ℝ$ -order cubic bipolar fuzzy Dombi weighted averaging ( $ℝ$ -CBFDWA) and $ℝ$ -order cubic bipolar fuzzy Dombi ordered weighted averaging ( $ℝ$ -CBFDOWA). In Section 6, an algorithm for new MCDM technique is developed based on proposed Dombi’s aggregation operators using cubic bipolar fuzzy information. In section 7, a numerical illustration is presented to discuss the practical application of proposed MCDM technique. Section 8 provides the conclusion of the research work.

2 Preliminaries

In the present section, we recall the definitions of bipolar fuzzy set (BFS), interval-valued bipolar fuzzy set (IVBFS) and cubic bipolar fuzzy set (CBFS). We recall the algebraic operations of CBFNs under $ℙ$ ( $ℝ$ )-order, score functions of CBFNs under $ℙ$ ( $ℝ$ )-order and accuracy function of CBFNs. Moreover, we prove some novel results for CBFNs.

Definition 2.1. [10] A bipolar fuzzy set (BFS) on the universe $\ddot{K}$ is an object of the form $ϑ = {(\ddot{k}, {\tilde{Δ}}_{ϑ}^{+} (\ddot{k}), {\tilde{Δ}}_{ϑ}^{-} (\ddot{k})) | \ddot{k} \in \ddot{K}}$ where the positive membership degree ${\tilde{Δ}}_{ϑ}^{+} (\ddot{k}) \in [0, 1]$ denotes the satisfaction degree of an element $\ddot{k}$ to the property corresponding to a BFS ϑ and the negative membership degree ${\tilde{Δ}}_{ϑ}^{-} (\ddot{k})) \in [- 1, 0]$ denotes the satisfaction degree of $\ddot{k}$ to some implicit counter property corresponding to a BFS ϑ, respectively and for every $\ddot{k} \in \ddot{K}$ . For the sake of simplicity, an ordered pair $({\tilde{Δ}}_{ϑ}^{+} (\ddot{k}), {\tilde{Δ}}_{ϑ}^{-} (\ddot{k}))$ can be considered as bipolar fuzzy number (BFN).

Definition 2.2. [15] An interval-valued bipolar fuzzy set (IVBFS) over the initial universe $\ddot{K}$ is defined as $\ddot{ϑ} = {(\ddot{k}, [{\tilde{ϒ}}_{ℓ \ddot{ϑ}}^{+} (\ddot{k}), {\tilde{ϒ}}_{u \ddot{ϑ}}^{+} (\ddot{k})], [{\tilde{ϒ}}_{ℓ \ddot{ϑ}}^{-} (\ddot{k}), {\tilde{ϒ}}_{u \ddot{ϑ}}^{-} (\ddot{k})]) | \ddot{k} \in \ddot{K}}$ where the positive membership degree $[{\tilde{ϒ}}_{ℓ \ddot{ϑ}}^{+} (\ddot{k}), {\tilde{ϒ}}_{u \ddot{ϑ}}^{+} (\ddot{k})] \subseteq [0, 1]$ depicts the satisfaction degree of an element $\ddot{k}$ to the property corresponding to an IVBFS ( $\ddot{ϑ}$ ) and the negative membership degree $[{\tilde{ϒ}}_{ℓ \ddot{ϑ}}^{-} (\ddot{k}), {\tilde{ϒ}}_{u \ddot{ϑ}}^{-} (\ddot{k})] \subseteq [- 1, 0]$ depicts the satisfaction degree of an element $\ddot{k}$ to some implicit counter property corresponding to the an IVBFS ( $\ddot{ϑ}$ ), respectively and for every $\ddot{k} \in \ddot{K}$ . An interval-valued bipolar fuzzy number (IVBFN) is written as $\ddot{ϑ} = ([{\tilde{ϒ}}_{ℓ \ddot{ϑ}}^{+}, {\tilde{ϒ}}_{u \ddot{ϑ}}^{+}], [{\tilde{ϒ}}_{ℓ \ddot{ϑ}}^{-}, {\tilde{ϒ}}_{u \ddot{ϑ}}^{-}])$ .

Definition 2.3. [16] A cubic set (CS) over the initial universe $\ddot{K}$ is defined as $\tilde{φ} = {(\ddot{k}, \bar{Θ} (\ddot{k}), \bar{Λ} (\ddot{k})) | \ddot{k} \in \ddot{K}}$ where $\bar{Θ}$ is an IVFS on $\ddot{K}$ and $\bar{Λ}$ is a FS on $\ddot{K}$ .

Definition 2.4. [45 –47] A cubic bipolar fuzzy set (CBFS) over the initial universe $\ddot{K}$ is defined as

$Ω = {(\ddot{k}, \ddot{ϑ} (\ddot{k}), ϑ (\ddot{k})) | \ddot{k} \in \ddot{K}}$

where $\ddot{ϑ}$ is an IVBFS on $\ddot{K}$ and ϑ is a BFS on $\ddot{K}$ . Thus, CBFS can be rewritten as

$Ω = {(\ddot{k}, [{\tilde{ϒ}}_{ℓ Ω}^{+} (\ddot{k}), {\tilde{ϒ}}_{u Ω}^{+} (\ddot{k})], [{\tilde{ϒ}}_{ℓ Ω}^{-} (\ddot{k}), {\tilde{ϒ}}_{u Ω}^{-} (\ddot{k})], {{\tilde{Δ}}_{Ω}^{+} (\ddot{k}), {\tilde{Δ}}_{Ω}^{-} (\ddot{k})}) | \ddot{k} \in \ddot{K}}$

where the intervals $[{\tilde{ϒ}}_{ℓ Ω}^{+} (\ddot{k}), {\tilde{ϒ}}_{u Ω}^{+} (\ddot{k})] \subseteq [0, 1]$ and $[{\tilde{ϒ}}_{ℓ Ω}^{-} (\ddot{k}), {\tilde{ϒ}}_{u Ω}^{-} (\ddot{k})] \subseteq [- 1, 0]$ represent the interval-valued positive and negative membership degrees, respectively and ${\tilde{Δ}}_{\ddot{Ω}}^{+} (\ddot{k}) \in [0, 1]$ and ${\tilde{Δ}}_{\ddot{Ω}}^{-} (\ddot{k}) \in [- 1, 0]$ represent the positive and negative membership, respectively, of an element $\ddot{k} \in \ddot{K}$ . The cubic bipolar fuzzy number (CBFN) is represented as $Ω = ([{\tilde{ϒ}}_{ℓ Ω}^{+}, {\tilde{ϒ}}_{u Ω}^{+}], [{\tilde{ϒ}}_{ℓ Ω}^{-}, {\tilde{ϒ}}_{u Ω}^{-}], {{\tilde{Δ}}_{Ω}^{+}, {\tilde{Δ}}_{Ω}^{-}})$ .

Definition 2.5. [46] Let $Ω_{i} = ([{\tilde{ϒ}}_{ℓ Ω_{i}}^{+}, {\tilde{ϒ}}_{u Ω_{i}}^{+}], [{\tilde{ϒ}}_{ℓ Ω_{i}}^{-}, {\tilde{ϒ}}_{u Ω_{i}}^{-}], {{\tilde{Δ}}_{Ω_{i}}^{+}, {\tilde{Δ}}_{{\ddot{Ω}}_{i}}^{-}})$ , (i = 1, 2, 3) be three cubic bipolar fuzzy numbers, and ω > 0, then their operations under $ℙ$ -order are defined as

$Ω_{1} \cup_{ℙ} Ω_{2} = ([max {{\tilde{ϒ}}_{ℓ Ω_{1}}^{+}, {\tilde{ϒ}}_{ℓ Ω_{2}}^{+}}, max {{\tilde{ϒ}}_{u Ω_{1}}^{+}, {\tilde{ϒ}}_{u Ω_{2}}^{+}}], [min {{\tilde{ϒ}}_{ℓ Ω_{1}}^{-}, {\tilde{ϒ}}_{ℓ Ω_{2}}^{-}}, min {{\tilde{ϒ}}_{u Ω_{1}}^{-}, {\tilde{ϒ}}_{u Ω_{2}}^{-}}], {max {{\tilde{Δ}}_{Ω_{1}}^{+}, {\tilde{Δ}}_{Ω_{2}}^{+}}, \min {{\tilde{Δ}}_{Ω_{1}}^{-}, {\tilde{Δ}}_{Ω_{2}}^{-}}})$

$Ω_{1} \cap_{ℙ} Ω_{2} = ([\min {{\tilde{ϒ}}_{ℓ Ω_{1}}^{+}, {\tilde{ϒ}}_{ℓ Ω_{2}}^{+}}, min {{\tilde{ϒ}}_{u Ω_{1}}^{+}, {\tilde{ϒ}}_{u Ω_{2}}^{+}}], [max {{\tilde{ϒ}}_{ℓ Ω_{1}}^{-}, {\tilde{ϒ}}_{ℓ Ω_{2}}^{-}}, max {{\tilde{ϒ}}_{u Ω_{1}}^{-}, {\tilde{ϒ}}_{u Ω_{2}}^{-}}], {min {{\tilde{Δ}}_{Ω_{1}}^{+}, {\tilde{Δ}}_{Ω_{2}}^{+}}, max {{\tilde{Δ}}_{Ω_{1}}^{-}, {\tilde{Δ}}_{Ω_{2}}^{-}}})$

$Ω_{1} \oplus_{ℙ} Ω_{2} = ([{\tilde{ϒ}}_{ℓ Ω_{1}}^{+} + {\tilde{ϒ}}_{ℓ Ω_{2}}^{+} - {\tilde{ϒ}}_{ℓ Ω_{1}}^{+} {\tilde{ϒ}}_{ℓ τ_{2}}^{+}, {\tilde{ϒ}}_{u Ω_{1}}^{+} + {\tilde{ϒ}}_{u Ω_{2}}^{+} - {\tilde{ϒ}}_{u Ω_{1}}^{+} {\tilde{ϒ}}_{u Ω_{2}}^{+}], [- ({\tilde{ϒ}}_{ℓ Ω_{1}}^{-} {\tilde{ϒ}}_{ℓ Ω_{2}}^{-}), - ({\tilde{ϒ}}_{u Ω_{1}}^{-} {\tilde{ϒ}}_{u Ω_{2}}^{-})], {{\tilde{Δ}}_{Ω_{1}}^{+} + {\tilde{Δ}}_{Ω_{2}}^{+} - {\tilde{Δ}}_{Ω_{1}}^{+} {\tilde{Δ}}_{Ω_{2}}^{+}, - ({\tilde{Δ}}_{Ω_{1}}^{-} {\tilde{Δ}}_{Ω_{2}}^{-})})$

$Ω_{1} \otimes_{ℙ} Ω_{2} = ([{\tilde{ϒ}}_{ℓ Ω_{1}}^{+} {\tilde{ϒ}}_{ℓ Ω_{2}}^{+}, {\tilde{ϒ}}_{u Ω_{1}}^{+} ϒ_{u Ω_{2}}^{+}], [- (| {\tilde{ϒ}}_{ℓ Ω_{1}}^{-} | + | {\tilde{ϒ}}_{ℓ Ω_{2}}^{-} | - ({\tilde{ϒ}}_{ℓ Ω_{1}}^{-} {\tilde{ϒ}}_{ℓ Ω_{2}}^{-})), - (| {\tilde{ϒ}}_{u Ω_{1}}^{-} | + | {\tilde{ϒ}}_{u Ω_{2}}^{-} | - ({\tilde{ϒ}}_{u Ω_{1}}^{-} {\tilde{ϒ}}_{u Ω_{2}}^{-}))], {{\tilde{Δ}}_{Ω_{1}}^{+} {\tilde{Δ}}_{Ω_{2}}^{+}, - (| {\tilde{Δ}}_{Ω_{1}}^{-} | + | {\tilde{Δ}}_{Ω_{2}}^{-} | - ({\tilde{Δ}}_{Ω_{1}}^{-} {\tilde{Δ}}_{Ω_{2}}^{-}))})$

$Ω_{1}^{ω} = ([({\tilde{ϒ}}_{ℓ Ω_{1}}^{+})^{ω}, ({\tilde{ϒ}}_{u Ω_{1}}^{+})^{ω}], [- (1 - (1 - | {\tilde{ϒ}}_{ℓ Ω_{1}}^{-} |)^{ω}), - (1 - (1 - | {\tilde{ϒ}}_{u Ω_{1}}^{-} |)^{ω})], {({\tilde{Δ}}_{Ω_{1}}^{+})^{ω}, - (1 - (1 - | {\tilde{Δ}}_{Ω_{1}}^{-} |)^{ω})})$

$ω Ω_{1} = ([1 - (1 - {\tilde{ϒ}}_{ℓ Ω_{1}}^{+})^{ω}, 1 - (1 - {\tilde{ϒ}}_{u Ω_{1}}^{+})^{ω}], [- | {\tilde{ϒ}}_{ℓ Ω_{1}}^{-} |^{ω}, - | {\tilde{ϒ}}_{u Ω_{1}}^{-} |^{ω}], {1 - (1 - {\tilde{Δ}}_{Ω_{1}}^{+})^{ω}, - | {\tilde{Δ}}_{Ω_{1}}^{-} |^{ω}})$

Definition 2.6. [46] Let $Ω_{i} = ([{\tilde{ϒ}}_{ℓ Ω_{i}}^{+}, {\tilde{ϒ}}_{u Ω_{i}}^{+}], [{\tilde{ϒ}}_{ℓ Ω_{i}}^{-}, {\tilde{ϒ}}_{u Ω_{i}}^{-}], {{\tilde{Δ}}_{Ω_{i}}^{+}, {\tilde{Δ}}_{Ω_{i}}^{-}})$ , (i = 1, 2, 3) be three cubic bipolar fuzzy numbers, and Ω > 0, then their operations under $ℝ$ -order are defined as

$Ω_{1} \cup_{ℝ} Ω_{2} = ([max {{\tilde{ϒ}}_{ℓ Ω_{1}}^{+}, {\tilde{ϒ}}_{ℓ Ω_{2}}^{+}}, max {{\tilde{ϒ}}_{u Ω_{1}}^{+}, {\tilde{ϒ}}_{u Ω_{2}}^{+}}], [min {{\tilde{ϒ}}_{ℓ Ω_{1}}^{-}, {\tilde{ϒ}}_{ℓ Ω_{2}}^{-}}, min {{\tilde{ϒ}}_{u Ω_{1}}^{-}, {\tilde{ϒ}}_{u Ω_{2}}^{-}}], {min {{\tilde{Δ}}_{Ω_{1}}^{+}, {\tilde{Δ}}_{Ω_{2}}^{+}}, max {{\tilde{Δ}}_{Ω_{1}}^{-}, {\tilde{Δ}}_{Ω_{2}}^{-}}})$

$Ω_{1} \cap_{ℝ} Ω_{2} = ([min {{\tilde{ϒ}}_{ℓ Ω_{1}}^{+}, {\tilde{ϒ}}_{ℓ Ω_{2}}^{+}}, min {{\tilde{ϒ}}_{u Ω_{1}}^{+}, {\tilde{ϒ}}_{u Ω_{2}}^{+}}], [max {{\tilde{ϒ}}_{ℓ Ω_{1}}^{-}, {\tilde{ϒ}}_{ℓ Ω_{2}}^{-}}, max {{\tilde{ϒ}}_{u Ω_{1}}^{-}, {\tilde{ϒ}}_{u Ω_{2}}^{-}}], {max {{\tilde{Δ}}_{Ω_{1}}^{+}, {\tilde{Δ}}_{Ω_{2}}^{+}}, min {{\tilde{Δ}}_{Ω_{1}}^{-}, {\tilde{Δ}}_{Ω_{2}}^{-}}})$

$Ω_{1} \oplus_{ℝ} Ω_{2} = ([{\tilde{ϒ}}_{ℓ Ω_{1}}^{+} + {\tilde{ϒ}}_{ℓ Ω_{2}}^{+} - {\tilde{ϒ}}_{ℓ Ω_{1}}^{+} {\tilde{ϒ}}_{ℓ τ_{2}}^{+}, {\tilde{ϒ}}_{u Ω_{1}}^{+} + {\tilde{ϒ}}_{u Ω_{2}}^{+} - {\tilde{ϒ}}_{u Ω_{1}}^{+} {\tilde{ϒ}}_{u Ω_{2}}^{+}], [- ({\tilde{ϒ}}_{ℓ Ω_{1}}^{-} {\tilde{ϒ}}_{ℓ Ω_{2}}^{-}), - ({\tilde{ϒ}}_{u Ω_{1}}^{-} {\tilde{ϒ}}_{u Ω_{2}}^{-})], {{\tilde{Δ}}_{Ω_{1}}^{+} {\tilde{Δ}}_{Ω_{2}}^{+}, - (| {\tilde{Δ}}_{Ω_{1}}^{-} | + | {\tilde{Δ}}_{Ω_{2}}^{-} | - ({\tilde{Δ}}_{Ω_{1}}^{-} {\tilde{Δ}}_{Ω_{2}}^{-}))})$

$Ω_{1} \otimes_{ℝ} Ω_{2} = ([{\tilde{ϒ}}_{ℓ Ω_{1}}^{+} {\tilde{ϒ}}_{ℓ Ω_{2}}^{+}, {\tilde{ϒ}}_{u Ω_{1}}^{+} {\tilde{ϒ}}_{u Ω_{2}}^{+}], [- (| {\tilde{ϒ}}_{ℓ Ω_{1}}^{-} | + | {\tilde{ϒ}}_{ℓ Ω_{2}}^{-} | - ({\tilde{ϒ}}_{ℓ Ω_{1}}^{-} {\tilde{ϒ}}_{ℓ Ω_{2}}^{-})), - (| {\tilde{ϒ}}_{u Ω_{1}}^{-} | + | {\tilde{ϒ}}_{u Ω_{2}}^{-} | - ({\tilde{ϒ}}_{u Ω_{1}}^{-} {\tilde{ϒ}}_{u Ω_{2}}^{-}))], {{\tilde{Δ}}_{Ω_{1}}^{+} + {\tilde{Δ}}_{Ω_{2}}^{+} - {\tilde{Δ}}_{Ω_{1}}^{+} {\tilde{Δ}}_{Ω_{2}}^{+}, - ({\tilde{Δ}}_{Ω_{1}}^{-} {\tilde{Δ}}_{Ω_{2}}^{-})})$

$Ω_{1}^{Ω} = ([({\tilde{ϒ}}_{ℓ Ω_{1}}^{+})^{Ω}, ({\tilde{ϒ}}_{u Ω_{1}}^{+})^{Ω}], [- (1 - (1 - | {\tilde{ϒ}}_{ℓ Ω_{1}}^{-} |)^{Ω}), - (1 - (1 - | {\tilde{ϒ}}_{u Ω_{1}}^{-} |)^{Ω})], {1 - (1 - {\tilde{Δ}}_{Ω_{1}}^{+})^{Ω}, - | {\tilde{Δ}}_{Ω_{1}}^{-} |^{Ω}})$

$Ω Ω_{1} = ([1 - (1 - {\tilde{ϒ}}_{ℓ Ω_{1}}^{+})^{Ω}, 1 - (1 - {\tilde{ϒ}}_{u Ω_{1}}^{+})^{Ω}], [- | {\tilde{ϒ}}_{ℓ Ω_{1}}^{-} |^{Ω}, - | {\tilde{ϒ}}_{u Ω_{1}}^{-} |^{Ω}], {({\tilde{Δ}}_{Ω_{1}}^{+})^{Ω}, - (1 - (1 - | {\tilde{Δ}}_{Ω_{1}}^{-} |)^{Ω})})$

Definition 2.7. [46] Let $Ω = ([{\tilde{ϒ}}_{ℓ Ω}^{+}, {\tilde{ϒ}}_{u Ω}^{+}], [{\tilde{ϒ}}_{ℓ Ω}^{-}, {\tilde{ϒ}}_{u Ω}^{-}], {{\tilde{Δ}}_{Ω}^{+}, {\tilde{Δ}}_{Ω}^{-}})$ be a CBFN. The complement of Ω can be defined as

$Ω^{c} = ([1 - {\tilde{ϒ}}_{u Ω}^{+}, 1 - {\tilde{ϒ}}_{ℓ Ω}^{+}], [- 1 + | {\tilde{ϒ}}_{u Ω}^{-} |, - 1 + | {\tilde{ϒ}}_{ℓ Ω}^{-} |], {1 - {\tilde{Δ}}_{Ω}^{+}, - 1 + | {\tilde{Δ}}_{Ω}^{-} |})$

Now, we propose the following theorem.

Theorem 2.8. Let

$Ω_{1} = ([{\tilde{ϒ}}_{ℓ Ω_{1}}^{+}, {\tilde{ϒ}}_{u Ω_{1}}^{+}], [{\tilde{ϒ}}_{ℓ Ω_{1}}^{-}, {\tilde{ϒ}}_{u Ω_{1}}^{-}], {{\tilde{Δ}}_{Ω_{1}}^{+}, {\tilde{Δ}}_{Ω_{1}}^{-}})$ ,

$Ω_{2} = ([{\tilde{ϒ}}_{ℓ Ω_{2}}^{+}, {\tilde{ϒ}}_{u Ω_{2}}^{+}], [{\tilde{ϒ}}_{ℓ Ω_{2}}^{-}, {\tilde{ϒ}}_{u Ω_{2}}^{-}], {{\tilde{Δ}}_{Ω_{2}}^{+}, {\tilde{Δ}}_{Ω_{2}}^{-}})$ and

$Ω = ([{\tilde{ϒ}}_{ℓ Ω}^{+}, {\tilde{ϒ}}_{u Ω}^{+}], [{\tilde{ϒ}}_{ℓ Ω}^{-}, {\tilde{ϒ}}_{u Ω}^{-}], {{\tilde{Δ}}_{Ω}^{+}, {\tilde{Δ}}_{Ω}^{-}})$ be three CBFNs, and let λ > 0, then the following ones are valid under $ℙ$ -order

(Ω^c) ^λ = (λΩ) ^c

λ (Ω^c) = (Ω^λ) ^c

$λ (Ω_{1} \cup_{ℙ} Ω_{2}) = λ Ω_{1} \cup_{ℙ} λ Ω_{2}$

$(Ω_{1} \cup_{ℙ} Ω_{2})^{λ} = Ω_{1}^{λ} \cup_{ℙ} Ω_{2}^{λ}$

$λ (Ω_{1} \cap_{ℙ} Ω_{2}) = λ Ω_{1} \cap_{ℙ} λ Ω_{2}$

$(Ω_{1} \cap_{ℙ} Ω_{2})^{λ} = Ω_{1}^{λ} \cap_{ℙ} Ω_{2}^{λ}$

Proof. We shall prove (i), (iii), (v) and (ii), (iv), (vi) can be proved analogously.

(i) $(Ω^{c})^{λ} = ([1 - {\tilde{ϒ}}_{u Ω}^{+}, 1 - {\tilde{ϒ}}_{ℓ Ω}^{+}], [- 1 + | {\tilde{ϒ}}_{u Ω}^{-} |, - 1 + | {\tilde{ϒ}}_{ℓ Ω}^{-} |], {1 - {\tilde{Δ}}_{Ω}^{+}, - 1 + | {\tilde{Δ}}_{Ω}^{-} |})^{λ}$

= $([(1 - {\tilde{ϒ}}_{u Ω}^{+})^{λ}, ((1 - {\tilde{ϒ}}_{ℓ Ω}^{+})^{λ}], [- (1 - (1 - (1 - | {\tilde{ϒ}}_{u Ω}^{-} |))^{λ}), - (1 - (1 - (1 - | {\tilde{ϒ}}_{ℓ Ω}^{-} |))^{λ})], {(1 - {\tilde{Δ}}_{Ω}^{+})^{λ}, - (1 - (1 - (1 - | {\tilde{Δ}}_{Ω}^{-} |))^{λ})})$

= $([(1 - {\tilde{ϒ}}_{u Ω}^{+})^{λ}, ((1 - {\tilde{ϒ}}_{ℓ Ω}^{+})^{λ}], [- 1 + (| {\tilde{ϒ}}_{u Ω}^{-} |)^{λ}, - 1 + (| {\tilde{ϒ}}_{ℓ Ω}^{-} |)^{λ}], {(1 - {\tilde{Δ}}_{Ω}^{+})^{λ}, - 1 + (| {\tilde{Δ}}_{Ω}^{-} |)^{λ}})$

$(λ τ)^{c} = ([(1 - {\tilde{ϒ}}_{u Ω}^{+})^{λ}, ((1 - {\tilde{ϒ}}_{ℓ Ω}^{+})^{λ}], [- 1 + (| {\tilde{ϒ}}_{u Ω}^{-} |)^{λ}, - 1 + (| {\tilde{ϒ}}_{ℓ Ω}^{-} |)^{λ}], {(1 - {\tilde{Δ}}_{Ω}^{+})^{λ}, - 1 + (| {\tilde{Δ}}_{Ω}^{-} |)^{λ}}) = (Ω^{c})^{λ}$ □

Proof. (iii) $λ (Ω_{1} \cup_{ℙ} Ω_{2}) = ([1 - (1 - max {{\tilde{ϒ}}_{ℓ Ω_{1}}^{+}, {\tilde{ϒ}}_{ℓ Ω_{2}}^{+}})^{λ}, 1 - (1 - max {{\tilde{ϒ}}_{u Ω_{1}}^{+}, {\tilde{ϒ}}_{u Ω_{2}}^{+}})^{λ}], [- | min {{\tilde{ϒ}}_{ℓ Ω_{1}}^{-}, {\tilde{ϒ}}_{ℓ Ω_{2}}^{-}} |^{λ}, - | min {{\tilde{ϒ}}_{u Ω_{1}}^{-}, {\tilde{ϒ}}_{u Ω_{2}}^{-}} |^{λ}], {1 - (1 - max {{\tilde{Δ}}_{Ω_{1}}^{+}, {\tilde{Δ}}_{Ω_{2}}^{+}})^{λ}, - | min {{\tilde{Δ}}_{Ω_{1}}^{-}, {\tilde{Δ}}_{Ω_{2}}^{-}} |^{λ}})$

$λ Ω_{1} \cup_{ℙ} λ Ω_{2} = ([max {1 - (1 - {\tilde{ϒ}}_{ℓ Ω_{1}}^{+})^{λ}, 1 - (1 - {\tilde{ϒ}}_{ℓ Ω_{2}}^{+})^{λ}}, max {1 - (1 - {\tilde{ϒ}}_{u Ω_{1}}^{+})^{λ}, 1 - (1 - {\tilde{ϒ}}_{u Ω_{2}}^{+})^{λ}}], [min {- | {\tilde{ϒ}}_{ℓ Ω_{1}}^{-} |^{λ}, - | {\tilde{ϒ}}_{ℓ Ω_{2}}^{-} |^{λ}}, min {- | {\tilde{ϒ}}_{u Ω_{1}}^{-} |^{λ}, - | {\tilde{ϒ}}_{u Ω_{2}}^{-} |^{λ}}], {max {1 - (1 - {\tilde{Δ}}_{Ω_{1}}^{+})^{λ}, 1 - (1 - {\tilde{Δ}}_{Ω_{2}}^{+})^{λ}}, min {- | {\tilde{Δ}}_{Ω_{1}}^{-} |^{λ}, - | {\tilde{Δ}}_{Ω_{2}}^{-} |^{λ}}})$

= $([1 - (1 - max {{\tilde{ϒ}}_{ℓ Ω_{1}}^{+}, {\tilde{ϒ}}_{ℓ Ω_{2}}^{+}})^{λ}, 1 - (1 - max {{\tilde{ϒ}}_{u Ω_{1}}^{+}, {\tilde{ϒ}}_{u Ω_{2}}^{+}})^{λ}], [- min {{\tilde{ϒ}}_{ℓ Ω_{1}}^{-}, {\tilde{ϒ}}_{ℓ Ω_{2}}^{-}} |^{λ}, - min {{\tilde{ϒ}}_{u Ω_{1}}^{-}, {\tilde{ϒ}}_{u Ω_{2}}^{-}} |^{λ}], {1 - (1 - max {{\tilde{Δ}}_{Ω_{1}}^{+}, {\tilde{Δ}}_{Ω_{2}}^{+}})^{λ}, - min {{\tilde{Δ}}_{Ω_{1}}^{-}, {\tilde{Δ}}_{Ω_{2}}^{-}} |^{λ}})$

= $λ (Ω_{1} \cup_{ℙ} Ω_{2})$ □

Proof. (v) $λ (Ω_{1} \cap_{ℙ} Ω_{2}) = ([1 - (1 - min {{\tilde{ϒ}}_{ℓ Ω_{1}}^{+}, {\tilde{ϒ}}_{ℓ Ω_{2}}^{+}})^{λ}, 1 - (1 - min {{\tilde{ϒ}}_{u Ω_{1}}^{+}, {\tilde{ϒ}}_{u Ω_{2}}^{+}})^{λ}], [- | max {{\tilde{ϒ}}_{ℓ Ω_{1}}^{-}, {\tilde{ϒ}}_{ℓ Ω_{2}}^{-}} |^{λ}, - | max {{\tilde{ϒ}}_{u Ω_{1}}^{-}, {\tilde{ϒ}}_{u Ω_{2}}^{-}} |^{λ}], {1 - (1 - min {{\tilde{Δ}}_{Ω_{1}}^{+}, {\tilde{Δ}}_{Ω_{2}}^{+}})^{λ}, - | max {{\tilde{Δ}}_{Ω_{1}}^{-}, {\tilde{Δ}}_{Ω_{2}}^{-}} |^{λ}})$

$λ Ω_{1} \cap_{ℙ} λ Ω_{2} = ([min {1 - (1 - {\tilde{ϒ}}_{ℓ Ω_{1}}^{+})^{λ}, 1 - (1 - {\tilde{ϒ}}_{ℓ Ω_{2}}^{+})^{λ}}, min {1 - (1 - {\tilde{ϒ}}_{u Ω_{1}}^{+})^{λ}, 1 - (1 - {\tilde{ϒ}}_{u Ω_{2}}^{+})^{λ}}], [max {- | ϒ_{ℓ Ω_{1}}^{-} |^{λ}, - | ϒ_{ℓ Ω_{2}}^{-} |^{λ}}, max {- | {\tilde{ϒ}}_{u Ω_{1}}^{-} |^{λ}, - | {\tilde{ϒ}}_{u Ω_{2}}^{-} |^{λ}}], {min {1 - (1 - {\tilde{Δ}}_{Ω_{1}}^{+})^{λ}, 1 - (1 - {\tilde{Δ}}_{Ω_{2}}^{+})^{λ}}, max {- | {\tilde{Δ}}_{Ω_{1}}^{-} |^{λ}, - | {\tilde{Δ}}_{Ω_{2}}^{-} |^{λ}}})$

= $([1 - (1 - min {{\tilde{ϒ}}_{ℓ Ω_{1}}^{+}, {\tilde{ϒ}}_{ℓ Ω_{2}}^{+}})^{λ}, 1 - (1 - min {{\tilde{ϒ}}_{u Ω_{1}}^{+}, {\tilde{ϒ}}_{u Ω_{2}}^{+}})^{λ}], [- | max {{\tilde{ϒ}}_{ℓ Ω_{1}}^{-}, {\tilde{ϒ}}_{ℓ Ω_{2}}^{-}} |^{λ}, - max {{\tilde{ϒ}}_{u Ω_{1}}^{-}, {\tilde{ϒ}}_{u Ω_{2}}^{-}} |^{λ}], {1 - (1 - min {{\tilde{Δ}}_{Ω_{1}}^{+}, {\tilde{Δ}}_{Ω_{2}}^{+}})^{λ}, - max {{\tilde{Δ}}_{Ω_{1}}^{-}, {\tilde{Δ}}_{Ω_{2}}^{-}} |^{λ}})$

= $λ (Ω_{1} \cap_{ℙ} Ω_{2})$ □

Theorem 2.9. Let $Ω_{1} = ([{\tilde{ϒ}}_{ℓ Ω_{1}}^{+}, {\tilde{ϒ}}_{u τ_{1}}^{+}], [{\tilde{ϒ}}_{ℓ Ω_{1}}^{-}, {\tilde{ϒ}}_{u Ω_{1}}^{-}], {{\tilde{Δ}}_{Ω_{1}}^{+}, {\tilde{Δ}}_{Ω_{1}}^{-}})$ ,

$Ω_{2} = ([{\tilde{ϒ}}_{ℓ Ω_{2}}^{+}, {\tilde{ϒ}}_{u Ω_{2}}^{+}], [{\tilde{ϒ}}_{ℓ Ω_{2}}^{-}, {\tilde{ϒ}}_{u Ω_{2}}^{-}], {{\tilde{Δ}}_{Ω_{2}}^{+}, {\tilde{Δ}}_{Ω_{2}}^{-}})$ and

(Ω^c) ^λ = (λΩ) ^c

λ (Ω^c) = (Ω^λ) ^c

$λ (Ω_{1} \cup_{ℝ} Ω_{2}) = λ Ω_{1} \cup_{ℝ} λ Ω_{2}$

$(Ω_{1} \cup_{ℝ} Ω_{2})^{λ} = Ω_{1}^{λ} \cup_{ℝ} Ω_{2}^{λ}$

$λ (Ω_{1} \cap_{ℝ} Ω_{2}) = λ Ω_{1} \cap_{ℝ} λ Ω_{2}$

$(Ω_{1} \cap_{ℝ} Ω_{2})^{λ} = Ω_{1}^{λ} \cap_{ℝ} Ω_{2}^{λ}$

Proof. Straightforward. □

Definition 2.10. [46] Let $Ω_{i} = ([{\tilde{ϒ}}_{ℓ Ω_{i}}^{+}, {\tilde{ϒ}}_{u Ω_{i}}^{+}], [{\tilde{ϒ}}_{ℓ Ω_{i}}^{-}, {\tilde{ϒ}}_{u Ω_{i}}^{-}], {{\tilde{Δ}}_{Ω_{i}}^{+}, {\tilde{Δ}}_{Ω_{i}}^{-}})$ , (i = 1, 2) be two CBFNs, then

Ω₁ = Ω₂ if

$[{\tilde{ϒ}}_{ℓ Ω_{1}}^{+}, {\tilde{ϒ}}_{u Ω_{1}}^{+}] = [{\tilde{ϒ}}_{ℓ Ω_{2}}^{+}, {\tilde{ϒ}}_{u Ω_{2}}^{+}]$ and $[{\tilde{ϒ}}_{ℓ Ω_{1}}^{-}, {\tilde{ϒ}}_{u Ω_{1}}^{-}] = [{\tilde{ϒ}}_{ℓ Ω_{2}}^{-}, {\tilde{ϒ}}_{u Ω_{2}}^{-}]$

${\tilde{Δ}}_{Ω_{1}}^{+} = {\tilde{Δ}}_{Ω_{2}}^{+}$ and ${\tilde{Δ}}_{Ω_{1}}^{-} = {\tilde{Δ}}_{Ω_{2}}^{-}$ .

$Ω_{1} \subseteq_{ℙ} Ω_{2}$ if

$[{\tilde{ϒ}}_{ℓ Ω_{1}}^{+}, {\tilde{ϒ}}_{u Ω_{1}}^{+}] \leq [{\tilde{ϒ}}_{ℓ Ω_{2}}^{+}, {\tilde{ϒ}}_{u Ω_{2}}^{+}]$ and $[{\tilde{ϒ}}_{ℓ Ω_{1}}^{-}, {\tilde{ϒ}}_{u Ω_{1}}^{-}] \geq [{\tilde{ϒ}}_{ℓ Ω_{2}}^{-}, {\tilde{ϒ}}_{u Ω_{2}}^{-}]$

${\tilde{Δ}}_{Ω_{1}}^{+} \leq {\tilde{Δ}}_{Ω_{2}}^{+}$ and ${\tilde{Δ}}_{Ω_{1}}^{-} \geq {\tilde{Δ}}_{Ω_{2}}^{-}$ .

$Ω_{1} \subseteq_{ℝ} Ω_{2}$ if

${\tilde{Δ}}_{Ω_{1}}^{+} \geq {\tilde{Δ}}_{Ω_{2}}^{+}$ and ${\tilde{Δ}}_{Ω_{1}}^{-} \leq {\tilde{Δ}}_{Ω_{2}}^{-}$ .

Definition 2.11. [45] For any CBFN Ω, the score function under $ℙ$ -order can be defined as

$S_{P} (Ω) = \frac{1}{6} ([{\tilde{ϒ}}_{ℓ Ω}^{+} + {\tilde{ϒ}}_{u Ω}^{+}] + [1 + {\tilde{ϒ}}_{ℓ Ω}^{-} + 1 + {\tilde{ϒ}}_{u Ω}^{-}] - {\tilde{Δ}}_{Ω}^{+} + 1 + {\tilde{Δ}}_{Ω}^{-})$ .

Similarly, the score function under $ℝ$ -order can be defined as $S_{R} (Ω) = \frac{1}{6} ([{\tilde{ϒ}}_{ℓ Ω}^{+} + {\tilde{ϒ}}_{u Ω}^{+}] + [1 + {\tilde{ϒ}}_{ℓ Ω}^{-} + 1 + {\tilde{ϒ}}_{u Ω}^{-}] + {\tilde{Δ}}_{Ω}^{+} - (1 + {\tilde{Δ}}_{Ω}^{-}))$ where $S_{P, R} (Ω) \in [0, 1]$ .

Definition 2.12. [45] For any CBFN Ω, the accuracy function can be defined as

$A (Ω) = \frac{1}{6} ([{\tilde{ϒ}}_{ℓ Ω}^{+} + {\tilde{ϒ}}_{u Ω}^{+}] - [{\tilde{ϒ}}_{ℓ Ω}^{-} + {\tilde{ϒ}}_{u Ω}^{-}] + {\tilde{Δ}}_{Ω}^{+} - {\tilde{Δ}}_{Ω}^{-})$ where $A (Ω) \in [0, 1]$ .

We use score function and accuracy function to compare two CBFNs.

(i) If $S_{P, R} (Ω_{1}) < S_{P, R} (Ω_{2})$ , we say that Ω₁ is smaller than Ω₂, and denote it by Ω₁ ≺ Ω₂.

(ii) If $S_{P, R} (Ω_{1}) > S_{P, R} (Ω_{2})$ , we say that Ω₁ is greater than Ω₂, and denote it by Ω₁ ≻ Ω₂.

(iii) If $S_{P, R} (Ω_{1}) = S_{P, R} (Ω_{2})$ , then

if $A (Ω_{1}) > A (Ω_{2})$ , then Ω₁ ≻ Ω₂

if $A (Ω_{1}) = A (Ω_{2})$ , then Ω₁ ∼ Ω₂

3 Dombi operations on CBFNs

3.1 Dombi operations

Dombi [34] established Dombi product and Dombi sum, which are special cases of $\hat{T}$ -norms and $\hat{T}$ -conorms, respectively.

Definition 3.1. Let ${\bar{ϱ}}_{1}$ and ${\bar{ϱ}}_{2}$ be any two real numbers such that $({\bar{ϱ}}_{1}, {\bar{ϱ}}_{2}) \in [0, 1] \times [0, 1]$ . Then the Dombi’s $\hat{T}$ -norms and $\hat{T}$ -conorms ( $\hat{S}$ -norm) for ${\bar{ϱ}}_{1}$ and ${\bar{ϱ}}_{2}$ can be defined as $\begin{matrix} \ddot{D} ({\bar{ϱ}}_{1}, {\bar{ϱ}}_{2}) & = & \frac{1}{1 + {[{(\frac{1 - {\bar{ϱ}}_{1}}{{\bar{ϱ}}_{1}})}^{\overset{˘}{$}} + {(\frac{1 - {\bar{ϱ}}_{2}}{{\bar{ϱ}}_{2}})}^{\overset{˘}{$}}]}^{\frac{1}{\overset{˘}{$}}}} \\ {\ddot{D}}^{*} ({\bar{ϱ}}_{1}, {\bar{ϱ}}_{2}) & = & 1 - \frac{1}{1 + {[{(\frac{{\bar{ϱ}}_{1}}{1 - {\bar{ϱ}}_{1}})}^{\overset{˘}{$}} + {(\frac{{\bar{ϱ}}_{2}}{1 - {\bar{ϱ}}_{2}})}^{\overset{˘}{$}}]}^{\frac{1}{\overset{˘}{$}}}} \end{matrix}$ with $\overset{˘}{$} \in [1, \infty)$ as an operational parameter.

Based on Dombi’s $\hat{T}$ -norms and $\hat{T}$ -conorms, we define Dombi’s operations on CBFNs as follows.

3.2 Dombi operations on CBFNs under $ℙ$ -order

Definition 3.2. Let ${\ddot{¥}}_{1} = ([{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}, {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}], [{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{-}, {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{-}], {{\tilde{Δ}}_{{\ddot{¥}}_{1}}^{+}, {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{-}})$ and

${\ddot{¥}}_{2} = ([{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{+}, {\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{+}], [{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{-}, {\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{-}], {{\tilde{Δ}}_{{\ddot{¥}}_{2}}^{+}, {\tilde{Δ}}_{{\ddot{¥}}_{2}}^{-}})$ be two CBFNs, and $\overset{˘}{ξ} \in [1, \infty)$ and λ > 0. Then the Dombi’s operations (under $ℙ$ -Order) on CBFNs are as follows

(i) ${\ddot{¥}}_{1} \oplus_{ℙ} {\ddot{¥}}_{2}$ =

$([1 - \frac{1}{1 + {{(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}} + {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}} + {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

$[\frac{- 1}{1 + {{(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{-} |})}^{\overset{˘}{ξ}} + {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{-} |})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{-} |})}^{\overset{˘}{ξ}} + {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{-} |})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${1 - \frac{1}{1 + {{(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{1}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}} + {(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{2}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{2}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{-} |})}^{\overset{˘}{ξ}} + {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{2}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{2}}^{-} |})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

(ii) ${\ddot{¥}}_{1} \otimes_{ℙ} {\ddot{¥}}_{2}$ =

$([\frac{1}{1 + {{(\frac{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}}{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}} + {(\frac{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{+}}{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{1}{1 + {{(\frac{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}}{{\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}} + {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

$[- 1 + \frac{1}{1 + {{(\frac{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{-} |}{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{-}})}^{\overset{˘}{ξ}} + {(\frac{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{-} |}{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{-}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}, - 1 + \frac{1}{1 + {{(\frac{| {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{-} |}{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{-}})}^{\overset{˘}{ξ}} + {(\frac{| {\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{-} |}{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{-}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${\frac{1}{1 + {{(\frac{1 - {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{+}}{{\tilde{Δ}}_{{\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}} + {(\frac{1 - {\tilde{Δ}}_{{\ddot{¥}}_{2}}^{+}}{{\tilde{Δ}}_{{\ddot{¥}}_{2}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}, - 1 + \frac{1}{1 + {{(\frac{| {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{-} |}{1 + {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{-}})}^{\overset{˘}{ξ}} + {(\frac{| {\tilde{Δ}}_{{\ddot{¥}}_{2}}^{-} |}{1 + {\tilde{Δ}}_{{\ddot{¥}}_{2}}^{-}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

(iii) $λ {\ddot{¥}}_{1}$ = $([1 - \frac{1}{1 + {{λ {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{λ {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

$[\frac{- 1}{1 + {{λ {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{λ {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{1} |}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}],$

${1 - \frac{1}{1 + {{λ {(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{1}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{λ {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

(iv) ${\ddot{¥}}_{1}^{λ}$ = $([\frac{1}{1 + {{λ {(\frac{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}}{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{1}{1 + {{λ {(\frac{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}}{{\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ , $[- 1 + \frac{1}{1 + {{λ {(\frac{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{-} |}{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, - 1 + \frac{1}{1 + {{λ {(\frac{| {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{-} |}{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ , ${\frac{1}{1 + {{λ {(\frac{1 - {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{+}}{{\tilde{Δ}}_{{\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, - 1 + \frac{1}{1 + {{λ {(\frac{| {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{-} |}{1 + {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

3.3 Dombi operations on CBFNs under $ℝ$ -order

Definition 3.3. Let ${\ddot{¥}}_{1} = ([{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}, {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}], [{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{-}, {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{-}], {{\tilde{Δ}}_{{\ddot{¥}}_{1}}^{+}, {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{-}})$ , ${\ddot{¥}}_{2} = ([{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{+}, {\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{+}], [{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{-}, {\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{-}], {{\tilde{Δ}}_{{\ddot{¥}}_{2}}^{+}, {\tilde{Δ}}_{{\ddot{¥}}_{2}}^{-}})$ be two CBFNs, whereas $\overset{˘}{ξ} \in [1, \infty)$ and λ > 0. Then the Dombi’s operations(under $ℝ$ -Order) between them are as under

(i) ${\ddot{¥}}_{1} \oplus_{ℝ} {\ddot{¥}}_{2}$ = $([1 - \frac{1}{1 + {{(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}} + {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}} + {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

(ii) ${\ddot{¥}}_{1} \otimes_{ℝ} {\ddot{¥}}_{2}$ = $([\frac{1}{1 + {{(\frac{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}}{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}} + {(\frac{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{+}}{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{1}{1 + {{(\frac{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}}{{\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}} + {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${1 - \frac{1}{1 + {{(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{1}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}} + {(\frac{μ_{{\ddot{¥}}_{2}}^{+}}{1 - μ_{{\ddot{¥}}_{2}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{-} |})}^{\overset{˘}{ξ}} + {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{2}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{2}}^{-} |})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

(iii) $λ {\ddot{¥}}_{1}$ =

$([1 - \frac{1}{1 + {{λ {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{λ {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${\frac{1}{1 + {{λ {(\frac{1 - {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{+}}{{\tilde{Δ}}_{{\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, - 1 + \frac{1}{1 + {{λ {(\frac{| {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{-} |}{1 + {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

$[- 1 + \frac{1}{1 + {{λ {(\frac{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{-} |}{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, - 1 + \frac{1}{1 + {{λ {(\frac{| {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{-} |}{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

4 $ℙ$ -order cubic bipolar fuzzy Dombi averaging aggregation operators

In this section, we introduce two Dombi weighted aggregation operators based on $ℙ$ -order Dombi operations of CBFNs including $ℙ$ -order cubic bipolar fuzzy Dombi weighted averaging operator ( $ℙ$ -CBFDWAO) and $ℙ$ -order cubic bipolar fuzzy Dombi ordered weighted averaging operator ( $ℙ$ -CBFDOWAO). Additionally, we discuss some significant characteristics of these two operators.

4.1 $ℙ$ -order cubic bipolar fuzzy Dombi weighted averaging ( $ℙ$ -CBFDWA) operator

Definition 4.1. Let ${\ddot{¥}}_{i} = ([{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}], [{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}], {{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}, {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}})$ , where i = (1, 2, . . . , t), be the collection of CBFNs. Then the $ℙ$ -order cubic bipolar fuzzy Dombi weighted averaging operator( $ℙ$ -CBFDWA) operator is a mapping $ℙ - CBFDWA : {\ddot{¥}}^{t} \to \ddot{¥}$ such that $ℙ$ -CBFDWA $({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{n})$ = ${\underset{i = 1}{\overset{t}{\oplus}}}_{ℙ} η_{i} {\ddot{¥}}_{i}$

where η = (η₁, η₂, . . . , η_t) is the weight vector of CBFNs ${\ddot{¥}}_{i}$ , i = 1, 2, . . . , t with η_i ∈ [0, 1] and $\sum_{i = 1}^{t} η_{i} = 1$ .

Theorem 4.2. Let $({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ be a family of CBFNs with the corresponding weight vector η = (η₁, η₂, . . . η_t) such that η_i ∈ [0, 1] and $\sum_{i = 1}^{t} η_{i} = 1$ , then the aggregated value of these CBFNs by using $ℙ$ -CBFDWA operator is again a CBFN and it can be defined as $ℙ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ = ${\underset{i = 1}{\overset{t}{\oplus}}}_{ℙ} η_{i} {\ddot{¥}}_{i}$ = $([1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ , $[\frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$ ...... (∗)

Proof. The proof is by mathematical induction on t. If t=2, conforming to the Dombi operations on CBFNs, we see that $ℙ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2})$ = $η_{1} {\ddot{¥}}_{1} \oplus_{ℙ} η_{2} {\ddot{¥}}_{2}$

= $([1 - \frac{1}{1 + {η_{1} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}} + η_{2} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {η_{1} {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}} + η_{2} {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

$[\frac{- 1}{1 + {η_{1} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{1}}^{-} |})}^{\overset{˘}{ξ}} + η_{2} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{2}}^{-} |})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {η_{1} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{1}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{1} |}^{-}})}^{\overset{˘}{ξ}} + η_{2} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{2}}^{-} |})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${1 - \frac{1}{1 + {η_{1} {(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{1}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{+}})}^{\overset{˘}{ξ}} + η_{2} {(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{2}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{2}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {η_{1} {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{1}}^{-} |})}^{\overset{˘}{ξ}} + η_{2} {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{2}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{2}}^{-} |})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}})$ = $([1 - \frac{1}{1 + {{\sum_{i = 1}^{2} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{\sum_{i = 1}^{2} η_{i} {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ , $[\frac{- 1}{1 + {{\sum_{i = 1}^{2} η_{i} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{2} η_{i} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{i} |}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${1 - \frac{1}{1 + {{\sum_{i = 1}^{2} η_{i} {(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{2} η_{i} {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

Thus, (*) holds for t=2. Suppose that (∗) holds for t=p, where p ∈ N, i.e. $ℙ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{p}) = {\underset{i = 1}{\overset{p}{\oplus}}}_{ℙ} η_{i} {\ddot{¥}}_{i}$ = $([1 - \frac{1}{1 + {{\sum_{i = 1}^{p} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{\sum_{i = 1}^{p} η_{i} {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

$[\frac{- 1}{1 + {{\sum_{i = 1}^{p} η_{i} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{p} η_{i} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{i} |}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${1 - \frac{1}{1 + {{\sum_{i = 1}^{p} η_{i} {(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{p} η_{i} {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

Now, if n = p + 1, then we have

$ℙ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{p}, {\ddot{¥}}_{p + 1}) = \oplus_{i = 1}^{p} η_{i} {\ddot{¥}}_{i} \oplus η_{p + 1} {\ddot{¥}}_{p + 1}$ = $([1 - \frac{1}{1 + {{\sum_{i = 1}^{p} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{\sum_{i = 1}^{p} η_{i} {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ , $[\frac{- 1}{1 + {{\sum_{i = 1}^{p} η_{i} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{p} η_{i} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{i} |}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${1 - \frac{1}{1 + {{\sum_{i = 1}^{p} η_{i} {(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}})}^{k}}}^{\frac{1}{k}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{p} η_{i} {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}}) \oplus ([1 - \frac{1}{1 + {{η_{p + 1} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{p + 1}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{s + 1}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{η_{s + 1} {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{p + 1}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{p + 1}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ , $[\frac{- 1}{1 + {{η_{p + 1} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{p + 1}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{p + 1}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{k}}}, \frac{- 1}{1 + {{η_{p + 1} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{p + 1}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{p + 1} |}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${1 - \frac{1}{1 + {{η_{p + 1} {(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{p + 1}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{p + 1}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{η_{p + 1} {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{p + 1}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{p + 1}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

= $([1 - \frac{1}{1 + {{\sum_{i = 1}^{p + 1} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{\sum_{i = 1}^{p + 1} η_{i} {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ , $[\frac{- 1}{1 + {{\sum_{i = 1}^{p + 1} η_{i} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{p + 1} η_{i} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{i} |}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${1 - \frac{1}{1 + {{\sum_{i = 1}^{p + 1} η_{i} {(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{p + 1} η_{i} {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

Therefore, for t = p + 1, theorem holds. Hence, (∗)holds for all t ∈ N. □

Example 4.3. Suppose that ${\ddot{¥}}_{1} = ([0.5, 0.7], [- 0.2, - 0.1], {0.65, - 0.17})$ ,

${\ddot{¥}}_{2} = ([0.3, 0.4], [- 0.8, - 0.6], {0.6, - 0.5})$ ,

${\ddot{¥}}_{3} = ([0.1, 0.4], [- 0.7, - 0.5], {0.5, - 0.3})$ ${\ddot{¥}}_{4} = ([0.6, 0.8], [- 0.4, - 0.2], {0.4, - 0.1})$ are four CBFNs and η = (0.3, 0.2, 0.3, 0.2) is the weight vector for these CBFNs. Take the value of operational parameter $\overset{˘}{ξ} = 4$ . Now,

$ℙ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, {\ddot{¥}}_{3}, {\ddot{¥}}_{4}) = {\underset{i = 1}{\overset{4}{\oplus}}}_{ℙ} η_{i} {\ddot{¥}}_{i}$

$[\frac{- 1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{i} |}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${1 - \frac{1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

= ([0.5173, 0.7358] , [-0.2519, - 0.1298] , {0.5989, - 0.1386} )

Now, we discuss some important properties of $ℙ$ -CBFDWA operator.

Theorem 4.4 (Idempotency)

Let ${\ddot{¥}}_{i} = ([{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}], [{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}], {{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}, {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}})$ , where i = (1, 2, . . . t), be a family of CBFNs such that ${\ddot{¥}}_{i} = \ddot{¥}$ ∀ (i = 1, 2, . . . , t), then

$ℙ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ = $\ddot{¥}$ .

Proof. Since ${\ddot{¥}}_{i} = \ddot{¥}$

for all (i = 1, 2, . . . , t), then by Eq (∗), we have $ℙ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ =

$([1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

$[\frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{i} |}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

= $([1 - \frac{1}{1 + {{(\frac{{\tilde{ϒ}}_{ℓ \ddot{¥}}^{+}}{1 - {\tilde{ϒ}}_{ℓ \ddot{¥}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{(\frac{{\tilde{ϒ}}_{u \ddot{¥}}^{+}}{1 - {\tilde{ϒ}}_{u \ddot{¥}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ , $[\frac{- 1}{1 + {{(\frac{1 + {\tilde{ϒ}}_{ℓ \ddot{¥}}^{-}}{| {\tilde{ϒ}}_{ℓ \ddot{¥}}^{-} |})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{(\frac{1 + {\tilde{ϒ}}_{u \ddot{¥}}^{-}}{| {\tilde{ϒ}}_{u \ddot{¥}}^{-} |})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${1 - \frac{1}{1 + {{(\frac{{\tilde{Δ}}_{\ddot{¥}}^{+}}{1 - {\tilde{Δ}}_{\ddot{¥}}^{+}})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{(\frac{1 + {\tilde{Δ}}_{\ddot{¥}}^{-}}{| {\tilde{Δ}}_{\ddot{¥}}^{-} |})}^{\overset{˘}{ξ}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

= $([1 - \frac{1}{1 + {(\frac{{\tilde{ϒ}}_{ℓ \ddot{¥}}^{+}}{1 - {\tilde{ϒ}}_{ℓ \ddot{¥}}^{+}})}}, 1 - \frac{1}{1 + {(\frac{{\tilde{ϒ}}_{u \ddot{¥}}^{+}}{1 - {\tilde{ϒ}}_{u \ddot{¥}}^{+}})}}]$ ,

$[\frac{- 1}{1 + {(\frac{1 + {\tilde{ϒ}}_{ℓ \ddot{¥}}^{-}}{| {\tilde{ϒ}}_{ℓ \ddot{¥}}^{-} |})}}, \frac{- 1}{1 + {(\frac{1 + {\tilde{ϒ}}_{u \ddot{¥}}^{-}}{| {\tilde{ϒ}}_{u \ddot{¥}}^{-} |})}}]$ ,

${1 - \frac{1}{1 + {(\frac{{\tilde{Δ}}_{\ddot{¥}}^{+}}{1 - {\tilde{Δ}}_{\ddot{¥}}^{+}})}}, \frac{- 1}{1 + {(\frac{1 + {\tilde{Δ}}_{\ddot{¥}}^{-}}{| {\tilde{Δ}}_{\ddot{¥}}^{-} |})}}})$

= $([{\tilde{ϒ}}_{ℓ \ddot{¥}}^{+}, {\tilde{ϒ}}_{u \ddot{¥}}^{+}], [{\tilde{ϒ}}_{ℓ \ddot{¥}}^{-}, {\tilde{ϒ}}_{u \ddot{¥}}^{-}], {{\tilde{Δ}}_{\ddot{¥}}^{+}, {\tilde{Δ}}_{\ddot{¥}}^{-}})$

= $\ddot{¥}$ □

Theorem 4.5 (Monotonicity)

${\ddot{¥}}_{i}^{*} = ([{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}^{*}}^{+}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}^{*}}^{+}], [{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}^{*}}^{-}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}^{*}}^{-}], {{\tilde{Δ}}_{{\ddot{¥}}_{i}^{*}}^{+}, {\tilde{Δ}}_{{\ddot{¥}}_{i}^{*}}^{-}})$ , i = (1, 2, . . . , t), be two huddles of CBFNs such that ${\ddot{¥}}_{i} \leq {\ddot{¥}}_{i}^{*}$ ∀ i, then

$ℙ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t}) \leq ℙ - CBFDWA ({\ddot{¥}}_{1}^{*}, {\ddot{¥}}_{2}^{*}, . . ., {\ddot{¥}}_{t}^{*})$

Proof. Since ${\ddot{¥}}_{i} \leq {\ddot{¥} i}^{*}$ for i = (1, 2, . . . , t), we have ${\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+} \leq {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}^{*}}^{+}$ $\Rightarrow \frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}} \leq \frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}^{*}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}^{*}}^{+}}$ $\Rightarrow 1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}} \leq 1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}^{*}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}^{*}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}$ $\Rightarrow \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \geq \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}^{*}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}^{'}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}$ $\Rightarrow 1 - \frac{1}{1 + {{\sum_{i = 1}^{n} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \leq 1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}^{*}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}^{*}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}$ In a similar manner, we can prove that $1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \leq 1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{i}^{*}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{i}^{*}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}$ ,

$\frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \geq \frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}^{*}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}^{*}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}$ and $\frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \geq \frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{i}^{*}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{i}^{*}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}$ ,

$1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \leq 1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{i}^{*}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{i}^{*}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}$ and $\frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \geq \frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{i}^{*}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{i}^{*}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}$

Hence, we conclude that

$ℙ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t}) \leq ℙ - CBFDWA ({\ddot{¥}}_{1}^{*}, {\ddot{¥}}_{2}^{*}, . . ., {\ddot{¥}}_{t}^{*})$ □

Theorem 4.6 (Boundedness)

Let ${\ddot{¥}}_{min} = min ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ and

${\ddot{¥}}_{max} = max ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ , then

${\ddot{¥}}_{\min} \leq ℙ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{n}) \leq {\ddot{¥}}_{max}$ .

Proof. Let ${\ddot{¥}}_{min} = min ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t}) = ([min_{i} ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}), min_{i} ({\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+})], [max_{i} ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}), max_{i} ({\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-})], {min_{i} ({\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}), max_{i} ({\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-})})$ and

${\ddot{¥}}_{max} = max ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t}) = ([max_{i} ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}), max_{i} ({\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+})], [min_{i} ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}), min_{i} ({\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-})], {max_{i} ({\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}), min_{i} ({\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-})})$ . Since,

$min ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}) \leq {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+} \leq max ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+})$

$\Rightarrow 1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{min ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+})}{1 - min ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+})})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}} \leq 1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}} \leq 1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{max ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+})}{1 - max ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+})})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}$

$\Rightarrow \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{min ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+})}{1 - min ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+})})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \geq \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \geq \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{max ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+})}{1 - max ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+})})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}$

$\Rightarrow 1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{min ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+})}{1 - min ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+})})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \leq 1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \leq 1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{max ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+})}{1 - max ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+})})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}$

In the same manner, the following inequalities can be proved

$1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{min ({\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+})}{1 - min ({\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+})})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \leq 1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \leq 1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{max ({\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+})}{1 - max ({\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+})})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}$

$\frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + min ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-})}{| min ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}) |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \leq \frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \leq \frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + max ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-})}{| max ({\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}) |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}$

$\frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + min ({\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-})}{| min ({\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}) |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \leq \frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \leq \frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + max ({\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-})}{| max ({\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}) |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}$

$1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{min ({\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+})}{1 - min ({\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+})})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \leq 1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \leq 1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{max ({\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+})}{1 - max ({\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+})})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}$

$\frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + min ({\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-})}{| min ({\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}) |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \leq \frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}} \leq \frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + max ({\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-})}{| max ({\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}) |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}$

Hence, we conclude that

${\ddot{¥}}_{\min} \leq ℙ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t}) \leq {\ddot{¥}}_{\max}$ □

4.2 $ℙ$ -order cubic bipolar fuzzy Dombi ordered weighted averaging ( $ℙ$ -CBFDOWA) operator

Definition 4.7. Let ${\ddot{¥}}_{i} = ([{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}], [{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}], {{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}, {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}})$ , where i = (1, 2, . . . t), be a huddle of CBFNs with the corresponding weight vector η = (η₁, η₂, . . . η_t) with η_i ∈ [0, 1] and $\sum_{i = 1}^{t} η_{i} = 1$ . Then the $ℙ$ -order cubic bipolar fuzzy Dombi ordered weighted averaging( $ℙ$ -CBFDOWA) operator is a mapping $ℙ - CBFDOWA : {\ddot{¥}}^{t} \to \ddot{¥}$ such that $ℙ - CBFDOWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t}) = \oplus_{i = 1}^{t} η_{i} {\ddot{¥}}_{σ (i)}$

where $({\ddot{¥}}_{σ (1)}, {\ddot{¥}}_{σ (2)}, . . ., {\ddot{¥}}_{σ (t)})$ is an arrangement of $({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ such that ${\ddot{¥}}_{σ (i - 1)} \geq {\ddot{¥}}_{σ (i)}$ ∀ (i = 1, 2, . . . , t).

Theorem 4.8. Let $({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ be a family of CBFNs with the corresponding weight vector η = (η₁, η₂, . . . η_t) such that η_i ∈ [0, 1] and $\sum_{i = 1}^{t} η_{i} = 1$ ,then the aggregated value of these CBFNs by using $ℙ - CBFDOWA$ operator is again a CBFN and it can be defined as

$ℙ - CBFDOWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{n})$ =

$([1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{σ (i)}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{σ (i)}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{σ (i)}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{σ (i)}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

$[\frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{σ (i)}}^{-}}{| {\tilde{ϒ}}_{ℓ τ_{σ (i)}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{σ (i)}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{σ (i)}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{σ (i)}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{σ (i)}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{n} η_{i} {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{σ (i)}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{σ (i)}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

.....(∗)

where $({\ddot{¥}}_{σ (1)}, {\ddot{¥}}_{σ (2)}, . . ., {\ddot{¥}}_{σ (n)})$ is an arrangement of $({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ such that ${\ddot{¥}}_{σ (i - 1)} \geq {\ddot{¥}}_{σ (i)}$ ∀ (i = 1, 2, . . . , t).

Example 4.9. Suppose that

${\ddot{¥}}_{1} = ([0.2, 0.35], [- 0.5, - 0.4], {0.5, - 0.2})$ ,

${\ddot{¥}}_{2} = ([0.4, 0.6], [- 0.3, - 0.1], {0.7, - 0.6})$ ,

${\ddot{¥}}_{3} = ([0.5, 0.6], [- 0.4, - 0.2], {0.3, - 0.2})$ ,

${\ddot{¥}}_{4} = ([0.6, 0.8], [- 0.6, - 0.4], {0.4, - 0.5})$

are four CBFNs and η = (0.3, 0.2, 0.3, 0.2) is the weight vector for these CBFNs. Take the value of operational parameter $\overset{˘}{ξ} = 4$ . Now, these four CBFNs can be aggregated by using $ℙ - CBFDOWA$ operator. First, we calculate score functions of ${\ddot{¥}}_{1}$ , ${\ddot{¥}}_{2}$ , ${\ddot{¥}}_{3}$ and ${\ddot{¥}}_{4}$ by using Eq(1) as follows

$S_{ℙ} ({\ddot{¥}}_{1}) = \frac{0.2 + 0.35 + 1 - 0.5 + 1 - 0.4 - 0.5 + 1 - 0.2}{6} = 0.3250$ ,

$S_{ℙ} ({\ddot{¥}}_{2}) = \frac{0.4 + 0.6 + 1 - 0.3 + 1 - 0.1 - 0.7 + 1 - 0.6}{6} = 0.3833$ , $S_{ℙ} ({\ddot{¥}}_{3}) = \frac{0.5 + 0.6 + 1 - 0.4 + 1 - 0.2 - 0.3 + 1 - 0.2}{6} = 0.5000$ , $S_{ℙ} ({\ddot{¥}}_{4}) = \frac{0.6 + 0.8 + 1 - 0.6 + 1 - 0.4 - 0.4 + 1 - 0.5}{6} = 0.4167$

This gives, $S_{ℙ} ({\ddot{¥}}_{3}) > S_{ℙ} ({\ddot{¥}}_{4}) > S_{ℙ} ({\ddot{¥}}_{2}) > S_{ℙ} ({\ddot{¥}}_{1})$ . So, we have ${\ddot{¥}}_{σ (1)} = {\ddot{¥}}_{3}$ , ${\ddot{¥}}_{σ (2)} = {\ddot{¥}}_{4}$ , ${\ddot{¥}}_{σ (3)} = {\ddot{¥}}_{2}$ and ${\ddot{¥}}_{σ (4)} = {\ddot{¥}}_{1}$ . Now,

$ℙ - CBFDOWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, {\ddot{¥}}_{3}, {\ddot{¥}}_{4}) = \oplus_{i = 1}^{4} η_{i} {\ddot{¥}}_{σ (i)}$

= $([1 - \frac{1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{σ (i)}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{σ (i)}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{σ (i)}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{σ (i)}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ , $[\frac{- 1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{σ (i)}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{σ (i)}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{σ (i)}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{σ (i)} |}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${1 - \frac{1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{{\tilde{Δ}}_{{\ddot{¥}}_{σ (i)}}^{+}}{1 - {\tilde{Δ}}_{{\ddot{¥}}_{σ (i)}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{1 + {\tilde{Δ}}_{{\ddot{¥}}_{σ (i)}}^{-}}{| {\tilde{Δ}}_{{\ddot{¥}}_{σ (i)}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

= ([0.5198, 0.7307] , [-0.3563, - 0.1294] , {0.6349, - 0.2291} )

One can easily prove the following properties for $ℙ - CBFDOWA$ operator.

Theorem 4.10. (Idempotency)

Let ${\ddot{¥}}_{i} = ([{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}], [{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}], {{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}, {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}})$ , where i = (1, 2, . . . t), be a huddle of CBFNs such that ${\ddot{¥}}_{i} = \ddot{¥}$ ∀ (i = 1, 2, . . . , t), then

$ℙ - CBFDOWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ = $\ddot{¥}$ .

Theorem 4.11. (Monotonicity)

$ℙ - CBFDOWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t}) \leq ℙ - CBFDOWA ({\ddot{¥}}_{1}^{*}, {\ddot{¥}}_{2}^{*}, . . ., {\ddot{¥}}_{t}^{*})$ .

Theorem 4.12. (Boundedness)

Let ${\ddot{¥}}_{min} = min ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ and

${\ddot{¥}}_{max} = max ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ , then

${\ddot{¥}}_{\min} \leq ℙ - CBFDOWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t}) \leq {\ddot{¥}}_{\max}$ .

5 $ℝ$ -order cubic bipolar fuzzy Dombi averaging aggregation operators

In the previous section, we discussed Dombi weighted averaging aggregation operators under $ℙ$ -order. In this section, we shall discuss the same operators under $ℝ$ -order,i.e., $ℝ$ -order cubic bipolar fuzzy Dombi weighted averaging operator $(ℝ - CBFDWAO)$ and $ℝ$ -order cubic bipolar fuzzy Dombi ordered weighted averaging operator $(ℝ - CBFDOWAO)$ .

5.1 $ℝ$ -order cubic bipolar fuzzy Dombi weighted averaging ( $ℝ$ -CBFDWA) operator

Definition 5.1. Let ${\ddot{¥}}_{i} = ([{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}], [{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}], {{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}, {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}})$ , where i = (1, 2, . . . t), be a huddle of CBFNs.Then the $ℝ$ -order cubic bipolar fuzzy Dombi weighted averaging( $ℝ$ -CBFDWA) operator is a mapping $ℝ - CBFDWA : {\ddot{¥}}^{t} \to \ddot{¥}$ such that

$ℝ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t}) = {\underset{i = 1}{\overset{t}{\oplus}}}_{ℝ} η_{i} {\ddot{¥}}_{i}$

where η = (η₁, η₂, . . . η_t) is the weight vector of CBFNs ${\ddot{¥}}_{i}$ (i=1,2,...,t) with η_i ∈ [0, 1] and $\sum_{i = 1}^{t} η_{i} = 1$ .

Theorem 5.2. Let $({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ be a family of CBFNs with the corresponding weight vector η = (η₁, η₂, . . . η_t) such that η_i ∈ [0, 1] and $\sum_{i = 1}^{t} η_{i} = 1$ ,then the agglomerated value of these CBFNs by using $ℝ - CBFDWA$ operator is again a CBFN and it can be established as

$ℝ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ =

$[\frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${\frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 - {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}}{{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, - 1 + \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{| {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-} |}{1 + {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

Example 5.3. Suppose that

${\ddot{¥}}_{1} = ([0.5, 0.7], [- 0.2, - 0.1], {0.65, - 0.17})$ ,

${\ddot{¥}}_{2} = ([0.3, 0.4], [- 0.8, - 0.6], {0.6, - 0.5})$ ,

${\ddot{¥}}_{3} = ([0.1, 0.4], [- 0.7, - 0.5], {0.5, - 0.3})$ ,

${\ddot{¥}}_{4} = ([0.6, 0.8], [- 0.4, - 0.2], {0.45, - 0.1})$

are four CBFNs and η = (0.3, 0.2, 0.3, 0.2) is the weight vector for these CBFNs. Take the value of operational parameter $\overset{˘}{ξ} = 4$ . Now,

$ℝ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, {\ddot{¥}}_{3}, {\ddot{¥}}_{4}) = {\underset{i = 1}{\overset{4}{\oplus}}}_{ℝ} η_{i} {\ddot{¥}}_{i}$

= $([1 - \frac{1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ , $[\frac{- 1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${\frac{1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{1 - {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}}{{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, - 1 + \frac{1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{| {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-} |}{1 + {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

= ([0.5173, 0.7358] , [-0.2519, - 0.1298] , {0.5130, - 0.4039})

The following properties hold for $ℝ$ -CBFDWA operator as well.

Theorem 5.4. (Idempotency) Let

${\ddot{¥}}_{i} = ([{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}], [{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}], {{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}, {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}})$ , where i = (1, 2, . . . t), be a huddle of CBFNs such that ${\ddot{¥}}_{i} = \ddot{¥}$ ∀ (i = 1, 2, . . . , t), then

$ℝ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . . {\ddot{¥}}_{t})$ = $\ddot{¥}$

Theorem 5.5. (Monotonicity) Let

$ℝ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . . {\ddot{¥}}_{t}) \leq ℝ - CBFDWA ({\ddot{¥}}_{1}^{*}, {\ddot{¥}}_{2}^{*}, . . ., {\ddot{¥}}_{t}^{*})$

Theorem 5.6. (Boundedness) Let

Let ${\ddot{¥}}_{min} = min ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ and

${\ddot{¥}}_{max} = max ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ , then

${\ddot{¥}}_{min} \leq ℝ - CBFDWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . . {\ddot{¥}}_{t}) \leq {\ddot{¥}}_{max}$ .

5.2 $ℝ$ -order cubic bipolar fuzzy Dombi ordered weighted averaging ( $ℝ$ -CBFDOWA) operator

Definition 5.7. Let ${\ddot{¥}}_{i} = ([{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}], [{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}], {{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}, {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}})$ , where i = (1, 2, . . . , t), be a huddle of CBFNs with the corresponding weight vector η = (η₁, η₂, . . . η_t) with η_i ∈ [0, 1] and $\sum_{i = 1}^{t} η_{i} = 1$ . Then the $ℝ$ -order cubic bipolar fuzzy Dombi ordered weighted averaging( $ℝ$ -CBFDOWA) operator is a mapping $ℝ - CBFDOWA : {\ddot{¥}}^{t} \to \ddot{¥}$ such that

$ℝ - CBFDOWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . . {\ddot{¥}}_{t}) = {\underset{i = 1}{\overset{t}{\oplus}}}_{ℝ} η_{i} {\ddot{¥}}_{σ (i)}$

Theorem 5.8. Let $({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ be a family of CBFNs with the corresponding weight vector η = (η₁, η₂, . . . , η_t) such that η_i ∈ [0, 1] and $\sum_{i = 1}^{t} η_{i} = 1$ ,then the agglomerated value of these CBFNs by using $ℝ - CBFDOWA$ operator is again a CBFN and it can be defined as $ℝ - CBFDOWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ $= ([1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{σ (i)}}^{+}}{1 - {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{σ (i)}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{{\tilde{ϒ}}_{u {\ddot{¥}}_{σ (i)}}^{+}}{1 - {\tilde{ϒ}}_{u {\ddot{¥}}_{σ (i)}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ , $[\frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{σ (i)}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{σ (i)}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{σ (i)}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{σ (i)}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ , ${\frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{1 - {\tilde{Δ}}_{{\ddot{¥}}_{σ (i)}}^{+}}{{\tilde{Δ}}_{{\ddot{¥}}_{σ (i)}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, - 1 + \frac{1}{1 + {{\sum_{i = 1}^{t} η_{i} {(\frac{| {\tilde{Δ}}_{{\ddot{¥}}_{σ (i)}}^{-} |}{1 + {\tilde{Δ}}_{{\ddot{¥}}_{σ (i)}}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

.......(∗)

Example 5.9. Suppose that

${\ddot{¥}}_{1} = ([0.2, 0.35], [- 0.5, - 0.4], {0.5, - 0.2})$ ,

${\ddot{¥}}_{2} = ([0.4, 0.6], [- 0.3, - 0.1], {0.7, - 0.6})$ ,

${\ddot{¥}}_{3} = ([0.5, 0.6], [- 0.4, - 0.2], {0.3, - 0.2})$ ,

${\ddot{¥}}_{4} = ([0.6, 0.8], [- 0.6, - 0.4], {0.4, - 0.5})$

are four CBFNs and η = (0.3, 0.2, 0.3, 0.2) ^T is the weight vector for these CBFNs. Take the value of operational parameter $\overset{˘}{ξ} = 4$ . Now, these four CBFNs can be aggregated by using $ℝ - CBFDOWA$ operator. First, we calculate score functions of ${\ddot{¥}}_{1}$ , ${\ddot{¥}}_{2}$ , ${\ddot{¥}}_{3}$ and ${\ddot{¥}}_{4}$ by using Eq(2) as follows

$S_{ℝ} ({\ddot{¥}}_{1}) = \frac{0.2 + 0.35 + 1 - 0.5 + 1 - 0.4 + 0.5 - (1 - 0.2)}{6} = 0.2250$ ,

$S_{ℝ} ({\ddot{¥}}_{2}) = \frac{0.4 + 0.6 + 1 - 0.3 + 1 - 0.1 + 0.7 - (1 - 0.6)}{6} = 0.4833$ ,

$S_{ℝ} ({\ddot{¥}}_{3}) = \frac{0.5 + 0.6 + 1 - 0.4 + 1 - 0.2 + 0.3 - (1 - 0.2)}{6} = 0.3333$ ,

$S_{ℝ} ({\ddot{¥}}_{4}) = \frac{0.6 + 0.8 + 1 - 0.6 + 1 - 0.4 + 0.4 - (1 - 0.5)}{6} = 0.3833$

This gives, $S_{ℝ} ({\ddot{¥}}_{2}) > S_{ℝ} ({\ddot{¥}}_{4}) > S_{ℝ} ({\ddot{¥}}_{3}) > S_{ℝ} ({\ddot{¥}}_{1})$ . So, we have ${\ddot{¥}}_{σ (1)} = {\ddot{¥}}_{2}$ , ${\ddot{¥}}_{σ (2)} = {\ddot{¥}}_{4}$ , ${\ddot{¥}}_{σ (3)} = {\ddot{¥}}_{3}$ and ${\ddot{¥}}_{σ (4)} = {\ddot{¥}}_{1}$ . Now,

$ℝ - CBFDOWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, {\ddot{¥}}_{3}, {\ddot{¥}}_{4}) = \oplus_{i = 1}^{4} η_{i} {\ddot{¥}}_{σ (i)}$

$[\frac{- 1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{1 + {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{σ (i)}}^{-}}{| {\tilde{ϒ}}_{ℓ {\ddot{¥}}_{σ (i)}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{1 + {\tilde{ϒ}}_{u {\ddot{¥}}_{σ (i)}}^{-}}{| {\tilde{ϒ}}_{u {\ddot{¥}}_{σ (i)}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${\frac{1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{1 - {\tilde{Δ}}_{{\ddot{¥}}_{σ (i)}}^{+}}{{\tilde{Δ}}_{{\ddot{¥}}_{σ (i)}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, - 1 + \frac{1}{1 + {{\sum_{i = 1}^{4} η_{i} {(\frac{| {\tilde{Δ}}_{{\ddot{¥}}_{σ (i)}}^{-} |}{1 + {\tilde{Δ}}_{{\ddot{¥}}_{σ (i)}}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

= ([0.5198, 0.7307] , [-0.3563, - 0.1294] , {0.3593, - 0.5339})

Three important characteristics of $ℝ$ -CBFDOWA operator are given by

Theorem 5.10. (Idempotency)

Let

${\ddot{¥}}_{i} = ([{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{+}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{+}], [{\tilde{ϒ}}_{ℓ {\ddot{¥}}_{i}}^{-}, {\tilde{ϒ}}_{u {\ddot{¥}}_{i}}^{-}], {{\tilde{Δ}}_{{\ddot{¥}}_{i}}^{+}, {\tilde{Δ}}_{{\ddot{¥}}_{i}}^{-}})$ , where i = (1, 2, . . . , t), be a huddle of CBFNs such that ${\ddot{¥}}_{i} = \ddot{¥}$ ∀ (i = 1, 2, . . . , t), then

$ℝ - CBFDOWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . . {\ddot{¥}}_{t})$ = $\ddot{¥}$

Theorem 5.11. (Monotonicity)

Let

$ℝ - CBFDOWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . . {\ddot{¥}}_{t}) \leq ℝ - CBFDOWA ({\ddot{¥}}_{1}^{*}, {\ddot{¥}}_{2}^{*}, . . ., {\ddot{¥}}_{t}^{*})$

Theorem 5.12. (Boundedness)

Let

Let ${\ddot{¥}}_{\min} = \min ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ and

${\ddot{¥}}_{\max} = \max ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . ., {\ddot{¥}}_{t})$ , then

${\ddot{¥}}_{\min} \leq ℝ - CBFDOWA ({\ddot{¥}}_{1}, {\ddot{¥}}_{2}, . . . {\ddot{¥}}_{t}) \leq {\ddot{¥}}_{\max}$ .

6 Application of CBFDWAA operators in solving MCDM problems

In this section, we propose a method based on CBFDWA operators to deal with MCDM problem under cubic bipolar fuzzy environment. For a given MCDM problem, suppose that $\tilde{Q} = {{\ddot{q}}_{1}, {\ddot{q}}_{2}, . . ., {\ddot{q}}_{m}}$ is the collection of $m$ feasible alternatives and $\tilde{O} = {{\ddot{o}}_{1}, {\ddot{o}}_{2}, . . ., {\ddot{o}}_{n}}$ is the collection of $n$ criteria under which the alternatives are to be evaluated. Let $$ = ($_{1}, $_{2}, . . ., $_{n})$ be the corresponding weight vector of the criteria with $$_{j} \in [0, 1]$ and $\sum_{j = 1}^{n} $_{j} = 1$ . The decision maker appraises the alternative ${\ddot{q}}_{i}$ $(i = 1, 2, . . ., m)$ with respect to the criterion ${\ddot{o}}_{j}$ $(j = 1, 2, . . ., n)$ and gives his appraisal in the form of CBFN,i.e. $ℏ_{ij} = ([{\tilde{ϒ}}_{ℓ_{ij}}^{+}, {\tilde{ϒ}}_{u_{ij}}^{+}], [{\tilde{ϒ}}_{ℓ_{ij}}^{-}, {\tilde{ϒ}}_{u_{ij}}^{-}], {{\tilde{Δ}}_{ij}^{+}, {\tilde{Δ}}_{ij}^{-}})$ , where $< [{\tilde{ϒ}}_{ℓ_{ij}}^{+}, {\tilde{ϒ}}_{u_{ij}}^{+}]; {\tilde{Δ}}_{ij}^{+} >$ and $< [{\tilde{ϒ}}_{ℓ_{ij}}^{-}, {\tilde{ϒ}}_{u_{ij}}^{-}]; {\tilde{Δ}}_{ij}^{-} >$ represent positive and negative membership degrees which provide positive information and negative information of the alternative ${\ddot{q}}_{i}$ w.r.t the criterion ${\ddot{o}}_{j}$ . Thus, we can construct a cubic bipolar fuzzy decision matrix (CBFDM) $H = (ℏ_{ij})_{m \times n}$ . Now, to handle this MCDM problem, we shall utilize the CBFDWA operators in the following way.

Algorithm

Step1 . Contrive the cubic bipolar fuzzy decision matrix (CBFDM) $H = (ℏ_{ij})_{m \times n}$ with the help of evaluation information given by the decision maker by means of CBFNs $ℏ_{ij}$ .

Step2 . Acquire the overall preference value $ℏ_{i}$ $(i = 1, 2, . . ., m)$ of the alternative ${\ddot{q}}_{i}$ $(i = 1, 2, . . ., m)$ w.r.t the criterion ${\ddot{o}}_{j}$ $(j = 1, 2, . . ., n)$ by utilizing the $ℙ$ -CBFDWA operator

$ℏ_{i} = ℙ - CBFDWA (ℏ_{i 1}, ℏ_{i 2}, . . ., ℏ_{in}) = {\underset{j = 1}{\overset{n}{\oplus}}}_{P} η_{j} ℏ_{ij}$

$[\frac{- 1}{1 + {{\sum_{j = 1}^{n} η_{j} {(\frac{1 + {\tilde{ϒ}}_{ℓ_{ij}}^{-}}{| {\tilde{ϒ}}_{ℓ_{ij}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{j = 1}^{n} η_{j} {(\frac{1 + {\tilde{ϒ}}_{u_{ij}}^{-}}{| {\tilde{ϒ}}_{u_{ij} |}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${1 - \frac{1}{1 + {{\sum_{j = 1}^{n} η_{j} {(\frac{{\tilde{Δ}}_{ij}^{+}}{1 - {\tilde{Δ}}_{ij}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{j = 1}^{n} η_{j} {(\frac{1 + {\tilde{Δ}}_{ij}^{-}}{| {\tilde{Δ}}_{ij}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$

or by utilizing $ℝ$ -CBFDWA operator

$ℏ_{i} = ℝ - CBFDWA (ℏ_{il}, ℏ_{i 2}, . . ., ℏ_{in}) = {\underset{j = 1}{\overset{n}{\oplus}}}_{R} η_{j} ℏ_{ij}$

= $([1 - \frac{1}{1 + {{\sum_{j = 1}^{n} η_{j} {(\frac{{\tilde{ϒ}}_{ℓ_{ij}}^{+}}{1 - {\tilde{ϒ}}_{ℓ_{ij}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, 1 - \frac{1}{1 + {{\sum_{j = 1}^{n} η_{j} {(\frac{{\tilde{ϒ}}_{u_{ij}}^{+}}{1 - {\tilde{ϒ}}_{u_{ij}}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ , $[\frac{- 1}{1 + {{\sum_{j = 1}^{n} η_{j} {(\frac{1 + {\tilde{ϒ}}_{ℓ_{ij}}^{-}}{| {\tilde{ϒ}}_{ℓ_{ij}}^{-} |})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, \frac{- 1}{1 + {{\sum_{j = 1}^{n} η_{j} {(\frac{1 + {\tilde{ϒ}}_{u_{ij}}^{-}}{| {\tilde{ϒ}}_{u_ij |}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}]$ ,

${\frac{1}{1 + {{\sum_{j = 1}^{n} η_{j} {(\frac{1 - {\tilde{Δ}}_{ij}^{+}}{{\tilde{Δ}}_{ij}^{+}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}, - 1 + \frac{1}{1 + {{\sum_{j = 1}^{n} η_{j} {(\frac{| {\tilde{Δ}}_{ij}^{-} |}{1 + {\tilde{Δ}}_{ij}^{-}})}^{\overset{˘}{ξ}}}}^{\frac{1}{\overset{˘}{ξ}}}}})$ .

Step3 . In case of $ℙ$ -order, work out the score function $S_{P} (ℏ_{i})$ of the overall preference values $ℏ_{i}$ $(i = 1, 2, . . ., m)$ . If the score function of two preference values $ℏ_{l}$ and $ℏ_{p}$ is same,i.e. $S_{P} (ℏ_{l})$ = $S_{P} (ℏ_{p})$ , then calculate the accuracy function $A (ℏ_{l})$ and $A (ℏ_{p})$ of these two preference values. Similarly, in case of $ℝ$ -order, compute the score function $S_{R} (ℏ_{i})$ of $ℏ_{i}$ $(i = 1, 2, . . ., m)$ . If the score function of two preference values $ℏ_{l}$ and $ℏ_{p}$ is same,i.e. $S_{R} (ℏ_{l})$ = $S_{R} (ℏ_{p})$ , then calculate the accuracy function $A (ℏ_{l})$ and $A (ℏ_{p})$ of these two preference values.

Step4 . Arrange the overall preference values $ℏ_{i}$ $(i = 1, 2, . . ., n)$ in descending order on the basis of their score and accuracy functions.

Step5 . Rank the alternatives. The ranking of the alternatives is contingent on the ranking of their corresponding preference values $ℏ_{i}$ .

Step6 . Choose the best alternative(s).

The flow chart of the algorithm is given in the Figure 1.

Fig. 1

Flow chart diagram of proposed algorithm.

6.1 Interpretative example

A project is a provisional undertaking to generate a specific product, service or outcome. A number of key success factors play a vital role in the success of the project. One significant factor is the oversight of an extraordinarily competent project manager. Therefore, it is important to choose the right project manager to carry out the project successfully. According to the field of a certain organization, a project manager with relevant qualification and expertise is hired. Suppose that a company wants to employ a project manager for its upcoming project. Among many candidates, only five candidates have been short listed by the decision expert on the basis of their educational background, past experience and reputation. So, let the set of alternatives (candidates) be $\ddot{Q} = {{\ddot{q}}_{1}, {\ddot{q}}_{2}, {\ddot{q}}_{3}, {\ddot{q}}_{4}, {\ddot{q}}_{5}}$ . Now, the decision maker evaluates each candidate ${\ddot{q}}_{i}$ $(i = 1, 2, . . ., 5)$ with respect to four criteria ${\ddot{o}}_{i}$ $(j = 1, 2, 3, 4)$ which are given with their detailed description in Table 1.

Table 1
Project manager selection criteria

Criterion Description of criterion

Leadership skills $({\ddot{o}}_{1})$ Communication skills, Interpersonal skills,

The ability to share a clear vision and inspire others

Management skills $({\ddot{o}}_{2})$ Project management, Budget management, Risk management,

Time management, Resource management

Personal qualities $({\ddot{o}}_{3})$ Goal-orientedness, Confidence, Creative skills

Technical capability $({\ddot{o}}_{4})$ Knowledge of the tools of relevant technology, Digital literacy

Criterion	Description of criterion
Leadership skills $({\ddot{o}}_{1})$	Communication skills, Interpersonal skills,
	The ability to share a clear vision and inspire others
Management skills $({\ddot{o}}_{2})$	Project management, Budget management, Risk management,
	Time management, Resource management
Personal qualities $({\ddot{o}}_{3})$	Goal-orientedness, Confidence, Creative skills
Technical capability $({\ddot{o}}_{4})$	Knowledge of the tools of relevant technology, Digital literacy

To demonstrate the importance of each criterion in decision making, the decision maker assigns a weight vector η = (0.25, 0.35, 0.25, 0.15) corresponding to the criteria ${\ddot{o}}_{i}$ $(j = 1, 2, 3, 4)$ . Now, the decision maker evaluates each candidate under every criterion and gives his assessment in the form of CBFNs. The cubic bipolar fuzzy decision matrix $H = (ℏ_{ij})_{5 \times 4}$ is presented in Table 2.

Table 2

Cubic Bipolar Fuzzy Decision Matrix (CBFDM)

Z	${\ddot{o}}_{1}$	${\ddot{o}}_{2}$
${\ddot{q}}_{1}$	([0.5,0.7],[-0.3,-0.2],{0.6,-0.2}	([0.4,0.6],[-0.7,-0.5],{0.3,-0.5}
${\ddot{q}}_{2}$	([0.3,0.4],[-0.7,-0.4],{0.7,-0.5}	([0.7,0.9],[-0.4,-0.2],{0.5,-0.3}
${\ddot{q}}_{3}$	([0.1,0.3],[-0.6,-0.3],{0.4,-0.5}	([0.5,0.7],[-0.3,-0.1],{0.8,-0.4}
${\ddot{q}}_{4}$	([0.2,0.4],[-0.5,-0.2],{0.3,-0.3}	([0.5,0.8],[-0.3,-0.1],{0.9,-0.1}
${\ddot{q}}_{5}$	([0.3,0.6],[-0.5,-0.3],{0.4,-0.5}	([0.6,0.8],[-0.4,-0.2],{0.5,-0.5}
	${\ddot{o}}_{3}$	${\ddot{o}}_{4}$
${\ddot{q}}_{1}$	([0.3,0.4],[-0.6,-0.4],{0.5,-0.5}	([0.7,0.8],[-0.4,-0.2],{0.7,-0.3}
${\ddot{q}}_{2}$	([0.1,0.3],[-0.5,-0.3],{0.4,-0.6}	([0.4,0.6],[-0.7,-0.6],{0.8,-0.7}
${\ddot{q}}_{3}$	([0.4,0.6],[-0.3,-0.2],{0.7,-0.4}	([0.2,0.5],[-0.6,-0.4],{0.3,-0.5}
${\ddot{q}}_{4}$	([0.1,0.3],[-0.6,-0.4],{0.4,-0.5}	([0.6,0.8],[-0.4,-0.2],{0.7,-0.3}
${\ddot{q}}_{5}$	([0.3,0.4],[-0.7,-0.5],{0.3,-0.8}	([0.7,0.9],[-0.2,-0.1],{0.6,-0.4}

In order to select the best candidate among all the candidates, we utilize $ℙ$ -CBFDWA and $ℝ$ -CBFDWA operator. Now, the above mentioned decision making problem can be solved by the method that was proposed in section 6 as follows:

Step1 . Consider the cubic bipolar fuzzy decision matrix (CBFDM) given in Table 2.

Step2 . For $\overset{˘}{ξ} = 1$ , we acquire the overall preference values $ℏ_{i}$ $(i = 1, 2, . . ., 5)$ of the alternatives ${\ddot{q}}_{i}$ $(i = 1, 2, . . ., 5)$ w.r.t the criterion ${\ddot{o}}_{i}$ $(j = 1, 2, 3, 4)$ by availing $ℙ$ -CBFDWA operator which are given below

ℏ₁ = ([0.4847, 0.6522] , [-0.4706, - 0.3008] , {0.5294, - 0.3390}),

ℏ₂ = ([0.5126, 0.7849] , [-0.5138, - 0.2892] , {0.6296, - 0.4352}),

ℏ₃ = ([0.3679, 0.5916] , [-0.3750, - 0.1678] , {0.6889, - 0.4348}),

ℏ₄ = ([0.3995, 0.6945] , [-0.4068, - 0.1633] , {0.7905, - 0.1875}),

ℏ₅ = ([0.5214, 0.7670] , [-0.4029, - 0.2128] , {0.4591, - 0.5298})

Step3 . We use Eq (1) to compute the score functions of the overall preference values $ℏ_{i}$ $(i = 1, 2, . . ., 5)$ as follows:

$S_{ℙ} (ℏ_{1}) = 0.4162$ , $S_{ℙ} (ℏ_{2}) = 0.4050$ , $S_{ℙ} (ℏ_{3}) = 0.3822$ , $S_{ℙ} (ℏ_{4}) = 0.4243$ and

$S_{ℙ} (ℏ_{5}) = 0.4464$

Step4 . We rank the overall preference values $ℏ_{i}$ $(i = 1, 2, . . ., 5)$ on the basis of their score functions $S_{ℙ} (ℏ_{i})$ $(i = 1, 2, . . ., 5)$ as ℏ₅ ≻ ℏ ₄ ≻ ℏ ₁ ≻ ℏ ₂ ≻ ℏ ₃.

Step5 . We rank the alternatives as ${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$ .

Step6 . The above ranking infers that ${\ddot{q}}_{5}$ is the most recommendable candidate among all.

Now, $ℙ$ -CBFDWA operator can be replaced with $ℝ$ -CBFDWA operator to solve the MCDM problem by following the same steps

Step1 . Consider the cubic bipolar fuzzy decision matrix (CBFDM) given in Table 2.

Step2 . For $\overset{˘}{ξ} = 1$ , we acquire the overall preference values $ℏ_{i}$ $(i = 1, 2, . . ., 5)$ of the alternatives ${\ddot{q}}_{i}$ $(i = 1, 2, . . ., 5)$ w.r.t the criterion ${\ddot{o}}_{i}$ $(j = 1, 2, 3, 4)$ by employing $ℝ$ -CBFDWA operator which are given below

ℏ₁ = ([0.4847, 0.6522] , [-0.4706, - 0.3008] , {0.4352, - 0.4209}),

ℏ₂ = ([0.5126, 0.7849] , [-0.5138, - 0.2892] , {0.5349, - 0.5294}),

ℏ₃ = ([0.3679, 0.5916] , [-0.3750, - 0.1678] , {0.5209, - 0.4444}),

ℏ₄ = ([0.3995, 0.6945] , [-0.4068, - 0.1633] , {0.4851, - 0.3152}),

ℏ₅ = ([0.5214, 0.7670] , [-0.4029, - 0.2128] , {0.4152, - 0.6296})

Step3 . We use Eq (2) to calculate the score functions of the overall preference values $ℏ_{i}$ $(i = 1, 2, . . ., 5)$ as follows:

$S_{ℝ} (ℏ_{1}) = 0.3703$ , $S_{ℝ} (ℏ_{2}) = 0.4265$ , $S_{ℝ} (ℏ_{3}) = 0.3970$ , $S_{ℝ} (ℏ_{4}) = 0.3874$ and

$S_{ℝ} (ℏ_{5}) = 0.4520$

Step4 . We rank the overall preference values on the basis of their score functions $S_{ℝ} (ℏ_{i})$ $(i = 1, 2, . . ., 5)$ as ℏ₅ ≻ ℏ ₂ ≻ ℏ ₃ ≻ ℏ ₄ ≻ ℏ ₁.

Step5 . We rank the alternatives as

${\ddot{q}}_{5} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1}$ .

Step6 . The above ranking infers that ${\ddot{q}}_{5}$ is the most recommendable candidate among all.

Note that the best alternative remains the same whether we use $ℙ$ -CBFDWA or $ℝ$ -CBFDWA operator.

6.2 Sensitivity analysis

To delineate the impact of the operational parameter $\overset{˘}{ξ}$ on MCDM results, we have used disparate values of $\overset{˘}{ξ}$ to rank the alternatives. Tables 3 and 4 display the values of the score function along with the ranking orders of the alternatives ${\ddot{q}}_{i}$ $(i = 1, 2, . . ., 5)$ in the range of $1 \leq \overset{˘}{ξ} \leq 5$ based on $ℙ$ -CBFDWA and $ℝ$ -CBFDWA operators.

Table 3
Ranking based on different operational parameters $\overset{˘}{ξ}$ of $ℙ$ -CBFDWA operator

$\overset{˘}{ξ}$ $S_{ℙ} (ℏ_{1})$ $S_{ℙ} (ℏ_{2})$ $S_{ℙ} (ℏ_{3})$ $S_{ℙ} (ℏ_{4})$ $S_{ℙ} (ℏ_{5})$ Ranking

1 0.4162 0.4050 0.3822 0.4243 0.4464 ${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$

2 0.4425 0.4344 0.3956 0.4456 0.4763 ${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$

3 0.4614 0.4487 0.4027 0.4586 0.4950 ${\ddot{q}}_{5} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$

4 0.4739 0.4568 0.4097 0.4667 0.5068 ${\ddot{q}}_{5} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$

5 0.4824 0.4619 0.4136 0.4722 0.5146 ${\ddot{q}}_{5} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$

$\overset{˘}{ξ}$	$S_{ℙ} (ℏ_{1})$	$S_{ℙ} (ℏ_{2})$	$S_{ℙ} (ℏ_{3})$	$S_{ℙ} (ℏ_{4})$	$S_{ℙ} (ℏ_{5})$	Ranking
1	0.4162	0.4050	0.3822	0.4243	0.4464	${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$
2	0.4425	0.4344	0.3956	0.4456	0.4763	${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$
3	0.4614	0.4487	0.4027	0.4586	0.4950	${\ddot{q}}_{5} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$
4	0.4739	0.4568	0.4097	0.4667	0.5068	${\ddot{q}}_{5} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$
5	0.4824	0.4619	0.4136	0.4722	0.5146	${\ddot{q}}_{5} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$

Table 4

Ranking based on different operational parameters $\overset{˘}{ξ}$ of $ℝ$ -CBFDWA operator

$\overset{˘}{ξ}$	$S_{ℝ} (ℏ_{1})$	$S_{ℝ} (ℏ_{2})$	$S_{ℝ} (ℏ_{3})$	$S_{ℝ} (ℏ_{4})$	$S_{ℝ} (ℏ_{5})$	Ranking
1	0.3703	0.4265	0.3970	0.3874	0.4520	${\ddot{q}}_{5} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1}$
2	0.3939	0.4578	0.4058	0.4086	0.4869	${\ddot{q}}_{5} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{3} ≻ {\ddot{q}}_{1}$
3	0.4110	0.4737	0.4114	0.4221	0.5090	${\ddot{q}}_{5} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{3} ≻ {\ddot{q}}_{1}$
4	0.4227	0.4830	0.4153	0.4310	0.5227	${\ddot{q}}_{5} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{3}$
5	0.4308	0.4894	0.4181	0.4370	0.5315	${\ddot{q}}_{5} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{3}$

From Table 3, it can be seen that different values of parameter $\overset{˘}{ξ}$ can change the ranking order formed on the basis of $ℙ$ -CBFDWA operator. When $1 \leq \overset{˘}{ξ} \leq 2$ , the ranking is ${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$ and the optimal alternative is ${\ddot{q}}_{5}$ . When $3 \leq \overset{˘}{ξ} \leq 5$ , then the corresponding ranking order is ${\ddot{q}}_{5} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$ and the best choice is ${\ddot{q}}_{5}$ . Similarly, from Table 4, it is clear that different values of parameter $\overset{˘}{ξ}$ can alter the ranking order formed on the basis of $ℝ$ -CBFDWA operator. When $\overset{˘}{ξ} = 1$ , the ranking is ${\ddot{q}}_{5} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1}$ and the most apt alternative is ${\ddot{q}}_{5}$ . When $2 \leq \overset{˘}{ξ} \leq 3$ , then the corresponding ranking order is ${\ddot{q}}_{5} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{3} ≻ {\ddot{q}}_{1}$ and the most desirable option is ${\ddot{q}}_{5}$ . When $4 \leq \overset{˘}{ξ} \leq 5$ , then the ranking is ${\ddot{q}}_{5} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{3}$ and the best one is ${\ddot{q}}_{5}$ .

From the above analysis, it is concluded that in this MCDM problem, both $ℙ$ -CBFDWA and $ℝ$ -CBFDWA operators are sensitive to $\overset{˘}{ξ}$ . Also, the $ℝ$ -CBFDWA operator is more responsive to $\overset{˘}{ξ}$ in comparison with $ℙ$ -CBFDWA operator. However, the best alternative remains the same irrespective of the choice of operator and the value of parameter $\overset{˘}{ξ}$ .

The effect of $\overset{˘}{ξ}$ on the ranking of alternatives under $ℙ$ -order and $ℝ$ -order is shown in the Figure 2 and Figure 3, respectively.

Fig. 2

Effect of $\overset{˘}{ξ}$ on ranking under $ℙ$ -order.

Fig. 3

Effect of $\overset{˘}{ξ}$ on ranking under $ℝ$ -order.

6.3 Comparability analysis

In this section, we shall compare our proposed cubic bipolar fuzzy Dombi averaging aggregation operators with some of the existing cubic bipolar fuzzy weighted averaging aggregation operators introduced in [45]. The comparison results are shown in Table 5.

Table 5
Decision results of MCDM problem with cubic bipolar information

Averaging Operators Ranking Optimal alternative

$ℙ$ -CBFWA [45]             ${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$ ${\ddot{q}}_{5}$

$ℝ$ -CBFWA [45]             ${\ddot{q}}_{5} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1}$ ${\ddot{q}}_{5}$

$ℙ$ -CBFOWA [45]           ${\ddot{q}}_{5} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{3} ≻ {\ddot{q}}_{2}$ ${\ddot{q}}_{5}$

$ℝ$ -CBFOWA [45]           ${\ddot{q}}_{5} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$ ${\ddot{q}}_{5}$

$ℙ$ -CBFDWA (Proposed) ${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$ ${\ddot{q}}_{5}$

$ℝ$ -CBFDWA (Proposed) ${\ddot{q}}_{5} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1}$ ${\ddot{q}}_{5}$

Averaging Operators	Ranking	Optimal alternative
$ℙ$ -CBFWA [45]	${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$	${\ddot{q}}_{5}$
$ℝ$ -CBFWA [45]	${\ddot{q}}_{5} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1}$	${\ddot{q}}_{5}$
$ℙ$ -CBFOWA [45]	${\ddot{q}}_{5} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{3} ≻ {\ddot{q}}_{2}$	${\ddot{q}}_{5}$
$ℝ$ -CBFOWA [45]	${\ddot{q}}_{5} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$	${\ddot{q}}_{5}$
$ℙ$ -CBFDWA (Proposed)	${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$	${\ddot{q}}_{5}$
$ℝ$ -CBFDWA (Proposed)	${\ddot{q}}_{5} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1}$	${\ddot{q}}_{5}$

The comparison results are shown in Table 6.

From Table 5, Table 6, Figure 2, Figure 3, and Figure 4, it is perceptible that the ranking order based on $ℙ$ -CBFWA( $ℝ$ -CBFWA) operator coincides with the ranking based on $ℙ$ -CBFDWA( $ℝ$ -CBFDWA) operator, whereas the ranking order based on $ℙ$ -CBFOWA( $ℝ$ -CBFOWA) operator is different from the ranking based on $ℙ$ -CBFDWA( $ℝ$ -CBFDWA) operator, however, the best alternative remains unchanged. Due to absence of parameter in the existing aggregation operators, our proposed cubic bipolar fuzzy Dombi averaging aggregation operators are more effective and reliable than existing aggregation operators.

Table 6

Comparison of decision results with other aggregation operators

Averaging Operators	Ranking	Optimal alternative
IFGAO (Xu and Yager [5])	${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$	${\ddot{q}}_{5}$
PFAO (Peng and Yuang [27])	${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$	${\ddot{q}}_{5}$
PCFAO (Khan et al.)	${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$	${\ddot{q}}_{5}$
IFDBMO (Liu et al. [35])	${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$	${\ddot{q}}_{5}$
q-ROFAO (Liu and Wang [38])	${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$	${\ddot{q}}_{5}$
BFDAO (Jana et al. [39])	${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$	${\ddot{q}}_{5}$
PDFAO (Akram et al. [40])	${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$	${\ddot{q}}_{5}$
SVNDWAO (Chen and Ye [42])	${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$	${\ddot{q}}_{5}$
NCDAO (Shi and Ye [44])	${\ddot{q}}_{5} ≻ {\ddot{q}}_{4} ≻ {\ddot{q}}_{1} ≻ {\ddot{q}}_{2} ≻ {\ddot{q}}_{3}$	${\ddot{q}}_{5}$

Final ranking of alternatives under $ℙ (ℝ)$ -order is shown with the help of pie chart in the Figure 4.

Fig. 4

Comparison of ranking under $ℙ (ℝ)$ -order.

7 Conclusion

To deal with vague and uncertain information a cubic bipolar fuzzy set is a new approach which is superior to existing bipolar fuzzy set and interval-valued bipolar fuzzy set because it contains triplet of bipolar fuzzy information with two types of operations, including $ℙ$ -order and $ℝ$ -order. This model has the ability to address two-sided contrasting features which provide vital information with a bipolar fuzzy number and an interval-valued bipolar fuzzy number simultaneously. The presence of a general parameter in Dombi’s operators ensures excellent flexibility to tackle vagueness. We extended Dombi’s operations to CBFNs with $ℙ$ -order and $ℝ$ -order. We introduced some cubic bipolar fuzzy averaging aggregation operators including, $ℙ$ -CBFDWA operator, $ℙ$ -CBFDOWA operator, $ℝ$ -CBFDWA operator and $ℝ$ -CBFDOWA operator. Additionally, some necessary properties of suggested operators like idempotency, boundedness and monotonicity are investigated. Moreover, we have proposed a robust MCDM method based on $ℙ$ -CBFDWA and $ℝ$ -CBFDWA operators to handle vagueness in MCDM problems using cubic bipolar fuzzy information. A numerical illustration is provided to interpret the effectiveness and useability of new MCDM technique and the impact of operational parameter on the decision-making outcomes is also analyzed. We observe that the ranking orders may be influenced by the use of different operational parameters, however, the optimal alternative remains the same. In the end, we give sensitivity analysis and comparative analysis of suggested Dombi’s aggregation operators with the existing aggregation operators to examine the rationality, efficiency, and reliability.

In future, we shall develop more aggregation operators including cubic bipolar fuzzy Dombi Bonferroni mean operators, cubic bipolar fuzzy Dombi Heronian mean operators, cubic bipolar fuzzy Dombi prioritized operators, cubic bipolar fuzzy Einstein prioritized operators, and cubic bipolar fuzzy Dombi Hamy mean operators.

Footnotes

Acknowledgment

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudi Arabia for funding this work through research groups program under grant number R.G. P-2/29/42.

References

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

Zadeh

L.A.

, Similarity relations and fuzzy ordering, Information Sciences 3(2) (1971), 177–200.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets andSystems 20(1) (1986), 87–96.

Atanassov

K.T.

, Intuitionistic fuzzy sets, In: Intuitionistic Fuzzy Sets, Studies in Fuzziness and Soft Computing, 35. Physica, Heidelberg 35 (1999), 1–137. https://doi.org/10.1007/978-3-7908-1870-31.

Z.S.

and Yager

R.R.

, Some geometric aggregation operators basedon intuitionistic fuzzy sets, International Journal of GeneralSystems 35 (2006), 417–433.

Yager

R.R.

, Pythagorean fuzzy subsets, Proceedings of the Joint IFSA World Congress and NAFIPS Annual Meeting (2013), 57–61.

Yager

R.R.

, Pythagorean membership grades in multicriteria decisionmaking, IEEE Transactions on Fuzzy Systems 22(4) (2014), 958–965.

Zhang

and Xu

, Extension of TOPSIS to multiple criteriadecision making with Pythagorean fuzzy sets, InternationalJournal of Intelligent Systems 29(12) (2014), 1061–1078.

Yager

R.R.

, Generalized Orthopair Fuzzy sets, IEEE Transactionson Fuzzy Systems 25(5) (2017), 1220–1230.

10.

Zhang

W.R.

, Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis, NAFIPS/IFIS/NASA94. Proceedings of the First International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference. The Industrial Fuzzy Control and Intellige (1994), 305–309.

11.

Zhang

W.R.

, Bipolar fuzzy sets, Proceedings of IEEE Inrenational Conference on Fuzzy Systems (1998), 835–840.

12.

Lee

K.M.

, Bipolar-valued fuzzy sets and their basic operations, Proceedings of Internatinal Conference, Bangkok, Thailand, (2000), 307–317.

13.

Lee

K.M.

, Comparison of interval-valued fuzzy sets, intuitionisticfuzzy sets, and bipolar-valued fuzzy sets, Journal of KoreanInstitute of Intelligent Systems 14(2) (2004), 125–129.

14.

Alcantud

J.C.R.

, Santos-García

, Peng

X.D.

and Zhan

, Dualextended hesitant fuzzy sets, Symmetry 11(5) (2019), 1–13.

15.

Wei

, Wei

and Gao

, Multiple attribute decision making withinterval-valued bipolar fuzzy information and their application toemerging technology commercialization evaluation, IEEE Access 6 (2018), 60930–60955.

16.

Jun

Y.B.

, Kim

C.S.

and Yang

K.O.

, Cubic sets, Annals of FuzzyMathematics and Informatics 4(1) (2012), 83–98.

17.

Aslam

, Abdullah

and Ullah

, Bipolar fuzzy soft set and itsapplication in decision making, Journal of Intelligent & FuzzySystems 27(2) (2014), 729–742.

18.

Mahmood

, Mehmood

and khan

, Multiple criteria decisionmaking based on bipolar-valued fuzzy set, Annals of FuzzyMathematics and Informatics 11(6) (2016), 1003–1009.

19.

Karaaslan

, Neutrosophic Soft Set with Applications in DecisionMaking, International Journal of Information Science andIntelligent System 4(2) (2015), 1–20.

20.

Naz

and Shabir

, On fuzzy bipolar soft sets, their algebraicstructures and applications, Journal of Intelligent & FuzzySystems 26(4) (2014), 1645–1656.

21.

Wei

, Alsaadi

F.E.

, Hayat

and Alsaedi

, Bipolar fuzzy Hamacheraggregation operators in multiple attribute decision making, International Journal of Fuzzy Systems 33(6) (2017), 1–12.

22.

Deli

, Ali

and Smarandache

, Bipolar neutrosophic sets and their application based on multi-criteria decision making problems, Proceedings of the 2015 International Conference on Advanced Mechatronic Systems, Beijing, China, (2015).

23.

Akram Shumaiza

and Arshad

, Bipolar fuzzy TOPSIS and bipolarfuzzy ELECTRE-I methods to diagnosis, Journal of Computationaland Applied Mathematics 39 (2019), 1–21.

24.

Yager

R.R.

, On ordered weighted averaging aggregation operators inmulti-criteria decision making, IEEE Transactions on Systems,Man and Cybernetics 18(1) (1988), 183–190.

25.

Z.S.

, Intuitionistic fuzzy aggreagtion operators, IEEETransactions on Fuzzy Systems 15(6) (2007), 1179–1187.

26.

Z.S.

and Cai

, Intuitionistic fuzzy information aggregation: theory and applications, Springer Heidelberg New York Dordrecht London, (2012).

27.

Peng

X.D.

and Yuang

, Fundamental properties of Pythagorean fuzzyaggregation operators, Fundamenta Informaticae 147(2016), 415–446.

28.

Khan

, Khan

M.S.A.

, Shahzad

and Abdullah

, Pythagorean cubicfuzzy aggregation operators and their apllication to multi-criteriadecision making problems, Journal of Intelligent & FuzzySystems 36(1) (2019), 595–607.

29.

Kaur

and Garg

, Multi-attribute decision-making based onBonferroni mean operators under cubic intuitionistic fuzzy setenvironment, Entropy 20(1) (2018), 1–26.

30.

Garg

, A new possibility degree measure for interval-valued q-rungorthopair fuzzy sets in decisionmaking, International Journalof Intelligent Systems 36(1) (2021), 526–557.

31.

Riaz

, Garg

, Farid

H.M.A.

and Chinram

, Multi-criteriadecision making based on bipolar picture fuzzy operators and newdistance measures, Computer Modeling in Engineering andSciences 127(2) (2021), 771–800.

32.

Riaz

, Pamucar

, Farid

H.M.A.

and Hashmi

M.R.

, q-Rung orthopair fuzzy prioritized aggregation operators and their application towards green supplier chain management, Symmetry 12(6) (2020), 1–33.

33.

Ashraf

, Abdullah

and Khan

, Fuzzy decision support modelingfor internet finance soft power eval uation based on sinetrigonometric Pythagorean fuzzy information, Journal of Ambient Intelligence and Humanized Computing 12(3) (2021), 3101–3119.

34.

Dombi

, A general class of fuzzy operators, the demorgan class offuzzy operators and fuzziness measures induced by fuzzy operators, Fuzzy Sets and Systems 8(2) (1982), 149–163.

35.

Liu

, Liu

and Chen

S.M.

, Some intuitionistic fuzzy Dombi Bonferroni mean operators and their application to multi-attribute group decision making, Journal of the Operational Research Society 69(1) (2017), 1–24.

36.

Liu

, Wang

and Liu

, Multiple attribute group decision makingbased on q-rung orthopair fuzzy Heronian mean operators, International Journal of Intelligent Systems 33(3) (2018), 2341–2363.

37.

Liu

, Wang

, Li

, Zhao

and Liu

, Q-rung orthopair fuzzy multiple attribute group decision-making method based on normalized bidirectional projection model and generalized knowledge-based entropy measure, Journal of Ambient Intelligence and Humanized Computing 12(8) (2021), 2715–2730.

38.

Liu

and Wang

, Some q-rung orthopair fuzzy aggregationoperators and their applications to multipleattribute decisionmaking, International Journal of Intelligent Systems 33(4) (2017), 1–22.

39.

Jana

, Pal

and Wang

J.Q.

, Bipolar fuzzy Dombi aggregationoperators and its application in multipleattribute decision-makingprocess, Journal of Ambient intelligence and HumanizedComputing 10 (2019), 3533–3549.

40.

Akram

, Dudek

W.A.

and Dar

J.M.

, Pythagorean Dombi fuzzyaggregation operators with application in multicriteriadecision-making, International Journal of Intelligent Systems 34(3) (2019), 3000–3019.

41.

Khan

A.A.

, Ashraf

, Abdullah

, Qiyas

, Luo

and Khan

S.U.

, Pythagorean fuzzy Dombi aggregation operators and their applicationin decision support system, Symmetry 11(3) (2019), 1–19.

42.

Chen

and Ye

, Some single-valued neutrosophic Dombi weightedaggregation operators for multiple attribute decision-making, Symmetry 9(6) (2017), 82.

43.

Riaz

and Hashmi

M.R.

, A novel approach to censuses process byusing Pythagorean m-polar fuzzy Dombi’s aggregation operators, Journal of Intelligent & Fuzzy Systems 38(2) (2020), 1977–1995.

44.

Shi

and Ye

, Dombi aggregation operators of neutrosophic cubicsets for multiple attribute decisionmaking, Algorithms 11(3) (2018), 29.

45.

Riaz

and Tehrim

S.T.

, Multi-attribute group decision making basedon cubic bipolar fuzzy information using averaging aggregationoperators, Jouranl of Intelligent & Fuzzy Systems 37(2) (2019), 2473–2494.

46.

Riaz

and Tehrim

S.T.

, Cubic bipolar fuzzy ordered weightedgeometric aggregation operators and their application using internaland external cubic bipolar fuzzy data, Computational andApplied Mathematics 38(2) (2019), 87.

47.

Riaz

and Tehrim

S.T.

, Cubic bipolar fuzzy set with application tomulti-criteria group decision making using geometric aggregationoperators, Soft Computing 24(16) (2020), 16111–16133.

Cubic bipolar fuzzy Dombi averaging aggregation operators with application to multi-criteria decision-making

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Dombi operations on CBFNs

3.1 Dombi operations

3.2 Dombi operations on CBFNs under ℙ -order

3.3 Dombi operations on CBFNs under ℝ -order

4 ℙ -order cubic bipolar fuzzy Dombi averaging aggregation operators

4.1 ℙ -order cubic bipolar fuzzy Dombi weighted averaging ( ℙ -CBFDWA) operator

4.2 ℙ -order cubic bipolar fuzzy Dombi ordered weighted averaging ( ℙ -CBFDOWA) operator

5 ℝ -order cubic bipolar fuzzy Dombi averaging aggregation operators

5.1 ℝ -order cubic bipolar fuzzy Dombi weighted averaging ( ℝ -CBFDWA) operator

5.2 ℝ -order cubic bipolar fuzzy Dombi ordered weighted averaging ( ℝ -CBFDOWA) operator

6 Application of CBFDWAA operators in solving MCDM problems

Footnotes

Acknowledgment

References

3.2 Dombi operations on CBFNs under $ℙ$ -order

3.3 Dombi operations on CBFNs under $ℝ$ -order

4 $ℙ$ -order cubic bipolar fuzzy Dombi averaging aggregation operators

4.1 $ℙ$ -order cubic bipolar fuzzy Dombi weighted averaging ( $ℙ$ -CBFDWA) operator

4.2 $ℙ$ -order cubic bipolar fuzzy Dombi ordered weighted averaging ( $ℙ$ -CBFDOWA) operator

5 $ℝ$ -order cubic bipolar fuzzy Dombi averaging aggregation operators

5.1 $ℝ$ -order cubic bipolar fuzzy Dombi weighted averaging ( $ℝ$ -CBFDWA) operator

5.2 $ℝ$ -order cubic bipolar fuzzy Dombi ordered weighted averaging ( $ℝ$ -CBFDOWA) operator