Trapezoidal cubic hesitant fuzzy aggregation operators and their application in group decision-making

Abstract

The aim of this paper is to define some new operation laws for trapezoidal cubic hesitant fuzzy (TrCHF) numbers and define some new aggregation operators. In order to extend these operational laws of the trapezoidal cubic hesitant fuzzy numbers (TrCHFNs), we constructed a series of aggregation operator so called trapezoidal cubic hesitant fuzzy averaging and geometric aggregation operators. Furthermore, these aggregation operators applied to group decision making. Finally, a numerical example is used to illustrate the validity of the proposed approach in group decision-making problems.

Keywords

Trapezoidal cubic hesitant fuzzy sets operational laws aggregation operator multiple attribute group decision-making numerical application

1 Introduction

Multiple attribute group decision making (MAGDM) is the process where the decision makers select the optimal alternative from all of the predefined alternatives by comparative analysis in terms of multiple attributes variables. MAGDM problems have successful applications in the management, scientific, political, and cultural and other fields. In the fact decision-making process, decision makers are often trapped in using real number to evaluate alternatives, because the objective things are difficult to describe, and people’s judgments are subjective and uncertain. We always face tasks and activities in which it is necessary to use decision making processes in our daily lives. Therefore, Atanassov [3] introduced the concept of intuitionistic fuzzy set (IFS) characterized by a membership function and a non-membership function, which is a generalization of the concept of fuzzy set [24]. The IFS has been proven to be highly useful to deal with uncertainty and vagueness. According to Zadeh [25], a conditional statement “If x = A then y = C” describes a relation between the two fuzzy variables x and y. He therefore suggests that the conditional statement should be represented by a fuzzy relation from U and V.

Chen el al. [5] generalized the concept of hesitant fuzzy set (HFS) to that of interval-valued hesitant fuzzy set (IVHFS) in which the membership degrees of an element to a given set are not exactly defined, but denoted by several possible interval values. Torra [19] proposed the hesitant fuzzy set which permitted the membership having a set of possible values and discussed the relationship between the hesitant fuzzy set and intuitionistic fuzzy set. The hesitant fuzzy set can be applied to many decision-making problems. He also proved that the operations he proposed are consistent with the ones of the intuitionistic fuzzy set when applied to the envelope of the hesitant fuzzy set. To get the optimal alternative in a decision making problem with multiple attributes and multiple persons, there are usually two ways: (1) aggregate the decision makers’ opinions under each attribute for alternatives, then aggregate the collective values of attributes for each alternative; (2) aggregate the attribute values given by the decision makers for each alternative, and then aggregate the decision makers’ opinions for each alternative. Hesitant Fuzzy Set (HFS), a speculation of fuzzy set, was proposed by Torra and Narukawa [20] which permits the membership of a component to a set speak by few conceivable qualities. HFSs are an exceptionally valuable instrument to express individual’s aversion in everyday life. Since its predominance for communicating instability, HFS has drawn more consideration from scientists. When people make a decision, they are usually hesitant and irresolute for one thing or another which makes it difficult to reach a final agreement. For example, two decision makers discuss the membership degree of an element x to a set A, and one wants to assign 0.4 but the other 0.7. Xia et al. [22] discussed the relationship between intuitionistic fuzzy set and hesitant fuzzy set, based on which we develop some operations and aggregation operators for hesitant fuzzy elements. Mahmood et al. [18] introduced cubic hesitant fuzzy set, defined internal (external) cubic hesitant fuzzy set, P(R)-union and P(R)-intersection, P(R)-addition and P(R)-multiplication of cubic hesitant fuzzy sets. By using the defined operations of cubic hesitant fuzzy sets the author proved their different results. The author defined R-weighted averaging and R-weighted geometric operators for cubic hesitant fuzzy sets and practiced it in multi-criteria decision-making problem.

Cubic set displayed by jun in [15]. Cubic sets are the theories of fuzzy sets and intuitionistic fuzzy sets, in which there are two depictions, one is used for the level of membership and other is used for the level of non-membership. The membership capacity is hold as interval while non-membership is altogether viewed as the customary fuzzy set.

Aliya et al. [8] developed the hamming distance for triangular cubic fuzzy number and weighted averaging operator. Aliya et al. [9] proposed the cubic TOPSIS method and grey relational analysis set. Aliya et al. [10] defined the triangular cubic fuzzy number and operational laws. The author developed the triangular cubic fuzzy hybrid aggregation (TCFHA) administrator to total all individual fuzzy choice structure provide by the decision makers into the aggregate cubic fuzzy decision matrix. Aliya et al. [1] defined the generalized triangular cubic linguistic hesitant fuzzy weighted geometric (GTCHFWG) operator, generalized triangular cubic linguistic hesitant fuzzy ordered weighted average (GTCLHFOWA) operator, generalized triangular cubic linguistic hesitant fuzzy ordered weighted geometric (GTCLHFOWG) operator, generalized triangular cubic linguistic hesitant fuzzy hybrid averaging (GTCLHFHA) operator and generalized triangular cubic linguistic hesitant fuzzy hybrid geometric (GTCLHFHG) operator. Aliya et al. [12] developed Trapezoidal linguistic cubic hesitant fuzzy TOPSIS method to solve the MCDM method based on trapezoidal linguistic cubic hesitant fuzzy TOPSIS method. Aliya et al. [11] define aggregation operators for triangular cubic linguistic hesitant fuzzy sets which include cubic linguistic fuzzy (geometric) operator, triangular cubic linguistic hesitant fuzzy weighted geometric (TCLHFWG) operator, triangular cubic linguistic hesitant fuzzy ordered weighted geometric (TCHFOWG) operator and triangular cubic linguistic hesitant fuzzy hybrid geometric (TCLHFHG) operator. Aliye et al. [13] defined the trapezoidal cubic fuzzyy weighted arithmetic averaging operator and weighted geometric averaging operator. Expected values, score function, and accuracy function of trapezoidal cubic fuzzy numbers are defined. Fahmi et al. [14] defined some Einstein operations on cubic fuzzy set (CFS) and develop three arithmetic averaging operators, which are cubic fuzzy Einstein weighted averaging (CFEWA) operator, cubic fuzzy Einstein ordered weighted averaging (CFEOWA) operator and cubic fuzzy Einstein hybrid weighted averaging (CFEHWA) operator, for aggregating cubic fuzzy data. Amin et al. [2] introduced the new concept of the trapezoidal cubic hesitant fuzzy TOPSIS method.

Zhang et al. [28] proposed a consensus-based group decision-making framework for FMEA with the aim of classifying FMs into several ordinal risk classes in which we assumed that FMEA participants provide their preferences in a linguistic way using possibility hesitant fuzzy linguistic information. Zhang et al. [29] proposed SNA-based consensus framework, a trust propagation and aggregation mechanism to yield experts’ weights from the social trust network is presented, and the obtained weights of experts are then integrated into the consensus-based MAGDM framework. Zhang et al. [26] investigated the 2-rank MAGDM problem under the multigranular linguistic context, and proposes a 2-rank consensus reaching framework with the minimum adjustments. Dong et al. [7] proposed that obtains the ranking of individual alternatives and a collective solution. Zhang et al. [30] extended to improve the additive consistency and impute the missing elements for incomplete hesitant fuzzy preference relation (HFPR). Zhang et al. [31] proposed a new procedure for group analytic hierarchy process to deal with multi-criteria group decision making problems. Yu et al. [23] proposed to calculate the gain, loss for an unbalanced HFLTS over another, special case of the unbalanced HFLTS, the formulae of gain and loss for a balanced HFLTS. Li et al. [16] proposed to improve the willingness of decision makers who follow the suggestions to revise their preferences in order to achieve a consensus in linguistic LSGDM problems. Liu et al. [17] proposed the minimum cost strategic weight manipulation model, which is achieved via optimization approach, with the mixed 0-1 linear programming model being proved appropriate in this context. Wu el al. [21] proposed a new linguistic group decision model called the maximum support degree model (MSDM), aiming at maximizing the support degree of the group opinion as well as guarantying the accuracy of the group opinion.

Despite having a bulk of related literature on the problem under consideration, the following aspects related to trapezoidal cubic hesitant fuzzy sets and their aggregation operators motivated the researchers to carry it an in depth inquiry into the current study.

The main advantages of the proposed operators are these aggregation operators provided more accurate and precious result as compare to the above mention operators.

We generalized the concept of trapezoidal cubic hesitant fuzzy sets, trapezoidal intuitionistic hesitant fuzzy sets and introduce the concept of trapezoidal cubic hesitant fuzzy sets. If we take only one element in the membership degree of the trapezoidal cubic hesitant fuzzy number, i.e. instead of interval we take a fuzzy number, then we get trapezoidal intuitionistic hesitant fuzzy numbers, similarly if we take membership degree as fuzzy number and non-membership degree equal to zero, then we get trapezoidal hesitant fuzzy numbers.

The objective of the study includes:

Propose trapezoidal cubic hesitant fuzzy sets, operational laws, score value and accuracy value of TrCHFSs.

Propose six aggregation operators, namely TrCHFWA, TrCHFWG, TrCHFOWA, TrCHFOWG, TrCHFHA and TrCHFHG Operators.

Establish MADM program approach based trapezoidal cubic hesitant fuzzy sets.

Provide illustrative examples of MADM program.

In order to testify the application of the developed method, we apply the trapezoidal cubic hesitant fuzzy sets in the decision making.

The initial decision matrix is composed of LVs. In order to fully consider the randomness and ambiguity of linguistic term, we convert LVs into the trapezoidal cubic hesitant fuzzy sets, and the decision matrix is transformed into the trapezoidal cubic hesitant fuzzy decision matrix.

The operator can fully express the uncertainty of the qualitative concept and trapezoidal cubic hesitant fuzzy operators can capture the interdependencies among any multiple inputs or attributes by a variable parameter. The aggregation operators can take into account the importance of the attribute weights. Nevertheless, sometimes, for some MAGDM problems, the weights of the attributes are important factors for decision process.

In this paper, a novel concept called trapezoidal cubic hesitant fuzzy variable which combines the trapezoidal hesitant fuzzy and intuitionistic hesitant fuzzy set is proposed. Then some new operators for aggregating intuitionistic trapezoidal cubic hesitant fuzzy information are proposed, such as trapezoidal cubic hesitant fuzzy weighted averaging (TrCHFWA) operator, trapezoidal cubic hesitant fuzzy weighted geometric (TrCHFWG) operator, trapezoidal cubic hesitant fuzzy ordered weighted averaging (TrCHFOWA) operator, trapezoidal cubic hesitant fuzzy ordered weighted geometric (TrCHFOWG) operator, trapezoidal cubic hesitant fuzzy hybrid averaging (TrCHFHA) operator and trapezoidal cubic hesitant fuzzy hybrid geometric (TrCHFHG) operator. Furthermore, novel methods to solve the MAGDM problems in which the attribute weights take the form of real numbers, attribute values take the form of trapezoidal cubic hesitant fuzzy information are developed based on the proposed operators. Finally, a numerical example of emergency logistics supplier selection is given to illustrate the applications of the developed method.

In this paper, trapezoidal cubic hesitant fuzzy numbers are introduced as cubic hesitant fuzzy numbers of a special type, which have appealing interpretations and can, be easily specified and implemented by the decision maker. The concept of the trapezoidal cubic hesitant fuzzy numbers and ranking method as well as applications are discussed in depth.

The paper is organized as follows: Section 2, we discuss some basic ideas to the fuzzy set, cubic set and trapezoidal cubic hesitant fuzzy set. In Section 3, we first develop some novel aggregation operators, such as trapezoidal cubic hesitant fuzzy weighted averaging (TrCHFWA) operator, trapezoidal cubic hesitant fuzzy weighted geometric (TrCHFWG) operator, trapezoidal cubic hesitant fuzzy ordered weighted averaging (TrCHFOWA) operator, trapezoidal cubic hesitant fuzzy ordered weighted geometric (TrCHFOWG) operator, trapezoidal cubic hesitant fuzzy hybrid averaging (TrCHFHA) operator and trapezoidal cubic hesitant fuzzy hybrid geometric (TrCHFHG) operator, for aggregating a collection of trapezoidal cubic hesitant fuzzy numbers (TrCHFNs). Section 4 develops an approach to group decision makings with trapezoidal cubic hesitant fuzzy data. Section 5, gives an example to illustrate the application of the developed method. Section 6, we propose the comparison method. The paper is concluded in Section 7.

2 Preliminaries

Definition 2.1. [24] Let H be a universe of discourse. The idea of fuzzy set was presented by Zadeh and defined as follows; $J = {\ddot{h}, Γ_{J} (\ddot{h}) | \ddot{h} \in H}$ . A fuzzy set in a set $\ddot{h}$ is defined Γ _J : H → I, is a membership function, $Γ_{J} (\ddot{h})$ denoted the degree of membership of the element $\ddot{h}$ to the set H, where I = [0, 1]. The collection of all fuzzy subsets of H is denoted by I ^H. Define a relation on I ^H as follows: $(\forall Γ, η \in I^{H}) (Γ ⩽ η \Leftrightarrow (\forall \ddot{h} \in \ddot{h}) (Γ (\ddot{h}) ⩽ η (\ddot{h}))) .$

Definition 2.2. [13] An intuitionistic fuzzy set A in H is given by A ={ 〈 h, Γ _A (h), η _A (h) 〉 | ∀ h ∈ H }. The Γ _A is represent membership function and η _A is represent non-membership function, where Γ _A : H → [0, 1] and η _A : H → [0, 1], with the conditions 0 ⩽ Γ _A (h) + η _A (h) ⩽1, ∀ h ∈ H . Let π (h) =1 - Γ _A (h) - η _A (h), ∀ h ∈ H, then π (h) is called the degree of indeterminacy of H to the set A.

Definition 2.3. [19, 20] Let H be a fixed set, a hesitant fuzzy set (HFS) on H is in terms of a function that when applied to H returns a subsets of [0,1], which can be represented as the following mathematical symbol: E ={ 〈h, Γ _E (h) 〉|h ∈ H } where Γ _E (h) is a set of values in [0,1], denoting the possible membership degrees of the element h ∈ H to the set E For convenience, we call Γ _E (h) a hesitant fuzzy element (HFE).

Definition 2.4. [15] Let H be a nonempty set. By a cubic set in H we mean a structure $F = {\ddot{h}, α (\ddot{h}), β (\ddot{h}) : \ddot{h} \in H}$ in which α is an IVF set in H and β is a fuzzy set in H A cubic set $F = {\ddot{h}, α (\ddot{h}), β (\ddot{h}) : \ddot{h} \in H}$ is simply denoted by F = 〈α, β〉. The collection of all cubic sets in $\ddot{h}$ is denoted by C ^H . A cubic set F = 〈α, β〉 in which $α (\ddot{h}) = 0$ And $β (\ddot{h}) = 1$ (resp. $α (\ddot{h}) = 1$ And $β (\ddot{h}) = 0$ for all $\ddot{h} \in H$ is denoted by 0 (resp. 1). A cubic set D = 〈λ, ξ〉 in which $λ (\ddot{h}) = 0$ and $ξ (\ddot{h}) = 0$ (resp. $λ (\ddot{h}) = 1$ and $ξ (\ddot{h}) = 1$ ) for all $\ddot{h} \in H$ is denoted by 0 (resp. 1).

Definition 2.5. [15] Let H be a non-empty set. A cubic set F = (C, λ) in H is said to be an internal cubic set if $C^{-} (\ddot{h}) ⩽ λ (\ddot{h}) ⩽ C^{+} (\ddot{h})$ for all $\ddot{h} \in H .$

Definition 2.6. [15] Let H be a non-empty set. A cubic set F = (C, λ) in H is said to be an external cubic set if $λ (\ddot{h}) \notin (C^{-} (\ddot{h}), C^{+} (\ddot{h}))$ for all $\ddot{h} \in H .$

1.1 Trapezoidal cubic hesitant fuzzy sets

Definition 2.1.1. Let $\tilde{b}$ be the trapezoidal cubic hesitant fuzzy number on the set of real numbers, its IVTrHFS is defined as:

λ_{\tilde{b}} (h) = {\begin{matrix} [\frac{(h - r)}{(s - r)}] [μ_{b}^{-}, μ_{b}^{+}] & r \leq h s \\ [μ_{b}^{-}, μ_{b}^{+}] & s \leq h t \\ \frac{(t - h)}{(d - s)} [μ_{b}^{-}, μ_{b}^{+}] & t \leq h u \\ 0 & h (r, u) or h (r, u) \end{matrix}

and its trapezoidal hesitant fuzzy set is

$Γ_{\tilde{b}} (h) = {\begin{matrix} s_{θ (h)}, [\frac{(h - r)}{(s - r)}] [η_{\tilde{b}}] & r \leq h s \\ [η_{\tilde{b}}] & s \leq h t \\ s_{θ (h)}, \frac{(t - h)}{(d - s)} [η_{\tilde{b}}] & t \leq h u \\ 0 & h (r, u) or h (r, u) \end{matrix}$ where $0 \leq λ_{\tilde{b}} (h) \leq 1, 0 \leq Γ_{\tilde{b}} (h) \leq 1$ and r, s, t, u are real numbers. Then the TrCHFN $\tilde{b}$ basically denoted by $b = {[r, s, t, u], 〈 [μ_{b}^{-}, μ_{b}^{+}], η_{\tilde{b}} 〉} .$ Further, the TrCHFN reduced to a TCHFN. If the TrCHFN $\tilde{b}$ is called a normal TCHFN denoted as $\tilde{b} = {[r, s, t, u]; 〈 [1, 1], 0 〉} .$ Therefore, the TrCHFN considered now can be regarded as generalized TrHCFN. Then $\tilde{b}$ is called trapezoidal cubic hesitant fuzzy number.

Definition 2.1.2. Let, $h = {\begin{matrix} [ɛ, \\ ξ, \\ ψ, \\ υ], \\ 〈 [Ω^{-}, \\ Ω^{+}], \\ Ω 〉 \end{matrix}}, h_{1} = {\begin{matrix} [ɛ_{1}, \\ ξ_{1}, \\ ψ_{1}, \\ υ_{1}], \\ 〈 [Ω_{1}^{-}, \\ Ω_{1}^{+}], \\ Ω_{1} 〉 \end{matrix}}$ and $h_{2} = {\begin{matrix} [ɛ_{2}, \\ ξ_{2}, \\ ψ_{2}, \\ υ_{2}], \\ 〈 [Ω_{2}^{-}, \\ Ω_{2}^{+}], \\ Ω_{2} 〉 \end{matrix}}$ be any three TrCHFNs. Then,

${\ddot{h}}^{c} = {α^{c} | α \in \ddot{h}} = \underset{μ^{-} \in Ω^{-}, μ^{+} \in Ω^{+}, η \in Ω}{⋃_{a \in ɛ, b \in ξ, c \in ψ, d \in υ,}}$ $[a, b, c, d], 〈 [μ_{b^{-}}, μ_{b^{+}}], η_{\tilde{b}} 〉;$ ${\ddot{h}}_{1} \cup {\ddot{h}}_{2} = \underset{μ_{1}^{+} \in Ω_{1}^{+}, μ_{2}^{+} \in Ω_{2}^{+}, η_{1} \in Ω_{1}, η_{2} \in Ω_{2}}{\underset{d_{2} \in υ_{2}, μ_{1}^{-} \in Ω_{1}^{-}, μ_{2}^{-} \in Ω_{2}^{-},}{\underset{c_{1} \in ψ_{1}, c_{2} \in ψ_{2}, d_{1} \in υ_{1},}{⋃_{a_{1} \in ɛ_{1}, a_{2} \in ɛ_{2}, b_{1} \in ξ_{1}, b_{2} \in ξ_{2},}}}}$ $\begin{matrix} {\begin{matrix} [a_{1} \cup a_{2}, b_{1} \cup b_{2}, c_{1} \cup c_{2}, d_{1} \cup d_{2}], \\ 〈 [μ_{{\ddot{h}}_{1}}^{-} \cup μ_{{\ddot{h}}_{2}}^{-}, μ_{{\ddot{h}}_{1}}^{+} \cup μ_{{\ddot{h}}_{2}}^{+}], η_{{\ddot{h}}_{1}} \cap η_{{\ddot{h}}_{2}} 〉 \end{matrix}}; \end{matrix}$ ${\ddot{h}}_{1} \cap {\ddot{h}}_{2} = \underset{μ_{1}^{+} \in Ω_{1}^{+}, μ_{2}^{+} \in Ω_{2}^{+}, η_{1} \in Ω_{1}, η_{2} \in Ω_{2}}{\underset{d_{2} \in υ_{2}, μ_{1}^{-} \in Ω_{1}^{-}, μ_{2}^{-} \in Ω_{2}^{-},}{\underset{c_{1} \in ψ_{1}, c_{2} \in ψ_{2}, d_{1} \in υ_{1},}{⋃_{a_{1} \in ɛ_{1}, a_{2} \in ɛ_{2}, b_{1} \in ξ_{1}, b_{2} \in ξ_{2},}}}}$ $\begin{matrix} {\begin{matrix} [a_{1} \cap a_{2}, b_{1} \cap b_{2}, c_{1} \cap c_{2}, d_{1} \cap d_{2}], \\ 〈 [μ_{{\ddot{h}}_{1}}^{-} \cap μ_{{\ddot{h}}_{2}}^{-}, μ_{{\ddot{h}}_{1}}^{+} \cap μ_{{\ddot{h}}_{2}}^{+}], η_{{\tilde{b}}_{1}} \cup η_{{\ddot{h}}_{2}} 〉 \end{matrix}}; \end{matrix}$

${\ddot{h}}_{1} \oplus {\ddot{h}}_{2} = \underset{μ_{1}^{+} \in Ω_{1}^{+}, μ_{2}^{+} \in Ω_{2}^{+}, η_{1} \in Ω_{1}, η_{2} \in Ω_{2}}{\underset{d_{2} \in υ_{2}, μ_{1}^{-} \in Ω_{1}^{-}, μ_{2}^{-} \in Ω_{2}^{-},}{\underset{c_{1} \in ψ_{1}, c_{2} \in ψ_{2}, d_{1} \in υ_{1},}{⋃_{a_{1} \in ɛ_{1}, a_{2} \in ɛ_{2}, b_{1} \in ξ_{1}, b_{2} \in ξ_{2},}}}}$

$\begin{matrix} {\begin{matrix} [a_{1} + a_{2} - a_{1} a_{2}, b_{1} + b_{2} - b_{1} b_{2}, \\ c_{1} + c_{2} - c_{1} c_{2}, d_{1} + d_{2} - d_{1} d_{2}], \\ 〈 [μ_{{\ddot{h}}_{1}}^{-} + μ_{{\ddot{h}}_{2}}^{-} - μ_{{\ddot{h}}_{1}}^{-} μ_{{\ddot{h}}_{2}}^{-}, μ_{{\ddot{h}}_{1}}^{+} + μ_{{\ddot{h}}_{2}}^{+} - μ_{{\ddot{h}}_{1}}^{+} μ_{{\ddot{h}}_{2}}^{+}], \\ η_{{\tilde{b}}_{1}} η_{{\tilde{b}}_{2}} 〉 \end{matrix}}; \end{matrix}$

$λ \ddot{h} = {λ h | a \in \ddot{h}} = \underset{μ^{-} \in Ω^{-}, μ^{+} \in Ω^{+}, η \in Ω}{⋃_{a \in ɛ, b \in ξ, c \in ψ, d \in υ,}}$

$\begin{matrix} {\begin{matrix} [1 - (1 - a)^{λ}], [1 - (1 - b)^{λ}], \\ [1 - (1 - c)^{λ}], [1 - (1 - d)^{λ}]; \\ 〈 [1 - (1 - μ_{\ddot{h}}^{-})^{λ}], [1 - (1 - μ_{\ddot{h}}^{+})^{λ}], \\ (η_{\ddot{h}})^{λ} 〉 \end{matrix}}; \end{matrix}$

${\ddot{h}}^{λ} = {α^{λ} | a \in \ddot{h}} = \underset{μ^{-} \in Ω^{-}, μ^{+} \in Ω^{+}, η \in Ω}{⋃_{a \in ɛ, b \in ξ, c \in ψ, d \in υ,}}$

$\begin{matrix} {\begin{matrix} [(a)^{λ}, (b)^{λ}, (c)^{λ}, (d)^{λ}], \\ 〈 [(μ_{\ddot{h}}^{-})^{λ}, (μ_{\ddot{h}}^{+})^{λ}], 1 - (1 - η_{\ddot{h}})^{λ} 〉 \end{matrix}} . \end{matrix}$

Example 2.1.3. Let $α = {\begin{matrix} [0.4, 0.6, \\ 0.8, 0.10], \\ 〈 [0.2, 0.4], \\ 0.3 〉 \end{matrix}}$ $α_{1} = {\begin{matrix} [0.2, 0.4, \\ 0.6, 0.8], \\ 〈 [0.4, 0.6], \\ 0.5 〉 \end{matrix}}$ and $α_{2} = {\begin{matrix} [0.9, 0.11, \\ 0.13, 0.15], \\ 〈 [0.11, 0.13], \\ 0.10 〉 \end{matrix}}$ be any three TrCHFNs. Then,

α ^c = 〈[0.4, 0.6, 0.8, 0.10], [0.2, 0.4], 0.3〉,

$α_{1} \cup α_{2} = {\begin{matrix} [0.2 \cup 0.9, 0.4 \cup 0.11, 0.6 \cup 0.13, \\ 0.8 \cup 0.15], 〈 [0.4 \cup 0.11, 0.6 \cup 0.13], 0.5 \cap 0.10 〉 \end{matrix}};$

$α_{1} \cap α_{2} = {\begin{matrix} [0.2 \cap 0.9, 0.4 \cap 0.11, 0.6 \cap 0.13, \\ 0.8 \cap 0.15, 〈 [0.4 \cap 0.11, 0.6 \cap 0.13], 0.5 \cup 0.10 〉 \end{matrix}};$ $α_{1} \oplus α_{2} = {\begin{matrix} [0.2 + 0.9 - (0.2) (0.9), 0.4 + 0.11 - (0.4) (0.11), \\ 0.6 + 0.13 - (0.6) (0.13), 0.8 + 0.15 - (0.8) (0.15)], \\ 〈 [0.4 + 0.11 - (0.4) 0 (0.11), 0.6 + 0.13 - (0.6) (0.13)], \\ (0.5) (0.10) 〉 \end{matrix}};$ $α_{1} \otimes α_{2} = {\begin{matrix} [(0.2) (0.9), (0.4) (0.11), (0.6) (0.13), \\ (0.8) (0.15)], [(0.4) (0.11), (0.6) (0.13)], \\ 0.5 + 0.10 - (0.5) (0.10) 〉 \end{matrix}};$ $λ = (0.25, 0.25, 0.25, 0.25)^{T}$ $λ α = {\begin{matrix} S_{3 \times 0.25}, [1 - (1 - 0.4)^{0.25}, 1 - (1 - 0.6)^{0.25}, \\ 1 - (1 - 0.8)^{0.25}, 1 - (1 - 0.10)^{0.25}], \\ 〈 [1 - (1 - 0.2)^{0.25}, 1 - (1 - 0.4)^{0.25}], \\ (0.3)^{0.25} 〉 \end{matrix}};$ $λ α = {\begin{matrix} [(0.4)^{0.25}, (0.6)^{0.25}, (0.8)^{0.25}, (0.10)^{0.25}], \\ 〈 [(0.2)^{0.25}, (0.4)^{0.25}], 1 - (1 - 0.3)^{0.25} 〉 \end{matrix}}$

Theorem 2.1.4. Let $h = {\begin{matrix} [ɛ, \\ ξ, \\ ψ, \\ υ], \\ 〈 [Ω^{-}, \\ Ω^{+}], \\ Ω 〉 \end{matrix}}, h_{1} = {\begin{matrix} [ɛ_{1}, \\ ξ_{1}, \\ ψ_{1}, \\ υ_{1}], \\ 〈 [Ω_{1}^{-}, \\ Ω_{1}^{+}], \\ Ω_{1} 〉 \end{matrix}}$ and $h_{2} = {\begin{matrix} [ɛ_{2}, \\ ξ_{2}, \\ ψ_{2}, \\ υ_{2}], \\ 〈 [Ω_{2}^{-}, \\ Ω_{2}^{+}], \\ Ω_{2} 〉 \end{matrix}}$ be any three TrCHFNs. Then, the score function $S (α) = 〈 \frac{{[a + b + c + d] + [μ^{-} + μ^{+}] - η}}{12} 〉,$ accuracy function $\ddot{h} (α) = 〈 \frac{{[a + b + c + d] + [μ^{-} + μ^{+}] + η}}{12}$ gave an order relation between two TrCHFNs α ₁ and α ₂ .

If s (α ₁) > s (α ₂), then s (α ₁) > s (α ₂).

If s (α ₁) = s (α ₂), then the following hold.

If H (α ₁) > H (α ₂), then α ₁ > α ₂.

If H (α ₁) = H (α ₂), then α ₁ = α ₂.

If H (α ₁) > H (α ₂), then α ₁ > α ₂.

Example 2.1.5. Let $\begin{matrix} {\ddot{h}}_{1} = {\begin{matrix} [0.4, 0.5, 0.6, 0.7], \\ 〈 [0.1, 0.3], 0.2 〉 \end{matrix}}, \\ {\ddot{h}}_{2} = {\begin{matrix} [0.8, 0.9, 0.10, 0.11], \\ 〈 [0.14, 0.16], 0.15 〉 \end{matrix}} \end{matrix}$ and ${\ddot{h}}_{2} = {\begin{matrix} [0.2, 0.4, 0.6, 0.8], \\ 〈 [0.10, 0.12], 0.11 〉 \end{matrix}}$ be three TrCHFNs. Then, the score function $S (h_{1}) = \frac{2.4}{12} = 0.2, S (h_{1}) = \frac{2.06}{12} = 0.1716, S (h_{3}) = \frac{2.11}{12} = 0.1758$ .

Accuracy function $H (h_{1}) = \frac{2.8}{12} = 0.2333, H (h_{2}) = \frac{2.36}{12} = 0.1966, H (h_{3}) = \frac{2.33}{12} = 0.1941$ .

Theorem 2.1.6. Let $h = {\begin{matrix} [ɛ, \\ ξ, \\ ψ, \\ υ], \\ 〈 [Ω^{-}, \\ Ω^{+}], \\ Ω 〉 \end{matrix}}, h_{1} = {\begin{matrix} [ɛ_{1}, \\ ξ_{1}, \\ ψ_{1}, \\ υ_{1}], \\ 〈 [Ω_{1}^{-}, \\ Ω_{1}^{+}], \\ Ω_{1} 〉 \end{matrix}}$ and $h_{2} = {\begin{matrix} [ɛ_{2}, \\ ξ_{2}, \\ ψ_{2}, \\ υ_{2}], \\ 〈 [Ω_{2}^{-}, \\ Ω_{2}^{+}], \\ Ω_{2} 〉 \end{matrix}}$ are three TrCHFEs and, then we have that

${\ddot{h}}_{1} \cup {\ddot{h}}_{2} = ({\ddot{h}}_{1} \cap {\ddot{h}}_{2})^{c},$

${\ddot{h}}_{1} \cap {\ddot{h}}_{2} = ({\ddot{h}}_{1} \cup {\ddot{h}}_{2})^{c}$ ,

$({\ddot{h}}^{c})^{λ} = (λ \ddot{h})^{c},$

$λ ({\ddot{h}}^{c}) = ({\ddot{h}}^{λ})^{c}$ ,

${\ddot{h}}_{1} \oplus {\ddot{h}}_{2} = ({\ddot{h}}_{1} \otimes {\ddot{h}}_{2})^{c},$

${\ddot{h}}_{1} + {\ddot{h}}_{2} = {\ddot{h}}_{2} + {\ddot{h}}_{1},$

${\ddot{h}}_{1} \otimes {\ddot{h}}_{2} = {\ddot{h}}_{2} \otimes {\ddot{h}}_{1},$

$λ ({\ddot{h}}_{1} \oplus {\ddot{h}}_{2}) = λ {\ddot{h}}_{1} \oplus λ {\ddot{h}}_{2}; λ ⩾ 0$

$λ_{1} {\ddot{h}}_{1} + {\ddot{h}}_{2} a_{1} = ({\ddot{h}}_{1} + {\ddot{h}}_{2}) a_{1}; λ_{1}, λ_{2} ⩾ 0;$

${\ddot{h}}_{1} \otimes {\ddot{h}}_{2} = ({\ddot{h}}_{1})^{λ_{1} + λ_{2}}, λ_{1}, λ_{2} ⩾ 0;$

${\ddot{h}}_{1} \otimes {\ddot{h}}_{2} = ({\ddot{h}}_{1} \otimes {\ddot{h}}_{2})^{λ_{1}}, λ_{1} ⩾ 0;$

${\ddot{h}}_{1} \otimes {\ddot{h}}_{2} = ({\ddot{h}}_{1} \oplus {\ddot{h}}_{2})^{c}$

Definition 2.1.7. For a TrCHFE $\ddot{h}$ , $s (\ddot{h}) = \sum_{\tilde{Ω} \in \ddot{h}} \tilde{Ω} / # \ddot{h}$ is called the score function of $\ddot{h}$ , where $# \ddot{h}$ is the number of the elements in $\ddot{h}$ . For two TrCHFEs ${\ddot{h}}_{1}$ and ${\ddot{h}}_{2}$ , if $s ({\ddot{h}}_{1}) > s ({\ddot{h}}_{2})$ , then ${\ddot{h}}_{1} > {\ddot{h}}_{2}$ , if $s ({\ddot{h}}_{1}) = s ({\ddot{h}}_{2})$ , then ${\ddot{h}}_{1} = {\ddot{h}}_{2}$ .

3 Aggregation trapezoidal cubic hesitant fuzzy information

In this section, we exhibit a series of operators for aggregating the trapezoidal cubic hesitant fuzzy information and investigate some desired properties of these operators.

3.1 The TrCHFWA and TrCHFWG Operators

Definition 3.1.1. Let ${\ddot{h}}_{i} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ are the collections of TrCHFEs and let τ = (τ ₁, τ ₂, τ _n) ^T be the weight vector of TrCHFEs ${\ddot{h}}_{i} (i = 1, 2, \dots, n)$ where τ _i ∈ [0, 1], $\sum_{i = 1}^{n} τ_{i} = 1 .$ The trapezoidal cubic hesitant fuzzy weighted averaging operator is a mapping ${\ddot{h}}^{n} \to \ddot{h}$ such that TrCHFWA $({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = \oplus_{i = 1}^{n} (τ_{i} {\ddot{h}}_{i})$ . If $τ = (\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})^{T}$ , then the TrCHFWA operator reduces to the trapezoidal cubic hesitant fuzzy averaging operator: ${TrCHFA ({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = \oplus_{i = 1}^{n} (\frac{1}{n} {\ddot{h}}_{i})} .$

Theorem 3.1.2. Let ${\ddot{h}}_{i} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ be a collection of TrCHFEs. Then the aggregated value is calculated by using the TrCHFWA operator is also a TrCHFE and

${\begin{matrix} TrCHFWA ({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = \oplus_{i = 1}^{n} (τ_{i} {\ddot{h}}_{i}) = \\ \underset{μ^{-} \in Ω^{-}, μ^{+} \in Ω^{+}, η \in Ω}{⋃_{a \in ɛ, b \in ξ, c \in ψ, d \in υ,}} \\ [\prod_{i = 1}^{n} (a_{i})^{τ_{i}}, \prod_{i = 1}^{n} (b_{i})^{τ_{i}}, \prod_{i = 1}^{n} (c_{i})^{τ_{i}}, \prod_{i = 1}^{n} (d_{i})^{τ_{i}}], \\ 〈 \begin{matrix} [\begin{matrix} [1 - \prod_{i = 1}^{n} (1 - μ_{i}^{-})^{τ_{i}}], \\ [1 - \prod_{i = 1}^{n} (1 - μ_{i}^{+})^{τ_{i}}] \end{matrix}], \\ \prod_{i = 1}^{n} (η_{i})^{τ_{i}} \end{matrix} 〉 \end{matrix}} .$

Definition 3.1.3. Let ${\ddot{h}}_{i} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ be a collection of TrCHFEs and let $\ddot{Ψ} = ({\ddot{Ψ}}_{1}, {\ddot{Ψ}}_{2}, \dots, {\ddot{Ψ}}_{n})^{T}$ be the weight vector of TrCHFEs ${\ddot{h}}_{i}$ (i = 1, 2, …, n), where ${\ddot{Ψ}}_{i} \in [0, 1]$ , $\sum_{i = 1}^{n} {\ddot{Ψ}}_{i} = 1 .$ The trapezoidal cubic hesitant fuzzy weighted geometric operator is a mapping ${\ddot{h}}^{n} \to \ddot{h}$ such that TrCHFWG $({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = \otimes_{i = 1}^{n} ({\ddot{h}}_{i}^{{\ddot{Ψ}}_{i}})$ . If $\ddot{Ψ} = (\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})^{T}$ , then the TrCHFWG operator reduces to the trapezoidal cubic hesitant fuzzy averaging operator: TrCHFG $({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = \otimes_{i = 1}^{n} ({\ddot{h}}_{i}^{\frac{1}{n}})$ .

Theorem 3.1.4. Let ${\ddot{h}}_{i} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ be a collection of TrCHFEs. Then the aggregated value is calculated by using the TrCHFWG operator is also a TrCHFE and

${\begin{matrix} TrCHFWG ({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = \\ \underset{μ^{-} \in Ω^{-}, μ^{+} \in Ω^{+}, η \in Ω}{⋃_{a \in ɛ, b \in ξ, c \in ψ, d \in υ,}} \\ [\prod_{i = 1}^{n} (a_{i})^{{\ddot{Ψ}}_{i}}, \prod_{i = 1}^{n} (b_{i})^{{\ddot{Ψ}}_{i}}, \prod_{i = 1}^{n} (c_{i})^{{\ddot{Ψ}}_{i}}, \prod_{i = 1}^{n} (d_{i})^{{\ddot{Ψ}}_{i}}] \\ 〈 \begin{matrix} [\prod_{i = 1}^{n} (μ^{-})^{{\ddot{Ψ}}_{i}}, \prod_{i = 1}^{n} (μ^{+})^{{\ddot{Ψ}}_{i}}], \\ \prod_{i = 1}^{n} (η)^{{\ddot{Ψ}}_{i}} \end{matrix} 〉 \end{matrix}} .$

Corollary 3.1.5. Suppose that ${\ddot{h}}_{i} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ is a collection of TrCHFEs and $\ddot{Ψ} = ({\ddot{Ψ}}_{1}, {\ddot{Ψ}}_{2}, \dots, {\ddot{Ψ}}_{n})^{T}$ is their weight vector with ${\ddot{Ψ}}_{i} \in [0, 1]$ and $\sum_{i = 1}^{n} {\ddot{Ψ}}_{i} = 1$ , then one has TrCHFWG $({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) ⩽$ TrCHFWA $({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n})$ .

Theorem 3.1.6. Let ${\ddot{h}}_{i} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ be a collection of TrCHFEs having the weight vector $\ddot{Ψ} = ({\ddot{Ψ}}_{1}, {\ddot{Ψ}}_{2}, \dots, {\ddot{Ψ}}_{n})^{T}$ such that ${\ddot{Ψ}}_{i} \in [0, 1]$ and $\sum_{i = 1}^{n} {\ddot{Ψ}}_{i} = 1,$ λ> 0; then TrCHFWG $({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) ⩽$ TrCHFWA $({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n})$ .

Example 3.1.7. Suppose that ${\ddot{h}}_{1} = {\begin{matrix} 〈 {[0.2, 0.4, \\ 0.6, 0.8], \\ [0.5, 0.8], \\ 0.6}, 〉 \end{matrix}},$ ${\ddot{h}}_{2} = {\begin{matrix} 〈 {[0.4, 0.8, \\ 0.10, 0.12], \\ [0.7, 0.9], \\ 0.8}, 〉 \end{matrix}}$ and ${\ddot{h}}_{3} = {\begin{matrix} 〈 {[0.1, 0.2, \\ 0.3, 0.4], \\ [0.8, 0.12], \\ 0.10}, 〉 \end{matrix}}$ , are three TrCLHFEs and τ = {0.18, 0.35, 0.47} is their weight vector. Then, by Definition above, we can obtain $\begin{matrix} TrCHFWA \\ λ ({\ddot{h}}_{1}, {\ddot{h}}_{2}, {\ddot{h}}_{3}) = {\begin{matrix} [0.2354, 0.5423, 0.3088, 0.43702], \\ 〈 [0.7281, 0.2278], 0.4595 〉 \end{matrix}}; \end{matrix}$ $\begin{matrix} TrCHFWA \\ ({\ddot{h}}_{1}, {\ddot{h}}_{2}, {\ddot{h}}_{3}) = {\begin{matrix} [[0.7646, 0.4677, 0.6912, 0.5629], \\ 〈 [0.2719, 0.7722], 0.5405 〉 \end{matrix}} \end{matrix} .$ score function $S (h_{1}) = \frac{2.7}{12} = 0.225, S (h_{2}) = \frac{2.23}{12} = 0.1858, s (h_{3}) = \frac{1.82}{12} = 0.1516$ .

3.2 The TrCHFOWA and TrCHFOWG operators

Definition 3.2.1. Let ${\ddot{h}}_{i} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ be a collection of TrCHFEs. The trapezoidal cubic hesitant fuzzy ordered weighted averaging operator is denoted and defined by the mapping ${\ddot{h}}^{n} \to \ddot{h}$ , such that TrCHFOWA $({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = \oplus_{i = 1}^{n} (τ_{i} {\ddot{h}}_{σ (i)}),$ where τ = (τ ₁, τ ₂, …, τ _n) ^T is the aggregation-associated vector of TrCHFEs, such that τ _i ∈ [0, 1] and $\sum_{i = 1}^{n} τ_{i} = 1 .$

Theorem 3.2.2. Let ${\ddot{h}}_{i} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ be a collection of TrCHFEs. Then the aggregated value is calculated by using the TrCHFOWA operator is also a TrCHFE and ${\begin{matrix} TrCHFOWA ({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = \underset{μ^{-} \in Ω^{-}, μ^{+} \in Ω^{+}, η \in Ω}{⋃_{a \in ɛ, b \in ξ, c \in ψ, d \in υ,}} \\ [1 - \prod_{i = 1}^{n} (1 - a_{α_{σ (i)}})^{τ_{i}}, 1 - \prod_{i = 1}^{n} (1 - b_{α_{σ (i)}})^{τ_{i}}, \\ 1 - \prod_{i = 1}^{n} (1 - c_{α_{σ (i)}})^{τ_{i}}, 1 - \prod_{i = 1}^{n} (1 - d_{α_{σ (i)}})^{τ_{i}}] \\ , [\prod_{i = 1}^{n} (μ_{α_{σ (i)}}^{-})^{τ_{i}}, \prod_{i = 1}^{n} (μ_{α_{σ (i)}}^{+})^{τ_{i}}], \prod_{i = 1}^{n} (η_{α_{σ (i)}})^{τ_{i}} \\ | α_{σ (1)}, α_{σ (2)}, \dots, α_{σ (n)} \in {\ddot{h}}_{α_{σ (n)}} \end{matrix}} .$

Definition 3.2.3. Let ${\ddot{h}}_{i} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ be a collection of TrCHFEs. The trapezoidal cubic hesitant fuzzy ordered weighted geometric operator is denoted and defined by the mapping ${\ddot{h}}^{n} \to \ddot{h}$ , such that TrCHFOWG $({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = \otimes_{i = 1}^{n} ({\ddot{h}}_{σ (i)}^{τ_{i}}) .$ Where τ = (τ ₁, τ ₂, …, τ _n) ^T is the aggregation-associated vector of TrCHFEs, such that τ _i ∈ [0, 1] and $\sum_{i = 1}^{n} τ_{i} = 1 .$

Theorem 3.2.4. Let ${\ddot{h}}_{i} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ be a collection of TrCHFEs. Then the aggregated value is calculated by using the TrCHFOWG operator is also a TrCHFE and ${\begin{matrix} TrCHFOWG ({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = \underset{μ^{-} \in Ω^{-}, μ^{+} \in Ω^{+}, η \in Ω}{⋃_{a \in ɛ, b \in ξ, c \in ψ, d \in υ,}} \\ [\prod_{i = 1}^{n} (a_{α_{σ (i)}})^{τ_{i}}, \prod_{i = 1}^{n} (b_{α_{σ (i)}})^{τ_{i}}, \prod_{i = 1}^{n} (c_{α_{σ (i)}})^{τ_{i}}, \prod_{i = 1}^{n} (d_{α_{σ (i)}})^{τ_{i}}], \\ [\prod_{i = 1}^{n} (μ_{α_{σ (i)}}^{-})^{τ_{i}}, \prod_{i = 1}^{n} (μ_{α_{σ (i)}}^{+})^{τ_{i}}], \\ (1 - \prod_{i = 1}^{n} (1 - η_{α_{σ (i)}})^{τ_{i}}) | α_{σ (1)}, α_{σ (2)}, . . ., α_{σ (n)} \in {\ddot{h}}_{α_{σ (n)}} \end{matrix}} .$

Theorem 3.2.5. Assume ${\ddot{h}}_{i} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ are the collections of TrCHFEs, λ > 0 and let τ = (τ ₁, τ ₂, …, τ _n) ^T be the aggregation associated vector such that τ _i ∈ [0, 1] and $\sum_{i = 1}^{n} τ_{i} = 1$ then TrCHFOWG $({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) \leq TrCHFOWA ({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) .$

Example 3.2.6. Suppose that ${\ddot{h}}_{1} = {\begin{matrix} [0.5, 0.7, \\ 0.9, 0.11], \\ 〈 [0.15, 0.17], \\ 0.16 〉 \end{matrix}}$ , ${\ddot{h}}_{2} = {\begin{matrix} [0.8, 0.10, \\ 0.12, 0.14], \\ 〈 [0.22, 0.24], \\ 0.23 〉 \end{matrix}}$ and ${\ddot{h}}_{3} = {\begin{matrix} [0.26, 0.28, \\ 0.30, 0.32], \\ 〈 [0.5, 0.7], \\ 0.9 〉 \end{matrix}}$ are three TrCHFEs. Then, we can define score function $s ({\ddot{h}}_{1}) = \frac{{[0.5 + 0.7 + 0.9 + 0.11] + 〈 [0.15 + 0.17] - 0.16 〉}}{12} = \frac{2.37}{12} = 0.1976;$ $S ({\ddot{h}}_{2}) = \frac{{[0.8 + 0.10 + 0.12 + 0.14] + 〈 [0.22 + 0.24] - 0.23 〉}}{12} = \frac{{[1.16] + 〈 [0.46] - 0.23 〉}}{12} = \frac{1.39}{12} = 0.1159;$ $S ({\ddot{h}}_{3}) = \frac{{[0.26 + 0.28 + 0.30 + 0.32] + 〈 [0.5 + 0.7] - 0.9 〉}}{12} = \frac{{[1.16] + 〈 [1.2] - 0.9 〉}}{12} = \frac{1.46}{12} = 0.1216 .$

3.3 The TrCHFHA and TrCHFHG operators

Definition 3.3.1. Let ${\ddot{h}}_{i} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ be the gathering of TrCHFEs. The trapezoidal cubic hesitant fuzzy hybrid averaging operator is defined by the mappin ${\ddot{h}}^{n} \to \ddot{h}$ such that, TrCHFHA $({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = \oplus_{i = 1}^{n} (τ_{i} {\ddot{h}}_{σ (i)}),$ where ${\ddot{h}}_{σ (i)}$ is the largest i th of ${\ddot{h}}_{k} = n τ_{i} {\ddot{h}}_{σ (i)}$ (k = 1, 2, …, n) and where τ = (τ ₁, τ ₂, …, τ _n) ^T is the weight vector of TrCHFEs, with τ _i ∈ [0, 1] and $\sum_{i = 1}^{n} τ_{i} = 1,$ and n is the balancing coefficient which plays a role of balance. Then we define the following aggregation operators, which are all based on the mapping ${\ddot{h}}^{n} \to \ddot{h}$ with an aggregation-associated vector τ = (τ ₁, τ ₂, …, τ _n) ^T such that τ _i ∈ [0, 1] and $\sum_{i = 1}^{n} τ_{i} = 1 .$

Theorem 3.3.2. Let ${\ddot{h}}_{i} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ be a collection of TrCHFEs. Then the aggregated value is calculated by using the TrCHFHA operator is also a TrCHFE and

${\begin{matrix} TrCHFHA ({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = \underset{μ^{-} \in Ω^{-}, μ^{+} \in Ω^{+}, η \in Ω}{⋃_{a \in ɛ, b \in ξ, c \in ψ, d \in υ,}} \\ [(1 - \prod_{i = 1}^{n} (1 - a_{α_{σ (i)}})^{τ_{i}}), (1 - \prod_{i = 1}^{n} (1 - b_{α_{σ (i)}})^{τ_{i}}), \\ (1 - \prod_{i = 1}^{n} (1 - c_{α_{σ (i)}})^{τ_{i}}), (1 - \prod_{i = 1}^{n} (1 - d_{α_{σ (i)}})^{τ_{i}})], \\ 〈 [(\prod_{i = 1}^{n} (μ_{α_{σ (i)}^{-}})^{τ_{i}}), (\prod_{i = 1}^{n} (μ_{α_{σ (i)}^{+}})^{τ_{i}})], (\prod_{i = 1}^{n} (η_{α_{σ (i)}})^{τ_{i}}) 〉 \\ | α_{σ (1)}, α_{σ (2)}, \dots, α_{σ (n)} \in {\ddot{h}}_{α_{σ (n)}} \end{matrix}} .$

Definition 3.3.3. For a collection of TrCHFEs ${\ddot{h}}_{i} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ , τ = (τ ₁, τ ₂, …, τ _n) ^T is the weight vector of TrCHFEs, with τ _i ∈ [0, 1] and $\sum_{i = 1}^{n} τ_{i} = 1,$ and n is the balancing coefficient which plays a role of balance. Then we define the following aggregation operators, which are all based on the mapping ${\ddot{h}}^{n} \to \ddot{h}$ with an aggregation-associated vector τ = (τ ₁, τ ₂, …, τ _n) ^T such that τ _i ∈ [0, 1] and $\sum_{i = 1}^{n} τ_{i} = 1 .$ The trapezoidal cubic hesitant fuzzy hybrid geometric operator is TrCHFHG $({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = \otimes_{i = 1}^{n} ({\ddot{h}}_{σ (i)})^{τ_{i}}$ , where ${\ddot{h}}_{σ (i)}$ is the largest i th of ${\ddot{h}}_{k} = ({\ddot{h}}_{σ (i)})^{n τ_{k}}$ (k = 1, 2, …, n) .

Theorem 3.3.4. Let ${\ddot{h}}_{i} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ be a collection of TrCHFEs. Then the aggregated value is calculated by using the TrCHFHG operator is also a TrCHFE and

${\begin{matrix} TrCHFHG ({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = \\ \underset{μ^{-} \in Ω^{-}, μ^{+} \in Ω^{+}, η \in Ω}{⋃_{a \in ɛ, b \in ξ, c \in ψ, d \in υ,}} \\ [(\prod_{i = 1}^{n} (a_{α_{σ (i)}})^{τ_{i}}), (\prod_{i = 1}^{n} (b_{α_{σ (i)}})^{τ_{i}}), \\ (\prod_{i = 1}^{n} (c_{α_{σ (i)}})^{τ_{i}}), (\prod_{i = 1}^{n} (d_{α_{σ (i)}})^{τ_{i}})] \\ 〈 \begin{matrix} [\begin{matrix} (1 - \prod_{i = 1}^{n} (1 - μ_{α_{σ (i)}^{-}})^{τ_{i}}), \\ (1 - \prod_{i = 1}^{n} (1 - μ_{α_{σ (i)}^{+}})^{τ_{i}}) \end{matrix}], \\ (1 - \prod_{i = 1}^{n} (1 - η_{α_{σ (i)}})^{τ_{i}}) \end{matrix} 〉 \\ | α_{σ (1)}, α_{σ (2)}, . . ., α_{σ (n)} \in {\ddot{h}}_{α_{σ (n)}} \end{matrix}} .$

4 An approach to multiple attribute group decision-making with trapezoidal cubic hesitant fuzzy information

In this section, we consume the suggested trapezoidal cubic hesitant fuzzy aggregation operators to develop an approach to multiple attribute group decision-making with trapezoidal cubic hesitant fuzzy information. First, a multiple attribute group decision-making with triangular cubic hesitant fuzzy information can be described as follows. Let Y = {Y ₁, Y ₂, …, Y _m} be a set of m alternatives, G = {G ₁, G ₂, …, G _n} a gathering of n attributes, whose weight vector is $\ddot{Ψ} = ({\ddot{Ψ}}_{1}, {\ddot{Ψ}}_{2}, \dots, {\ddot{Ψ}}_{n})^{T}$ , with ${\ddot{Ψ}}_{i} \in [0, 1]$ , i = 1, 2, …, n and $\sum_{i = 1}^{n} {\ddot{Ψ}}_{i} = 1,$ and let D = {D ₁, D ₂, …, D _l} be a set of l decision makers, whose weight vector is τ = (τ ₁, τ ₂, …, τ _n) ^T, with τ _k ∈ [0, 1], k = 1, 2, …, l, and $\sum_{k = 1}^{l} τ_{k} = 1$ . Let $R^{k} = (r_{ij}^{k})_{m \times n}$ be the trapezoidal cubic hesitant fuzzy decision matrix, where $r_{ij}^{k} = {γ_{ij}^{k} | γ_{ij}^{k} \in r_{ij}^{k}} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ is an TrCHFE given by the decision maker D _k ∈ D, where ${\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ indicates the possible interval value trapezoidal hesitant fuzzy set range that the alternative Y _i ∈ Y satisfies the attribute G _j ∈ G, while $η_{\tilde{b}}$ indicates the possible trapezoidal hesitant fuzzy set range that the alternative Y _i ∈ Y does not satisfy the attribute G _j ∈ G. In general, there are benefit attributes (i.e., the bigger the attribute values, the better) and cost attributes (i.e., the smaller the attribute values, the better) in a multiple attribute group decision-making problem. In such cases, we transform the attribute values of cost type into the attribute values of benefit type. That is, normalize the trapezoidal cubic hesitant fuzzy decision matrix $R^{k} = (r_{ij}^{k})_{m \times n}$ a corresponding trapezoidal cubic hesitant fuzzy decision matrix $A_{ij}^{k} = a_{ij}^{k} = {α_{ij}^{k} | α_{ij}^{k} \in A_{ij}^{k}} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ , where $a_{ij}^{k} {\begin{matrix} r_{ij}^{k} for benefit attribute, G_{j} \\ (r_{ij}^{k})^{c} for cost attribute, G \end{matrix}$ i = 1, 2, …, m, j = 1, 2, …, n, k = 1, 2, …, l, where $r_{ij}^{k}$ is the complement of $(r_{ij}^{k})^{c}$ such that

$(r_{ij}^{k})^{c} = {γ_{ij}^{k} | γ_{ij}^{k} \in r_{ij}^{k}} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}} .$

In the following, we utilize the proposed operators to develop an approach to multiple attribute group decision making with trapezoidal cubic hesitant fuzzy information, which includes the following steps.

Step 1. Construct the decision matrix $(R_{ij}^{k}) = (r_{ij}^{k})_{m \times n}$ into a normalized matrix $A_{ij}^{k} = a_{ij}^{k} = {α_{ij}^{k} | α_{ij}^{k} \in A_{ij}^{k}} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}} .$

Step 2. Utilize the TrCHFWA operator ${\begin{matrix} TrCHFWA ({\ddot{h}}_{1}, {\ddot{h}}_{2}, . . ., {\ddot{h}}_{n}) = \\ \oplus_{i = 1}^{n} (τ_{i} {\ddot{h}}_{i}) = \underset{μ^{-} \in Ω^{-}, μ^{+} \in Ω^{+}, η \in Ω}{⋃_{a \in ɛ, b \in ξ, c \in ψ, d \in υ,}} \\ [\prod_{i = 1}^{n} (a_{i})^{τ_{i}}, \prod_{i = 1}^{n} (b_{i})^{τ_{i}}, \prod_{i = 1}^{n} (c_{i})^{τ_{i}}, \prod_{i = 1}^{n} (d_{i})^{τ_{i}}], \\ 〈 \begin{matrix} [\begin{matrix} [1 - \prod_{i = 1}^{n} (1 - μ_{b^{-}})^{τ_{i}}], \\ [1 - \prod_{i = 1}^{n} (1 - μ_{b^{+}})^{τ_{i}}] \end{matrix}], \\ \prod_{i = 1}^{n} (η_{\tilde{bi}})^{τ_{i}} \end{matrix} 〉 \end{matrix}}$ to aggregate all the individual Trapezoidal cubic hesitant fuzzy decision matrix $(A_{ij}^{k}) = (α_{ij}^{k})_{m \times n}$ into the collective Trapezoidal cubic hesitant fuzzy decision matrix $A_{ij}^{k} = a_{ij}^{k} = {α_{ij}^{k} | α_{ij}^{k} \in A_{ij}^{k}} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$

Step 3. Utilize the TrCHFHA operator ${\begin{matrix} TrCHFHA ({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = \underset{μ^{-} \in Ω^{-}, μ^{+} \in Ω^{+}, η \in Ω}{⋃_{a \in ɛ, b \in ξ, c \in ψ, d \in υ,}} \\ [(1 - \prod_{i = 1}^{n} (1 - a_{α_{σ (i)}})^{τ_{i}}), (1 - \prod_{i = 1}^{n} (1 - b_{α_{σ (i)}})^{τ_{i}}), \\ (1 - \prod_{i = 1}^{n} (1 - c_{α_{σ (i)}})^{τ_{i}}), (1 - \prod_{i = 1}^{n} (1 - d_{α_{σ (i)}})^{τ_{i}})], \\ 〈 [(\prod_{i = 1}^{n} (μ_{α_{σ (i)}^{-}})^{τ_{i}}), (\prod_{i = 1}^{n} (μ_{α_{σ (i)}^{+}})^{τ_{i}})], (\prod_{i = 1}^{n} (η_{α_{σ (i)}})^{τ_{i}}) 〉 \\ | α_{σ (1)}, α_{σ (2)}, \dots, α_{σ (n)} \in {\ddot{h}}_{α_{σ (n)}} \end{matrix}}$ to aggregate all the preference values a _ij (j = 1, 2, …, n) in the i th line of A, and then derive the collective overall preference value $A_{ij}^{k} = a_{ij}^{k} = {α_{ij}^{k} | α_{ij}^{k} \in A_{ij}^{k}} = {\begin{matrix} [ɛ_{i}, ξ_{i}, \\ ψ_{i}, υ_{i}], \\ 〈 [Ω_{i}^{-}, \\ Ω_{i}^{+}], \\ Ω_{i} 〉 \end{matrix}}$ of the alternative Y _i (i = 1, 2, …, m), where $\ddot{ξ} = ({\ddot{ξ}}_{1}, {\ddot{ξ}}_{2}, \dots, {\ddot{ξ}}_{n})^{T}$ is the associated weight vector of the TrCHFHA operator, with ${\ddot{ξ}}_{j} \in [0, 1], j = 1, 2, \dots, n,$ and $\sum_{j = 1}^{n} ξ_{j} = 1$ .

Step 4. Calculate the score values s (a _i) (i = 1, 2, …, m) of a (i = 1, 2, …, m): $s (a_{i}) = \frac{\frac{1}{12} [a + b + c + d], 〈 [μ_{b^{-}} + μ_{b^{+}}] - η_{\tilde{b}} 〉}{# (a_{i})} .$

Step 5. Get the priority of the alternatives Y _i (i = 1, 2, …, m) by ranking s (a _i) (i = 1, 2, …, m) .

5 The application of the developed approach in group decision-making problems

Global environmental concern is a reality, and an increasing attention is focusing on the green production in various industries. A car company is desirable to select the most appropriate green supplier for one of the key elements in its manufacturing process. After pre-evaluation, three suppliers (B _i = 1, 2, 3) are remained as alternatives for further evaluation. Three criteria are considered as

C ₁ : Invention quality; C ₂ : Knowledge capability; C ₃ : Pollution control, (who’s weighting vector τ = (0.34, 0.26, 0.40) ^T) . This company is a group of DMs form three consultancy department: D ₁ is form the production, D ₂ is from purchasing department; D ₃ is from engineering department. After the information procurement and factual treatment, the evaluations of the alternatives with respect to attributes can be represented by CLVs appeared in Tables 1–3. Assume the decision-makers.

Step 1. Calculate the trapezoidal cubic hesitant fuzzy decision matrices three

Table 1
Trapezoidal cubic hesitant fuzzy decision matrix

	C ₁	C ₂	C ₃
B ₁	${\begin{matrix} [0.2, 0.4, 0.6, 0.8]; \\ 〈 [0.10, 0.12], 0.11 〉 \end{matrix}}$	${\begin{matrix} [0.4, 0.5, 0.7, 0.8]; \\ 〈 [0.14, 0.16], 0.15 〉 \end{matrix}}$	${\begin{matrix} [0.1, 0.2, 0.3, 0.4]; \\ 〈 [0.8, 0.10], 0.9 〉 \end{matrix}}$
B ₂	${\begin{matrix} [0.3, 0.4, 0.5, 0.6]; \\ 〈 [0.9, 0.11], 0.10 〉 \end{matrix}}$	${\begin{matrix} [0.2, 0.4, 0.6, 0.8]; \\ 〈 [0.10, 0.12], 0.11 〉 \end{matrix}}$	${\begin{matrix} [0.4, 0.5, 0.7, 0.8]; \\ 〈 [0.14, 0.16], 0.15 〉 \end{matrix}}$
B ₃	${\begin{matrix} [0.1, 0.2, 0.3, 0.4]; \\ 〈 [0.8, 0.10], 0.9 〉 \end{matrix}}$	${\begin{matrix} [0.3, 0.4, 0.5, 0.6]; \\ 〈 [0.9, 0.11], 0.10 〉 \end{matrix}}$	${\begin{matrix} [0.2, 0.4, 0.6, 0.8]; \\ 〈 [0.10, 0.12], 0.11 〉 \end{matrix}}$

Table 2

Trapezoidal cubic hesitant fuzzy decision matrix

	C ₁	C ₂	C ₃
B ₁	${\begin{matrix} [0.3, 0.4, 0.5, 0.6]; \\ 〈 [0.9, 0.11], 0.10 〉 \end{matrix}}$	${\begin{matrix} [0.1, 0.2, 0.3, 0.4]; \\ 〈 [0.8, 0.10], 0.9 〉 \end{matrix}}$	${\begin{matrix} [0.1, 0.2, 0.3, 0.4]; \\ 〈 [0.8, 0.10], 0.9 〉 \end{matrix}}$
B ₂	${\begin{matrix} [0.2, 0.4, 0.6, 0.8]; \\ 〈 [0.10, 0.12], 0.11 〉 \end{matrix}}$	${\begin{matrix} [0.4, 0.5, 0.7, 0.8]; \\ 〈 [0.14, 0.16], 0.15 〉 \end{matrix}}$	${\begin{matrix} [0.3, 0.4, 0.5, 0.6]; \\ 〈 [0.9, 0.11], 0.10 〉 \end{matrix}}$
B ₃	${\begin{matrix} [0.4, 0.5, 0.7, 0.8]; \\ 〈 [0.14, 0.16], 0.15 〉 \end{matrix}}$	${\begin{matrix} [0.2, 0.4, 0.6, 0.8]; \\ 〈 [0.10, 0.12], 0.11 〉 \end{matrix}}$	${\begin{matrix} [0.2, 0.4, 0.6, 0.8]; \\ 〈 [0.10, 0.12], 0.11 〉 \end{matrix}}$

Table 3

Trapezoidal cubic hesitant fuzzy decision matrix

	C ₁	C ₂	C ₃
B ₁	${\begin{matrix} [0.4, 0.5, 0.7, 0.8]; \\ 〈 [0.14, 0.16], 0.15 〉 \end{matrix}}$	${\begin{matrix} [0.3, 0.4, 0.5, 0.6]; \\ 〈 [0.9, 0.11], 0.10 〉 \end{matrix}}$	${\begin{matrix} [0.3, 0.4, 0.5, 0.6]; \\ 〈 [0.9, 0.11], 0.10 〉 \end{matrix}}$
B ₂	${\begin{matrix} [0.2, 0.4, 0.6, 0.8]; \\ 〈 [0.10, 0.12], 0.11 〉 \end{matrix}}$	${\begin{matrix} [0.4, 0.5, 0.7, 0.8]; \\ 〈 [0.14, 0.16], 0.15 〉 \end{matrix}}$	${\begin{matrix} [0.1, 0.2, 0.3, 0.4]; \\ 〈 [0.8, 0.10], 0.9 〉 \end{matrix}}$
B ₃	${\begin{matrix} [0.3, 0.4, 0.5, 0.6]; \\ 〈 [0.9, 0.11], 0.10 〉 \end{matrix}}$	${\begin{matrix} [0.1, 0.2, 0.3, 0.4]; \\ 〈 [0.8, 0.10], 0.9 〉 \end{matrix}}$	${\begin{matrix} [0.2, 0.4, 0.6, 0.8]; \\ 〈 [0.10, 0.12], 0.11 〉 \end{matrix}}$

Step 2. Using TrCHFWA operators and τ = (0.35, 0.40, 0.25) ^T

Table 4

Trapezoidal cubic hesitant fuzzy weighted averaging operator

	C ₁	C ₂	C ₃
B ₁	${\begin{matrix} [0.3173, 0.4512, \\ 0.6264, 0.7647] \\ ; 〈 \begin{matrix} [0.5916 \\ 0.1369 \\ , 0.1061 \end{matrix} 〉 \end{matrix}}$	${\begin{matrix} [0.3223, 0.4349, \\ 0.5941, 0.7031] \\ ; 〈 \begin{matrix} [0.8031 \\ 0.1465 \\ , 0.1787 \end{matrix} 〉 \end{matrix}}$	${\begin{matrix} [0.3122, 0.2128, \\ 0.2964, 0.3839] \\ ; 〈 \begin{matrix} [0.7485 \\ 0.0785 \\ , 0.5334 \end{matrix} 〉 \end{matrix}}$
B ₂	${\begin{matrix} [0.2449, 0.4151, \\ 0.5868, 0.7947] \\ ; 〈 \begin{matrix} [0.5851 \\ 0.1221 \\ , 0.0952 \end{matrix} 〉 \end{matrix}}$	${\begin{matrix} [0.3922, 0.5317, \\ 0.7354, 0.8551] \\ ; 〈 \begin{matrix} [0.1502 \\ 0.1735 \\ , 0.0906 \end{matrix} 〉 \end{matrix}}$	${\begin{matrix} [0.2158, 0.3000, \\ 0.4307, 0.5319] \\ ; 〈 \begin{matrix} [0.6378 \\ 0.0943 \\ , 0.3408 \end{matrix} 〉 \end{matrix}}$
B ₃	${\begin{matrix} [0.2885, 0.3931, \\ 0.5456, 0.6545] \\ ; 〈 \begin{matrix} [0.7587 \\ 0.1294 \\ , 0.2216 \end{matrix} 〉 \end{matrix}}$	${\begin{matrix} [0.2397, 0.3922, \\ 0.5445, 0.7031] \\ ; 〈 \begin{matrix} [0.6341 \\ 0.1395 \\ , 0.1578 \end{matrix} 〉 \end{matrix}}$	${\begin{matrix} [0.1541, 0.3182, \\ 0.4970, 0.7009] \\ ; 〈 \begin{matrix} [0.0759 \\ 0.0914 \\ , 0.1911 \end{matrix} 〉 \end{matrix}}$

Step 3. Utilize the TrCHFHA operator ξ = (0.50, 0.25, 0.25) ^T

Table 5

Trapezoidal cubic hesitant fuzzy hybrid averaging operator

B ₁	{ [0.2764, 0.3949, 0.5531, 0.6827];〈 [0.6985, 0.1251], 0.1809 〉 }
B ₂	{ [0.2779, 0.4213, 0.5995, 0.7687];〈 [0.5202, 0.1284], 0.1293 〉 }
B ₃	{ [0.2446, 0.3749, 0.5336, 0.6791];〈 [0.6254, 0.1226], 0.1961 〉 }

Step 4. Calculate the score values s (a _i) (i = 1, 2, …, m) of a (i = 1, 2, …, m):TrCHFHA score value s (a ₁) =0.1021, s (a ₂) =0.0894, s (a ₃) =0.0842 .

Accuracy Function

$H (a_{1}) = 0.1596, H (a_{2}) = 0.1341, H (a_{3}) = 0.1441 .$

6 Comparison analysis

In order to verify the validity and effectiveness of the proposed approach, a comparative study is conducted using the methods of interval-valued intuitionistic hesitant fuzzy number [27] and hesitant triangular intuitionistic fuzzy number [6], which are special cases of TrCHFNs, to the same illustrative example.

6.1 A comparison analysis with the existing MCDM method interval-valued intuitionistic hesitant fuzzy number

An interval-valued intuitionistic hesitant fuzzy number can be considered as a special case of trapezoidal cubic hesitant fuzzy numbers when there is the only element in membership and non-membership degree [27]. For comparison, the interval-valued intuitionistic hesitant fuzzy number can be transformed to the trapezoidal cubic hesitant fuzzy number (TrCHFN) by calculating the average value of the membership and nonmember ship degrees. After transformation, the interval-valued intuitionistic hesitant fuzzy number is given in Table 6.

Table 6
Interval-valued intuitionistic hesitant fuzzy number

${\ddot{c}}_{1}$ ${\ddot{c}}_{2}$ ${\ddot{c}}_{3}$

${\ddot{A}}_{1}$ ${\begin{matrix} [0.2, 0.4], \\ [0.10, 0.12] \end{matrix}}$ ${\begin{matrix} [0.4, 0.5], \\ [0.14, 0.16] \end{matrix}}$ ${\begin{matrix} 〈 [0.1, 0.2], \\ [0.8, 0.10] \end{matrix}}$

${\ddot{A}}_{2}$ ${\begin{matrix} [0.3, 0.5], \\ [0.9, 0.11] \end{matrix}}$ ${\begin{matrix} [0.2, 0.4], \\ [0.10, 0.12] \end{matrix}}$ ${\begin{matrix} [0.4, 0.7], \\ [0.14, 0.16] \end{matrix}}$

${\ddot{A}}_{3}$ ${\begin{matrix} [0.1, 0.3], \\ [0.8, 0.10] \end{matrix}}$ ${\begin{matrix} [0.3, 0.4], \\ [0.9, 0.11] \end{matrix}}$ ${\begin{matrix} [0.2, 0.4], \\ [0.10, 0.12] \end{matrix}}$

	${\ddot{c}}_{1}$	${\ddot{c}}_{2}$	${\ddot{c}}_{3}$
${\ddot{A}}_{1}$	${\begin{matrix} [0.2, 0.4], \\ [0.10, 0.12] \end{matrix}}$	${\begin{matrix} [0.4, 0.5], \\ [0.14, 0.16] \end{matrix}}$	${\begin{matrix} 〈 [0.1, 0.2], \\ [0.8, 0.10] \end{matrix}}$
${\ddot{A}}_{2}$	${\begin{matrix} [0.3, 0.5], \\ [0.9, 0.11] \end{matrix}}$	${\begin{matrix} [0.2, 0.4], \\ [0.10, 0.12] \end{matrix}}$	${\begin{matrix} [0.4, 0.7], \\ [0.14, 0.16] \end{matrix}}$
${\ddot{A}}_{3}$	${\begin{matrix} [0.1, 0.3], \\ [0.8, 0.10] \end{matrix}}$	${\begin{matrix} [0.3, 0.4], \\ [0.9, 0.11] \end{matrix}}$	${\begin{matrix} [0.2, 0.4], \\ [0.10, 0.12] \end{matrix}}$

Step 1. Calculate the IVIHFWA operator ω = (0.3, 0.2, 0.5) ^T

Table 7

Interval-valued intuitionistic hesitant fuzzy weighted averaging operator

${\ddot{A}}_{1}$	{, [0.2225, 0.3482]} {, [0.2598, 0.1531]}
${\ddot{A}}_{2}$	{[0.1959, 0.3821], [0.4169, 0.2917]}
${\ddot{A}}_{3}$	{[0.2900, 0.4980], [0.2683, 0.0363]}

Step 2. Calculate the score value s ₁ = 0.0789, s ₂ = -0.0653, s ₃ = 0.2417 .

Step 3. Rank all the alternatives. According to the ranking of score function S (z _i), the ranking is s ₃ > s ₁ > s ₂.

The ranking of all alternatives s ₃ > s ₁ > s ₂ and s ₃ is the best selection. Obviously, the ranking is derived from the method proposed by Zhang [27], is different from the result of the proposed method. The main reasons are that an interval-valued intuitionistic hesitant fuzzy number only consider the triangular number, membership degrees of an element and nonmember ship degrees, which may result in information interval-valued intuitionistic hesitant fuzzy number are not equal.

6.2 A comparison analysis with the existing MCDM method hesitant triangular intuitionistic fuzzy number

The hesitant triangular intuitionistic fuzzy number can be considered as a special case of trapezoidal cubic hesitant fuzzy numbers when there is the only element in membership and non-membership degree [6]. For comparison, the hesitant triangular intuitionistic fuzzy number can be transformed into the trapezoidal cubic hesitant fuzzy number (TrCHFN) by calculating the average value of the membership and nonmember ship degrees. After transformation, the hesitant triangular intuitionistic fuzzy number is given in Table 8.

Table 8
Hesitant triangular intuitionistic fuzzy number

${\ddot{c}}_{1}$ ${\ddot{c}}_{2}$ ${\ddot{c}}_{3}$

${\ddot{A}}_{1}$ ${\begin{matrix} [0.2, 0.4, 0.6], \\ [0.9, 0.11], \end{matrix}}$ ${\begin{matrix} [0.1, 0.2, 0.3], \\ [0.8, 0.10] \end{matrix}}$ ${\begin{matrix} [0.3, 0.4, 0.5], \\ [0.9, 0.11] \end{matrix}}$

${\ddot{A}}_{2}$ ${\begin{matrix} [0.4, 0.6, 0.8], \\ [0.10, 0.12] \end{matrix}}$ ${\begin{matrix} [0.4, 0.5, 0.7], \\ [0.14, 0.16] \end{matrix}}$ ${\begin{matrix} [0.1, 0.2, 0.3], \\ [0.8, 0.10], \end{matrix}}$

${\ddot{A}}_{3}$ ${\begin{matrix} [0.3, 0.4, 0.5], \\ [0.9, 0.11] \end{matrix}}$ ${\begin{matrix} [0.2, 0.3, 0.4], \\ [0.8, 0.9], \end{matrix}}$ ${\begin{matrix} [0.4, 0.6, 0.8], \\ [0.10, 0.12] \end{matrix}}$

Step 1. Calculate the HTIFWG operator ω = (0.5, 0.3, 0.2) ^T

Table 9

Hesitant triangular intuitionistic fuzzy weighted geometric operator

${\ddot{A}}_{1}$	{, [0.0774, 0.1788, 0.3], [0.8049, 0.1556]}
${\ddot{A}}_{2}$	{[0.2892, 0.4299, 0.5855], [0.2598, 0.1150]}
${\ddot{A}}_{3}$	{[0.4742, 0.5908, 0.6931], [0.5908, 0.3991]}

Step 2. Calculate the score value

Table 10

Comparison analysis with existing methods

Method	Ranking
trapezoidal cubic hesitant fuzzy number	s ₁ > s ₂ > s ₃
Interval-valued intuitionistic hesitant	s ₃ > s ₁ > s ₂
fuzzy number [27]
hesitant triangular intuionistic fuzzy number [6]	s ₁ > s ₃ > s ₂

s ₁ = 0.1203, s ₂ = 0.0629, s ₃ = 0.1123 .

Step 3. Rank all the alternatives. According to the ranking of score function S (z _i), the ranking is s ₁ > s ₃ > s ₂. The ranking of all alternatives s ₁ > s ₃ > s ₂ and s ₁ is the best selection. Obviously, the ranking is derived from the method proposed by Chen et al. [6], is different from the result of the proposed method. The main reasons are that hesitant triangular intuitionistic fuzzy number only considers the triangular number, membership degrees of an element and nonmember ship degrees, which may result in information hesitant triangular intuitionistic fuzzy number, are not equal.

The following advantages of our proposal can be summarized on the basis of the above comparison analyses. Trapezoidal cubic hesitant fuzzy number (TrCHFN) are very suitable for illustrating uncertain or fuzzy information in MCDM problems because the membership and non-membership degrees can be two sets of several possible values, which cannot be achieved by interval-valued intuitionistic hesitant fuzzy number and intuitionistic triangular hesitant fuzzy number. On the bases of basis operations, aggregation operators and comparison method of trapezoidal cubic hesitant fuzzy number (TrCHFN) can be also used to process interval-valued intuitionistic hesitant fuzzy number and intuitionistic triangular hesitant fuzzy number after slight adjustments, because trapezoidal cubic hesitant fuzzy number (TrCHFN) can be considered as the generalized form of interval-valued intuitionistic hesitant fuzzy number and intuitionistic triangular hesitant fuzzy number. The defined operations of trapezoidal cubic hesitant fuzzy number (TrCHFN) give us more accurate than the existing operators. From the above case study and the comparative analysis, the particular advantages of the proposed trapezoidal cubic hesitant fuzzy approach are summarized as below:

Both criteria weights and performance evaluations of alternatives take the form of trapezoidal cubic hesitant fuzzy aggregation operator, which can represent the fuzziness and uncertainty of decision makers’ assessment information accurately. Moreover, the reliability of the evaluation values provided by experts can be considered with the Global environmental of trapezoidal cubic hesitant fuzzy. By using the prospect theory, the developed model can not only take the bounded rationality of experts under risk and uncertainty conditions into account, but also fully include different risk preferences in the face of gains and losses. According to the basic idea of the hesitant, the proposed approach can obtain the optimal alternative as well as compromise solutions in uncertain emergency conditions, which is more consistent with practical DM situations.

7 Conclusion

In this paper, we first develop the concept of trapezoidal cubic hesitant fuzzy sets, deliberate they are some basic properties and develop some operational laws for trapezoidal cubic hesitant fuzzy elements. We discuss the score value and accuracy value of the trapezoidal cubic hesitant fuzzy number. Then, we focus on trapezoidal cubic hesitant fuzzy information aggregation techniques and exhibit a series of trapezoidal cubic hesitant fuzzy aggregation operators. Moreover, we apply the developed aggregation operators to multiple attribute group decision-making with trapezoidal cubic hesitant fuzzy information. Finally, we provide illustrative examples of MADM to demonstrate the application and effectiveness of the established method. In group decision-making problems, the experts usually come from different fields and have different backgrounds and levels of knowledge, they usually have diverging opinions. In the future, we extend to define some generalized aggregation operators for TrCHFNs. We further define some new t-norm and t-conorm operators for TrCHFNs. We will also establish a relationship between TrCHFNs and rough set theory.

For future works, it is recommended to extend the proposed decision making approach through considering more complex characteristics in the emergency decision making (EDM) process, such as the priority level among experts and the interdependent relationship between criteria. Second, we have utilized the expectation function to transform the group trapezoidal cubic hesitant fuzzy number evaluations of experts into crisp values. Although the sound effectiveness and performance of the proposed approach, we should not ignore the potential information loss in the transformation process. Therefore, it is interesting to investigate the influence of the transformation process on EDM in the future. Additionally, the algorithms for improving the reliability and accuracy of EDM in the large group context and the heterogeneous information environment should be developed, which is the aim for the future research.

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Trapezoidal cubic hesitant fuzzy aggregation operators and their application in group decision-making

Abstract

Keywords

1 Introduction

2 Preliminaries

1.1 Trapezoidal cubic hesitant fuzzy sets

3 Aggregation trapezoidal cubic hesitant fuzzy information

3.1 The TrCHFWA and TrCHFWG Operators

3.2 The TrCHFOWA and TrCHFOWG operators

3.3 The TrCHFHA and TrCHFHG operators

4 An approach to multiple attribute group decision-making with trapezoidal cubic hesitant fuzzy information

5 The application of the developed approach in group decision-making problems

Table 1 Trapezoidal cubic hesitant fuzzy decision matrix

6.1 A comparison analysis with the existing MCDM method interval-valued intuitionistic hesitant fuzzy number

References

Table 1
Trapezoidal cubic hesitant fuzzy decision matrix