Complex hesitant fuzzy sets and its applications in multiple attributes decision-making problems

Abstract

Decision-makers (DMs) usually face many obstacles to give the right decision, multiplicity of them highlights a problem to represent a set of potential values to assign a collective membership degree of an object to a set for several DM’s opinions. However, a hesitant fuzzy set (HFS) deals with such problems. The complexity appears in DM’s opinion which can be changed for the same object but with different times/phases. Each of them has a set of potential values in different times/phases of an object. In this paper, the periodicity of hesitant fuzzy information is studied and applied by extending the range of HFS from [0, 1] to the unit disk in the complex plane to provide more ability for illustrating the full meaning of information to overcome the obstacles in decision making in the mathematical model. Moreover, the advantage of complex hesitant fuzzy set (CHFS) is that the amplitude and phase terms of CHFSs can represent hesitant fuzzy information, some basic operations on CHFS are also presented and we study its properties, in addition, several aggregation operators under CHFS are introduced, also, the relation between CHFS and complex intuitionistic fuzzy sets (CIFS) are presented. Finally, an efficient algorithm with a consistent process and an application in multiple attributes decision-making (MADM) problems are presented to show the effectiveness of the presented approach by using CHFS aggregation operators.

Keywords

Hesitant fuzzy set complex fuzzy sets complex intuitionistic fuzzy sets complex hesitant fuzzy sets CIF aggregation operators CHF aggregation operators

1 Introduction

Decision making is an important and useful field in our daily life, due to the rapid difficulties of representing objects without losing the full meaning of information systems, decision-makers face difficulties in giving appropriate choices under different circumstances, fuzzy set theory [39], intuitionistic fuzzy set theory [8], hesitant fuzzy set theory [31], and other theories had been presented to deal with decision-making problems. (See, [1 , and 45]).

In 1965, Zadeh [39] introduced the concept of fuzzy set (FS), where its membership degree lies in [0, 1], many researchers used FS in decision-making problems [21 , 45]. In [37 , 43], authors proposed and contributed new and different models to deal with decision-making problems and multiple criteria decision making (MCDM) problems, later on, Atanassov [8] introduced the concept of intuitionistic fuzzy set (IFS) by adding membership degree to FS, however, the advantage appeared in the concept of IFS is getting wider range information that may represent belongingness, non-belongingness, and hesitancy degrees, numerous applications were enhanced and employed the notion of IFS in decision making [12 , 32].

In 2009, Torra and Narukawa [31] introduced the concept of hesitant fuzzy set (HFS), to solve the problem of having a set of possible values to assign two or more values for an object to a set, for example, if three decision-makers (DMs) argue about the membership degree of an object to a set, and the first DM wants to assign 0.5, the second DM wants to assign 0.7, and the third DM wants to assign 0.9, also, they had shown IFS is not appropriate to present this kind of situation, the HFS is considered as an extension of FS, and encompassed IFS, the membership degree that an object in a set expressed by numerous potential values, and can convey the hesitant information more comprehensively than other uncertainty sets. In 2011, Xia and Xu [35] defined the concept of hesitant fuzzy element (HFE), which can be considered as the basic unit of an HFS, and it is also a simple and effective tool used to express the decision maker’s hesitant preferences in the process of decision making. Xia and Xu [34] discussed two ways to have the optimal alternative in a MADM and multiple persons, there are “(1) Aggregate the decision-makers (DMs)’ opinions under each attribute for alternatives, and then aggregate the collective values of attributes for each alternative; (2) Aggregate the attribute values given by the DMs for each alternative, and then aggregate the DMs opinions for each alternative”, moreover, in the references [14 , 44] researchers have done lots of results and studies on distance, similarity, entropy, aggregation, and clustering analysis, decision making and correlation measures with hesitant fuzzy information.

The uncertainties present in the fuzzy information and its extensions are handled with degrees of membership that lie in real numbers [0, 1], which may lose some useful information and therefore affect the decision results, a modification to these is the idea of extending the range of membership function from [0, 1] to the unit disk in complex plane handles the two-dimensional information in a single set, this idea has been widely addressed by numerous researchers [3–5 , 24–27].

The mentioned idea was started by Ramote et al. research [18] in 2002 by introducing the concept of a complex fuzzy set (CFS). The novelty of CFS performed in additional dimension membership. The CFS is reduced to a traditional fuzzy set without the phase term [18]. The mentioned idea has its ability to represent uncertainty and periodicity information at the same time. They detected the phase of degrees to differentiate the different meanings in different phases for analogous data. The uncertainty and periodicity semantics can be denoted by amplitude and phase terms, of the complex numbers, these terms can be fuzzy sets [25], the significance of phase feature of complex-valued membership function separates between the work of [18] and the work of [17], the complex fuzzy membership grade can be represented in polar form and Cartesian form with two fuzzy components [25].

In 2012 and 2013 Alkouri and Salleh introduced the notions of complex intuitionistic fuzzy sets and their relations (CIFS and CIFR) [3 –5]. CIFS extends the range of both membership and non-membership functions to the complex plane. They studied the distance, basic operations, and relations under the CIFS notion with appropriate application in decision-making. Garg and Rani [10] generalized the concept of CIFSs to a higher stage and illustrated good multi-criteria decision-making in the complex realm. Rani and Garg [20] introduced some distance measure of CIFS and their applications in the decision-making process. In 2018, Garg and Rani [19] established some power aggregation operators under the CIFS environment, furthermore, a multi-criteria decision-making (MCDM) approach was obtained under the CIFS environment, several research papers have been prepared in handling the information values using diverse aggregation operators with applications in MCDM process under the CIFS environment [9–11 , 20].

Respecting the previous studies and their appearance to the field, a hesitant fuzzy set is considered as a generalization to both fuzzy sets and intuitionistic fuzzy sets. Also, HFS came to solve some limitations and problems in decision-making fields, where, both FS and IFS could not solve as illustrated in section 1.Comparably, the complex hesitant fuzzy set has the priority to be presented and studied in this stage. Especially, after Ramote et al. and Alkouri and Salleh introduced their notions of complex fuzzy sets and complex intuitionistic fuzzy sets, respectively. Also, CHFS has inherited the advantages and ability to solve the limitation in both FS and IFS besides the periodic nature earned from complex numbers. This gives a highlight for CHFS to be a generalization of both CFS and CIFS besides what we have proved in the present paper. In this study, the main problem is how to represent changeable hesitant fuzzy information for the same object at different times/ phases without losing the full meaning of hesitant fuzzy information in an appropriate and occupied mathematical structure. However, DM’s opinion can be changeable and give dissimilar membership values of the same object in different times/phases. Therefore, this study aims to use two variables to represent uncertainty and periodicity semantics in hesitant fuzzy form. For example, suppose we have three models of cars: Mazda, Mercedes, and Toyota. But every year the car factories produce the same model but with slight disparate enhancement. Regarding hesitant fuzzy information, Each DM has a set of potential values to assign two or more values for a model of car, which could be represented by hesitant fuzzy sets, but there is no other variable to be represented the production date of cars and their enhancement factor. Surely, Mercedes 2020, 2019, 2018, and 2017 have different Specifications. Thus, the decision-makers will confuse to give the same membership value for the Mercedes car (see the application in section 6). Therefore, to deal with this issue, we extend the range of HFS from [0, 1] to the unit disk in the complex plane to get the ability to represent the uncertainty and periodicity semantics in the same mathematical structure. So, a mathematical structure for CHFS, its properties, operators, aggregations operators, and algorithm are introduced. This structure is represented in a proper and operative application by extending HFS to the realm of a complex number. Also, problems that measure the periodic variation in hesitant fuzzy information at several types of phase, which convey hesitant information are addressed. Also, a general of the complex hesitant fuzzy element to represent uncertainty and periodicity semantics simultaneously by applying the range of hesitant fuzzy elements in complex geometry and identified the phase of degrees as a hesitant fuzzy set is introduced to differentiate the different meanings in different phases for analogous data. In the last section, a comparison study between CHFS and some related notions is presented and summarized.

2 Preliminaries

Definition 2.1. [39] A fuzzy set A in a universe of discourse U is characterized by a membership function μ_A (x) that takes values in the interval [0,1].

Definition 2.2. [31] Let X be a reference set, then a hesitant fuzzy set on X is defined as a function g when applied to X returns a subset of values in the unit interval: $g : X \to p ([0, 1])$ (2.1)

A hesitant fuzzy set may be also constructed from a set of fuzzy sets.

Definition 2.3. [31] The lower and upper bound of a hesitant fuzzy element g are defined as: $g^{-} = \min {γ : γ \in g}$ (2.2)

$g^{+} = \max {γ : γ \in g}$ (2.3)

Definition 2.4. [31] The complement, union, and intersection of a hesitant fuzzy element g are defined as:

$g^{c} = \underset{μ \in g}{\cup} {1 - γ}$ (2.4)

$g_{1} \cup g_{2} = \underset{μ_{1} \in g_{1}, μ_{2} ɛ g_{2}}{\cup} \max {γ_{1}, γ_{2}}$ (2.5)

$g_{1} \cap g_{2} = \underset{μ_{1} \in g_{1}, μ_{2} ɛ g_{2}}{\cup} \min {γ_{1}, γ_{2}}$ (2.6)

Definition 2.5. [31] Let g be a hesitant fuzzy element, the Atanassov’s intuitionistic fuzzy set, A_env (g), is defined as the envelope of g, where A_env (g) is represented as:

$A_{env} (g) = {g^{-}, 1 - g^{+}}$ (2.7)

Definition 2.6. [31] Let A ={ g₁, g₂, …, g_n } be a set of n HFEs, υ a function on A, υ : [0, 1] ⁿ → [0, 1], then

$υ_{A} = \underset{γ ɛ {g_{1} \times g_{2 \times \dots \times} g_{n}}}{\cup} {υ (γ)}$ (2.8)

Definition 2.7. [34] Let g_j (j = 1, 2, …, n) be a collection of HFEs. A hesitant fuzzy weighted averaging (HFWA) operator is a mapping ⊖ⁿ→ ⊖ such that

$\begin{matrix} HFWA (g_{1}, g_{2}, \dots, g_{n}) = \oplus_{j = 1}^{n} w_{j} g_{j} \\ = \underset{γ_{1} \in g_{1}, γ_{2} \in g_{2}, \dots, γ_{n} \in g_{n}}{\cup} {1 - \prod_{j = 1}^{n} {(1 - γ_{j})}^{w_{j}}} \end{matrix}$ (2.9)

Where w = (w₁, w₂, …, w_n) ^T is the weight vector of g_j (j = 1, 2, …, n) with w_j ∈ [0, 1], j = 1, 2, …, n, and $\sum_{j = 1}^{n} w_{j} = 1$ . Notice that if $w = {(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$

then the HFWA operator reduces to the hesitant fuzzy averaging (HFA) operator:

$\begin{matrix} HFA (g_{1}, g_{2}, \dots, g_{n}) = \frac{1}{n} \oplus_{j = 1}^{n} g_{j} \\ = \underset{γ_{n} ɛ g_{n}}{\cup} {1 - \prod_{j = 1}^{n} {(1 - γ_{j})}^{\frac{1}{n}}} \end{matrix}$ (2.10)

Definition 2.8. [34] Let g_j = (j = 1, 2, …, n) be a collection of HFEs and let HFWG: ⊖ⁿ→ ⊖, if

$\begin{matrix} HFWG (g_{1}, g_{2}, \dots, g_{n}) = \frac{1}{n} \oplus g_{j}^{w_{j}} \\ = \underset{γ_{n} ɛ g_{n}}{\cup} {\prod_{j = 1}^{n} γ_{j}^{w_{j}}} \end{matrix}$ (2.11)

then HFWG is called a hesitant fuzzy weighted geometric (HFWG) operator,

where w = (w₁, w₂, …, w_n) ^T is the weight vector of g_j (j = 1, 2, …, n) with w_j ∈ [0.1] , j = 1, 2, …, n and $\sum_{j = 1}^{n} w_{j} = 1$ .

In the case where, $w = {(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$ , the HFWG operator reduces to the hesitant fuzzy geometric (HFG) operator: $\begin{matrix} HFG (g_{1}, g_{2}, \dots, g_{n}) = \oplus_{j = 1}^{n} g_{j}^{\frac{1}{n}} \\ = \underset{γ_{n} \in g_{n}}{\cup} {\prod_{j = 1}^{n} γ_{n}^{\frac{1}{n}}} \end{matrix}$ (2.12)

Definition 2.9. [34] For a collection of the HFEs g_j (j = 1, 2, …, n), a generalized hesitant fuzzy weighted averaging (GHFWA_λ) operator is a mapping GHFWA: ⊖ⁿ→ ⊖ such that $\begin{matrix} {GHFWA}_{λ} (g_{1}, g_{2}, \dots, g_{n}) = {(\oplus_{j = 1}^{n} w_{j} g_{j}^{λ})}^{\frac{1}{λ}} \\ = \underset{γ_{n} ɛ g_{n}}{\cup} {{(1 - \prod_{j = 1}^{n} {(1 - r_{j}^{λ})}^{w_{j}})}^{\frac{1}{λ}}} \end{matrix}$ (2.13)

where w = (w₁, w₂, …, w_n) ^T is the weight vector of g_j (j = 1, 2, …, n) with w_j ∈ [0.1] , j = 1, 2, …, n and $\sum_{j = 1}^{n} w_{j} = 1$ .

If λ = 1, then the GHFWA operator reduces to the HFWA operator; If $w = {(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$ then the GHFWA operator reduces to the GHFA operator.

Definition 2.10. [35] Let g_i (i = 1, 2, …, n) be a collection of HFEs which has the weight vector w = (w₁, w₂, …, w_n) ^t such that w_j = [0, 1], j = 1, 2, …, n, and $\sum_{j = 1}^{n} w_{j} = 1$ , λ > 0. A generalized hesitant fuzzy weighted geometric (GHFWG) operator is a mapping ⊖ⁿ→ ⊖, defined by $\begin{matrix} {GHFWG}_{λ} (g_{1}, g_{2}, \dots, g_{n}) = \frac{1}{λ} (\otimes_{j = 1}^{n} {(λ g_{j})}^{w_{j}}) \\ = \underset{γ_{n} ɛ g_{n}}{\cup} {1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - γ_{j})}^{λ})}^{w_{j}})}^{\frac{1}{λ}}} \end{matrix}$ (2.14)

If λ = 1, then the GHFWG_λ operator becomes the HFWG operator; If $w = {(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$ , then the GHFWG_λ operator reduces to the GHFG operator.

Definition 2.11. [18] A complex fuzzy complement of A is defined as follows: $\begin{matrix} \bar{A} = {(x, μ_{\bar{A}} (x)) : x \in U} \\ = {(x, r_{\bar{A}} (x) . e^{{iw}_{\bar{A}} (x)}) : x \in U} \end{matrix}$ (2.15)

where $r_{\bar{A}} (x) = 1 - r_{A} (x)$ and $w_{\bar{A}} (x) = w_{A} (x) = 2 π - w_{A} (x) = π + w_{A} (x)$ .

Definition 2.12. [3] A complex Atanassov’s intuitionistic fuzzy set A, defined on a universe of discourse U, is characterized by membership and non-membership functions μ_A (x) and γ_A (x), respectively, that assign to any element x ∈ U a complex-valued grade of both membership and non-membership in A.

By definition, the values of μ_A (x), γ_A (x), and their sum lies within the unit circle in the complex plane, and with the form μ_A (x) = r_A (x) . e^{iαw_{μ
_A}(x)} for a membership function in A and γ_A (x) = k_A (x) . e^{iαw_{γ
_A}(x)} for a non-membership function in A, where $i = \sqrt{- 1}$ , each of r_A (x) and k_A (x) are real-valued and both belong to the interval [0, 1] such that 0 ⩽ r_A (x) + k_A (x) ⩽ 1, also w_{μ
_A} (x) and w_{γ
_A} (x) are real-valued. We represent the CAIFS A as

A ={ 〈x, μ_A (x) , γ_A (x) 〉 : xɛU }, where μ_A (x) : $U \to {a : a \in C, | a | ⩽ 1}, γ_{A} (x) : U \to {a^{'} : a^{'} \in C, | a^{'} | ⩽ 1}$ $| μ_{A} (x) + γ_{A} (x) | ⩽ 1 .$

Definition 2.13. [3] If A = {〈x, r_A (x) . e^{iω_{r
_A}(x)}, k_A (x) . e^{iω_{k
_A}(x)}〉} and B = {〈x, r_B (x) . e^{iω_{r
_B}(x)}, k_B (x) . e^{iω_{k
_B}(x)}〉} are CAIFSs in a universe of discourse U, then

A ⊂ B if and only if r_A (x) < r_B (x) and k_A (x) > k_B (x), for amplitude terms and the phase terms (arguments) ω_{r
_A} (x) < ω_{r
_B} (x) and ω_{k
_A} (x) > ω_{k
_B} (x), for all x ∈ U.

A = B if and only if r_A (x) = r_B (x) and k_A (x) = k_B (x), for amplitude terms and the phase terms (arguments) ω_{r
_A} (x) = ω_{r
_B} (x) and ω_{k
_A} (x) = ω_{k
_B} (x), for all x ∈ U.

Let CAIFS (U) be the set of all complex Atanassov’s intuitionistic fuzzy sets on U, then

The standard S-norm: A ∪ B is given by $\begin{matrix} A \cup B = {(x, \max (r_{A} (x), r_{B} (x)) . e^{i α \max (ω_{r_{A}} (x), ω_{r_{B}} (x))}, \\ \min (k_{A} (x), k_{B} (x)) . e^{i α \min (ω_{k_{A}} (x), ω_{k_{B}} (x))}) : x \in U}, \end{matrix}$ (2.16)

this union is called the basic CIF union.

The standard T-norm: A ∩ B is given by: $\begin{matrix} A \cap B = {(x, \min (r_{A} (x), r_{B} (x)) . e^{i α \min (ω_{r_{A}} (x), ω_{r_{B}} (x))}, \\ \max (k_{A} (x), k_{B} (x)) . e^{i α \max (ω_{k_{A}} (x), ω_{k_{B}} (x))}) : x \in U}, \end{matrix}$ (2.17)

This intersection is called the basic CIF intersection.

Let A ={ 〈x, r_A (x) . e^{iω_{r
_A}(x)}, k_A (x) . e^{iω_{k
_A}(x)}〉 : x ∈ U } be a complex intuitionistic fuzzy set. Define the complement of c (A), as $\begin{matrix} c (A) = \bar{A} \\ = {〈 x, k_{A} (x) . e^{i α ω_{k_{A}} (x)}, r_{A} (x) . e^{i α ω_{r_{A}} (x)} 〉 : x \in U} \\ = {〈 x, k_{A} (x) . e^{i α ω_{\bar{A}}^{k} (x)}, r_{A} (x) . e^{i α ω_{\bar{A}}^{r} (x)} 〉 : x \in U}, \end{matrix}$ (2.18)

Where $ω_{\bar{A}} (x) = ω_{A} (x), 2 π - ω_{A} (x)$ , or ω_A (x) + π.

3 Complex hesitant fuzzy sets and elements

In this section, we introduce the formal definition of complex hesitant fuzzy sets and some of their operations, the Score function and deviation degree of a complex hesitant fuzzy element, and the rank of the CHFEs.

Definition 3.1. Let K be a set, a complex hesitant fuzzy element on W is in terms of a function that when applied to returns a subset of the unit disk in the complex plane. We may express the CHFEs by a $K = {〈 w, g_{K} (w) = r_{K} (w) . e^{i α θ_{K} (w)} 〉 : w \in W}$ (3.1)

Where $i = \sqrt{- 1}$ , r_K (w) and θ_K (w) are a fuzzy set of values in [0, 1], denoting the possible amplitude and phase membership degrees of the element w ∈ W to the set K. We call g_K (w) = r_K (w) . e^iαθ_K(w) complex hesitant fuzzy elements and € the set of all complex hesitant fuzzy elements (CHFEs).

The scaling factor α ∈ (0, 2π] is used to restrict the performance of the phases within (0, 2π), and the unit disk. Hence, the phase terms θ_K (w) represent the HF information, and phase term values in this representation case belong to the interval [0, 1]. So, in this paper, α will be measured as 2π.

Example 3.1. Suppose the three models of cars are w₁ : Mazda, w₂ : Mercedes, and w₃ : Toyota. Let W ={ w₁, w₂, w₃ } be a set three models of cars, g_k (w₁) = { 0.5, 0.2 } . e^{i2π{0.4,0.1}}, g_k (w₂) = { 0.2, 0.6, 0.3 } . e^{i2π{0.1,0.3}}, g_k (w₃) = { 0.2, 0.6 } . e^{i2π{0.1,0.3,0.4}} be the CHFEs of w_j (j = 1, 2, 3) to a set K, respectively. Then $K = {〈 w_{1}, {0.5, 0.2} . e^{i 2 π {0.4, 0.1}} 〉,$ $〈 w_{2}, {0.2, 0.6, 0.3} . e^{i 2 π {0.1, 0.3}} 〉,$ $〈 w_{3}, {0.2, 0.6,} . e^{i 2 π {0.1, 0.3, 0.4}} 〉}$

is a CHFS. Normally every year each factory produces the same model but with slight disparate enhancement. Regarding hesitant fuzzy information, Each DM has a set of potential values to assign two or more values for a model of car, which could be represented by hesitant fuzzy sets in the form of amplitude term such as r (w₁) ={ 0.5, 0.2 } for the model “Mazda”. Surely, Mazda 2020, 2019, 2018, or 2017 have different Specifications. So, the DMs assign to the phase term the values θ (w₁) ={ 0.4, 0.1 } for “Mazda” production date and their enhancement factor. Similarly, DM may assign values for w₂andw₃. Therefore, we get the ability to represent the uncertainty and periodicity semantics in the same mathematical structure.

Note 3.1. Some special CHFEs for w in W gave.

Complex empty set: g = { 0 } e^iα{0}, denoted by O_c.

Complex full set: g = { 1 } e^iα{1}, denoted by I_c.

Complex complete ignorance: g = [0, 1] . e^iα[0,1] = U_c.

Complex nonsense set: g = Ø _c.

Some deep clarifications on these special CHFEs can be made. As obtained in Definition 3.1, the CHFS conveys the set W to a subset of [0, 1] e^iα[0,1] by using the function g, therefore, if g conveys no value, then we declare that g is a complex nonsense set, similarly, if it conveys the set [0, 1] e^iα[0,1], then we call it complex complete ignorance, specifically, if it conveys individual value w ∈ [0, 1] e^iα[0,1], then it makes sense because the single value w ∈ [0, 1] e^iα[0,1], for instance, when γ = 0e^iα(0), which means the complex membership degree is zero, we call it the complex empty set, although if w = 1e^iα(1), then we call it the complex full set.

Note 3.2. The difference between the CHFS and the CIFS, CFS, and FS embodied in (1) we should not consider the complex empty set as the set with no values, and (2) we also should not consider the complex full set as the set of all possible values.

The interpretations of the special CHFEs in decision making can be illustrated as follows: Suppose a company is using CHFS to evaluate alternatives of a specific product and has a team of (DMs) to give this evaluation from different backgrounds, the complex empty set denotes all DMs who disapprove both the alternative and its phase, the complex full set conveys that all DMs agree with both the alternative and its phase. The complex complete ignorance denotes all DMs have no idea for both the alternative and its phase, and the complex nonsense set implies nonsense.

Now, to compare the CHFEs, we define the score function under CHFE as follows:

Definition 3.2. For a CHFE g, $\begin{matrix} s (g) = \frac{1}{2} [s (g_{r}) + s (g_{θ})] \\ = \frac{1}{2} [\frac{1}{l_{g_{r}}} \sum_{r \in g_{r}} r + \frac{1}{l_{g_{θ}}} \sum_{θ \in g_{θ}} θ] \end{matrix}$ (3.2)

is called the score of g, where l_{g
_r} and l_{g
_θ} are the number of the elements in g_r and g_θ, respectively, for two CHFEs g₁ and g₂, if s (g₁) > s (g₂), then g₁ is greater than g₂, denoted by g₁ > g₂; if s (g₁) = s (g₂), then g₁ is indifferent to g₂, denoted by g₁ ∼ g₂.

Example 3.2. Let g₁ = {0.7, 0.2, 0.4} . e^{i2π{0.2,0.8,0.9}}, g₂ = {0.7, 0.3} . e^{i2π{0.2,0.8,0.5}} and g₃ = {0.8, 0.4, 0.5} . e^{i2π{0.6,0.7}} be three CHFEs, then by Definition 3.2, we evaluate $s (g_{1}) = \frac{1}{2} [\frac{0.7 + 0.2 + 0.4}{3} + \frac{0.2 + 0.8 + 0.9}{3}] = 0.5333$ $s (g_{2}) = \frac{1}{2} [\frac{0.7 + 0.3}{2} + \frac{0.2 + 0.8 + 0.5}{3}] = 0.5$ $s (g_{3}) = \frac{1}{2} [\frac{0.8 + 0.4 + 0.5}{3} + \frac{0.6 + 0.7}{2}] = 0.6083$

Then s (g₃) > s (g₁) > s (g₂), which indicates that g₃ > g₁ > g₂.

s (g) value associated with the average value of all elements and its phases in g conveying the average opinion and average opinion’s phase of DMs, the better the CHFE g is associated to the bigger s (g) value. However, in the case of indifference between two CHFS, we should use a different comparison method to distinguish two CHFEs.

Definition 3.3. For a CHFE g, we define the deviation degree σ (g) of g as follows: $\begin{matrix} σ (g) = \\ \frac{1}{2} [{(\frac{1}{l_{g_{r}}} \sum_{r \in g_{r}} {(r - s (g_{r}))}^{2})}^{\frac{1}{2}} + {(\frac{1}{l_{g_{θ}}} \sum_{θ \in g_{θ}} {(θ - s (g_{θ}))}^{2})}^{\frac{1}{2}}] \end{matrix}$ (3.3)

σ (g) is the average of the standard variance between the mean values of s (g_r) and s (g_θ). Thus the value of σ (g) represents the average deviation degree between all values in both g_r and g_θ of CHFE g and their mean value. Subsequently, we present a new comparison method and ranking of two CHFEs, inspired by Definition 3.2 and definition 3.3, below:

Definition 3.4. Let g₁ and g₂ be two CHFEs, S (g₁) and S (g₂) the scores of g₁ and g₂, respectively, and σ (g₁) and σ (g₂) the deviation degrees of g₁ and g₂, respectively, we have:

If S (g₁) < S (g₂), then g₁ < g₂

Suppose S (g₁) = S (g₂), we have

If σ (g₁) = σ (g₂), then g₁ = g₂

If σ (g₁) < σ (g₂), then g₁ < g₂

If (g₁) > σ (g₂), then g₁ > g₂.

Example 3.3. Let g₁ = {0.8, 0.6, 0.7} . e^{i2π{0.6,0.4,0.5}}, g₂ = {0.3, 0.5} . e^{i2π{0.9,0.8,0.7}} and g₃ = {0.5, 0.8, 0.5} . e^{i2π{0.5,0.7}} be three CHFEs, respectively, since s (g₁) =0.6, s (g₂) =0.6 and s (g₃) =0.6 thus σ (g₁) =0.0133, σ (g₂) =0.00833 and σ (g₃) =0.015, so σ (g₂) < σ (g₁) < σ (g₃) therefore g₂ < g₁ < g₃.

Definition 3.5. Let g be a CHFE, then $\begin{matrix} σ^{'} (g) = \\ \frac{1}{2} [\frac{1}{l_{g_{r}}} \sqrt{\sum_{r_{j}, r_{m} ɛ g_{r}} {(r_{i} - r_{m})}^{2}} + \frac{1}{l_{g_{θ}}} \sqrt{\sum_{θ_{j}, θ_{m} ɛ g_{θ}} {(θ_{i} - θ_{m})}^{2}}] \end{matrix}$ (3.4)

is called the average of the deviation degree of g, which reflects the average of the standard deviation among all pairs of both g_r and g_θ of elements in the CHFE g.

4 Basic operations and relations on CHFE

In this section, we present three basic operations under the notion of CHFEs, namely, CHF complement, CHF union, and CHF intersection, then we propose the notion of the envelope of CHFE, which is reduced to CIFN [3] and then, discuss further relationships between CHFEs and the concept of CIFNs.

Definition 4.1. Let g₁ and g₂ be two CHFEs. Then we define the three operations on them, as follows:

$g^{c} = {\underset{r \in g_{r}}{\cup} {1 - r}} e^{i α {\underset{θ \in g_{θ}}{\cup} {1 - θ}}}$ .

$g_{1} \cup g_{2} = {\underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} max {r_{1}, r_{2}}} e^{i α {\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} max {θ_{1}, θ_{2}}}}$ .

$g_{1} \cap g_{2} = {\underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} min {r_{1}, r_{2}}} e^{i α {\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} min {θ_{1}, θ_{2}}}}$ .

Given a CIFN [3] (μ = re^{iαθ
_r}, γ = ke^{iαθ
_k}), its corresponding CHFE is straightforward: g = [re^{iαθ
_r}, (1 - k) e^iα(1-θ_k)], if r ≠ 1 - k and θ_r ≠ 1 - θ_k, the construction of the CIFN from the CHFE is not easy when the CHFE contains more than one number for each w ∈ W, so, we showed that the envelope of CHFE is a CIFN, expressed in the following definition:

Definition 4.2. Let g be a CHFE. The CIFN β_env (g) is defined as the envelope of g, where β_env (g) is represented as $(g^{-} = g_{r}^{-} . e^{i α g_{θ}^{-}}, g^{+} = (1 - g_{r}^{+}) . e^{i α (1 - g_{θ}^{+})})$ , with $g_{r}^{-} e^{i α g_{θ}^{-}} = min {r | r ɛ g_{r}^{-}} . e^{i α \min {θ | θ ɛ g_{θ}^{-}}}$ and $g_{r}^{+} e^{i α g_{θ}^{+}} = \max {r | r ɛ g_{r}^{+}} . e^{i α \max {θ | θ ɛ g_{θ}^{+}}}$ are the lower and the upper bounds, respectively.

The following theorems discuss further relationships between CHFEs and CIFNs:

Theorem 4.1. Let g & g₁ & g₂ be CHFS and CIFN, β_env (g) be the envelope of g, then

β_env (g^c) = (β_env (g)) ^c.

β_env (g₁ ∪ g₂) = β_env (g₁) ∪ β_env (g₂).

β_env (g₁ ∩ g₂) = β_env (g₁) ∩ β_env (g₂).

Note 4.1. Theorem 4.1 is also held when g₁ & g₂ define intervals.

Proof. (1) $β_{env} (g^{c}) = β_{env} (\underset{r \in g_{r}, θ \in g_{θ}}{\cup} {(1 - r) e^{i α (1 - θ)}})$

$= \underset{r \in g_{r}, θ \in g_{θ}}{\cup} (g^{-}, 1 - g^{+})$

$= \underset{r \in g_{r}, θ \in g_{θ}}{\cup} (\begin{matrix} min {1 - r | r ɛ g_{r}^{-}} e^{i α [\min {1 - θ | θ ɛ g_{θ}^{-}}]}, \\ 1 - \max {1 - r | r ɛ g_{r}^{+}} e^{i α [1 - \max {1 - θ | θ ɛ g_{θ}^{+}}]} \end{matrix})$

$= \underset{r \in g_{r}, θ \in g_{θ}}{\cup} (\begin{matrix} 1 - max {r | r ɛ g_{r}^{-}} e^{i α [1 - \max {θ | θ ɛ g_{θ}^{-}}}, \\ \min {r | r ɛ g_{r}^{+}} e^{i α [\min {θ | θ ɛ g_{θ}^{+}}]} \end{matrix})$ $= \underset{r \in g_{r}, θ \in g_{θ}}{\cup} (1 - g^{+}, g^{-}) = {(α_{env} (g))}^{c} = R . H . S .$

(2) $L . H . S . = β_{env} (g_{1} \cup g_{2}) =$ $β_{env} (\underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} \max {r_{1}, r_{2}} e^{i α (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} \max {θ_{1}, θ_{2}})}) =$ $〈 \min (\cup_{r_{j}} {\max {r_{1}, r_{2}} | r_{j} ɛ {(g_{r}^{-})}_{j}})$ $. e^{i α [\min {\cup_{θ_{j}} \max {θ_{1}, θ_{2}} | θ_{j} ɛ {(g_{θ}^{-})}_{j}}]}$ $1 - (\max {\cup_{r_{j}} \max {r_{1}, r_{2}} | r_{j} ɛ {(g_{r}^{+})}_{j}})$ $. e^{i α [1 - (\max {\cup_{θ_{j}} \max {θ_{1}, θ_{2}} | θ_{j} ɛ {(g_{θ}^{+})}_{j}})]} 〉$ $R . H . S . = β_{env} (g_{1}) \cup β_{env} (g_{2})$ $= 〈 g_{r_{1}}^{-} e^{i α g_{θ_{1}}^{-}}, (1 - g_{r_{1}}^{+}) e^{i α (1 - g_{θ_{1}}^{+})} 〉$ $\cup 〈 g_{r_{2}}^{-} e^{i α g_{θ_{2}}^{-}}, (1 - g_{r_{2}}^{+}) e^{i α (1 - g_{θ_{2}}^{+})} 〉$ $= 〈 min {r_{1} | r_{1} ɛ g_{r_{1}}^{-}} e^{i α [\min {θ_{1} | θ_{1} ɛ g_{θ_{1}}^{-}}]},$ $1 - max {r_{1} | r_{1} ɛ g_{r_{1}}^{-}} e^{i α (1 - \max {θ_{1} | θ_{1} ɛ g_{θ_{1}}^{-}}} 〉$ $\cup < min {r_{2} | r_{2} ɛ g_{r_{2}}^{-}} e^{i α [\min {θ_{2} | θ_{2} ɛ g_{θ_{2}}^{-}}]},$ $1 - max {r_{2} | r_{2} ɛ g_{r_{2}}^{-}} e^{i α (1 - \max {θ_{2} | θ_{2} ɛ g_{θ_{2}}^{-}})} >$ $= 〈 max {\min {r_{1} | r_{1} ɛ g_{r_{1}}^{-}}, \min {r_{2} | r_{2} ɛ g_{r_{2}}^{-}}}$ $. e^{i α [\max {\min {θ_{1} | θ_{1} ɛ g_{θ_{1}}^{-}}, \min {θ_{2} | θ_{2} ɛ g_{θ_{2}}^{-}}}]},$ $min {1 - \max {r_{1} | r_{1} ɛ g_{r_{1}}^{-}}, 1 - \max {r_{2} | r_{2} ɛ g_{r_{2}}^{-}}}$ $. e^{i α [\min {1 - \max {θ_{1} | θ_{1} ɛ g_{θ_{1}}^{-}}, 1 - \max {θ_{2} | θ_{2} ɛ g_{θ_{2}}^{-}}}]} 〉$ $= 〈 \min (\cup_{r \in g_{r}} {\max {r_{1}, r_{2}} | r_{j} ɛ {(g_{r}^{-})}_{j}})$ $. e^{i α \min {\cup_{θ \in g_{θ}} \max {θ_{1}, θ_{2}} | θ_{j} ɛ {(g_{θ}^{-})}_{j}}},$ $1 - \max {\cup_{r \in g_{r}} \max {r_{1}, r_{2}} | r_{j} ɛ {(g_{r}^{+})}_{j}}$ $. e^{i α [1 - (\max {\cup_{θ \in g_{θ}} \max {θ_{1}, θ_{2}} | θ_{j} ɛ {(g_{θ}^{+})}_{j}})]} 〉$ $= β_{env} (g_{1} \cup g_{2}) = L . H . S .$

(3) By using Definition (4.1(3)) and the definition (2.13), the proof is similar to the proof of (2).

Theorem 4.2. Let g₁ and g₂ be two CHFEs with g (x) being a nonempty convex set for all w in W, i.e., g₁ and g₂ are complex intuitionistic fuzzy numbers (CIFNs), then

$g_{1}^{c}$ is equivalent to the CIFN basic complement.

g₁ ∩ g₂ is equivalent to the CIFN basic intersection.

g₁ ∪ g₂ is equivalent to the CIFN basic union.

Proof. The proof is Straight forward by using definitions (2.12) and definition (3.1)

We define some operations and their properties for CHFEs Based on the relationship between CHFEs and CIFNs as follows:

Definition 4.3. Let g, g₁ and g₂ be CHFEs, and λ > 0, then

g^λ ={ ∪ _{r∈g_r} (r^λ) . e^{iα(∪_{θ∈g_θ}θ^λ)} }.

$λ g = {\underset{r \in g_{r}}{\cup} (1 - {(1 - r)}^{λ}) . e^{i α (\cup_{θ \in g_{θ}} (1 - {(1 - θ)}^{λ}))}}$ .

$g_{1} \oplus g_{2} = {\underset{r_{j} \in {(g_{r})}_{j}}{\cup} (r_{1} + r_{2} - r_{1} r_{2}) e^{i α (\cup_{θ_{j} \in {(g_{θ})}_{j}} (θ_{1} + θ_{2} - θ_{1} θ_{2}))}}$ .

$g_{1} \otimes g_{2} = {\underset{r_{j} \in {(g_{r})}_{j}}{\cup} (r_{1} r_{2}) e^{i α (\cup_{θ_{j} \in {(g_{θ})}_{j}} (θ_{1} θ_{2}))}}$ .

Notice that the operations in Definition 4.3 are held if g, g₁, g₂ are CIFNs.

Theorem 4.3. Let g, g₁ and g₂ be CHFEs, and λ > 0, then

β_env (g^λ) = (β_env (g)) ^λ

β_env (λg) = λ (β_env (g))

β_env (g₁ ⊕ g₂) = β_env (g₁) ⊕ β_env (g₂)

β_env (g₁ ⊗ g₂) = β_env (g₁) ⊗ β_env (g₂)

Proof. If g, g₁, g₂ are HFEs, then

L. H. S. of (1) is $\begin{matrix} β_{env} (g^{λ}) = \\ (β_{env} (\underset{r \in g_{r}}{\cup} r^{λ})) . e^{i α (α_{env} (\underset{θ \in g_{θ}}{\cup} θ^{λ}))} \\ = ({(g_{r}^{-})}^{λ}, 1 - {(g_{r}^{+})}^{λ}) . e^{i α ({(g_{θ}^{-})}^{λ}, 1 - {(g_{θ}^{+})}^{λ})} \end{matrix}$

While the R . H . S . = (β_env (g_r)) ^λ $= {(g_{r}^{-}, 1 - g_{r}^{+})}^{λ} . e^{i α {(g_{θ}^{-}, 1 - g_{θ}^{+})}^{λ}}$ $= ({(g_{r}^{-})}^{λ}, 1 - {(1 - (1 - g_{r}^{+}))}^{λ})$ $. e^{i α ({(g_{θ}^{-})}^{λ}, 1 - {(1 - (1 - g_{θ}^{+}))}^{λ})}$ $= ({(g_{r}^{-})}^{λ}, 1 - {(g_{r}^{+})}^{λ}) . e^{i α ({(g_{θ}^{-})}^{λ}, 1 - {(g_{θ}^{+})}^{λ})}$

The proof is similar to the proof of (1)

The L.H.S of (3) $(β_{env} (g_{1} \oplus g_{2}))$ $= (β_{env} {r_{1} + r_{2} - r_{1} r_{2} : r_{j} \in {(g_{r})}_{j}})$ $. e^{i α (α_{env} {θ_{1} + θ_{2} - θ_{1} θ_{2} : θ_{j} \in {(g_{θ})}_{j}})}$ $= (\begin{matrix} {(g_{r})}_{1}^{-} + {(g_{r})}_{2}^{-} - {(g_{r})}_{1}^{-} {(g_{r})}_{2}^{-}, \\ 1 - ({(g_{r})}_{1}^{+} + {(g_{r})}_{2}^{+} - {(g_{r})}_{1}^{+} {(g_{r})}_{2}^{+}) \end{matrix})$ $. e^{i α ({(g_{θ})}_{1}^{-} + {(g_{θ})}_{2}^{-} - {(g_{θ})}_{1}^{-} {(g_{θ})}_{2}^{-}, 1 - ({(g_{θ})}_{1}^{+} + {(g_{θ})}_{2}^{+} - {(g_{θ})}_{1}^{+} {(g_{θ})}_{2}^{+}))}$ $= (\begin{matrix} {(g_{r})}_{1}^{-} + {(g_{r})}_{2}^{-} - {(g_{r})}_{1}^{-} {(g_{r})}_{2}^{-}, \\ (1 - {(g_{r})}_{1}^{+}) (1 - {(g_{r})}_{2}^{+}) \end{matrix})$ $. e^{i α ({(g_{θ})}_{1}^{-} + {(g_{θ})}_{2}^{-} - {(g_{θ})}_{1}^{-} {(g_{θ})}_{2}^{-}, (1 - {(g_{θ})}_{1}^{+}) (1 - {(g_{θ})}_{2}^{+}))}$

The R.H.S of (3) $β_{env} (g_{1}) \oplus β_{env} (g_{2}) =$ $({(g_{r})}_{1}^{-}, 1 - {(g_{r})}_{1}^{+}) \oplus ({(g_{r})}_{2}^{-}, 1 - {(g_{r})}_{2}^{+})$ $. e^{i α ({(g_{θ})}_{1}^{-}, 1 - {(g_{θ})}_{1}^{+}) \oplus ({(g_{θ})}_{2}^{-}, 1 - {(g_{θ})}_{2}^{+})}$ $= (\begin{matrix} {(g_{r})}_{1}^{-} + {(g_{r})}_{2}^{-} - {(g_{r})}_{1}^{-} {(g_{r})}_{2}^{-}, \\ (1 - {(g_{r})}_{1}^{+}) (1 - {(g_{r})}_{2}^{+}) \end{matrix})$ $. e^{i α ({(g_{θ})}_{1}^{-} + {(g_{θ})}_{2}^{-} - {(g_{θ})}_{1}^{-} {(g_{θ})}_{2}^{-}, (1 - {(g_{θ})}_{1}^{+}) (1 - {(g_{θ})}_{2}^{+}))}$

The proof is similar to the proof of (3)

Theorem 4.4. If g, g₁, g₂, are CHFE’s then

$g_{1}^{c} \cup g_{2}^{c} = {(g_{1} \cap g_{2})}^{c} .$

$g_{1}^{c} \cap g_{2}^{c} = {(g_{1} \cup g_{2})}^{c} .$

(g^c) ^λ = (λg) ^c .

λ (g^c) = (g^λ) ^c .

$g_{1}^{c} \oplus g_{2}^{c} = {(g_{1} \otimes g_{2})}^{c}$ .

$g_{1}^{c} \otimes g_{2}^{c} = {(g_{1} \oplus g_{2})}^{c}$ .

Proof.

$g_{1}^{c} \cup g_{2}^{c} =$ $\underset{r_{j} \in {(g_{r})}_{j}}{\cup} \max {1 - r_{1}, 1 - r_{2}}$ $. e^{i α (\underset{θ_{j} \in {(g_{θ})}_{j}}{\cup} \max {1 - θ_{1}, 1 - θ_{2}})}$ $= \underset{r_{j} \in {(g_{r})}_{j}}{\cup} {1 - \min {r_{1}, r_{2}}}$ $. e^{i α (\underset{θ_{j} \in {(g_{θ})}_{j}}{\cup} {1 - \min {θ_{1}, θ_{2}}})} = {(g_{1} \cap g_{2})}^{c}$

The proof is similar to the proof of (1)

${(g^{c})}^{λ} = \underset{r_{j} \in {(g_{r})}_{j}}{\cup} {{(1 - r)}^{λ}} . e^{i α (\underset{θ_{j} \in {(g_{θ})}_{j}}{\cup} {{(1 - θ)}^{λ}})}$ $= {(\underset{r_{j} \in {(g_{r})}_{j}}{\cup} {1 - {(1 - r)}^{λ}})}^{c}$ $. e^{i α {(\underset{θ_{j} \in {(g_{θ})}_{j}}{\cup} {1 - {(1 - θ)}^{λ}})}^{c}} = {(λ g)}^{c}$

The proof is similar to the proof of (3)

$g_{1}^{c} \oplus g_{2}^{c} =$ $(\underset{r_{j} \in {(g_{r})}_{j}}{\cup} {(1 - r_{1}) + (1 - r_{2}) - (1 - r_{1}) (1 - r_{2})})$ $. e^{(\underset{θ_{j} \in {(g_{θ})}_{j}}{\cup} {(1 - θ_{1}) + (1 - θ_{2}) - (1 - θ_{1}) (1 - θ_{2})})}$ $= (\underset{r_{j} \in {(g_{r})}_{j}}{\cup} {1 - r_{1} r_{2}})$ $. e^{(\underset{θ_{j} \in {(g_{θ})}_{j}}{\cup} {1 - θ_{1} θ_{2}})} = {(g_{1} \otimes g_{2})}^{c}$

The proof is similar to the proof of (5)

Theorem 4.5. Let g, g₁ and g₃ be three CHFEs, λ > 0, then $λ (g_{1} \oplus g_{2}) = λ g_{1} \oplus λ g_{2}$ ${(g_{1} \otimes g_{2})}^{λ} = g_{1}^{λ} \otimes g_{2}^{λ}$

Proof.

(1) $\begin{matrix} λ (g_{1} \oplus g_{2}) = \\ λ (\underset{r_{j} \in {(g_{r})}_{j}}{\cup} {r_{1} + r_{2} - r_{1} r_{2}}) \\ . e^{i α (λ (\underset{θ_{j} \in {(g_{θ})}_{j}}{\cup} {θ_{1} + θ_{2} - θ_{1} θ_{2}}))} \\ = \underset{r_{j} \in {(g_{r})}_{j}}{\cup} {1 - (1 - {(r_{1} + r_{2} - r_{1} r_{2})}^{λ})} \\ . e^{i α (\underset{θ_{j} \in {(g_{θ})}_{j}}{\cup} {1 - (1 - {(θ_{1} + θ_{2} - θ_{1} θ_{2})}^{λ})})} \\ = \underset{r_{j} \in {(g_{r})}_{j}}{\cup} {1 - (1 - {(1 - (1 - r_{1}) (1 - r_{2}))}^{λ})} \\ . e^{i α (\underset{θ_{j} \in {(g_{θ})}_{j}}{\cup} {1 - (1 - {(1 - (1 - θ_{1}) (1 - θ_{2}))}^{λ})})} \\ = \underset{r_{j} \in {(g_{r})}_{j}}{\cup} {1 - ({(1 - r_{1})}^{λ} {(1 - r_{2})}^{λ})} \\ . e^{i α (\underset{θ_{j} \in {(g_{θ})}_{j}}{\cup} {1 - ({(1 - θ_{1})}^{λ} {(1 - θ_{2})}^{λ})})} (4.1) \end{matrix}$

and $λ g_{1} \oplus λ g_{2}$ $= \underset{r_{j} \in {(g_{r})}_{j}}{\cup} {1 - {(1 - r_{1})}^{λ} + 1$ $- {(1 - r_{2})}^{λ} - (1 - {(1 - r_{1})}^{λ}) (1 - {(1 - r_{2})}^{λ})}$ $. e^{i α (\underset{θ_{j} \in {(g_{θ})}_{j}}{\cup} {1 - {(1 - θ_{1})}^{λ} + 1 - {(1 - θ_{2})}^{λ} - (1 - {(1 - θ_{1})}^{λ}) (1 - {(1 - θ_{2})}^{λ})})}$ }} $= \underset{r_{j} \in {(g_{r})}_{j}}{\cup} {1 - {(1 - r_{1})}^{λ} {(1 - r_{2})}^{λ}}$ $. e^{i α (\underset{θ_{j} \in {(g_{θ})}_{j}}{\cup} {1 - {(1 - θ_{1})}^{λ} {(1 - θ_{2})}^{λ}})}$

so we have λ (g₁ ⊕ g₂) = λg₁ ⊕ λg₂

(2) The proof is similar to the proof of (1).

5 Aggregation operators of CHFS and their properties

As illustrated in the previous section, the relation between CHFEs and CIFNs was investigated. Rani and Garg [20] introduced lots of aggregation operators for aggregating CIFNs, such as CIF power averaging, CIF weighted power averaging, CIF ordered weighted power averaging, CIF power geometric, CIF weighted power geometric, and CIF ordered weighted power geometric, also, some aggregation operators have been developed by Xia and Xu [34] for aggregating HFS. In this section, we extend some aggregation operators of HFE to complex fields which are useful for aggregating CHFE and investigate some of their properties.

Definition 5.1. Let A ={ g₁, g₂, …, g_n } be a set of n CHFEs, define the function υ : [0, 1] ⁿe^iα[0,1]ⁿ → [0, 1] e^iα[0,1] on A as: $\begin{matrix} υ_{A} = \underset{r ɛ {{(g_{r})}_{1} \times {(g_{r})}_{2 \times \dots \times} {(g_{r})}_{n}}}{\cup} {υ (r)} \\ . e^{i α (\cup_{θ ɛ {{(g_{θ})}_{1} \times {(g_{θ})}_{2 \times \dots \times} {(g_{θ})}_{n}}} {υ (θ)})} \end{matrix}$ (5.1) allows for us to define affective aggregation operators for CHFEs, and study their needed properties:

Definition 5.2. Let g_j (j = 1, 2, …, n) be a collection of CHFEs. A complex hesitant fuzzy weighted averaging (CHFWA) operator is giving by:

$\begin{matrix} CHFWA (g_{1}, g_{2}, \dots, g_{n}) = \oplus_{j = 1}^{n} {(w_{r})}_{j} {(g_{r})}_{j} . e^{i α (\oplus_{j = 1}^{n} {(w_{θ})}_{j} {(g_{θ})}_{j})} \\ = \underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {1 - \prod_{j = 1}^{n} {(1 - r_{j})}^{{(w_{r})}_{j}}} \\ . e^{i α (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {1 - \prod_{j = 1}^{n} {(1 - θ_{j})}^{{(w_{θ})}_{j}}})} \end{matrix}$ (5.2)

Where w_r = ((w_r) ₁, (w_r) ₂, …, (w_r) _n) ^T and w_θ = ((w_θ) ₁, (w_θ) ₂, …, (w_θ) _n) ^T is the weight vector of g_j (j = 1, 2, …, n), with (w_r) _jand (w_θ) _j ∈ [0, 1], j = 1, 2, …, n, and $\sum_{j = 1}^{n} {(w_{r})}_{j} = \sum_{j = 1}^{n} {(w_{θ})}_{j} = 1$ . Notice that if $w_{r} = w_{θ} = {(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$ , then the CHFWA operator reduces to the complex hesitant fuzzy averaging (CHFA) operator:

$\begin{matrix} CHFA (g_{1}, g_{2}, \dots, g_{n}) = (\frac{1}{n} \oplus_{j = 1}^{n} g_{j}) e^{i α (\frac{1}{n} \oplus_{j = 1}^{n} g_{j})} \\ = \underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {1 - \prod_{j = 1}^{n} {(1 - r_{j})}^{\frac{1}{n}}} \\ . e^{i α (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {1 - \prod_{j = 1}^{n} {(1 - θ_{j})}^{\frac{1}{n}}})} \end{matrix}$ (5.3)

Definition 5.3. Let g_j = (j = 1, 2, …, n) be a collection of CHFEs and let CHFWG be a complex hesitant fuzzy weighted geometric operator if $\begin{matrix} CHFWG (g_{1}, g_{2}, \dots, g_{n}) \\ = (\frac{1}{n} \otimes_{j = 1}^{n} g_{j}^{{(w_{r})}_{j}}) . e^{i α (\frac{1}{n} \otimes_{j = 1}^{n} g_{j}^{{(w_{r})}_{j}})} \\ = (\underset{r_{j} ɛ g_{j}}{\cup} {\prod_{j = 1}^{n} r_{j}^{{(w_{r})}_{j}}}) . e^{i α (\underset{θ_{j} ɛ g_{j}}{\cup} {\prod_{j = 1}^{n} θ_{j}^{{(w_{θ})}_{j}}})} \end{matrix}$ (5.4)

where w_r = ((w_r) ₁, (w_r) ₂, …, (w_r) _n) ^T and w_θ = ((w_θ) ₁, (w_θ) ₂, …, (w_θ) _n) ^T is the weight vector of g_j (j = 1, 2, …, n) with (w_r) _jand (w_θ) _j ∈ [0, 1] , j = 1, 2, …, n and $\sum_{j = 1}^{n} {(w_{r})}_{j} = \sum_{j = 1}^{n} {(w_{θ})}_{j} = 1$ .

In the case where $w_{r} = w_{θ} = {(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$ , the CHFWA operator reduces to the complex hesitant fuzzy geometric (CHFG) operator: $\begin{matrix} CHFG (g_{1}, g_{2}, \dots, g_{n}) = \otimes_{j = 1}^{n} {(g_{r})}_{j}^{\frac{1}{n}} . e^{i α (\otimes_{j = 1}^{n} {(g_{θ})}_{j}^{\frac{1}{n}})} \\ = \underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {\prod_{j = 1}^{n} r_{j}^{\frac{1}{n}}} . e^{i α (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {\prod_{j = 1}^{n} r_{θ}^{\frac{1}{n}}})} \end{matrix}$ (5.5)

Lemma 5.1. Let r_j, θ_j > 0, (λ_r) _jand (λ_θ) _j > 0, j = 1, 2, …, n, and $\sum_{j = 1}^{n} {(λ_{r})}_{j} = \sum_{j = 1}^{n} {(λ_{θ})}_{j} = 1$ , then $\prod_{j = 1}^{n} r_{j}^{{(λ_{r})}_{j}} . e^{i α (\prod_{j = 1}^{n} θ_{j}^{{(λ_{θ})}_{j}})} ⩽ \sum_{j = 1}^{n} {(λ_{r})}_{j} r_{j} . e^{i α (\sum_{j = 1}^{n} {(λ_{θ})}_{j} θ_{j})}$ (5.6)

with equality if and only if r₁ = r₂ = … = r_n, and θ₁ = θ₂ = … = θ_n.

Theorem 5.1. Assume that g_j (j = 1, 2, …, n) is a collection of CHFEs, whose weight vector is w_r = ((w_r) ₁, (w_r) ₂, …, (w_r) _n) ^T and w_θ = ((w_θ) ₁, (w_θ) ₂, …, (w_θ) _n) ^T, with (w_r) _jand (w_θ) _j ∈ [0, 1] , j = 1, 2, …, n and $\sum_{j = 1}^{n} {(w_{r})}_{j} = \sum_{j = 1}^{n} {(w_{θ})}_{j} = 1$ ., then $CHFWG (g_{1}, g_{2}, \dots, g_{n}) ⩽ CHFWA (g_{1}, g_{2}, \dots, g_{n})$ (5.7)

Proof. By using Lemma 5.1, we get $\begin{matrix} \prod_{j = 1}^{n} r_{j}^{{(w_{r})}_{j}} . e^{i α (\prod_{j = 1}^{n} θ_{j}^{{(w_{θ})}_{j}})} \\ ⩽ \sum_{j = 1}^{n} {(w_{r})}_{j} r_{j} . e^{i α (\sum_{j = 1}^{n} {(w_{θ})}_{j} θ_{j})} \\ = (1 - \sum_{j = 1}^{n} {(w_{r})}_{j} (1 - r_{j})) e^{i α (1 - \sum_{j = 1}^{n} {(w_{θ})}_{j} (1 - θ_{j}))} \\ ⩽ (1 - \prod_{j = 1}^{n} {(1 - r_{j})}^{{(w_{r})}_{j}}) . e^{i α . (1 - \prod_{j = 1}^{n} {(1 - θ_{j})}^{{(w_{θ})}_{j}})} \end{matrix}$ (5.8)

For any r_jɛ (g_r) _j, θ_j (g_r) _j (j = 1, 2, …, n), which implies that

$\otimes_{j = 1}^{n} g_{j}^{w_{j}} ⩽ \oplus_{j = 1}^{n} w_{j} g_{j}$ .for j = 1, 2, …, n.

Definition 5.4. For a collection of the CHFEs g_j (j = 1, 2, …, n), a complex generalized hesitant fuzzy weighted averaging (CGHFWA) operator is a mapping CGHFWA: ⊖ⁿe^⊖ⁿ → ⊖ e^⊖ such that $\begin{matrix} {CGHFWA}_{λ} (g_{1}, g_{2}, \dots, g_{n}) \\ = {(\oplus_{j = 1}^{n} {(w_{r})}_{j} {(g_{r})}_{j}^{λ})}^{\frac{1}{λ}} . e^{i α ({(\oplus_{j = 1}^{n} {(w_{θ})}_{j} {(g_{θ})}_{j}^{λ})}^{\frac{1}{λ}})} \\ = \underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {{(1 - \prod_{j = 1}^{n} {(1 - r_{j}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}}} \\ . e^{i α (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {{(1 - \prod_{j = 1}^{n} {(1 - θ_{j}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}}})} \end{matrix}$ (5.9) where w_r = ((w_r) ₁, (w_r) ₂, …, (w_r) _n) ^T and w_θ = ((w_θ) ₁, (w_θ) ₂, …, (w_θ) _n) ^T is the weight vector of g_j (j = 1, 2, …, n) with (w_r) _j and (w_θ) _j ∈ [0.1] , j = 1, 2, …, n and $\sum_{j = 1}^{n} {(w_{r})}_{j} = \sum_{j = 1}^{n} {(w_{θ})}_{j} = 1$ . Notice that, if λ = 1, then the CGHFWA operator reduces to the CHFWA operator, also, If $w_{r} = w_{θ} = {(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$ then the CGHFWA operator reduces to the CGHFA operator.

Theorem 5.2 Let g_j (j = 1, 2, …, n) be a collection of CHFEs having the weight vector w_r = ((w_r) ₁, (w_r) ₂, …, (w_r) _n) ^T and w_θ = ((w_θ) ₁, (w_θ) ₂, …, (w_θ) _n) ^T such that (w_r) _j and (w_θ) _j ∈ [0, 1] , j = 1, 2, …, n and $\sum_{j = 1}^{n} w_{j} = 1$ , λ > 0, then $CHFWG (g_{1}, g_{2}, \dots, g_{n}) ⩽ {CGHFWA}_{λ} (g_{1}, g_{2}, \dots, g_{n})$ (5.10)

Proof. For any r_jɛ (g_r) _j, θ_j (g_r) _j for (j = 1, 2, …, n), based on Lemma 5.1, we have $\begin{matrix} \otimes_{j = 1}^{n} g_{j}^{w_{j}} = (\prod_{j = 1}^{n} {(r_{j}^{λ})}^{{(w_{r})}_{j}}) . e^{i α (\prod_{j = 1}^{n} {(θ_{j}^{λ})}^{{(w_{θ})}_{j}})} \\ ⩽ {(\sum_{j = 1}^{n} {(w_{r})}_{j} r_{j}^{λ})}^{\frac{1}{λ}} . e^{i α {(\sum_{j = 1}^{n} {(w_{θ})}_{j} θ_{j}^{λ})}^{\frac{1}{λ}}} \\ = {(1 - \sum_{j = 1}^{n} {(w_{r})}_{j} (1 - r_{j}^{λ}))}^{\frac{1}{λ}} . e^{i α {(1 - \sum_{j = 1}^{n} {(w_{θ})}_{j} (1 - θ_{j}^{λ}))}^{\frac{1}{λ}}} \\ ⩽ {(1 - \prod_{j = 1}^{n} {(1 - r_{j}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}} . e^{i α {(1 - \prod_{j = 1}^{n} {(1 - θ_{j}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}}} \\ = \oplus_{j = 1}^{n} w_{j} g_{j}^{\frac{1}{λ}} \end{matrix}$ (5.11)

From Theorem 5.1, we can conclude that the values obtained by the CHFWG operator are not bigger than the values obtained by the CGHFWA operator for any λ > 0.

Definition 5.5. Let g_i (i = 1, 2, …, n) be a collection of CHFEs having the weight vector w_r = ((w_r) ₁, (w_r) ₂, …, (w_r) _n) ^T and w_θ = ((w_θ) ₁, (w_θ) ₂, …, (w_θ) _n) ^T such that (w_r) _j and (w_θ) _j ∈ [0, 1], j = 1, 2, …, n, and $\sum_{j = 1}^{n} {(w_{r})}_{j} = \sum_{j = 1}^{n} {(w_{θ})}_{j} = 1$ , λ > 0. A complex generalized hesitant fuzzy weighted geometric (CGHFWG) operator is a mapping ⊖ⁿ . e^⊖ⁿ → ⊖ . e^⊖, defined by $\begin{matrix} {CGHFWG}_{λ} (g_{1}, g_{2}, \dots, g_{n}) = \frac{1}{λ} (\otimes_{j = 1}^{n} {(λ {(g_{r})}_{j})}^{{(w_{r})}_{j}}) . e^{i α (\frac{1}{λ} (\otimes_{j = 1}^{n} {(λ {(g_{θ})}_{j})}^{{(w_{θ})}_{j}}))} \\ = \underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - r_{j})}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}}} \\ . e^{i α (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - θ_{j})}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}}})} \end{matrix}$ (5.12)

Notice that, if λ = 1, then the CGHFWG operator becomes the CHFWG operator; and If $w_{r} = w_{θ} = {(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$ , then the CGHFWG operator reduces to the CGHFG operator.

Theorem 5.3. For a collection of CHFEs g_j (j = 1, 2, …, n), w_r = ((w_r) ₁, (w_r) ₂, …, (w_r) _n) ^T and w_θ = ((w_θ) ₁, (w_θ) ₂, …, (w_θ) _n) ^T is the weight vector such that (w_r) _j and (w_θ) _j ∈ [0, 1], j = 1, 2, …, n, and $\sum_{j = 1}^{n} {(w_{r})}_{j} = \sum_{j = 1}^{n} {(w_{θ})}_{j} = 1$ , λ > 0, then ${CGHFWG}_{λ} (g_{2}, g_{2}, \dots, g_{n}) ⩽ {CHFWA}_{λ} (g_{1}, g_{2}, \dots, g_{n})$ (5.13)

Proof. Let r_jɛ (g_r) _j, θ_j (g_r) _j for (j = 1, 2, …, n), based on Lemma 5.1, we have: $\begin{matrix} \frac{1}{λ} (\otimes_{j = 1}^{n} {(λ g_{j})}^{w_{j}}) = \\ 1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - r_{j})}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}} \\ . e^{i α (1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - θ_{j})}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}})} \\ ⩽ 1 - {(1 - \sum_{j = 1}^{n} w_{j} (1 - {(1 - r_{j})}^{λ}))}^{\frac{1}{λ}} \\ . e^{i α (1 - {(1 - \sum_{j = 1}^{n} w_{j} (1 - {(1 - θ_{j})}^{λ}))}^{\frac{1}{λ}})} \\ = 1 - {(1 - \sum_{j = 1}^{n} {(w_{r})}_{j} {(1 - r_{j})}^{λ})}^{\frac{1}{λ}} \\ . e^{i α (1 - {(1 - \sum_{j = 1}^{n} {(w_{θ})}_{j} {(1 - θ_{j})}^{λ})}^{\frac{1}{λ}})} \end{matrix}$ $\begin{matrix} ⩽ 1 - {(\prod_{j = 1}^{n} {(1 - r_{j})}^{{(w_{r})}_{j} λ})}^{\frac{1}{λ}} . e^{i α (1 - {(\prod_{j = 1}^{n} {(1 - θ_{j})}^{{(w_{θ})}_{j} λ})}^{\frac{1}{λ}})} \\ = 1 - \prod_{j = 1}^{n} {(1 - r_{j})}^{{(w_{r})}_{j}} . e^{i α (1 - \prod_{j = 1}^{n} {(1 - θ_{j})}^{{(w_{θ})}_{j}})} \\ = \oplus_{j = 1}^{n} w_{j} g_{j} \end{matrix}$ (5.14)

Notice that gives us the result that the values obtained by the CGHFWG operator are not bigger than the ones obtained by the CHFWA operator, no matter how the parameter λ > 0 changes.

Example 5.1. Let g₁ = { 0.6, 0.4, 0.3 } . e^{i2π{0.4,0.7,0.5}} and g₂ = { 0.7, 0.1 } . e^{i2π{0.3,0.2}} be two CHFEs, and let w_r = (0.2, 0.5) ^T and w_θ = (0.3, 0.6) be their weight vector, where λ = 4, then we have: ${CGHFWA}_{1} (g_{1}, g_{2}) = CHFWA (g_{1}, g_{2}) = \oplus_{j = 1}^{2} (w_{j} g_{j})$ $= \underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {1 - \prod_{j = 1}^{n} {(1 - r_{j})}^{{(w_{r})}_{j}}}$ $. e^{i α (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {1 - \prod_{j = 1}^{n} {(1 - θ_{j})}^{{(w_{θ})}_{j}}})}$ $= {\begin{matrix} 1 - {(1 - 0.6)}^{0.2} \times {(1 - 0.7)}^{0.5}, 1 - {(1 - 0.6)}^{0.2} \times {(1 - 0.1)}^{0.5}, \\ 1 - {(1 - 0.4)}^{0.2} \times {(1 - 0.7)}^{0.5}, 1 - {(1 - 0.4)}^{0.2} \times {(1 - 0.1)}^{0.5}, \\ 1 - {(1 - 0.3)}^{0.2} \times {(1 - 0.7)}^{0.5}, 1 - {(1 - 0.3)}^{0.2} \times {(1 - 0.1)}^{0.5} \end{matrix}}$ $. e^{i 2 π {\begin{matrix} 1 - {(1 - 0.4)}^{0.3} \times {(1 - 0.3)}^{0.6}, 1 - {(1 - 0.4)}^{0.3} \times {(1 - 0.2)}^{0.6}, 1 - {(1 - 0.7)}^{0.3} \times {(1 - 0.3)}^{0.6}, \\ 1 - {(1 - 0.7)}^{0.3} \times {(1 - 0.2)}^{0.6}, 1 - {(1 - 0.5)}^{0.3} \times {(1 - 0.3)}^{0.6}, 1 - {(1 - 0.5)}^{0.3} \times {(1 - 0.2)}^{0.6} \end{matrix}}}$ ${0.5440, 0.2102, 0.5055, 0.1435, 0.4900, 0.1166}$ $. e^{i 2 π {0.3074, 0.2496, 0.4374, 0.3905, 0.3442, 0.2895}}$ ${CGHFWA}_{4} (g_{1}, g_{2}) = {(\oplus_{j = 1}^{2} w_{j} g_{j}^{4})}^{\frac{1}{4}}$ $= \underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {{(1 - \prod_{j = 1}^{n} {(1 - r_{j}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}}}$ $. e^{i α (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {{(1 - \prod_{j = 1}^{n} {(1 - θ_{j}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}}})}$ $= {\begin{matrix} (1 - (1 - 0 . 6^{4})^{0.2} \times (1 - 0 . 7^{4})^{0.5})^{\frac{1}{4}}, \\ (1 - (1 - 0 . 6^{4})^{0.2} \times (1 - 0 . 1^{4})^{0.5})^{\frac{1}{4}}, \\ (1 - (1 - 0 . 4^{4})^{0.2} \times (1 - 0 . 7^{4})^{0.5})^{\frac{1}{4}}, \\ (1 - (1 - 0 . 4^{4})^{0.2} \times (1 - 0 . 1^{4})^{0.5})^{\frac{1}{4}}, \\ (1 - (1 - 0 . 3^{4})^{0.2} \times (1 - 0 . 7^{4})^{0.5})^{\frac{1}{4}}, \\ (1 - (1 - 0 . 3^{4})^{0.2} \times (1 - 0 . 1^{4})^{0.5})^{\frac{1}{4}}, \end{matrix}}$ $. e^{i 2 π {\begin{matrix} \begin{matrix} (1 - (1 - 0 . 4^{4})^{0.3} \times (1 - 0 . 3^{4})^{0.6})^{\frac{1}{4}}, (1 - (1 - 0 . 4^{4})^{0.3} \times (1 - 0 . 2^{4})^{0.6})^{\frac{1}{4}}, \\ (1 - (1 - 0 . 7^{4})^{0.3} \times (1 - 0 . 3^{4})^{0.6})^{\frac{1}{4}}, (1 - (1 - 0 . 7^{4})^{0.3} \times (1 - 0 . 2^{4})^{0.6})^{\frac{1}{4}}, \\ (1 - (1 - 0 . 5^{4})^{0.3} \times (1 - 0 . 3^{4})^{0.6})^{\frac{1}{4}}, (1 - (1 - 0 . 5^{4})^{0.3} \times (1 - 0 . 2^{4})^{0.6})^{\frac{1}{4}}, \end{matrix} \end{matrix}}}$ $= {0.6245, 0.070, 0.6037, 0.2688, 0.6001, 0.2023}$ $. e^{i 2 π {0.3349, 0.3054, 0.5376, 0.5318, 0.3934, 0.3766}}$ ${CGHFWG}_{1} (g_{1}, g_{2}) =$ $CHFWG (g_{1}, g_{2}) = \oplus_{j = 1}^{2} g_{j}^{w_{j}}$ $= (\underset{r_{j} ɛ g_{j}}{\cup} {\prod_{j = 1}^{n} r_{j}^{{(w_{r})}_{j}}}) . e^{i α (\underset{θ_{j} ɛ g_{j}}{\cup} {\prod_{j = 1}^{n} θ_{j}^{{(w_{θ})}_{j}}})}$ $= {\begin{matrix} 0 . 6^{0.2} \times 0 . 7^{0.5}, 0 . 6^{0.2} \times 0 . 1^{0.5}, 0 . 4^{0.2} \times 0 . 7^{0.5}, \\ 0 . 4^{0.2} \times 0 . 1^{0.5}, 0 . 3^{0.2} \times 0 . 7^{0.5}, 0 . 3^{0.2} \times 0 . 1^{0.5} \end{matrix}}$ $. e^{i 2 π {\begin{matrix} 0 . 4^{0.3} \times 0 . 3^{0.6}, 0 . 4^{0.3} \times 0 . 2^{0.6}, 0 . 7^{0.3} \times 0 . 3^{0.6}, \\ 0 . 7^{0.3} \times 0 . 2^{0.6}, 0 . 5^{0.3} \times 0 . 3^{0.6}, 0 . 5^{0.3} \times 0 . 2^{0.6} \end{matrix}}}$ $= {0.7554, 0.2855, 0.6966, 0.2633, 0.6576, 0.2486}$ $. e^{i 2 π {0.3689, 0.2892, 0.4363, 0.3421, 0.3944, 0.3092}}$ ${CGHFWG}_{4} (g_{1}, g_{2}) = \frac{1}{4} (\oplus_{j = 1}^{2} {(4 g_{j})}^{w_{j}})$ $= \underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - r_{j})}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}}}$ $. e^{i α (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - θ_{j})}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}}})}$ $= {\begin{matrix} 1 - (1 - (1 - (1 - 0.6)^{4})^{0.2} \times (1 - (1 - 0.7)^{4})^{0.5})^{\frac{1}{4}}, \\ 1 - (1 - (1 - (1 - 0.6)^{4})^{0.2} \times (1 - (1 - 0.1)^{4})^{0.5})^{\frac{1}{4}}, \\ 1 - (1 - (1 - (1 - 0.4)^{4})^{0.2} \times (1 - (1 - 0.7)^{4})^{0.5})^{\frac{1}{4}}, \\ 1 - (1 - (1 - (1 - 0.4)^{4})^{0.2} \times (1 - (1 - 0.1)^{4})^{0.5})^{\frac{1}{4}}, \\ 1 - (1 - (1 - (1 - 0.3)^{4})^{0.2} \times (1 - (1 - 0.7)^{4})^{0.5})^{\frac{1}{4}}, \\ 1 - (1 - (1 - (1 - 0.3)^{4})^{0.2} \times (1 - (1 - 0.1)^{4})^{0.5})^{\frac{1}{4}}, \end{matrix}}$ $. e^{i 2 π {\begin{matrix} 1 - (1 - (1 - (1 - 0.4)^{4})^{0.3} \times (1 - (1 - 0.3)^{4})^{0.6})^{\frac{1}{4}}, \\ 1 - (1 - (1 - (1 - 0.4)^{4})^{0.3} \times (1 - (1 - 0.2)^{4})^{0.6})^{\frac{1}{4}}, \\ 1 - (1 - (1 - (1 - 0.7)^{4})^{0.3} \times (1 - (1 - 0.3)^{4})^{0.6})^{\frac{1}{4}}, \\ 1 - (1 - (1 - (1 - 0.7)^{4})^{0.3} \times (1 - (1 - 0.2)^{4})^{0.6})^{\frac{1}{4}}, \\ 1 - (1 - (1 - (1 - 0.5)^{4})^{0.3} \times (1 - (1 - 0.3)^{4})^{0.6})^{\frac{1}{4}}, \\ 1 - (1 - (1 - (1 - 0.5)^{4})^{0.3} \times (1 - (1 - 0.2)^{4})^{0.6})^{\frac{1}{4}}, \end{matrix}}}$ $= {0.6902, 0.1966, 0.5793, 0.1904, 0.5108, 0.1833}$ $. e^{i 2 π {0.3429, 0.2594, 0.3736, 0.2773, 0.3596, 0.2693}}$

Theorem 5.4. Let g_j (j = 1, 2, …, n) be a collection of CHFEs with the weight vector w_r = ((w_r) ₁, (w_r) ₂, …, (w_r) _n) ^T and w_θ = ((w_θ) ₁, (w_θ) ₂, …, (w_θ) _n) ^T such that (w_r) _j and (w_θ) _j ∈ [0, 1], j = 1, 2, …, n, and $\sum_{j = 1}^{n} {(w_{r})}_{j} = \sum_{j = 1}^{n} {(w_{θ})}_{j} = 1$ , λ > 0, then

$\begin{matrix} (\oplus_{j = 1}^{n} {(w_{r})}_{j} {(g_{r})}_{j}^{c}) . e^{i α (\oplus_{j = 1}^{n} {(w_{θ})}_{j} {(g_{θ})}_{j}^{c})} = \\ {(\otimes_{j = 1}^{n} {(g_{r})}_{j}^{{(w_{r})}_{j}})}^{c} . e^{i α {(\otimes_{j = 1}^{n} {(g_{θ})}_{j}^{{(w_{θ})}_{j}})}^{c}} \end{matrix}$

$\begin{matrix} (\otimes_{j = 1}^{n} {({(g_{r})}_{j}^{c})}^{{(w_{r})}_{j}}) . e^{i α (\otimes_{j = 1}^{n} {(g_{j}^{c})}^{{(w_{θ})}_{j}})} = \\ {(\oplus_{j = 1}^{n} {(w_{r})}_{j} {(g_{r})}_{j})}^{c} . e^{i α {(\oplus_{j = 1}^{n} {(w_{θ})}_{j} {(g_{θ})}_{j})}^{c}} \end{matrix}$

${(\oplus_{j = 1}^{n} {(w_{r})}_{j} {({(g_{r})}_{j}^{c})}^{λ})}^{\frac{1}{λ}} . e^{i α {(\oplus_{j = 1}^{n} {(w_{θ})}_{j} {({(g_{θ})}_{j}^{c})}^{λ})}^{\frac{1}{λ}}}$ $= {(\frac{1}{λ} (\otimes_{j = 1}^{n} {(λ {(g_{r})}_{j})}^{{(w_{r})}_{j}}))}^{c} . e^{i α {(\frac{1}{λ} (\otimes_{j = 1}^{n} {(λ {(g_{θ})}_{j})}^{{(w_{θ})}_{j}}))}^{c}}$

$\frac{1}{λ} (\otimes_{j = 1}^{n} {(λ {(g_{r})}_{j}^{c})}^{{(w_{r})}_{j}}) . e^{i α (\frac{1}{λ} (\otimes_{j = 1}^{n} {(λ {(g_{θ})}_{j}^{c})}^{{(w_{θ})}_{j}}))}$ $= {({(\oplus_{j = 1}^{n} ({(w_{r})}_{j} {(g_{r})}_{j}^{c}))}^{\frac{1}{λ}})}^{c} . e^{i α {({(\oplus_{j = 1}^{n} ({(w_{θ})}_{j} {(g_{θ})}_{j}^{c}))}^{\frac{1}{λ}})}^{c}}$

Proof.

$(\oplus_{j = 1}^{n} ({(w_{r})}_{j} {(g_{r})}_{j}^{c})) . e^{i α (\oplus_{j = 1}^{n} ({(w_{θ})}_{j} {(g_{θ})}_{j}^{c}))}$ $= (\underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {1 - \prod_{j = 1}^{n} {(r_{j})}^{{(w_{r})}_{j}}})$ $. e^{i α (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {1 - \prod_{j = 1}^{n} {(θ_{j})}^{{(w_{θ})}_{j}}})}$ $= {(\underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {\prod_{j = 1}^{n} {(r_{j})}^{{(w_{r})}_{j}}})}^{c} . e^{i α {(\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {\prod_{j = 1}^{n} {(θ_{j})}^{{(w_{θ})}_{j}}})}^{c}}$ $= {(\otimes_{j = 1}^{n} {(g_{r})}_{j}^{{(w_{r})}_{j}})}^{c} . e^{i α {(\otimes_{j = 1}^{n} {(g_{θ})}_{j}^{{(w_{θ})}_{j}})}^{c}}$

$(\otimes_{j = 1}^{n} {({(g_{r})}_{j}^{c})}^{{(w_{r})}_{j}}) . e^{i α (\otimes_{j = 1}^{n} {({(g_{θ})}_{j}^{c})}^{{(w_{θ})}_{j}})}$ $= \underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {\prod_{j = 1}^{n} {(1 - r_{j})}^{{(w_{r})}_{j}}}$ $. e^{i α (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {\prod_{j = 1}^{n} {(1 - θ_{j})}^{{(w_{θ})}_{j}}})}$ $= {(\underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {1 - \prod_{j = 1}^{n} {(1 - r_{j})}^{{(w_{r})}_{j}}})}^{c}$ $. e^{i α {(\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {1 - \prod_{j = 1}^{n} {(1 - θ_{j})}^{{(w_{θ})}_{j}}})}^{c}}$ $= {(\oplus_{j = 1}^{n} {(w_{r})}_{j} {(g_{r})}_{j})}^{c} . e^{i α {(\oplus_{j = 1}^{n} {(w_{θ})}_{j} {(g_{θ})}_{j})}^{c}}$

${(\oplus_{j = 1}^{n} {(w_{r})}_{j} {({(g_{r})}_{j}^{c})}^{λ})}^{\frac{1}{λ}} . e^{i α {(\oplus_{j = 1}^{n} {(w_{θ})}_{j} {(g_{j}^{c})}^{λ})}^{\frac{1}{λ}}} =$ $(\underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {{(1 - \prod_{j = 1}^{n} {(1 - {(1 - r_{j})}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}}})$ $. e^{i α (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {{(1 - \prod_{j = 1}^{n} {(1 - {(1 - θ_{j})}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}}})}$ $= {(\underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - r_{j})}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}}})}^{c}$ $. e^{i α {(\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - θ_{j})}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}}})}^{c}}$ $= {(\frac{1}{λ} (\otimes_{j = 1}^{n} {(λ {(g_{r})}_{j})}^{{(w_{r})}_{j}}))}^{c} . e^{i α {(\frac{1}{λ} (\otimes_{j = 1}^{n} {(λ {(g_{θ})}_{j})}^{{(w_{θ})}_{j}}))}^{c}}$

$(\frac{1}{λ} (\otimes_{j = 1}^{n} {(λ {(g_{r})}_{j}^{c})}^{{(w_{r})}_{j}})) . e^{i α (\frac{1}{λ} (\otimes_{j = 1}^{n} {(λ {(g_{θ})}_{j}^{c})}^{{(w_{θ})}_{j}}))}$ $= (\underset{r_{j} ɛ g_{j}}{\cup} {(1 - {(1 - \prod_{j = 1}^{n} {(1 - r_{j}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}})})$ $. e^{i α (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {(1 - {(1 - \prod_{j = 1}^{n} {(1 - θ_{j}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}})})}$ ${(\underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {{(1 - \prod_{j = 1}^{n} {(1 - r_{j}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}}})}^{c}$ $. e^{i α {(\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {{(1 - \prod_{j = 1}^{n} {(1 - θ_{j}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}}})}^{c}}$ $= {({(\oplus_{j = 1}^{n} ({(w_{r})}_{j} {(g_{r})}_{j}^{c}))}^{\frac{1}{λ}})}^{c} . e^{i α {({(\oplus_{j = 1}^{n} ({(w_{θ})}_{j} {(g_{θ})}_{j}^{c}))}^{\frac{1}{λ}})}^{c}}$

Theorem 5.5. Let g_j (j = 1, 2, …, n) be a collection of CHFEs associated with the weight vector w_r = ((w_r) ₁, (w_r) ₂, …, (w_r) _n) ^T and w_θ = ((w_θ) ₁, (w_θ) ₂, …, (w_θ) _n) ^T such that (w_r) _j and (w_θ) _j ∈ [0, 1], j = 1, 2, …, n, and $\sum_{j = 1}^{n} {(w_{r})}_{j} = \sum_{j = 1}^{n} {(w_{θ})}_{j} = 1$ , λ > 0, then

$(β_{env} (\oplus_{j = 1}^{n} {(w_{r})}_{j} {(g_{r})}_{j})) . e^{i α (β_{env} (\oplus_{j = 1}^{n} {(w_{θ})}_{j} {(g_{θ})}_{j}))} = (\oplus_{j = 1}^{n} {(w_{r})}_{j} β_{env} {(g_{r})}_{j}) . e^{i α (\oplus_{j = 1}^{n} {(w_{θ})}_{j} β_{env} {(g_{θ})}_{j})}$

$(β_{env} (\otimes_{j = 1}^{n} {(g_{r})}_{j}^{{(w_{r})}_{j}})) . e^{i α (β_{env} (\otimes_{j = 1}^{n} {(g_{θ})}_{j}^{{(w_{θ})}_{j}}))} = (\otimes_{j = 1}^{n} {(w_{r})}_{j} β_{env} {(g_{r})}_{j}) . e^{i α (\otimes_{j = 1}^{n} {(w_{θ})}_{j} β_{env} {(g_{θ})}_{j})}$

$(β_{env} ({(\oplus_{j = 1}^{n} {(w_{r})}_{j} {(g_{r})}_{j}^{λ})}^{\frac{1}{λ}})) . e^{i α (β_{env} ({(\oplus_{j = 1}^{n} {(w_{θ})}_{j} {(g_{θ})}_{j}^{λ})}^{\frac{1}{λ}}))} = {(\oplus_{j = 1}^{n} {(w_{r})}_{j} {(β_{env} {(g_{r})}_{j})}^{λ})}^{\frac{1}{λ}} . e^{i α {(\oplus_{j = 1}^{n} {(w_{θ})}_{j} {(β_{env} {(g_{θ})}_{j})}^{λ})}^{\frac{1}{λ}}}$

$(β_{env} (\frac{1}{λ} (\otimes_{j = 1}^{n} {(λ {(g_{r})}_{j})}^{{(w_{r})}_{j}}))) . e^{i α (β_{env} (\frac{1}{λ} (\otimes_{j = 1}^{n} {(λ {(g_{θ})}_{j})}^{{(w_{θ})}_{j}})))} = (\frac{1}{λ} (\otimes_{j = 1}^{n} {(λ β_{env} {(g_{r})}_{j})}^{{(w_{r})}_{j}})) . e^{i α (\frac{1}{λ} (\otimes_{j = 1}^{n} {(λ β_{env} {(g_{θ})}_{j})}^{{(w_{θ})}_{j}}))}$

Proof.

$(β_{env} (\oplus_{j = 1}^{n} {(w_{r})}_{j} {(g_{r})}_{j})) . e^{(β_{env} (\oplus_{j = 1}^{n} {(w_{θ})}_{j} {(g_{θ})}_{j}))}$ $= (β_{env} (\underset{r_{j} ɛ (g_{r})_{j}_{j}}{\cup} {1 - \prod_{j = 1}^{n} {(1 - r_{j})}^{{(w_{r})}_{j}}}))$ $. e^{i α (β_{env} (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {1 - \prod_{j = 1}^{n} {(1 - θ_{j})}^{{(w_{θ})}_{j}}}))}$ $= (\begin{matrix} 1 - \prod_{j = 1}^{n} {(1 - {(g_{r})}_{j}^{-})}^{{(w_{r})}_{j}} \\ , 1 - (1 - \prod_{j = 1}^{n} {(1 - {(g_{r})}_{j}^{+})}^{{(w_{r})}_{j}}) \end{matrix})$ $. e^{i α (1 - \prod_{j = 1}^{n} {(1 - {(g_{θ})}_{j}^{-})}^{{(w_{θ})}_{j}}, 1 - (1 - \prod_{j = 1}^{n} {(1 - {(g_{θ})}_{j}^{+})}^{{(w_{θ})}_{j}}))}$ $= (\begin{matrix} 1 - \prod_{j = 1}^{n} {(1 - {(g_{r})}_{j}^{-})}^{{(w_{r})}_{j}}, \\ \prod_{j = 1}^{n} {(1 - {(g_{r})}_{j}^{+})}^{{(w_{r})}_{j}} \end{matrix})$ $. e^{i α (1 - \prod_{j = 1}^{n} {(1 - {(g_{θ})}_{j}^{-})}^{{(w_{θ})}_{j}}, \prod_{j = 1}^{n} {(1 - {(g_{θ})}_{j}^{+})}^{{(w_{θ})}_{j}})}$ $= (\oplus_{j = 1}^{n} ({(w_{r})}_{j} ({(g_{r})}_{j}^{-}, 1 - {(g_{r})}_{j}^{+})))$ $. e^{i α (\oplus_{j = 1}^{n} ({(w_{θ})}_{j} (g_{j}^{-}, 1 - g_{j}^{+})))}$ $= (\oplus_{j = 1}^{n} {(w_{r})}_{j} β_{env} {(g_{r})}_{j}) . e^{i α (\oplus_{j = 1}^{n} {(w_{θ})}_{j} β_{env} {(g_{θ})}_{j})}$

$(β_{env} (\otimes_{j = 1}^{n} {(g_{r})}_{j}^{{(w_{r})}_{j}})) . e^{i α (β_{env} (\otimes_{j = 1}^{n} {(g_{θ})}_{j}^{{(w_{θ})}_{j}}))}$ $= β_{env} (\underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {\prod_{j = 1}^{n} r_{j}^{{(w_{r})}_{j}}}) . e^{i α (β_{env} (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {\prod_{j = 1}^{n} θ_{j}^{{(w_{θ})}_{j}}}))}$ $= (\prod_{j = 1}^{n} {({(g_{r})}_{j}^{-})}^{{(w_{r})}_{j}}, 1 - \prod_{j = 1}^{n} {({(g_{r})}_{j}^{+})}^{{(w_{r})}_{j}})$ $. e^{i α (\prod_{j = 1}^{n} {({(g_{θ})}_{j}^{-})}^{{(w_{θ})}_{j}}, 1 - \prod_{j = 1}^{n} {({(g_{θ})}_{j}^{+})}^{{(w_{θ})}_{j}})}$ $= (\begin{matrix} \prod_{j = 1}^{n} {({(g_{r})}_{j}^{-})}^{{(w_{r})}_{j}}, \\ 1 - \prod_{j = 1}^{n} {(1 - (1 - {(g_{r})}_{j}^{+}))}^{{(w_{r})}_{j}} \end{matrix})$ $. e^{i α (\prod_{j = 1}^{n} {({(g_{θ})}_{j}^{-})}^{{(w_{θ})}_{j}}, 1 - \prod_{j = 1}^{n} {(1 - (1 - {(g_{θ})}_{j}^{+}))}^{{(w_{θ})}_{j}})}$ $= (\otimes_{j = 1}^{n} {({(g_{r})}_{j}^{-}, 1 - {(g_{r})}_{j}^{+})}^{{(w_{r})}_{j}})$ $. e^{i α ((\otimes_{j = 1}^{n} {({(g_{θ})}_{j}^{-}, 1 - {(g_{θ})}_{j}^{+})}^{{(w_{θ})}_{j}}))}$ $= (\otimes_{j = 1}^{n} {(w_{r})}_{j} β_{env} {(g_{r})}_{j}) . e^{i α (\otimes_{j = 1}^{n} {(w_{θ})}_{j} β_{env} {(g_{θ})}_{j})}$

$β_{env} ({(\oplus_{j = 1}^{n} {(w_{r})}_{j} {(g_{r})}_{j}^{λ})}^{\frac{1}{λ}})$ $. e^{i α (β_{env} ({(\otimes_{j = 1}^{n} {(w_{θ})}_{j} {({(g_{θ})}_{j})}^{λ})}^{\frac{1}{λ}}))}$ $= β_{env} (\underset{r_{j} ɛ {(g_{r})}_{j}}{\oplus} {{(1 - \prod_{j = 1}^{n} {(1 - r_{j}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}}})$ $. e^{i α (β_{env} (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {{(1 - \prod_{j = 1}^{n} {(1 - θ_{j}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}}}))}$ $= (\begin{matrix} {(1 - \prod_{j = 1}^{n} {(1 - {({(g_{r})}_{j}^{-})}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}}, \\ 1 - {(1 - \prod_{j = 1}^{n} {(1 - {({(g_{r})}_{j}^{+})}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}} \end{matrix})_$ $. e^{i α ({(1 - \prod_{j = 1}^{n} {(1 - {({(g_{θ})}_{j}^{-})}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}}, 1 - {(1 - \prod_{j = 1}^{n} {(1 - {({(g_{θ})}_{j}^{+})}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}})}$

$= (\begin{matrix} {(1 - \prod_{j = 1}^{n} {(1 - {({(g_{r})}_{j}^{-})}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}}, \\ 1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - (1 - {(g_{r})}_{j}^{+}))}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}} \end{matrix})$ $. e^{({(1 - \prod_{j = 1}^{n} {(1 - {({(g_{θ})}_{j}^{-})}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}}, 1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - (1 - {(g_{θ})}_{j}^{+}))}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}})}$ $= {(\oplus_{j = 1}^{n} {(w_{r})}_{j} {({(g_{r})}_{j}^{-}, 1 - {(g_{r})}_{j}^{+})}^{λ})}^{\frac{1}{λ}}$ $. e^{i α {(\oplus_{j = 1}^{n} {(w_{θ})}_{j} {({(g_{θ})}_{j}^{-}, 1 - {(g_{θ})}_{j}^{+})}^{λ})}^{\frac{1}{λ}}}$ $= {(\oplus_{j = 1}^{n} {(w_{r})}_{j} {(β_{env} {(g_{r})}_{j})}^{λ})}^{\frac{1}{λ}} . e^{i α {(\oplus_{j = 1}^{n} {(w_{θ})}_{j} {(β_{env} {(g_{θ})}_{j})}^{λ})}^{\frac{1}{λ}}}$

$β_{env} (\frac{1}{λ} (\otimes_{j = 1}^{n} {(λ {(g_{r})}_{j})}^{{(w_{r})}_{j}}))$ $. e^{i α (β_{env} (\frac{1}{λ} (\otimes_{j = 1}^{n} {(λ {(g_{θ})}_{j})}^{{(w_{θ})}_{j}})))} =$ $β_{env} (\underset{r_{j} ɛ {(g_{r})}_{j}}{\cup} {1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - r_{j})}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}}})$ $. e^{i α (β_{env} (\underset{θ_{j} ɛ {(g_{θ})}_{j}}{\cup} {1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - θ_{j})}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}}}))}$ $= (\begin{matrix} 1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - {(g_{r})}_{j}^{-})}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}}, \\ {(1 - \prod_{j = 1}^{n} {(1 - {(1 - {(g_{r})}_{j}^{+})}^{λ})}^{{(w_{r})}_{j}})}^{\frac{1}{λ}} \end{matrix})$ $. e^{(1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - {(g_{θ})}_{j}^{-})}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}}, {(1 - \prod_{j = 1}^{n} {(1 - {(1 - {(g_{θ})}_{j}^{+})}^{λ})}^{{(w_{θ})}_{j}})}^{\frac{1}{λ}})}$ $= \frac{1}{λ} (\prod_{j = 1}^{n} {(λ ({(g_{r})}_{j}^{-}, 1 - {(g_{r})}_{j}^{+}))}^{{(w_{r})}_{j}})$ $. e^{i α (\frac{1}{λ} (\prod_{j = 1}^{n} {(λ ({(g_{θ})}_{j}^{-}, 1 - {(g_{θ})}_{j}^{+}))}^{{(w_{θ})}_{j}}))}$ $= (\frac{1}{λ} \prod_{j = 1}^{n} {(λ β_{env} {(g_{r})}_{j})}^{{(w_{r})}_{j}}) . e^{i α (\frac{1}{λ} \prod_{j = 1}^{n} {(λ β_{env} {(g_{θ})}_{j})}^{{(w_{θ})}_{j}})}$

6 Algorithm and some applications on MADM problems

Alkouri and Salleh [4] introduced an application by applying a selection model from Song et al paper [51]. They generalized the selection operation to cover two-dimensional problems, namely, model of cars and production date of the cars by using CAIFS. In 2014, Xu [50] proposed in his book several methods for the DM to select the best choice depending on different hesitant fuzzy aggregation operators, which involves some algorithm steps from several literatures such as [46; 47; 48; 49]. In this section, adapted the method on the Xu and Xia [52], we extend the following decision-making methods to be suitable with the uncertainty and periodicity situation that the degrees of an alternative satisfies to an attribute are introduced by numerous potential values for both amplitude and phase terms which can be reproduced a CHFE.

Let A_l (l = 1, 2, 3, ..., n) be n alternatives and m attributes x_j (j = 1, 2, 3, ..., m) with the attribute amplitude weight vector (w_r) _j = (w_r1, w_r2, w_r3, …, w_rm) and attribute phase weight vector (w_θ) _j = (w_θ1, w_θ2, w_θ3, ..., w_θm) such that (w_r) _jand (w_θ) _j ∈ [0, 1], j = 1, 2, 3, ..., m, $\sum_{j = 1}^{m} {(w_{r})}_{j} = 1 \sum_{j = 1}^{m} {(w_{θ})}_{j} = 1$ . Suppose that a decision organization is authorized to provide all the possible degrees that the alternative A_l satisfies the attribute x_j, denoted by a CHFE g_lj.

By using the CHFWA (CHFWG) operator, we recommend an algorithm for the DM to choose the best choice with CHF information, which involves the following steps:

Step 1. Build the CHF decision matrix. The DM decides the appropriate attributes and its phases for each alternative, then he evaluates the given information concerning the attributes and its phases of the alternatives in the form of CHFEs, when he compared the alternatives over attributes and its phase, the customer may have many possible values according to the sub-attributes, therefore, it is natural to set out all the possible evaluations which are represented as CHFE, for an alternative under particular attributes and its phase specified by him. According to the customer favorites, the customer also decides the waited degrees w_j (j = 1, 2, . . . , m) for the related attributes. By adapting Xu and Xia [52] In the manner of aggregation, we possibly pact with the benefit type attributes or The cost type attributes, where the bigger the preference values of benefit type is the better; and the smaller the preference values of the cost type is the better. In some cases that combine two types, we may alter the preference values of the cost type attributes by the benefit type attributes. Then (G_lj) _n×m can be converted into the matrix Q = (q_lj) _n×m, where $q_{lj} = \cup_{t_{lj} \in q_{lj}} = {\begin{matrix} {(q_{r})_{lj}} e^{i α {(q_{θ})_{lj}}}, for benefit attripute x_{j} \\ {1 - (q_{r})_{lj}} e^{i α {1 - (q_{θ})_{lj}}}, for cost attripute x_{j} \end{matrix},$ $l = 1, 2, . . ., n, andj = 1, 2, . . ., m .$

Step 2. Apply the CHFWA operator (Definition 5.2) or the CHFWG operator (Definition 5.3) to obtain the CHFEs g_j (j = 1, 2, …, m) for the alternatives and their phases A_l (l = 1, 2, …, n).

Step 3. Calculate the scores s (g_j) (j = 1, 2, …, m) of g_j (j = 1, 2, …, m) by Definition 3.2 and the deviation degrees σ (g_j) (j = 1, 2, …, m) of g_j (j = 1, 2, …, m) by Definition 3.3.

Step 4. Rank A_l (l = 1, 2, …, n) by using the values s (g_j) (j = 1, 2, …, m) and σ (g_j) (j = 1, 2, …, m) of g_j (j = 1, 2, …, m).

Now, we apply the above algorithm’s steps in the following application in MADM problems.

Application. Suppose that a customer/ company wants to buy a car. Each model of car has a different production date (different phase), thus, each alternative will have different values according to different phases (production dates)(adapted from Alkouri and Salleh 2013a [4]), suppose there are three alternatives with different production dates A_l (l = 1, 2, 3) under consideration, also, suppose there are three attributes, which are x₁: quality, x₂: overall cost, and x₃: the appearance of the car, which consists of several sub-attributes as follows:

Quality consists of x₁₁: Remedy for quality problems, x₁₂: Safety, and x₁₃: Engine condition.

Overall cost consists of x₂₁: Repairs costs, x₂₂; Fuel consumption, and x₂₃: Product price.

Appearance consists of x₃₁: Body corrosion ratio; x₃₂: color, and x₃₃: Design. Both x₁, and x₃ are the benefit-type attributes and x₂ is the cost-type attribute.

But all attributes have mentioned above will be affected if we change the production date for the same model of cars. However, the decision made by some experts, so their decision depend on their knowledge and information about the problem domain. Also, the benefit and cost types attribute will be represented by the phase of production date and it will have possible phase values which are given by experts. Naturally, the price of any type of car will be increased for the current year comparing with the previous year. Therefore, it is fitting for the customer/ company to denote his/her favorite estimates in CHFEs to preserve the existing evaluation information sufficiently, which are characterized in the CHF decision matrix G (g_ij) _3×3 (see Table 1).

Table 1
CHF decision matrix G (g_lj) _3×3

x ₁ x ₂ x ₃

A ₁ {0.6, 0.7 } . e^{i2π{0.2,0.5,0.7}} {0.3, 0.4, 0.8 } . e^{i2π{0.4,0.7}} {0.1, 0.2, 0.4 } . e^{i2π{0.2,0.4}}

A ₂ {0.5, 0.8 } . e^{i2π{0.6,0.8}} {0.6, 0.8, 0.9 } . e^{i2π{0.5,0.6,0.8}} {0.3, 0.5 } . e^{i2π{0.4,0.5,0.7}}

A ₃ {0.4, 0.5, 0.7 } . e^{i2π{0.2,0.6,0.7}} {0.5, 0.8 } . e^{i2π{0.7,0.9}} {0.4, 0.5, 0.6 } . e^{i2π{0.7,0.8}}

	x ₁	x ₂	x ₃
A ₁	{0.6, 0.7 } . e^{i2π{0.2,0.5,0.7}}	{0.3, 0.4, 0.8 } . e^{i2π{0.4,0.7}}	{0.1, 0.2, 0.4 } . e^{i2π{0.2,0.4}}
A ₂	{0.5, 0.8 } . e^{i2π{0.6,0.8}}	{0.6, 0.8, 0.9 } . e^{i2π{0.5,0.6,0.8}}	{0.3, 0.5 } . e^{i2π{0.4,0.5,0.7}}
A ₃	{0.4, 0.5, 0.7 } . e^{i2π{0.2,0.6,0.7}}	{0.5, 0.8 } . e^{i2π{0.7,0.9}}	{0.4, 0.5, 0.6 } . e^{i2π{0.7,0.8}}

Let w_r = (0.7, 0.2, 0.1) ^T and w_θ = (0.3, 0.6, 0.1) ^T which are given by the customer, be the weight information of these attributes and their phases, respectively. To choose the most wanted car, here we employ the CHFWA operator to gain the CHFEs q_j (form Table 2) for the cars A_l (l = 1, 2, 3). Now, we calculate A₁

$q_{1} = CHFWA (q_{11}, q_{12}, q_{13})$ $= CHFWA ({0.6, 0.7} . e^{i 2 π {0.2, 0.5, 0.7}}, {0.7, 0.6, 0.2} . e^{i 2 π {0.6, 0.3}}$ $, {0.1, 0.2, 0.4} . e^{i 2 π {0.2, 0.4}})$ $= \underset{\oplus_{j = 1}^{3} {(w_{r})}_{j}, r_{j}}{\cup} {1 - \prod_{j = 1}^{3} {(1 - r_{j})}^{{(w_{r})}_{j}}}$ $. e^{i α {\underset{\oplus_{j = 1}^{3} {(w_{θ})}_{j}, θ_{j}}{\cup} {1 - \prod_{j = 1}^{3} {(1 - θ_{j})}^{{(w_{θ})}_{j}}}}}$

For the amplitude term, $\underset{\oplus_{i = 1}^{3} {(w_{r})}_{j}, {(q_{r})}_{j}}{\cup} {1 - \prod_{j = 1}^{3} {(1 - {(q_{r})}_{j})}^{{(w_{r})}_{j}}}$ $= {\begin{matrix} 1 - (1 - 0.6)^{0.7} \times (1 - 0.7)^{0.2} \times (1 - 0.1)^{0.1}, \\ 1 - (1 - 0.6)^{0.7} \times (1 - 0.7)^{0.2} \times (1 - 0.2)^{0.1}, \\ 1 - (1 - 0.6)^{0.7} \times (1 - 0.7)^{0.2} \times (1 - 0.4)^{0.1}, \\ 1 - (1 - 0.6)^{0.7} \times (1 - 0.6)^{0.2} \times (1 - 0.1)^{0.1}, \\ 1 - (1 - 0.6)^{0.7} \times (1 - 0.6)^{0.2} \times (1 - 0.2)^{0.1}, \\ 1 - (1 - 0.6)^{0.7} \times (1 - 0.6)^{0.2} \times (1 - 0.4)^{0.1}, \\ 1 - (1 - 0.6)^{0.7} \times (1 - 0.2)^{0.2} \times (1 - 0.1)^{0.1}, \\ 1 - (1 - 0.6)^{0.7} \times (1 - 0.2)^{0.2} \times (1 - 0.2)^{0.1}, \\ 1 - (1 - 0.6)^{0.7} \times (1 - 0.2)^{0.2} \times (1 - 0.4)^{0.1}, \\ 1 - (1 - 0.7)^{0.7} \times (1 - 0.7)^{0.2} \times (1 - 0.1)^{0.1}, \\ 1 - (1 - 0.7)^{0.7} \times (1 - 0.7)^{0.2} \times (1 - 0.2)^{0.1}, \\ 1 - (1 - 0.7)^{0.7} \times (1 - 0.7)^{0.2} \times (1 - 0.4)^{0.1}, \\ 1 - (1 - 0.7)^{0.7} \times (1 - 0.6)^{0.2} \times (1 - 0.1)^{0.1}, \\ 1 - (1 - 0.7)^{0.7} \times (1 - 0.6)^{0.2} \times (1 - 0.2)^{0.1}, \\ 1 - (1 - 0.7)^{0.7} \times (1 - 0.6)^{0.2} \times (1 - 0.4)^{0.1}, \\ 1 - (1 - 0.7)^{0.7} \times (1 - 0.2)^{0.2} \times (1 - 0.1)^{0.1}, \\ 1 - (1 - 0.7)^{0.7} \times (1 - 0.2)^{0.2} \times (1 - 0.2)^{0.1}, \\ 1 - (1 - 0.7)^{0.7} \times (1 - 0.2)^{0.2} \times (1 - 0.4)^{0.1}, \end{matrix}}$ $= {\begin{matrix} 0.5905, 0.5953, 0.6067, 0.5662, 0.5713, 0.5834, \\ 0.5017, 0.5075, 0.5215, 0.6652, 0.6691, 0.6785, \\ 0.6453, 0.6495, 0.6594, 0.5926, 0.5974, 0.6088 \end{matrix}}$ .

For the phase term, $\underset{\oplus_{i = 1}^{3} {(w_{θ})}_{j}, {(q_{θ})}_{j}}{\cup} {1 - \prod_{j = 1}^{3} {(1 - {(q_{θ})}_{j})}^{{(w_{θ})}_{j}}} =$ $= {\begin{matrix} 1 - (1 - 0.2)^{0.3} \times (1 - 0.6)^{0.6} \times (1 - 0.2)^{0.1}, \\ 1 - (1 - 0.2)^{0.3} \times (1 - 0.6)^{0.6} \times (1 - 0.4)^{0.1} \\ 1 - (1 - 0.2)^{0.3} \times (1 - 0.3)^{0.6} \times (1 - 0.2)^{0.1} \\ 1 - (1 - 0.2)^{0.3} \times (1 - 0.3)^{0.6} \times (1 - 0.4)^{0.1} \\ 1 - (1 - 0.5)^{0.3} \times (1 - 0.6)^{0.6} \times (1 - 0.2)^{0.1} \\ 1 - (1 - 0.5)^{0.3} \times (1 - 0.6)^{0.6} \times (1 - 0.4)^{0.1} \\ 1 - (1 - 0.5)^{0.3} \times (1 - 0.3)^{0.6} \times (1 - 0.2)^{0.1} \\ 1 - (1 - 0.5)^{0.3} \times (1 - 0.3)^{0.6} \times (1 - 0.4)^{0.1} \\ 1 - (1 - 0.7)^{0.3} \times (1 - 0.6)^{0.6} \times (1 - 0.2)^{0.1} \\ 1 - (1 - 0.7)^{0.3} \times (1 - 0.6)^{0.6} \times (1 - 0.4)^{0.1} \\ 1 - (1 - 0.7)^{0.3} \times (1 - 0.3)^{0.6} \times (1 - 0.2)^{0.1} \\ 1 - (1 - 0.7)^{0.3} \times (1 - 0.3)^{0.6} \times (1 - 0.4)^{0.1} \end{matrix}}$ $= {\begin{matrix} 0.4722, 0.4872, 0.2616, 0.2825, 0.5416, 0.5546, \\ 0.3587, 0.3769, 0.6067, 0.6179, 0.4498, 0.4654 \end{matrix}}$

Hence, q₁ = CHFWA (q₁₁, q₁₂, q₁₃) $= {\begin{matrix} 0.5905, 0.5953, 0.6067, 0.5662, 0.5713, 0.5834, \\ 0.5017, 0.5075, 0.5215, 0.6652, 0.6691, 0.6785, \\ 0.6453, 0.6495, 0.6594, 0.5926, 0.5974, 0.6088 \end{matrix}}$ $. e^{i 2 π {\begin{matrix} 0.4722, 0.4872, 0.2616, 0.2825, 0.5416, 0.5546, \\ 0.3587, 0.3769, 0.6067, 0.6179, 0.4498, 0.4654 \end{matrix}}}$ $A_{1} = q_{1} = CHFWA (q_{11}, q_{12}, q_{13})$ $= {0.60055} . e^{i 2 π {0.45626}}$

Similarly, we calculate A₂ : q₂ = CHFWA (q₂₁, q₂₂, q₂₃) = { 0.57813 } . e^{i2π{0.65734}}, and A₃ : q₃ = CHFWA (q₃₁, q₃₂, q₃₃) = { 0.50473 } . e^{i2π{0.39518}}. Then, by step 3, we calculate the scores s (q_j) (j = 1, 2, 3) and the deviation degrees σ (q_j) (j = 1, 2, 3) as follows:

$s (q) = \frac{1}{2} [s (q_{r}) + s (q_{θ})] = \frac{1}{2} [\frac{1}{l_{q_{r}}} \sum_{r \in q_{r}} r + \frac{1}{l_{q_{θ}}} \sum_{θ \in q_{θ}} θ]$ , where l_{q
_r} and l_{q
_θ} are the number of the elements in q_r and q_θ, respectively.

$s (q_{1}) = \frac{1}{2} [0.60055 + 0.45626] = 0.528405 .$ ,

$s (q_{2}) = \frac{1}{2} [0.57813 + 0.65734] = 0.617735$ , and

$s (q_{3}) = \frac{1}{2} [0.50473 + 0.39518] = 0.449955$ .

Finally, by step 4, we rank A_l (l = 1, 2, 3) by using the values s (q_j) and σ (q_j) of q_j (j = 1, 2, 3) as follows. $A_{3} < A_{1} < A_{2}$

Therefore, the best decision is to buy a car A₂.

Now, we calculate CHFWG aggregation operator as follows: $CHFWG (g_{1}, g_{2}, \dots, g_{n})$ $= (\frac{1}{n} \otimes_{j = 1}^{n} g_{j}^{{(w_{r})}_{j}}) . e^{i α (\frac{1}{n} \otimes_{j = 1}^{n} g_{j}^{{(w_{r})}_{j}})}$ $= (\underset{r_{j} ɛ g_{j}}{\cup} {\prod_{j = 1}^{n} r_{j}^{{(w_{r})}_{j}}}) . e^{i α (\underset{θ_{j} ɛ g_{j}}{\cup} {\prod_{j = 1}^{n} θ_{j}^{{(w_{θ})}_{j}}})}$

For the amplitude term, $(\underset{r_{j} ɛ g_{j}}{\cup} {\prod_{j = 1}^{3} {(q_{r})}_{j}^{{(w_{r})}_{j}}})$ $= {\begin{matrix} (0.6)^{0.7} \times (0.7)^{0.2} \times (0.1)^{0.1}, & (0.6)^{0.7} \times (0.7)^{0.2} \times (0.2)^{0.1}, \\ (0.6)^{0.7} \times (0.7)^{0.2} \times (0.4)^{0.1}, & (0.6)^{0.7} \times (0.6)^{0.2} \times (0.1)^{0.1}, \\ (0.6)^{0.7} \times (0.6)^{0.2} \times (0.2)^{0.1}, & (0.6)^{0.7} \times (0.6)^{0.2} \times (0.4)^{0.1}, \\ (0.6)^{0.7} \times (0.2)^{0.2} \times (0.1)^{0.1}, & (0.6)^{0.7} \times (0.2)^{0.2} \times (0.2)^{0.1}, \\ (0.6)^{0.7} \times (0.2)^{0.2} \times (0.4)^{0.1}, & (0.7)^{0.7} \times (0.7)^{0.2} \times (0.1)^{0.1}, \\ (0.7)^{0.7} \times (0.7)^{0.2} \times (0.2)^{0.1}, & (0.7)^{0.7} \times (0.7)^{0.2} \times (0.4)^{0.1}, \\ (0.7)^{0.7} \times (0.6)^{0.2} \times (0.1)^{0.1}, & (0.7)^{0.7} \times (0.6)^{0.2} \times (0.2)^{0.1}, \\ (0.7)^{0.7} \times (0.6)^{0.2} \times (0.4)^{0.1}, & (0.7)^{0.7} \times (0.2)^{0.2} \times (0.1)^{0.1}, \\ (0.7)^{0.7} \times (0.2)^{0.2} \times (0.2)^{0.1}, & (0.7)^{0.7} \times (0.2)^{0.2} \times (0.4)^{0.1}, \end{matrix}}$ $= {\begin{matrix} 0.5173, 0.5544, 0.5942, . 5016, 0.5376, 0.5762, \\ 0.4026, 0.4315, 0.4625, 0.5762, 0.6176, 0.6619, \\ 0.5587, 0.5988, 0.6418, 0.4485, 0.4807, 0.5152 \end{matrix}} .$

For the phase term, $(\underset{θ_{j} ɛ g_{j}}{\cup} {\prod_{j = 1}^{3} {(q_{θ})}_{j}^{{(w_{θ})}_{j}}})$ $= {\begin{matrix} (0.2)^{0.3} \times (0.6)^{0.6} \times (0.2)^{0.1}, & (0.2)^{0.3} \times (0.6)^{0.6} \times (0.4)^{0.1}, \\ (0.2)^{0.3} \times (0.3)^{0.6} \times (0.2)^{0.1}, & (0.2)^{0.3} \times (0.3)^{0.6} \times (0.4)^{0.1}, \\ (0.5)^{0.3} \times (0.6)^{0.6} \times (0.2)^{0.1}, & (0.5)^{0.3} \times (0.6)^{0.6} \times (0.4)^{0.1}, \\ (0.5)^{0.3} \times (0.3)^{0.6} \times (0.2)^{0.1}, & (0.5)^{0.3} \times (0.3)^{0.6} \times (0.4)^{0.1}, \\ (0.7)^{0.3} \times (0.6)^{0.6} \times (0.2)^{0.1}, & (0.7)^{0.3} \times (0.6)^{0.6} \times (0.4)^{0.1}, \\ (0.7)^{0.3} \times (0.3)^{0.6} \times (0.2)^{0.1}, & (0.7)^{0.3} \times (0.3)^{0.6} \times (0.4)^{0.1}, \end{matrix}}$ $= {\begin{matrix} 0.3866, 0.4144, 0.2551, 0.2734, 0.5090, 0.5455, \\ 0.3358, 0.3599, 0.5630, 0.6034, 0.3715, 0.3981 \end{matrix}}$

Hence, q₁ = CHFWG (q₁₁, q₁₂, q₁₃) $\begin{matrix} A_{1} = q_{1} = CHFWG (q_{11}, q_{12}, q_{13}) \\ = {0.5376} . e^{i 2 π {0.4179}} \end{matrix}$

Similarly, we calculate A₂ : q₂ = CHFWA (q₂₁, q₂₂, q₂₃) = { 0.4881 } . e^i2π{0.4529}, and A₃ : q₃ = CHFWA (q₃₁, q₃₂, q₃₃) = { 0.4760 } . e^i2π{0.2831}. Then, by step 3, we calculate the scores s (q_j) (j = 1, 2, 3) and the deviation degrees σ (q_j) (j = 1, 2, 3) as follows:

$s (q_{1}) = \frac{1}{2} [0.5376 + 0.4179] = 0.4881 .$ ,

$s (q_{2}) = \frac{1}{2} [0.4881 + 0.4529] = 0.4705$ , and

$s (q_{3}) = \frac{1}{2} [0.4760 + 0.2831] = 0.37955$ .

Finally, by step 4, we rank A_l (l = 1, 2, 3) by using the values s (q_j) and σ (q_j) of q_j (j = 1, 2, 3) as follows. $A_{3} < A_{2} < A_{1}$

Therefore, the best decision is to buy a car A₂.

Table 3 is indicated that the ranking of the alternatives is not similar for the CHFWA and CHFWG operators. Also, the score obtained by CHFWA operator become greater than the values obtained by the CHFWG for the same values of aggregation.

Table 2

CHF decision matrix Q (q_lj) _3×3

	x ₁	x ₂	x ₃
A ₁	{0.6, 0.7 } . e^{i2π{0.2,0.5,0.7}}	{0.7, 0.6, 0.2 } . e^{i2π{0.6,0.3}}	{0.1, 0.2, 0.4 } . e^{i2π{0.2,0.4}}
A ₂	{0.5, 0.8 } . e^{i2π{0.6,0.8}}	{0.3, 0.5 } . e^{i2π{0.4,0.5,0.7}}
A ₃	{0.4, 0.5, 0.7 } . e^{i2π{0.2,0.6,0.7}}	{0.5, 0.2 } . e^{i2π{0.3,0.1}}	{0.4, 0.5, 0.6 } . e^{i2π{0.7,0.8}}

By studying Tables 3 and 4, we discover that the ranking of alternatives obtained by the CHFWA operator is different from the HFWA operators. But for CHFWG and HFWG have same rank. The value of score obtained by CHFWA and CHFWG operators are different from the score value of HFWA and HFWG.

Table 3

Score values obtained by the CHFWA & CHFWG operators and the ranking of alternatives

	A ₁	A ₂	A ₃	Ranking
CHFWA	0.5284	0.6178	0.4499	A₃ < A₁ < A₂
CHFWG	0.4778	0.4705	0.3796	A₃ < A₂ < A₁

Table 4

Score values obtained by the HFWA & HFWG operators and the ranking of alternatives. (Phase term = 0)

	A ₁	A ₂	A ₃	Ranking
HFWA	0.6006	0.5781	0.5047	A₃ < A₂ < A₁
HFWG	0.5378	0.4881	0.4760	A₃ < A₂ < A₁

Table 5

A comparative study among different uncertainty sets (FS, IFS, HFS, CFS, and CIFS) with CHFS

	FS	IFS	HFS	CFS	CIFS	CHFS
Domain	Universe of discourse	Universe of discourse	Universe of discourse	Universe of discourse	Universe of discourse	Universe of discourse
Range	Single-valued in [0, 1]	Two values in [0, 1]	Several values in [0, 1]	One value in [0, 1] e^iα[0,1]	Two values in [0, 1] e^iα[0,1]	Several Values in [0, 1] e^iα[0,1]
Uncertainty measurement	√	√	√	√	√	√
Fluctuation measurement	√	√	√	√	√	√
Unit disk	X	X	X	√ s	√	√
Periodic measurement	X	X	X	√	√	√
Belingingness membership of Amplitude term	Single valued in [0, 1]	Single valued in [0, 1]	Several valued in [0, 1]	Single valued in [0, 1]	two valued in [0, 1]	Several valued in [0, 1]
Non-Belingingness membership of amplitude term	X	Single valued in [0, 1]	X	X	Single valued in [0, 1]	X
Belingingness membership of phase term	X	X	X	Single valued in [0, 1]	Single valued in [0, 1]	Several valued in [0, 1]
Non-Belingingness membership of phase term	X	X	X	X	Single valued in [0, 1]	X
Ability to convey hesitant information	X	Yes, for amplitude term only	Yes, for amplitude term only.	X	Yes, for both amplitude and phase terms.	Yes, for both amplitude and phase terms.
Constraint existence	X	√	X	X	√	X
Inclousion Relation	FS genralsied from the crisp sets.	Every IFS can be formed FS but the vise versa in not true	Every HFS can be formed FS and IFS but the vise versa in not true	Every CFS can be formed FS but the vise versa in not true	Every CIFS can be formed CFS and IFS but the vise versa in not true	Every CHFS can be formed FS, IFS, CFS and CIFS but the vise versa in not true.

7 Comparative study

CHFS is a generalization of CIFS as illustrated in Definition 4.2, by introducing the envelope of CHFS as a set of CIFS. In literature, relations between CIFS and each of CFS, IFS, and FS were comprehensively studied and introduced (see, [Alkouri and Salle, 2012, Ramote et. al. 2002,]). Also, the relation between HFS and each of FS and IFS was illustrated in [Tora and Narkawa, 2009]. Table 5 is introduced and summarized some comparative studies between several related uncertainty concepts and CHFS. The inclusion relation between CHFS and other uncertainty sets can be formalized as, “Every CHFS can be formed as FS, IFS, CFS, and CIFS but the vice versa is not true”. Specifically, If CHFS represents two periodic values by two experts then this forms a CIF information. If CHFS loses the periodic or the phase term is equal to 0 for all x ∈ X, then this form HF information. Other’s relations bond CIFS, CFS, IFS, FS, and HFS can be found in Table 5.

8 Conclusion

The complex hesitant fuzzy set presents information in more detail comparing with the hesitant fuzzy set’s approach by hiring hesitant fuzzy set to the phase term in the complex numbers. The gains of the complex hesitant fuzzy set may be determined by its ability to characterize problems. Not only the problem appears from a variety of DMs that give a set of potential values to assign a collective membership degree of an object to a set but also when each of them has a set of potential values in different times/phases of an object, subsequently, the collective membership degree will have a different meaning in different times/phases, thus, to deal with these issues, we introduced the concept of CHFS and their properties with range lies in the unit disk in the complex plane. CHFS can be applicable in several decision-making problems to select the optimal solution as CHFS uses two variables to represent the information instead of one variable. Some limitations of CHFS may appeared in representing a membership value for a collection of approximate descriptions of an object. This limitation can be covered by incorporating CHFS and the concept of the soft set as a future research. CHFS has no ability to represent the truth, intermediate, and falsity information as in complex neutrosophic set. So the future combination may be considered to introduce the concept of complex neutrosophic hesitant fuzzy set.

9 Conflicts of interest

The authors declare no conflicts of interest.

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