Linguistic interval-valued intuitionistic fuzzy copula power aggregation operators for multiattribute group decision making

Abstract

For the sake of better handle the imprecise and uncertain information in decision making problems(DMPs), linguistic interval-valued intuitionistic fuzzy numbers(LIVIFNs) based aggregation operators (AOS) are proposed by combining extended Copulas (ECs), extended Co-copulas (ECCs), power average operator and linguistic interval-valued intuitionistic fuzzy information (LIVIFI). First of all, ECs and ECCs, some specifics of ECs and ECCs, score and accuracy functions of LIVIFNs are gained. Then, based on ECs and ECCs, several aggregation operators are proposed to aggregate LIVIFI, which can offer decision makers (DMs) desirable generality and flexibility. In addition, the desired properties of proposed AOS are discussed. Last but not least, a MAGDM approach is constructed based on proposed AOs; Consequently, the effectiveness of the proposed approach is verified by a numerical example, and then the advantages are showed by comparing with other approaches.

Keywords

linguistic interval-valued intuitionistic fuzzy set (LIVIFS)Extended Copulas Extended Co-copulas PA operator multi-attribute group decision making(MAGDM)

1 Introduction

In actual life, multi-attribute decision making (MADM) has been applied in many fields such as real estate evaluation, mine site selection, investment scheme selection, etc. However, in the face of complex, unknown and uncertain DMPs, it is not realistic to be only evaluated by a single decision maker(DM), because personal decision-making ability and knowledge reserve are limited in front of such problems. Therefore, on the basis of MADM, scholars put forward MAGDM in which multiple experts negotiate to sort and choose the best alternative.

When dealing with evaluation problems, people prefer to describe evaluation information with accurate value. However, because of the fuzziness of human thinking, and the complex and changeable factors of decision making environment, it is difficult to meet the needs by using accurate number to describe decision making information. Generally, decision makers tend to use fuzzy language such as “excellent”, “good”, “bad” and fuzzy degree words such as “extraordinary”, “general”, “special” to express the evaluation value. Therefore, fuzzy set(FS) [1] has been widely used in the MAGDM problem after Zadeh first introduced it [2]. Up to now, on the basis of FS, scholars put forward more than twenty types of FSs [3], such as hesitant fuzzy set, intuitionistic fuzzy set(IFS) [4], etc. Among them, IFS and interval-valued IFS (IVIFS) [5] are more studied. By adding a nonmembership degree (NMD) into consideration, IFS and IVIFS are the powerful extension of FS. Since the introduction of IFS, the theories and applications of IFS have been studied comprehensively, including operational rules [6 –8], aggregation operators (AOs) [9 –11], score and accuracy functions [12, 13], fuzzy calculus [14, 15], similarity and distance measures [16, 17] and so on.

Although IFS and IVIFS have gained importance and popularity in MAGDM, its studies are restricted to tackle some uncertain information in quantitative environments. There exists such a kind of DMPs in which the decision makers (DMs) give their opinions on alternatives not easily in a quantitative form, but it can be easily described by linguistic variables (LVs) [18, 19]. For example, selecting a proper car from a certain number of alternatives is a typical decision problem in daily life. In this problem, factors affecting the car selection such as space size, engine performance, appearance, price, may be described by LVs like "poor", "medium", and "good". Under this condition, IFS and IVIFS are no longer suitable. In view of this, Xu [20, 21] introduced linguistic term set (LTS) and continuous linguistic term set (CLTS) respectively. Since the introduction of LTS (CLTs), some extended linguistic fuzzy sets have been developed, Such as, linguistic hesitant fuzzy set (LHFS) [22, 23], linguistic neutrosophic sets [24, 25], linguistic intuitionistic fuzzy sets (LIFS) [26], linguistic Pythagorean fuzzy set [29] and so on. As an important extended linguistic fuzzy set, LIFS is drawing much more attention. Chen [26] first introduced the concept of LIFS by combing LTS and IFS in which the MD and NMD are expressed by the LTS. Since then, the theories and applications on LIFS are also all comprehensively studied. For example, Liu [27, 28] proposed power Bonferroni operators and partitioned Heronian means operators based on LIFNs. Li et al. [30] developed a set of new operational rules and entropy for LIFSs. Peng [31] defined linguistic intuitionistic Frank improved weight Heronian mean operators (LIFFIWHM). Arora et al. [32] defined a prioritized linguistic intuitionistic fuzzy AO (LIFPWA) and applied it to MADM approach, which is more general and flexible to tackle DMPs. Besides, some MADM (MAGDM) approaches have been built based on LIFS or LIFN [33 –42].

In the above car selection decision problem, if the decision maker is not sure whether the description of a certain performance is "good" or "very good", but it is certain that it is between "good" and "very good". In this case, it is obviously more reasonable to describe it with linguistic interval-valued. In order to solve the above problem, Garg and Kumar [43] introduced linguistic interval-valued IFS (LIVIFS). LIVIFS is a more general form of LIFS. In LIVIFS, MD and NMD are expressed by the interval LVs (ILVs). When the upper and lower bounds of ILVs are equal, LIVIFS degenerates to LIFS. Based on LIVIFS, Garg and Kumar [43] presented a weighted average operator(LIVAIFWA_ω), an ordered weighted average operator (LIVAIFOWA_ω), and a hybrid average (LIVAIFHA_ω) operator. Kumar and Garg [44] proposed a prioritised weighted averaging operator. Liu and Qin [45] presented a weighted Maclaurin symmetric mean operator (LIVIFWMSM). Combined with power Muirhead mean operator, Qin et al. [46] developed a weighted aggregation operator(LIVIFAWPMM). Besides, some MADM (MAGDM) approaches have been built based on LIVIFS [47 –52].

But there are some limitations among these AOs: firstly, in practical decision-making problems, the aggregation of attribute values is a complicated process, in which a set of general and versatile AOs is needed. However, in the matter of most AOs, the operational rules are established in the light of t-norms (TNs) and t-conorms (TCs), the above mentioned AOs are only gotten by a special TNs and TCs, i. e. algebraic TN and algebraic TC. Secondly, these AOs are assembled based on the DM’s preference and attributes are independent of each other. In addition, for different attributes, the values need to be evaluated by domain experts combined with professional knowledge and personal preferences. So the data itself is subjective, and at the same time, it is inevitable that some experts have a strong preference for a candidate, which means extreme attribute values can be provided by some biased experts [53].

Among various of TNs and TCs, copulas and co-copulas are classical examples of TNs and TCs. Copulas [54] can not only reflect the dependence among variables, but also prevent information losing in the aggregation process. Copulas are functions that connect the joint distribution function with their respective edge distribution functions, which can describe the correlation between variables, so some people call it a connection function [54, 55]. As a tool for describing the dependence mechanism between variables, the copula function contains almost all the dependence information of random variables, especially when it is impossible to determine whether the traditional linear correlation coefficient can correctly measure the correlation between variables. There are basically two types of copula, Archimedean copula and Gaussian copula. In this paper, we only discuss Archimedean copula.

There are two distinguishing features of copulas: (1) Just as TNs and TCs, copulas and co-copulas are flexible because DMs can select different types of copulas to define the operations under fuzzy environment, and the results obtained from these operations are closed; (2) Copula function is flexible to capture the correlations among attributes in DMPs. Based on two obvious characteristic, copula have been applied to some DMPs. For example, Nelsen [54] applied copulas in aggregation function. Tao et al. [55] extended copulas to IFS and applied it to DMPs. In the light of Archimedean copula, Tao [56] studied a new computational model for unbalanced linguistic variables. Chen [57] defined new AOs in linguistic neutrosophic set based on Copulas and applied them to solve DMPs.

The existed AOs provide the most commonly used way to aggregate LIVIFS, but it lacks a unique way in practical applications. What is it the form of AO on the basis of copula function and LIVIFS? Considering Power average(PA) operator has the ability to reduce the irrational data provided by biased DMs, what is the form of generalized weighted PA operator? What are the differences between copula-based AOs and existed AOs? SO the goal and motivation of the present work are to synthesize ECs (ECCs), power operator, and LIVIFS and to develop some MAGDM approach with LIVIFI. The following three aspects should be addressed included in this work: (1) to define novel operational rules of LIVIFNs based on ECs and ECCs which can reflect the relevance, (2) to construct corresponding AOs regarding the proposed operational rules, (3) to develop a novel decision approach for MAGDM with LIVIFI. In these three main aspects, constructing novel operational rules is the core issue. Based on the above-analyzed, this work focuses on developing linguistic interval-valued intuitionisitic fuzzy Copula(LIVIFCA) aggregation operators with LIVIFI. The goals of this work are:

(1) to propose a new version of Copulas and Co-copulas by extending the domain and the range of Copulas and Co-copulas from [0, 1] to [0, t] (t > 0).

(2) to define a new operation rules of LIVIFNs based on ECs and ECCs.

(3) to propose a family of new AOs for managing LIVIFI by combing proposed new operational rules and PA operator.

(4) to propose a novel decision approach for MAGDM with LIVIFI and investigate some special cases.

In order to achieve the above goals, the organization is as follows. We firstly focused on new version of Copulas and Co-copulas which can tackle the linguistic information in Section 2. In Section 3, we introduce the LIVIFCA based on ECs and ECCs together with their properties. In Section 4, weighted averaging PA operator, and generalized weighted PA averaging operator are proposed based on LIVIFNs. The algorithm of MAGDM with LIVIFI based on LIVIFCA is contructed in Section 5. Case analysis will be carried out and some advantages of the proposed MAGDM approach based on LIVIFCA operators are analysed in Section 6 and the conclusion will be obtained in Section 7.

2 Preliminaries

In this section, firstly some basic concepts related to IFS and DIFS are reviewed, along with an overview of the evidential reasoning algorithm, which are the basis of the present work.

Definition 2.1. [43] Assume X is a universe of domain, S_[0,t] be a continuous LTS. A LIVIFSs A in X is depicted as

$A = (x, s_{μ_{A} (x)}, s_{ν_{A} (x)}) ∣ x \in X$ (1) where $s_{μ A (x)} = [s_{μ_{A}^{L} (x)}, s_{μ_{A}^{U} (x)}]$ , $s_{ν A (x)} = [s_{ν_{A}^{L} (x)}, s_{ν_{A}^{U} (x)}]$ are all subset of [s₀, s_t] and represent of linguistic MD and NMD of x to A respectively. For any $x \in X, s_{μ_{A}^{U} (x)} + s_{ν_{A}^{U} (x)} \leq s_{t}$ . The pair $([s_{μ_{A}^{L}}, s_{μ_{A}^{U}}], [s_{ν_{A}^{L}}, s_{ν_{A}^{U}}])$ is called LIVIFNs.

For convenience, we set α = ([s_a, s_b] , [s_c, s_d]), where s_a, s_b, s_c, s_d ∈ S_[0,t], and also [s_a, s_b] ∈ [s₀, s_t] , [s_c, s_d] ∈ [s₀, s_t] , b + d ≤ t.

Definition 2.2. [43] Let α = ([s_a, s_b] , [s_c, s_d]) be a LIVIFNs, score function and accuracy function of α is depicted as $S (α) = s_{(2 t + a - c + b - d) / 4}$ (2) $H (α) = s_{(a + b + c + d) / 2}$ (3) Then, for any two different LIVIFNs α₁ and α₂, we have,

(1) If S (α₁) < S (α₂), then α₁ < α₂;

(2) If S (α₁) = S (α₂) and H (α₁) = H (α₂), then α₁ < α₂.

Most existed AOs are built based on TN and TC. It is a pity that the range and the domain of TN and TC must be in [0, 1], under the circumstance, this type of AOs may be invalid in some real DMPs in which the decision information are expressed by LVs. In linguistic intuitionisitic decision making, if the MD and NMD of a LIVIFN can be transformed into [0, t], where t > 0, then all operation rules can be extended to [0, t]. Furthermore, some decision making approaches can be extended to deal with LIVIFI. For this purpose, the extended TN and extended TC are introduced by Liu [58]:

Definition 2.3. [58] An extended t-norm $T$ is a mapping from [0, t] ² to [0, t], if $T$ fulfils the followings: for all c, d, e ∈ [0, t],

(T1) $T (c, t) = c$ ;

(T2) $T (c, d) = T (d, c)$ ;

(T3) $T (c, T (d, e)) = T (T (c, d), e)$ .

If T just satisfies (T1), then T is called a semi-copula. With the help of extended TNs and extended TCs, we first introduce the concept of extended Copulas (ECs) and extended Co-copulas (ECCs) in order to handle some DMPs with LIVIFI.

Definition 2.4. A binary function $ℂ : [0, t]^{2} \to [0, t]$ is called an EC if $ℂ$ fulfills the conditions: for all c, d, c′, d′ ∈ [0, t],

(C1) $ℂ (c, d) + ℂ (c', d') \geq ℂ (c, d') + ℂ (c', d)$ ;

(C2) $ℂ (c, 0) = ℂ (0, c) = 0$ ;

(C3) $ℂ (c, t) = ℂ (t, c) = c$ .

Definition 2.5. Let ϱ : [0, t] → [0, + ∞) and ψ : [0, + ∞) → [0, t]. If ϱ satisfies continuity, strict decline and ϱ (t) =0. For all (c, d) ∈ [0, t] ², a Copula $ℂ$ is said to be extended Copulas (ECs) if ϱ, ψ satisfy:

$ψ (c) = {\begin{matrix} ϱ^{- 1} (c), & c \in [0, ϱ (0)]; \\ 0, & c \in [ϱ (0), + \infty) . \end{matrix}$ (4) and

$ℂ (c, d) = ψ (ϱ (c) + ϱ (d)) .$ (5) The generator ϱ of an ECs if a mapping from [0, t] to R⁺ and ϱ^-1 is the mapping from R⁺ to [0, t] with ϱ (0) =+ ∞ and ϱ (t) =0. According to Genest et al. [59], the $ℂ$ can be rewritten as

$ℂ (c, d) = ϱ^{- 1} (ϱ (c) + ϱ (d)) .$ (6)

Definition 2.6. Let $ℂ$ be an ECs, for all (c, d) ∈ [0, t] ², the new function

$ℂ^{*} (c, d) = t - ℂ (t - c, t - d),$ (7) is called extended Co-copulas (ECCs).

From the definition of ECCs, we have $ℂ^{*}$ is bounded; But it not an ECs. For example, for all d ∈ [0, t], $ℂ^{*} (t, d) = t - ℂ (t - t, t - c) = t,$ it follows that $ℂ^{*}$ don’t satisfy (C3).

In what follows, all ECs are all EACs if not specified. In order to introduce some new operations for LIFNs based on ECs and ECCs mentioned above, following conclusion is given firstly.

Theorem 2.1. For all c₁, c₂, d₁, d₂ ∈ [0, t], if c_i + d_i ≤ t (i = 1, 2), then $0 \leq ℂ (c_{1}, c_{2}) + ℂ^{*} (d_{1}, d_{2}) \leq t$ .

Proof. It follows easily from the definitions of ECs and ECCs that $0 \leq ℂ (c_{1}, c_{2}) + ℂ^{*} (d_{1}, d_{2})$ . So we just need to prove $ℂ (c_{1}, c_{2}) + ℂ^{*} (d_{1}, d_{2}) \leq t$ .

It follows from the definitions of EC and ECC that $\begin{matrix} ℂ (c_{1}, c_{2}) + ℂ^{*} (d_{1}, d_{2}) \\ = ℂ (c_{1}, c_{2}) + (t - ℂ (t - d_{1}, t - d_{2})) \\ = (ϱ^{- 1} (ϱ (c_{1}) + ϱ (c_{2}))) \\ + t - (ϱ^{- 1} (ϱ (t - d_{1}) + ϱ (1 - d_{2}))) \end{matrix}$

As ϱ is strictly decreasing and c_i + d_i ≤ t (i = 1, 2), it follows that $\begin{matrix} ϱ (c_{1}) + ϱ (c_{2}) \geq ϱ (t - d_{1}) + ϱ (t - d_{1}) . \end{matrix}$ Therefore $\begin{matrix} ϱ^{- 1} (ϱ (c_{1}) + ϱ (c_{2})) \leq ϱ^{- 1} (ϱ (t - d_{1}) + ϱ (t - d_{1})) \end{matrix}$ So, we have

$\begin{matrix} ℂ (c_{1}, c_{2}) + ℂ^{*} (d_{1}, d_{2}) \\ = (ϱ^{- 1} (ϱ (c_{1}) + ϱ (c_{2}))) + t - ϱ^{- 1} (ϱ (t - d_{1}) + ϱ (t - d_{2})) \\ \leq (ϱ^{- 1} (ϱ (c_{1}) + ϱ (c_{2}))) + t - (ϱ^{- 1} (ϱ (c_{1}) + ϱ (c_{2}))) = t \end{matrix}$

When generator ϱ (c) is different, several common formulas are shown in Table 1.

Table 1

The influence of parameters θ on the rank of alternatives

Type	generator ϱ (c)	EC and ECC	Condition
Gumble	$ϱ (c) = {(- \ln (\frac{c}{t}))}^{θ}$	$ℂ (c, d) = {te}^{- {({(- \ln (\frac{c}{t}))}^{θ} + {(- \ln (\frac{d}{t}))}^{θ})}^{\frac{1}{θ}}}$	θ ≥ 1
		$ℂ^{*} (c, d) = t - {te}^{- {({(- \ln (\frac{t - c}{t}))}^{θ} + {(- \ln (\frac{t - d}{t}))}^{θ})}^{\frac{1}{θ}}}$
Clayton	$ϱ (c) = {(\frac{c}{t})}^{- θ} - 1$	$ℂ (c, d) = t {({(\frac{c}{t})}^{- θ} + {(\frac{d}{t})}^{- θ} - 1)}^{- \frac{1}{θ}}$	θ ≠ 0
		$ℂ^{*} (c, d) = t - t {({(\frac{t - c}{t})}^{- θ} + {(\frac{t - d}{t})}^{- θ} - 1)}^{- \frac{1}{θ}}$
Frank	$ϱ (c) = \ln (\frac{e^{- \frac{θ c}{t}} - 1}{e^{- θ} - 1})$	$ℂ (c, d) = (- \frac{t}{θ}) \ln [\frac{(e^{- \frac{θ c}{t}} - 1) (e^{- \frac{θ d}{t}} - 1)}{e^{- θ} - 1} + 1]$	θ ≠ 0
		$ℂ^{*} (c, d) = t + \frac{t}{θ} \ln [\frac{(e^{- \frac{θ (t - c)}{t}} - 1) (e^{- \frac{θ (t - d)}{t}} - 1)}{e^{- θ} - 1} + 1]$
Ali-Mikhail-Haq	$ϱ (c) = \ln (\frac{t - θ (t - c)}{c})$	$ℂ (c, d) = \frac{tcd}{t^{2} - θ (t - c) (t - d)}$	θ ∈ [-1, 1)
		$ℂ^{*} (c, d) = t - \frac{t (t - c) (t - d)}{t^{2} - θ cd}$
Joe	$ϱ (c) = - \ln (1 - {(1 - \frac{c}{t})}^{θ})$	$ℂ (c, d) = t - \frac{{(t^{θ} ({(t - c)}^{θ} + {(t - d)}^{θ}) - {(t - c)}^{θ} {(t - d)}^{θ})}^{\frac{1}{θ}}}{t}$	θ ≥ 1
		$ℂ^{*} (c, d) = t {(c^{θ} + d^{θ} - {(\frac{cd}{t})}^{θ})}^{\frac{1}{θ}}$

In this following, we will give a new version of operational rules based on ECs and ECCs.

Definition 2.7. Let α₁ = ([s_{a
₁}, s_{b
₁}] , [s_{c
₁}, s_{d
₁}]) and α₂ = ([s_{a
₂}, s_{b
₂}] , [s_{c
₂}, s_{d
₂}]) be two LIVIFNs, the novel operational rules of LIVIFNs are given as follows:

$\begin{matrix} α_{1} \oplus_{ℂ} α_{2} = ([s_{ℂ^{*} (a_{1}, a_{2})}, s_{ℂ^{*} (b_{1}, b_{2})}], [s_{ℂ (c_{1}, c_{2})}, s_{ℂ (d_{1}, d_{2})}]) \\ = ([s_{t - (ϱ^{- 1} (ϱ (t - a_{1}) + ϱ (t - a_{2})))}, s_{t - (ϱ^{- 1} (ϱ (t - b_{1}) + ϱ (t - b_{2})))}], \\ [s_{ϱ^{- 1} (ϱ (c_{1}) + ϱ (c_{2}))}, s_{ϱ^{- 1} (ϱ (d_{1}) + ϱ (d_{2}))}]) \end{matrix}$ $\begin{matrix} α_{1} \otimes_{ℂ} α_{2} = ([s_{ℂ (a_{1}, a_{2})}, s_{ℂ (b_{1}, b_{2})}], [s_{ℂ^{*} (c_{1}, c_{2})}, s_{ℂ^{*} (d_{1}, d_{2})}]) \\ = ([s_{ϱ^{- 1} (ϱ (a_{1}) + ϱ (a_{2}))}, s_{ϱ^{- 1} (ϱ (b_{1}) + ϱ (b_{2}))}], \\ [s_{t - (ϱ^{- 1} (ϱ (t - c_{1}) + ϱ (t - c_{2})))}, s_{t - (ϱ^{- 1} (ϱ (t - d_{1}) + ϱ (t - d_{2})))}]) \end{matrix}$

It is easy to verify that $\oplus_{ℂ}, \otimes_{ℂ}$ satisfy associative law, that is, for all three LIVIFNs A, B, C,

$(A \oplus_{ℂ} B) \oplus_{ℂ} C = A \oplus_{ℂ} (B \oplus_{ℂ} C)$ ;

$(A \otimes_{ℂ} B) \otimes_{ℂ} C = A \otimes_{ℂ} (B \otimes_{ℂ} C)$ .

Theorem 2.2. Let α = ([s_a, s_b] , [s_c, s_d]) be a LIVIFNs, for n ∈ N^*, we have nα is still a LIVIFNs and $\begin{matrix} n α = & ([s_{t - (ϱ^{- 1} (n ϱ (t - a)))}, s_{t - (ϱ^{- 1} (n ϱ (t - b)))}], \\ [s_{ϱ^{- 1} (n ϱ (c))}, s_{ϱ^{- 1} (n ϱ (d))}]) \end{matrix}$ (8) where $n α = {\overset{︷}{α \oplus_{ℂ} α \oplus_{ℂ} \dots \oplus_{ℂ} α}}^{n}$ .

Proof. It is easy to obtain from Theorem 1 that nα is a LIVIFNs. Now we only prove that Eq.(18) holds for n ∈ N^*. when n = 1, $\begin{matrix} 1 α & = ([s_{t - (ϱ^{- 1} (ϱ (t - a)))}, s_{t - (ϱ^{- 1} (ϱ (t - b)))}], \\ [s_{ϱ^{- 1} (ϱ (c))}, s_{ϱ^{- 1} (ϱ (d))}]) \\ = ([s_{a}, s_{b}], [s_{c}, s_{d}]) = α \end{matrix}$ Presume Eq. (8) holds for n = k, i. e., $\begin{matrix} k α & = ([s_{t - (ϱ^{- 1} (k ϱ (t - a)))}, s_{t - (ϱ^{- 1} (k ϱ (t - b)))}], \\ [s_{ϱ^{- 1} (k ϱ (c))}, s_{ϱ^{- 1} (k ϱ (d))}]) \end{matrix}$ When n = k + 1, we have

$\begin{matrix} (k + 1) α = k α \oplus_{ℂ} α \\ = ([s_{t - (ϱ^{- 1} (k ϱ (t - a)))}, s_{t - (ϱ^{- 1} (k ϱ (t - b)))}], \\ [s_{ϱ^{- 1} (k ϱ (c))}, s_{ϱ^{- 1} (k ϱ (d))}]) \oplus_{ℂ} ([s_{a}, s_{b}], [s_{c}, s_{d}]) \\ = ([s_{t - (ϱ^{- 1} (ϱ (t - (t - (ϱ^{- 1} (k ϱ (t - a)))))) + ϱ (t - a))}, \\ s_{t - (ϱ^{- 1} (ϱ (t - (t - (ϱ^{- 1} (k ϱ (t - b)))))) + ϱ (t - b))}], \\ [s_{ϱ^{- 1} (k ϱ (c) + ϱ (c))}, s_{ϱ^{- 1} (k ϱ (d) + ϱ (d))}]) \\ = ([s_{t - (ϱ^{- 1} ((k ϱ (t - a)) + ϱ (t - a)))}, s_{t - (ϱ^{- 1} ((k ϱ (t - b)) + ϱ (t - b)))}], \\ [s_{ϱ^{- 1} (k ϱ (c) + ϱ (c))}, s_{ϱ^{- 1} (k ϱ (d) + ϱ (d))}]) \\ = ([s_{t - (ϱ^{- 1} ((k + 1) ϱ (t - a)))}, s_{t - (ϱ^{- 1} ((k + 1) ϱ (t - b)))}], \\ [s_{ϱ^{- 1} ((k + 1) ϱ (c))}, s_{ϱ^{- 1} ((k + 1) ϱ (d))}]) . \end{matrix}$

So Eq. (8) holds for all n ∈ N^*.□

Similarly, the following theorems can be obtained easily.

Theorem 2.3. Let α = ([s_a, s_b] , [s_c, s_d]) be a LIVIFNs, for all n ∈ N^*, we have αⁿ is still a LIVIFNs and $\begin{matrix} α^{n} = & ([s_{ϱ^{- 1} (n ϱ (a))}, s_{ϱ^{- 1} (n ϱ (b))}], \\ [s_{t - (ϱ^{- 1} (n ϱ (t - c)))}, s_{t - (ϱ^{- 1} (n ϱ (t - d)))}]) \end{matrix}$ (9) where $α^{n} = {\overset{︷}{α \otimes_{ℂ} α \otimes_{ℂ} \dots \otimes_{ℂ} α}}^{n}$ .

Based on Theorem 2.2 and Theorem 2.3, for all λ > 0, we can define the following operations: $\begin{matrix} λ α = & ([s_{t - (ϱ^{- 1} (λ ϱ (t - a)))}, s_{t - (ϱ^{- 1} (λ ϱ (t - b)))}], \\ [s_{ϱ^{- 1} (λ ϱ (c))}, s_{ϱ^{- 1} (λ ϱ (d))}]); \\ α^{λ} = & ([s_{ϱ^{- 1} (λ ϱ (a))}, s_{ϱ^{- 1} (λ ϱ (b))}], \\ [s_{t - (ϱ^{- 1} (λ ϱ (t - c)))}, s_{t - (ϱ^{- 1} (λ ϱ (t - d)))}]) . \end{matrix}$ It is easy to verify that the operational laws hold, for all three LIVIFNs α, α₁, α₂ and λ, λ₁, λ₂ > 0, $\begin{matrix} (3) λ_{1} α \oplus_{ℂ} λ_{2} α = (λ_{1} + λ_{2}) α; \\ (4) α_{1}^{λ} \otimes_{ℂ} α_{2}^{λ} = {(α_{1} \otimes_{ℂ} α_{2})}^{λ}; \\ (5) α_{1}^{λ} \otimes_{ℂ} α_{2}^{λ} = α^{λ_{1} + λ_{2}} . \end{matrix}$

Theorem 2.4. Let α₁ = ([s_a1, s_b1] , [s_c1, s_d1]) and α₂ = ([s_a2, s_b2] , [s_c2, s_d2]) be two LIVIFNs, then for all λ > 0, $α_{1} \oplus_{ℂ} α_{2}$ , $α_{1} \otimes_{ℂ} α_{2}$ , λα₁, $α_{1}^{λ}$ are all LIVIFNs.

3 The LIVIFCWA aggregation operators

In this part, we will give a family of linguistic interval-valued intuitionistic fuzzy copula weighted averaging (LIVIFCWA) operator by combining the LIVIFNs, ECs and ECCs introduced in Section 2.

3.1 Some discussions on LIVIFCWA

Definition 3.1. Assume α_t is family of LIVIFNs. The LIVIFCWA operator is a mapping from Ωⁿ to Ω defined by

$LIVIFCWA (α_{1}, α_{1}, \dots, α_{n}) = \oplus_{i = 1}^{n} w_{i} α_{i}$ (10) where w_i be the weight vector (WV) of α_i (i = 1, 2, ⋯ , n), and 0 ≤ w_k ≤ 1, $\sum_{k = 1}^{n} w_{k} = 1$ .

Theorem 3.1. Let α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) is family of LIVAIFNs, then the aggregated value based on the LIVIFCWA operator is also an LIVAIFN, and

$LIVIFCWA (α_{1}, α_{1}, \dots, α_{n})$ (11) $\begin{matrix} = ([s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - a_{i})))}, s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - b_{i})))}], \\ [s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (c_{i}))}, s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (d_{i}))}]) \end{matrix}$

Proof. We apply mathematical induction to prove Theorem 3.1.

(1)When n = 1, Theorem 3.1 is obvious.

(2)Presume Theorem 3.1 holds when n = l,i.e,

$\begin{matrix} LIVIFCWA (α_{1}, α_{2}, \dots, α_{l}) \\ = ([s_{t - (ϱ^{- 1} (\sum_{i = 1}^{l} w_{i} ϱ (t - a_{i})))}, s_{t - (ϱ^{- 1} (\sum_{i = 1}^{l} w_{i} ϱ (t - b_{i})))}], \\ [s_{ϱ^{- 1} (\sum_{i = 1}^{l} w_{i} ϱ (c_{i}))}, s_{ϱ^{- 1} (\sum_{i = 1}^{l} w_{i} ϱ (d_{i}))}]) . \end{matrix}$

So, when n = l + 1, we have

$\begin{matrix} LIVIFCWA (a_{1}, a_{2}, \dots, a_{l + 1}) \\ = {LIVIFCAA}_{w} (a_{1}, a_{2}, \dots, a_{l}) \oplus_{ℂ} w_{l + 1} α_{l + 1} \\ = ([s_{t - (ϱ^{- 1} (\sum_{i = 1}^{l} w_{i} ϱ (t - a_{i})))}, s_{t - (ϱ^{- 1} (\sum_{i = 1}^{l} w_{i} ϱ (t - b_{i})))}], \\ [s_{ϱ^{- 1} (\sum_{i = 1}^{l} w_{i} ϱ (c_{i}))}, s_{ϱ^{- 1} (\sum_{i = 1}^{l} w_{i} ϱ (d_{i}))}]) \\ \oplus_{ℂ} ([s_{t - (ϱ^{- 1} (w_{l + 1} ϱ (t - a_{l + 1})))}, s_{t - (ϱ^{- 1} (w_{l + 1} ϱ (t - b_{l + 1})))}], \\ [s_{ϱ^{- 1} (w_{l + 1} ϱ (c_{l + 1}))}, s_{ϱ^{- 1} (w_{l + 1} ϱ (d_{l + 1}))}]), \\ = ([s_{t - (ϱ^{- 1} (\sum_{i = 1}^{l + 1} w_{i} ϱ (t - a_{i})))}, s_{t - (ϱ^{- 1} (\sum_{i = 1}^{l + 1} w_{i} ϱ (t - b_{i})))}], \\ [s_{ϱ^{- 1} (\sum_{i = 1}^{l + 1} w_{i} ϱ (c_{i}))}, s_{ϱ^{- 1} (\sum_{i = 1}^{l + 1} w_{i} ϱ (d_{i}))}]) \end{matrix}$

Thus, theorem 3.1 holds for all n ∈ N^*.

In what follows, we will give some properties of LIVIFCWA operators.

Theorem 3.2. Let α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) be a collection of LIVIFNs, then

$\begin{matrix} LIVIFCWA (λ α_{1}, λ α_{1}, \dots, λ α_{n}) \\ = λ \cdot LIVIFCWA (α_{1}, α_{1}, \dots, α_{n}) \end{matrix}$ (12)

Proof. According to Theorems 2.1 and 2.2, we have $\begin{matrix} λ α_{i} = & ([s_{t - (ϱ^{- 1} (λ ϱ (t - a_{i})))}, s_{t - (ϱ^{- 1} (λ ϱ (t - b_{i})))}], \\ [s_{ϱ^{- 1} (λ ϱ (c_{i}))}, s_{ϱ^{- 1} (λ ϱ (d_{i}))}]) \end{matrix}$ therefore,

LIVIFCWA (λα₁, λα₁, …, λα_n) = ([s_a, s_b] , [s_c, s_d]) where

$a = t - (ϱ^{- 1} (λ \sum_{i = 1}^{n} w_{i} ϱ (t - a_{i})))$ ;

$b = t - (ϱ^{- 1} (λ \sum_{i = 1}^{n} w_{i} ϱ (t - b_{i})))$ ;

$c = ϱ^{- 1} (λ \sum_{i = 1}^{n} w_{i} ϱ (c_{i})); d = ϱ^{- 1} (λ \sum_{i = 1}^{n} w_{i} ϱ (d_{i}))$ . Furthermore, $\begin{matrix} λ \cdot & LIVIFCWA (α_{1}, α_{1}, \dots, α_{n}) \\ = λ ([s_{u_{L}}, s_{u_{U}}], [s_{v_{L}}, s_{v_{U}}]) \\ = ([s_{μ_{L}}, s_{μ_{U}}], [s_{ν_{L}}, s_{ν_{U}}]) \end{matrix}$ where

$u_{L} = t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - a_{i})))$ ;

$u_{U} = t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - b_{i})))$ ;

$v_{L} = ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (c_{i})); v_{U} = ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (d_{i}));$

$μ_{L} = t - (ϱ^{- 1} (λ \sum_{i = 1}^{n} w_{i} ϱ (t - a_{i})))$ ;

$μ_{U} = t - (ϱ^{- 1} (λ \sum_{i = 1}^{n} w_{i} ϱ (t - b_{i})))$ ;

$ν_{L} = ϱ^{- 1} (λ \sum_{i = 1}^{n} w_{i} ϱ (c_{i})); ν_{U} = ϱ^{- 1} (λ \sum_{i = 1}^{n} w_{i} ϱ (d_{i}))$ . □

Theorem 3.3. Let α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) be a collection of LIVIFNs, then

$\begin{matrix} LIVIFCWA (α_{1} \oplus_{ℂ} α, α_{2} \oplus_{ℂ} α, \dots, α_{n} \oplus_{ℂ} α) \\ = LIVIFCWA (α_{1}, α_{1}, \dots, α_{n}) \oplus_{ℂ} α \end{matrix}$ (13)

Proof. $\begin{matrix} α_{i} \oplus_{ℂ} α = & ([s_{t - (ϱ^{- 1} (ϱ (t - a_{i}) + ϱ (t - a)))}, \\ s_{t - (ϱ^{- 1} (ϱ (t - b_{i}) + ϱ (t - b)))}], \\ [s_{ϱ^{- 1} (ϱ (c_{i}) + ϱ (c))}, s_{ϱ^{- 1} (ϱ (d_{i}) + ϱ (d))}]) \end{matrix}$

$\begin{matrix} LIVIFCWA (α_{1} \oplus_{ℂ} α, & α_{2} \oplus_{ℂ} α, \dots, α_{n} \oplus_{ℂ} α \\ = ([s_{a}, s_{b}], [s_{c}, s_{d}]) \end{matrix}$ where

$a = t - (ϱ^{- 1} (λ \sum_{i = 1}^{n} w_{i} ϱ (t - a_{i}) + ϱ (t - a)))$ ;

$c = ϱ^{- 1} (λ \sum_{i = 1}^{n} w_{i} ϱ (c_{i}) + ϱ (c))$ ;

$b = t - (ϱ^{- 1} (λ \sum_{i = 1}^{n} w_{i} ϱ (t - b_{i}) + ϱ (t - b)))$ ;

$d = ϱ^{- 1} (λ \sum_{i = 1}^{n} w_{i} ϱ (d_{i}) + ϱ (d))$ .

and

$\begin{matrix} LIVIFCWA (α_{1}, α_{1}, \dots, α_{n}) \oplus_{ℂ} α \\ = ([s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - a_{i})))}, s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - b_{i})))}], \\ [s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (c_{i}))}, s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (d_{i}))}]) \\ \oplus_{ℂ} ([s_{a}, s_{b}], [s_{c}, s_{d}]) \\ = ([s_{t - (ϱ^{- 1} (ϱ (t - (t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - a)))))) + ϱ (t - a))}, \\ s_{t - (ϱ^{- 1} (ϱ (t - (t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - b)))))) + ϱ (t - b))}], \\ [s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (c) + ϱ (c))}, s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (d) + ϱ (d))}]) \\ = ([s_{t - (ϱ^{- 1} ((\sum_{i = 1}^{n} w_{i} ϱ (t - a)) + ϱ (t - a)))}, \\ s_{t - (ϱ^{- 1} ((\sum_{i = 1}^{n} w_{i} ϱ (t - b)) + ϱ (t - b)))}], \\ [s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (c) + ϱ (c))}, s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (d) + ϱ (d))}]) \end{matrix}$

Therefore, we can see that Theorem 3.3 is held.□

Proposition 3.1. Let α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) be a collection of LIVIFNs, then

$\begin{matrix} LIVIFCWA (λ α_{1} \oplus_{ℂ} α, λ α_{2} \oplus_{ℂ} α, \dots, λ α_{n} \oplus_{ℂ} α) \\ = λ {LIVIFCAA}_{w} (α_{1}, α_{1}, \dots, α_{n}) \oplus_{ℂ} α \end{matrix}$ (14)

Theorem 3.4. Let α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) be a collection of LIVIFNs, if α_i = α = ([s_a, s_b] , [s_c, s_d])

$LIVIFCWA (α_{1}, α_{2}, \dots, α_{n}) = α$ (15)

Proof. If α_i = α = ([s_a, s_b] , [s_c, s_d]) for i = 1, ⋯ , n, then

$\begin{matrix} LIVIFCWA (α_{1}, \dots, α_{n}) \\ = & ([s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - a)))}, s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - b)))}], \\ [s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (c))}, s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (d))}]) \\ = & ([s_{t - (ϱ^{- 1} (ϱ (t - a)))}, s_{t - (ϱ^{- 1} (ϱ (t - b)))}], \\ [s_{ϱ^{- 1} (ϱ (c))}, s_{ϱ^{- 1} (ϱ (d))}]) \\ = & ([s_{a}, s_{b}], [s_{c}, s_{d}]) \\ = & α . \end{matrix}$

□

Theorem 3.5. Let α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) and β_i = ([s_{τ
_i}, s_{θ
_i}] , [s_{η
_i}, s_{ν
_i}]) be a collection of LIVIFNs, if a_i ≤ τ_i, b_i ≤ θ_i, c_i ≥ η_i, d_i ≥ ν_i for all i, then

$\begin{matrix} LIVIFCWA & (α_{1}, \dots, α_{n}) \\ \leq LIVIFCWA (β_{1}, \dots, β_{n}) \end{matrix}$ (16)

Proof. On the one hand, since a_i ≤ τ_i, b_i ≤ θ_i, c_i ≥ η_i, d_i ≥ ν_i for all i, we have t - a_i ≥ t - τ_i, t - b_i ≥ t - θ_i. As ϱ and ϱ^-1 are monotonicity decreasing, thus ϱ (t - a_i) ≤ ϱ (t - τ_i), and ϱ (t - b_i) ≤ ϱ (t - θ_i), furthermore, $\begin{matrix} (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - a_{i}))) \geq (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - τ_{i}))) \\ (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - b_{i}))) \geq (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - θ_{i}))) \end{matrix}$ And so

$\begin{matrix} t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - a_{i}))) \leq t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - τ_{i}))) \\ t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - b_{i}))) \leq t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - θ_{i}))) \end{matrix}$

On the other hand, as c_i ≥ η_i, d_i ≥ ν_i, we have

$\begin{matrix} ϱ (c_{i}) \leq ϱ (η_{i}), ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (c_{i})) \geq ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (η_{i})) \end{matrix}$

similarly, we have $\begin{matrix} ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (d_{i})) \geq ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (ν_{i})) \end{matrix}$ Therefore, Theorem 3.5 is kept.□

Theorem 3.6. Let α^- = ([s_{a
^-}, s_{b
^-}] , [s_{c
^-}, s_{d
^-}]) and α⁺ = ([s_{a
⁺}, s_{b
⁺}] , [s_{c
⁺}, s_{d
⁺}]), where $a^{-} = min_{i} {a_{i}}, b^{-} = min_{i} {b_{i}}, c^{-} = max_{i} {c_{i}}, d^{-} = max_{i} {d_{i}}, a^{+} = max_{i} {a_{i}}, b^{+} = max_{i} {b_{i}}, c^{+} = min_{i} {c_{i}}, d^{+} = min_{i} {d_{i}}$ , then $\begin{matrix} α^{-} \leq LIVIFCWA (α_{1}, \dots, α_{n}) \leq α^{+} \end{matrix}$

Proof. According to Theorem 3.6, we can acquire LIVIFCWA (α^-, ⋯ , α^-) ≤ LIVIFCWA (α₁, ⋯ , α_n) ≤ LIVIFCWA (α⁺, ⋯ , α⁺)

According to Theorem 3.4, we have, LIVIFCWA (α^-, ⋯ , α^-) = α^- and LIVIFCWA (α⁺, ⋯ , α⁺) = α + Accordingly, α^- ≤ LIVIFCWA (α₁, ⋯ , α_n) ≤ α⁺

Theorem 3.7. Let α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) and β_i = ([s_{τ
_i}, s_{θ
_i}] , [s_{η
_i}, s_{ν
_i}]) be a collection of LIVIFNs, and 0 ≤ w_k ≤ 1, $\sum_{k = 1}^{n} w_{k} = 1$ , then

$\begin{matrix} LIVIFCWA (α_{1} \oplus_{ℂ} β_{1}, \dots, α_{n} \oplus_{ℂ} β_{n}) \\ = LIVIFCWA (α_{1}, \dots, α_{n}) \oplus_{ℂ} LIVIFCWA (β_{1}, \dots, β_{n}) \end{matrix}$

Proof. Since $\begin{matrix} α_{i} \oplus_{ℂ} β_{i} = & ([s_{t - (ϱ^{- 1} (ϱ (t - a_{i}) + ϱ (t - τ_{i})))}, \\ s_{t - (ϱ^{- 1} (ϱ (t - b_{i}) + ϱ (t - θ_{i})))}], \\ [s_{ϱ^{- 1} (ϱ (c_{i}) + ϱ (η_{i}))}, s_{ϱ^{- 1} (ϱ (d_{i}) + ϱ (ν_{i}))}]) \end{matrix}$ then we have,

$\begin{matrix} LIVIFCWA (α_{1} \oplus_{ℂ} β_{1}, \dots, α_{n} \oplus_{ℂ} β_{n}) \\ = ([s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ ((ϱ^{- 1} (ϱ (t - a_{i}) + ϱ (t - τ_{i}))))))}, \\ s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ ((ϱ^{- 1} (ϱ (t - b_{i}) + ϱ (t - θ_{i}))))))}], \\ [s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (ϱ^{- 1} (ϱ (c_{i}) + ϱ (η_{i}))))}, \\ s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (ϱ^{- 1} (ϱ (d_{i}) + ϱ (ν_{i}))))}]) \\ = ([s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} (ϱ (t - a_{i}) + ϱ (t - τ_{i}))))}, \\ s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} (ϱ (t - b_{i}) + ϱ (t - θ_{i}))))}], \\ [s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ((ϱ (c_{i}) + ϱ (η_{i})))}, s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ((ϱ (d_{i}) + ϱ (ν_{i})))}]) \end{matrix}$

Besides,

$\begin{matrix} LIVIFCWA (α_{1}, \dots, α_{n}) \oplus_{ℂ} LIVIFCWA (β_{1}, \dots, β_{n}) \\ = & ([s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - a_{i})))}, s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - b_{i})))}], \\ [s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (c_{i}))}, s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (d_{i}))}]) \\ \oplus_{ℂ} ([s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - τ_{i})))}, s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - θ_{i})))}], \\ [s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (η_{i}))}, s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (ν_{i}))}]) \\ = & ([s_{t - (ϱ^{- 1} (ϱ (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - a_{i}))) + ϱ (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - τ_{i})))))}, \\ s_{t - (ϱ^{- 1} (ϱ (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - b_{i}))) + ϱ (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (t - θ_{i})))))}], \\ [s_{ϱ^{- 1} (ϱ (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (c_{i}))) + ϱ (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (η_{i}))))}, \\ s_{ϱ^{- 1} (ϱ (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (d_{i}))) + ϱ (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ϱ (ν_{i}))))}]) \\ = & ([s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} (ϱ (t - a_{i}) + ϱ (t - τ_{i}))))}, \\ s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} (ϱ (t - b_{i}) + ϱ (t - θ_{i}))))}], \\ [s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ((ϱ (c_{i}) + ϱ (η_{i})))}, s_{ϱ^{- 1} (\sum_{i = 1}^{n} w_{i} ((ϱ (d_{i}) + ϱ (ν_{i})))}]) \end{matrix}$

Thus, Theorem 3.7 is kept.□

In this subsection, we will discussions on LIVIFCWA, and some special cases of LIVIFCWA will be given when the the generators of ECs take different form.

3.2 Some different types of LIVIFCWA

Let α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]).

(Case 3-1) When the generator $ϱ (c) = {(- \ln (\frac{c}{t}))}^{θ}$ , where $ϱ^{- 1} (c) = {te}^{- c^{\frac{1}{θ}}}$ and θ ≥ 1. The LIVIFCWA reduces to the following: $\begin{matrix} {LIVIFCWA}_{G} (α_{1}, \dots, α_{n}) = ([s_{a}, s_{b}], [s_{c}, s_{d}]) . \end{matrix}$ where

$a = t - {te}^{- {(\sum_{k = 1}^{n} w_{k} {(- \ln (\frac{t - a_{k}}{t}))}^{θ})}^{\frac{1}{θ}}}$ ;

$b = t - {te}^{- {(\sum_{k = 1}^{n} w_{k} {(- \ln (\frac{t - a_{k}}{t}))}^{θ})}^{\frac{1}{θ}}}$ ;

$c = {te}^{- {(\sum_{k = 1}^{n} w_{k} {(- \ln (\frac{c_{k}}{t}))}^{θ})}^{\frac{1}{θ}}}$ ;

$d = {te}^{- {(\sum_{k = 1}^{n} w_{k} {(- \ln (\frac{d_{k}}{t}))}^{θ})}^{\frac{1}{θ}}}$ .

(Case 3-2) When the generator $ϱ (c) = {(\frac{c}{t})}^{- θ} - 1$ , where $ϱ^{- 1} (c) = t (c + 1)^{- \frac{1}{θ}}$ , where θ ≥ -1 and θ ≠ 0. The LIVIFCWA reduces to the following: $\begin{matrix} {LIVIFCWA}_{C} (α_{1}, \dots, α_{n}) \\ = ([s_{a}, s_{b}], [s_{c}, s_{d}]) . \end{matrix}$ where

$a = t - t {(\sum_{k = 1}^{n} w_{k} ((\frac{t - a_{k}}{t})^{- θ} - 1) + 1)}^{- \frac{1}{θ}}$ ;

$b = t - t {(\sum_{k = 1}^{n} w_{k} ((\frac{t - b_{k}}{t})^{- θ} - 1) + 1)}^{- \frac{1}{θ}}$ ;

$c = t {(\sum_{k = 1}^{n} (w_{k} (\frac{c_{k}}{t})^{- θ} - 1) + 1)}^{- \frac{1}{θ}}$ ;

$d = t {(\sum_{k = 1}^{n} (w_{k} (\frac{d_{k}}{t})^{- θ} - 1) + 1)}^{- \frac{1}{θ}}$ .

(Case 3-3) When the generator $ϱ (c) = \ln (\frac{e^{- \frac{θ c}{t}} - 1}{e^{- θ} - 1})$ ,

where $ϱ^{- 1} (c) = (- \frac{t}{θ}) \ln (e^{c} (e^{- θ} - 1) + 1)$ and θ ≠ 0. The LIVIFCWA reduces to the following: $\begin{matrix} {LIVIFCWA}_{F} (α_{1}, α_{2}, \dots, α_{n}) = ([s_{a}, s_{b}], [s_{c}, s_{d}]) . \end{matrix}$ where

$a = t + \frac{t}{θ} \ln (\prod_{k = 1}^{n} {(\frac{e^{- θ (\frac{t - a_{k}}{t})} - 1}{e^{- θ} - 1})}^{w_{k}} (e^{- θ} - 1) + 1)$ ;

$b = t + \frac{t}{θ} \ln (\prod_{k = 1}^{n} {(\frac{e^{- θ (\frac{t - b_{k}}{t})} - 1}{e^{- θ} - 1})}^{w_{k}} (e^{- θ} - 1) + 1)$ ;

$c = - \frac{t}{θ} \ln (\prod_{k = 1}^{n} {(\frac{e^{- θ (\frac{c_{k}}{t})} - 1}{e^{- θ} - 1})}^{w_{k}} (e^{- θ} - 1) + 1)$ ;

$d = - \frac{t}{θ} \ln (\prod_{k = 1}^{n} {(\frac{e^{- θ (\frac{d_{k}}{t})} - 1}{e^{- θ} - 1})}^{w_{k}} (e^{- θ} - 1) + 1)$ .

(Case 3-4) When the generator $ϱ (c) = \ln (\frac{t - θ (t - c)}{c})$ , where $ϱ^{- 1} (c) = \frac{t (1 - θ)}{e^{c} - θ}$ , θ ∈ [-1, 1). The LIVIFCWA reduces to the following:

$\begin{matrix} {LIVIFCWA}_{AHM} (α_{1}, α_{2}, \dots, α_{n}) = ([s_{a}, s_{b}], [s_{c}, s_{d}]) . \end{matrix}$

where

$a = \frac{t [\prod_{k = 1}^{n} {((t - θ a_{k}))}^{w_{k}} - \prod_{k = 1}^{n} {(t - a_{k})}^{w_{k}}]}{\prod_{k = 1}^{n} {((t - θ a_{k}))}^{w_{k}} - θ \prod_{k = 1}^{n} {((t - a_{k}))}^{w_{k}}}$ ;

$b = \frac{t [\prod_{k = 1}^{n} {((t - θ b_{k}))}^{w_{k}} - \prod_{k = 1}^{n} {(t - b_{k})}^{w_{k}}]}{\prod_{k = 1}^{n} {((t - θ b_{k}))}^{w_{k}} - θ \prod_{k = 1}^{n} {((t - b_{k}))}^{w_{k}}}$ ;

$c = \frac{(1 - θ) t \prod_{k = 1}^{n} {(c_{k})}^{w_{k}}}{\prod_{k = 1}^{n} {((t - θ (t - c_{k})))}^{w_{k}} - θ \prod_{k = 1}^{n} {(c_{k})}^{w_{k}}}$ ;

$d = \frac{(1 - θ) t \prod_{k = 1}^{n} {(d_{k})}^{w_{k}}}{\prod_{k = 1}^{n} {((t - θ (t - d_{k})))}^{w_{k}} - θ \prod_{k = 1}^{n} {(d_{k})}^{w_{k}}}$ .

(Case 3-5) When generator $ϱ (c) = - \ln (1 - {(1 - \frac{c}{t})}^{θ})$ , where $ϱ^{- 1} (c) = t - t (1 - e^{- c})^{\frac{1}{θ}}$ , where θ ≥ 1. The LIVIFCWA reduces to the following: $\begin{matrix} {LIVIFCWA}_{J} (α_{1}, α_{2}, \dots, α_{n}) = ([s_{a}, s_{b}], [s_{c}, s_{d}]) . \end{matrix}$ where

$a = t - t {(1 - \prod_{k = 1}^{n} {(1 - {(\frac{a_{k}}{t})}^{θ})}^{w_{k}})}^{\frac{1}{θ}}$ ;

$b = t - t {(1 - \prod_{k = 1}^{n} {(1 - {(\frac{b_{k}}{t})}^{θ})}^{w_{k}})}^{\frac{1}{θ}}$ ;

$c = t {(1 - \prod_{k = 1}^{n} {(1 - {(1 - \frac{c_{k}}{t})}^{θ})}^{w_{k}})}^{\frac{1}{θ}}$ ;

$d = t {(1 - \prod_{k = 1}^{n} {(1 - {(1 - \frac{d_{k}}{t})}^{θ})}^{w_{k}})}^{\frac{1}{θ}}$ .

4 The linguistic interval-valued intuitionistic fuzzy copula weighted power averaging operators

4.1 The PA operator

Definition 4.1. Let α_t is family of LIVIFNs. The LIVIFCWPA operator is a mapping from Ωⁿ to Ω defined by $LIVIFCWPA (α_{1}, α_{1}, \dots, α_{n}) = \oplus_{i = 1}^{n} γ_{i} α_{i}$ (17) where $γ_{i} = \frac{ω_{i} (1 + T (α_{i}))}{\sum_{i = 1}^{n} ω_{i} (1 + T (α_{i})}$ , ω_i be the weight vector of α_i (i = 1, 2, ⋯ , n), and 0 ≤ ω_i ≤ 1, $\sum_{i = 1}^{n} ω_{k} = 1$ , $T (α_{j}) = \sum_{i = 1, i \neq j}^{n} Sup (α_{j}, α_{i})$ .

Theorem 4.1. Let α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) is family of LIVIFNs, then the aggregated value based on the LIVIFCWPA operator is also an LIVIFN, and

$LIVIFCWPA (α_{1}, α_{1}, \dots, α_{n}) \oplus_{i = 1}^{n} γ_{i} α_{i}$ (18) $\begin{matrix} = ([s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} γ_{i} ϱ (t - a_{i})))}, s_{t - (ϱ^{- 1} (\sum_{i = 1}^{n} γ_{i} ϱ (t - b_{i})))}], \\ [s_{ϱ^{- 1} (\sum_{i = 1}^{n} γ_{i} ϱ (c_{i}))}, s_{ϱ^{- 1} (\sum_{i = 1}^{n} γ_{i} ϱ (d_{i}))}]) \end{matrix}$ The proof is similar to Theorem 3.1, so we omitted it here. Obviously, the presented LIVIFCWPA operators still possess the theorems proposed in Theorem 3.2-3.7.

4.2 Generalized weighted PA operator

Definition 4.2. Assume α_t is family of LIVIFNs. The GLIVIFCWPA operator is a mapping from Ωⁿ to Ω defined by $GLIVIFCWPA (α_{1}, \dots, α_{n}) = (\oplus_{i = 1}^{n} γ_{i} α_{i}^{q})^{\frac{1}{q}}$ (19) where γ_i, ω_i, T (α_i) is the same as in definition 4.1.

Theorem 4.2. Assume α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) is family of LIVAIFNs, then the aggregated value based on the GLIVIFCWPA operator is also an LIVIFN, and $\begin{matrix} GLIVIFCWPA (α_{1}, & \dots, α_{n}) (\oplus_{i = 1}^{n} γ_{i} α_{i}^{q})^{\frac{1}{q}} \\ = ([s_{a}, s_{b}], [s_{c}, s_{d}]) \end{matrix}$ (20)

$\begin{matrix} a = ϱ^{- 1} (\frac{1}{q} ϱ (t - ϱ^{- 1} (\sum_{i = 1}^{n} γ_{i} ϱ (t - ϱ^{- 1} (q ϱ (a_{i})))))); \\ b = ϱ^{- 1} (\frac{1}{q} ϱ (t - ϱ^{- 1} (\sum_{i = 1}^{n} γ_{i} ϱ (t - ϱ^{- 1} (q ϱ (b_{i})))))); \\ c = t - ϱ^{- 1} (\frac{1}{q} ϱ (t - ϱ^{- 1} (\sum_{i = 1}^{n} γ_{i} ϱ (t - ϱ^{- 1} (q ϱ (t - c_{i})))))); \\ d = t - ϱ^{- 1} (\frac{1}{q} ϱ (t - ϱ^{- 1} (\sum_{i = 1}^{n} γ_{i} ϱ (t - ϱ^{- 1} (q ϱ (t - d_{i})))))) . \end{matrix}$

Proof. Since $α_{i} \oplus_{ℂ} β_{i} = ([s_{t - (ρ^{- 1} (ρ (t - a_{i}) + ρ (t - τ_{i})))}, s_{t - (ρ^{- 1} (ρ (t - b_{i}) + ρ (t - θ_{i})))}], [s_{ρ^{- 1} (ρ (c_{i}) + ρ (η_{i}))}, s_{ρ^{- 1} (ρ (d_{i}) + ρ (ν_{i}))}])$ $\begin{matrix} γ_{i} α_{i}^{q} = & ([s_{t - ϱ^{- 1} (γ_{i} ϱ (t - ϱ^{- 1} (q ϱ (a_{i}))))}, \\ s_{t - ϱ^{- 1} (γ_{i} ϱ (t - ϱ^{- 1} (q ϱ (b_{i}))))}], \\ [s_{ϱ^{- 1} (γ_{i} ϱ (t - ϱ^{- 1} (q ϱ (t - c_{i}))))}, \\ s_{ϱ^{- 1} (γ_{i} ϱ (t - ϱ^{- 1} (q ϱ (t - d_{i}))))}]) \end{matrix}$ $\begin{matrix} \oplus_{i = 1}^{n} γ_{i} α_{i}^{q} = & ([s_{t - ϱ^{- 1} (\sum_{i = 1}^{n} γ_{i} ϱ (t - ϱ^{- 1} (q ϱ (a_{i}))))}, \\ s_{t - ϱ^{- 1} (\sum_{i = 1}^{n} γ_{i} ϱ (t - ϱ^{- 1} (q ϱ (b_{i}))))}], \\ [s_{ϱ^{- 1} (\sum_{i = 1}^{n} γ_{i} ϱ (t - ϱ^{- 1} (q ϱ (t - c_{i}))))}, \\ s_{ϱ^{- 1} (\sum_{i = 1}^{n} γ_{i} ϱ (t - ϱ^{- 1} (q ϱ (t - d_{i}))))}]) \end{matrix}$ Then, we have

$\begin{matrix} (\oplus_{i = 1}^{n} γ_{i} α_{i}^{q})^{\frac{1}{q}} = ([s_{a}, s_{b}], [s_{c}, s_{d}]) \\ a = ϱ^{- 1} (\frac{1}{q} ϱ (t - ϱ^{- 1} (\sum_{i = 1}^{n} γ_{i} ϱ (t - ϱ^{- 1} (q ϱ (a_{i})))))); \\ b = ϱ^{- 1} (\frac{1}{q} ϱ (t - ϱ^{- 1} (\sum_{i = 1}^{n} γ_{i} ϱ (t - ϱ^{- 1} (q ϱ (b_{i})))))); \\ c = t - ϱ^{- 1} (\frac{1}{q} ϱ (t - ϱ^{- 1} (\sum_{i = 1}^{n} γ_{i} ϱ (t - ϱ^{- 1} (q ϱ (t - c_{i})))))); \\ d = t - ϱ^{- 1} (\frac{1}{q} ϱ (t - ϱ^{- 1} (\sum_{i = 1}^{n} γ_{i} ϱ (t - ϱ^{- 1} (q ϱ (t - d_{i})))))) . \end{matrix}$

(Case 4-1) When the generator $ϱ (c) = {(- \ln (\frac{c}{t}))}^{θ}$ , where $ϱ^{- 1} (c) = {te}^{- c^{\frac{1}{θ}}}$ and θ ≥ 1. The GLIVIFCWPA reduces to the following: $\begin{matrix} {GLIVIFCWPA}_{G} (α_{1}, \dots, α_{n}) = ([s_{a}, s_{b}], [s_{c}, s_{d}]) . \end{matrix}$

where $a = t (\frac{η_{a}}{t})^{{\frac{1}{q}}^{\frac{1}{θ}}}; η_{a} = t - {te}^{- {(\sum_{k = 1}^{n} w_{k} {(- \ln (\frac{t - ζ}{t}))}^{θ})}^{\frac{1}{θ}}}$ ;

$b = t (\frac{η_{b}}{t})^{{\frac{1}{q}}^{\frac{1}{θ}}}; η_{b} = t - {te}^{- {(\sum_{k = 1}^{n} w_{k} {(- \ln (\frac{t - ζ}{t}))}^{θ})}^{\frac{1}{θ}}}$ ;

$c = t - t (\frac{t - η_{c}}{t})^{{\frac{1}{q}}^{\frac{1}{θ}}}; η_{c} = {te}^{- {(\sum_{k = 1}^{n} w_{k} {(- \ln (\frac{ζ_{c}}{t}))}^{θ})}^{\frac{1}{θ}}}$ ;

$d = t - t (\frac{t - η_{d}}{t})^{{\frac{1}{q}}^{\frac{1}{θ}}}; η_{d} = {te}^{- {(\sum_{k = 1}^{n} w_{k} {(- \ln (\frac{ζ_{d}}{t}))}^{θ})}^{\frac{1}{θ}}}$ ;

$ζ_{a} = t (\frac{a_{k}}{t})^{q^{\frac{1}{θ}}}; ζ_{b} = t (\frac{b_{k}}{t})^{q^{\frac{1}{θ}}};$

$ζ_{c} = t - t (\frac{t - c_{k}}{t})^{q^{\frac{1}{θ}}}; ζ_{d} = t - t (\frac{t - d_{k}}{t})^{q^{\frac{1}{θ}}}$ ;

(Case 4-2) When the generator $ϱ (c) = {(\frac{c}{t})}^{- θ} - 1$ , where $ϱ^{- 1} (c) = t (c + 1)^{- \frac{1}{θ}}$ , where θ ≥ -1 and θ ≠ 0. The GLIVIFCWPA reduces to the following: $\begin{matrix} {GLIVIFCWPA}_{C} (α_{1}, \dots, α_{n}) = ([s_{a}, s_{b}], [s_{c}, s_{d}]) . \end{matrix}$ where $a = t {(\frac{1}{q} ((\frac{η_{a}}{t})^{- θ} - 1) + 1)}^{- \frac{1}{θ}}$ ;

$b = t {(\frac{1}{q} ((\frac{η_{b}}{t})^{- θ} - 1) + 1)}^{- \frac{1}{θ}}$ ;

$c = t - t {(\frac{1}{q} ((\frac{t - η_{c}}{t})^{- θ} - 1) + 1)}^{- \frac{1}{θ}}$ ;

$d = t - t {(\frac{1}{q} ((\frac{t - η_{d}}{t})^{- θ} - 1) + 1)}^{- \frac{1}{θ}}$ ;

$η_{a} = t - t {(\sum_{k = 1}^{n} w_{k} ((\frac{t - ζ_{a}}{t})^{- θ} - 1) + 1)}^{- \frac{1}{θ}}$ ;

$ζ_{a} = t {(q ((\frac{a_{k}}{t})^{- θ} - 1) + 1)}^{- \frac{1}{θ}}$ ;

$η_{b} = t - t {(\sum_{k = 1}^{n} w_{k} ((\frac{t - ζ_{b}}{t})^{- θ} - 1) + 1)}^{- \frac{1}{θ}}$ ;

$ζ_{b} = t {(q ((\frac{b_{k}}{t})^{- θ} - 1) + 1)}^{- \frac{1}{θ}}$ ;

$η_{c} = t {(\sum_{k = 1}^{n} (w_{k} (\frac{{zeta}_{c}}{t})^{- θ} - 1) + 1)}^{- \frac{1}{θ}}$ ;

$ζ_{c} = t - t {(q ((\frac{t - c_{k}}{t})^{- θ} - 1) + 1)}^{- \frac{1}{θ}}$ ;

$η_{d} = t {(\sum_{k = 1}^{n} (w_{k} (\frac{{zeta}_{d}}{t})^{- θ} - 1) + 1)}^{- \frac{1}{θ}};$

$ζ_{d} = t - t {(q ((\frac{t - d_{k}}{t})^{- θ} - 1) + 1)}^{- \frac{1}{θ}}$ .

(Case 4-3) When the generator $ϱ (c) = \ln (\frac{e^{- \frac{θ c}{t}} - 1}{e^{- θ} - 1})$ ,

where $ϱ^{- 1} (c) = (- \frac{t}{θ}) \ln (e^{c} (e^{- θ} - 1) + 1)$ and θ ≠ 0. The GLIVIFCWPA reduces to the following: $\begin{matrix} {GLIVIFCWPA}_{F} (α_{1}, \dots, α_{n}) = ([s_{a}, s_{b}], [s_{c}, s_{d}]) . \end{matrix}$ where { $a = (- \frac{t}{θ}) \ln ({(\frac{e^{- \frac{θ η_{a}}{t}} - 1}{e^{- θ} - 1})}^{\frac{1}{q}} (e^{- θ} - 1) + 1)$ ;

$c = t - (- \frac{t}{θ}) \ln ({(\frac{e^{- \frac{θ (t - η_{c})}{t}} - 1}{e^{- θ} - 1})}^{\frac{1}{q}} (e^{- θ} - 1) + 1)$ ;

$b = (- \frac{t}{θ}) \ln ({(\frac{e^{- \frac{θ η_{b}}{t}} - 1}{e^{- θ} - 1})}^{\frac{1}{q}} (e^{- θ} - 1) + 1)$ ;

$d = t - (- \frac{t}{θ}) \ln ({(\frac{e^{- \frac{θ (t - η_{d})}{t}} - 1}{e^{- θ} - 1})}^{\frac{1}{q}} (e^{- θ} - 1) + 1)$ ;

$ζ_{a} = (- \frac{t}{θ}) \ln ({(\frac{e^{- \frac{θ a_{k}}{t}} - 1}{e^{- θ} - 1})}^{q} (e^{- θ} - 1) + 1)$ ;

$η_{a} = t + \frac{t}{θ} \ln (\prod_{k = 1}^{n} {(\frac{e^{- θ (\frac{t - ζ_{a}}{t})} - 1}{e^{- θ} - 1})}^{w_{k}} (e^{- θ} - 1) + 1)$ ;

$ζ_{b} = (- \frac{t}{θ}) \ln ({(\frac{e^{- \frac{θ b_{k}}{t}} - 1}{e^{- θ} - 1})}^{q} (e^{- θ} - 1) + 1)$ ;

$η_{b} = t + \frac{t}{θ} \ln (\prod_{k = 1}^{n} {(\frac{e^{- θ (\frac{t - ζ_{b}}{t})} - 1}{e^{- θ} - 1})}^{w_{k}} (e^{- θ} - 1) + 1)$ ;

$ζ_{c} = t - (- \frac{t}{θ}) \ln ({(\frac{e^{- \frac{θ (t - c_{k})}{t}} - 1}{e^{- θ} - 1})}^{q} (e^{- θ} - 1) + 1)$ ;

$η_{c} = - \frac{t}{θ} \ln (\prod_{k = 1}^{n} {(\frac{e^{- θ (\frac{ζ_{c}}{t})} - 1}{e^{- θ} - 1})}^{w_{k}} (e^{- θ} - 1) + 1)$ ;

$ζ_{d} = t - (- \frac{t}{θ}) \ln ({(\frac{e^{- \frac{θ (t - d_{k})}{t}} - 1}{e^{- θ} - 1})}^{q} (e^{- θ} - 1) + 1)$ ;

$η_{d} = - \frac{t}{θ} \ln (\prod_{k = 1}^{n} {(\frac{e^{- θ (\frac{ζ_{d}}{t})} - 1}{e^{- θ} - 1})}^{w_{k}} (e^{- θ} - 1) + 1)$ .

(Case 4-4) When the generator $ϱ (c) = \ln (\frac{t - θ (t - c)}{c})$ , where $ϱ^{- 1} (c) = \frac{t (1 - θ)}{e^{c} - θ}$ , θ ∈ [-1, 1). The GLIVIFCWPA reduces to the following: $\begin{matrix} {GLIVIFCWPA}_{AHM} (α_{1}, \dots, α_{n}) \\ = ([s_{a}, s_{b}], [s_{c}, s_{d}]) . \end{matrix}$ where $a = \frac{t (1 - θ)}{(\frac{t - θ (t - η_{a}}{η_{a}})^{\frac{1}{q}} - θ}; b = \frac{t (1 - θ)}{(\frac{t - θ (t - η_{b}}{η_{b}})^{\frac{1}{q}} - θ}$ ;

$c = t - \frac{t (1 - θ)}{(\frac{t - θ η_{c})}{t - η_{c}})^{\frac{1}{q}} - θ}; d = t - \frac{t (1 - θ)}{(\frac{t - θ η_{d})}{t - η_{d}})^{\frac{1}{q}} - θ}$ ;

$ζ_{a} = \frac{t (1 - θ)}{(\frac{t - θ (t - a_{k})}{a_{k}})^{q} - θ}; ζ_{b} = \frac{t (1 - θ)}{(\frac{t - θ (t - b_{k})}{b_{k}})^{q} - θ}$ ;

$ζ_{c} = t - \frac{t (1 - θ)}{(\frac{t - θ c_{k})}{t - c_{k}})^{q} - θ}; ζ_{d} = t - \frac{t (1 - θ)}{(\frac{t - θ d_{k})}{t - d_{k}})^{q} - θ}$ ;

$η_{a} = \frac{t [\prod_{k = 1}^{n} {((t - θ ζ_{a}))}^{w_{k}} - \prod_{k = 1}^{n} {(t - ζ_{a})}^{w_{k}}]}{\prod_{k = 1}^{n} {((t - θ ζ_{a}))}^{w_{k}} - θ \prod_{k = 1}^{n} {((t - ζ_{a}))}^{w_{k}}}$ ;

$η_{b} = \frac{t [\prod_{k = 1}^{n} {((t - θ ζ_{b}))}^{w_{k}} - \prod_{k = 1}^{n} {(t - ζ_{b})}^{w_{k}}]}{\prod_{k = 1}^{n} {((t - θ ζ_{b}))}^{w_{k}} - θ \prod_{k = 1}^{n} {((t - ζ_{b}))}^{w_{k}}}$ ;

$η_{c} = \frac{(1 - θ) t \prod_{k = 1}^{n} {(ζ_{c})}^{w_{k}}}{\prod_{k = 1}^{n} {((t - θ (t - ζ_{c})))}^{w_{k}} - θ \prod_{k = 1}^{n} {(ζ_{c})}^{w_{k}}}$ ;

$η_{d} = \frac{(1 - θ) t \prod_{k = 1}^{n} {(ζ_{d})}^{w_{k}}}{\prod_{k = 1}^{n} {((t - θ (t - ζ_{d})))}^{w_{k}} - θ \prod_{k = 1}^{n} {(ζ_{d})}^{w_{k}}}$ ;

(Case 4-5) When generator $ϱ (c) = - \ln (1 - {(1 - \frac{c}{t})}^{θ})$ , where $ϱ^{- 1} (c) = t - t (1 - e^{- c})^{\frac{1}{θ}}$ , where θ ≥ 1. The GLIVIFCWPA reduces to the following: $\begin{matrix} {GLIVIFCWPA}_{J} (α_{1}, \dots, α_{n}) \\ = ([s_{a}, s_{b}], [s_{c}, s_{d}]) . \end{matrix}$ where $a = t - t {(1 - {(1 - (1 - \frac{η_{a}}{t})^{θ})}^{\frac{1}{q}})}^{\frac{1}{θ}}$ ;

$b = t - t {(1 - {(1 - (1 - \frac{η_{b}}{t})^{θ})}^{\frac{1}{q}})}^{\frac{1}{θ}}$ ;

$c = t {(1 - {(1 - (1 - \frac{t - η_{c}}{t})^{θ})}^{\frac{1}{q}})}^{\frac{1}{θ}}$ ;

$d = t {(1 - {(1 - (1 - \frac{t - η_{d}}{t})^{θ})}^{\frac{1}{q}})}^{\frac{1}{θ}}$ ;

$ζ_{a} = t - t {(1 - {(1 - (1 - \frac{a_{k}}{t})^{θ})}^{q})}^{\frac{1}{θ}};$ $η_{a} = t - t {(1 - \prod_{k = 1}^{n} {(1 - {(\frac{ζ_{a}}{t})}^{θ})}^{w_{k}})}^{\frac{1}{θ}}$ ;

$ζ_{b} = t - t {(1 - {(1 - (1 - \frac{b_{k}}{t})^{θ})}^{q})}^{\frac{1}{θ}}$ ;

$η_{b} = t - t {(1 - \prod_{k = 1}^{n} {(1 - {(\frac{ζ_{b}}{t})}^{θ})}^{w_{k}})}^{\frac{1}{θ}}$ ;

$ζ_{c} = t {(1 - {(1 - (1 - \frac{t - c_{k}}{t})^{θ})}^{q})}^{\frac{1}{θ}}$ ;

$η_{c} = t {(1 - \prod_{k = 1}^{n} {(1 - {(1 - \frac{ζ_{c}}{t})}^{θ})}^{w_{k}})}^{\frac{1}{θ}}$ ;

$ζ_{d} = t {(1 - {(1 - (1 - \frac{t - d_{k}}{t})^{θ})}^{q})}^{\frac{1}{θ}}$ ;

$η_{c} = t {(1 - \prod_{k = 1}^{n} {(1 - {(1 - \frac{ζ_{d}}{t})}^{θ})}^{w_{k}})}^{\frac{1}{θ}}$ ;

(Remark 1). if we assign q = 1, then the GLIVIFCWPA operator will degenerate into LIVIFCWPA operator.

5 LIMADM approach

In this part, we will give a approach for MAGDM. In general, a MAGDM problem consists of the following parts: (1) Alternative set: Ξ = {Ψ₁, ⋯ , Ψ_m}; (2)Attribute (Criteria) set: A = {a₁, ⋯ , a_n}; (3)WV of attribute: W = (w₁, ⋯ , w_n) ^T satisfies w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . (4) D = {D₁, D₂, ⋯ , D_p} be the set of DMs.

DMs evaluate the attribute value of alternative Ψ_i under the attribute a_j can be expressed by a LIVIFNs: $γ_{ij}^{k} = ([s_{a_{ij}^{k}}, s_{b_{ij}^{k}}], [s_{c_{ij}^{k}}, s_{d_{ij}^{k}}])$ . After that, an algorithm and process of MAGDM will be designed and given as follows:

Step 1. A revised decision matrix ${\hat{R}}^{k} = {({\hat{γ}}_{ij}^{k})}_{m \times n}$ is obtain by normalizing the original decision matrix R in term of Eq.(25):

${\hat{γ}}_{ij}^{k} = {\begin{matrix} s_{([s_{a_{ij}^{k}}, s_{b_{ij}^{k}}], [s_{c_{ij}^{k}}, s_{d_{ij}^{k}}])}, for benefit type \\ s_{([s_{c_{ij}^{k}}, s_{d_{ij}^{k}}], [s_{a_{ij}^{k}}, s_{b_{ij}^{k}}])}, for cost type . \end{matrix}$ (21)Step 2. Aggregating all attribute values ${\hat{γ}}_{ij}^{k} (i = 1, \dots, m; j = 1, \dots, n; k = 1, \dots, p;)$ to a comprehensive value $Z_{i}^{k}$ by GLIVIFCWPA operator.

Step 3. Calculate the supports $Sup (Z_{i}^{k}, Z_{i}^{t}) = 1 - | K_{F} (Z_{i}^{k}) - K_{F} (Z_{i}^{t}) |$ (22) $K_{F} (Z_{i}^{k}) =$ $\sqrt{(\sum_{m = a}^{d} (m_{i}^{k})^{2} + (a_{i}^{k} + c_{i}^{k})^{2} + (b_{i}^{k} + d_{i}^{k})^{2}) / 4}$ (23) Step 4. Calculate $T (Z_{i}^{k})$ and weights $w_{i}^{k}$ $T (Z_{i}^{k}) = \sum_{t = 1, k \neq t}^{p} Sup (Z_{i}^{k}, Z_{i}^{t})$ (24) $w_{i}^{k} = \frac{λ_{k} (1 + T (Z_{i}^{k}))}{\sum_{k = 1}^{p} λ_{k} (1 + T (Z_{i}^{k}))}$ (25) where $w_{i}^{k} \geq 0$ , and $\sum_{k = 1}^{p} w_{i}^{k} = 1$

Step 5. Aggregating all γ_ij = (s_{α
_ij}, s_{β
_ij}) (j = 1, ⋯ , n) in the ith low of the decision matrix ${({\hat{γ}}_{ij})}_{m \times n}$ into collective values γ_i = (s_{α
_i}, s_{β
_i}) (i = 1, ⋯ , m) of the alternatives A_i, by proposed linguistic intuitionistic aggregation operators.

Step 6. Ranking the alternative Ψ_i (i = 1, ⋯ , m), and selecting the desirable one in term of comparison rules.

The flow chart of the Novel MAGDM method is shown in Figure 1.

Fig. 1

Flow chart of MAGDM.

6 Case analysis

This example is from [15]. In the selection of companies for investment in the rural areas. There are four companies Ψ₁, Ψ₂, Ψ₃, Ψ₄ taken as in the form of the alternatives. The following four attributes (a₁, ⋯ , a₄) should be considered: a₁: Project cost; a₂: Technical capability; a₃: Financial status; a₄: Company background.

The expert use LVs $S = {s_{0} =$ extremely poor, s₁ = very poor, s₂ = poor, s₃ = slightly poor, s₄ = fair, s₅ = slightly good, s₆ = good, s₇ = very good, s₈ = extremely good } to evaluate the companies in the term of LIVIFNs. The decision matrix can be found in Table 2.

Table 2
D ecision matrix R^k (k = 1, 2, 3)

a ₁ a ₂ a ₃ a ₄

Ψ ₁ ([s₅, s₆] , [s₁, s₂]) ([s₄, s₆] , [s₁, s₁]) ([s₄, s₅] , [s₂, s₃]) ([s₆, s₇] , [s₁, s₁])

R ¹ Ψ ₂ ([s₃, s₅] , [s₂, s₃]) ([s₅, s₆] , [s₁, s₂]) ([s₂, s₄] , [s₃, s₄]) ([s₃, s₄] , [s₂, s₃])

Ψ ₃ ([s₅, s₆] , [s₁, s₂]) ([s₅, s₆] , [s₁, s₂]) ([s₃, s₅] , [s₂, s₃]) ([s₃, s₅] , [s₁, s₃])

Ψ ₄ ([s₄, s₅] , [s₂, s₃]) ([s₁, s₃] , [s₃, s₄]) ([s₃, s₅] , [s₁, s₃]) ([s₆, s₇] , [s₁, s₁])

Ψ ₁ ([s₂, s₄] , [s₁, s₃]) ([s₄, s₅] , [s₁, s₂]) ([s₄, s₅] , [s₁, s₃]) ([s₃, s₆] , [s₁, s₂])

R ² Ψ ₂ ([s₃, s₅] , [s₁, s₃]) ([s₁, s₂] , [s₁, s₄]) ([s₂, s₃] , [s₃, s₄]) ([s₃, s₅] , [s₁, s₃])

Ψ ₃ ([s₃, s₄] , [s₁, s₂]) ([s₃, s₆] , [s₁, s₂]) ([s₂, s₅] , [s₂, s₃]) ([s₃, s₄] , [s₂, s₃])

Ψ ₄ ([s₄, s₅] , [s₁, s₂]) ([s₃, s₃] , [s₃, s₅]) ([s₃, s₃] , [s₂, s₃]) ([s₄, s₆] , [s₁, s₁])

Ψ ₁ ([s₂, s₄] , [s₁, s₂]) ([s₂, s₃] , [s₁, s₄]) ([s₃, s₅] , [s₂, s₃]) ([s₅, s₇] , [s₁, s₁])

R ³ Ψ ₂ ([s₁, s₄] , [s₂, s₃]) ([s₄, s₅] , [s₁, s₂]) ([s₂, s₄] , [s₁, s₃]) ([s₃, s₄] , [s₂, s₄])

Ψ ₃ ([s₂, s₃] , [s₁, s₅]) ([s₃, s₅] , [s₁, s₂]) ([s₃, s₅] , [s₁, s₃]) ([s₃, s₅] , [s₂, s₃])

Ψ ₄ ([s₃, s₄] , [s₂, s₃]) ([s₁, s₂] , [s₃, s₄]) ([s₃, s₅] , [s₁, s₂]) ([s₅, s₆] , [s₁, s₁])

		a ₁	a ₂	a ₃	a ₄
	Ψ ₁	([s₅, s₆] , [s₁, s₂])	([s₄, s₆] , [s₁, s₁])	([s₄, s₅] , [s₂, s₃])	([s₆, s₇] , [s₁, s₁])
R ¹	Ψ ₂	([s₃, s₅] , [s₂, s₃])	([s₅, s₆] , [s₁, s₂])	([s₂, s₄] , [s₃, s₄])	([s₃, s₄] , [s₂, s₃])
	Ψ ₃	([s₅, s₆] , [s₁, s₂])	([s₅, s₆] , [s₁, s₂])	([s₃, s₅] , [s₂, s₃])	([s₃, s₅] , [s₁, s₃])
	Ψ ₄	([s₄, s₅] , [s₂, s₃])	([s₁, s₃] , [s₃, s₄])	([s₃, s₅] , [s₁, s₃])	([s₆, s₇] , [s₁, s₁])
	Ψ ₁	([s₂, s₄] , [s₁, s₃])	([s₄, s₅] , [s₁, s₂])	([s₄, s₅] , [s₁, s₃])	([s₃, s₆] , [s₁, s₂])
R ²	Ψ ₂	([s₃, s₅] , [s₁, s₃])	([s₁, s₂] , [s₁, s₄])	([s₂, s₃] , [s₃, s₄])	([s₃, s₅] , [s₁, s₃])
	Ψ ₃	([s₃, s₄] , [s₁, s₂])	([s₃, s₆] , [s₁, s₂])	([s₂, s₅] , [s₂, s₃])	([s₃, s₄] , [s₂, s₃])
	Ψ ₄	([s₄, s₅] , [s₁, s₂])	([s₃, s₃] , [s₃, s₅])	([s₃, s₃] , [s₂, s₃])	([s₄, s₆] , [s₁, s₁])
	Ψ ₁	([s₂, s₄] , [s₁, s₂])	([s₂, s₃] , [s₁, s₄])	([s₃, s₅] , [s₂, s₃])	([s₅, s₇] , [s₁, s₁])
R ³	Ψ ₂	([s₁, s₄] , [s₂, s₃])	([s₄, s₅] , [s₁, s₂])	([s₂, s₄] , [s₁, s₃])	([s₃, s₄] , [s₂, s₄])
	Ψ ₃	([s₂, s₃] , [s₁, s₅])	([s₃, s₅] , [s₁, s₂])	([s₃, s₅] , [s₁, s₃])	([s₃, s₅] , [s₂, s₃])
	Ψ ₄	([s₃, s₄] , [s₂, s₃])	([s₁, s₂] , [s₃, s₄])	([s₃, s₅] , [s₁, s₂])	([s₅, s₆] , [s₁, s₁])

6.1 Determining optimal company

Example 1. In this section, we use G-GLIVIFCWA operators to solve this MAGDM problem, where q = 1, θ = 1.

let λ = (0.243, 0.514, 0.243) be WV of the three experts, and w = (0.4, 0.25, 0.2, 0.15) be WV of the attributes.

Step 1. Since all attributes are all of the same type, the normalization procedure is omitted.

Step 2. Employ the proposed G-GLIVIFCWA operator to aggregate the evaluation values of each attribute into the integrated matrices $Z_{i}^{k}$ , as shown in Table 3.

Table 3
Integrated decision matrix $Z_{i}^{k}$

Z ¹ Z ² Z ³

Z ₁ ([s_4.7869, s_6.0452] , [s_1.1487, s_1.6438]) ([s_3.1356, s_4.8327] , [s_1.0000, s_2.5508]) ([s_2.7861, s_4.7567] , [s_1.1487, s_2.3246])

Z ₂ ([s_3.4360, s_5.0020] , [s_1.8239, s_2.8714]) ([s_2.3592, s_4.0486] , [s_1.2457, s_3.4146]) ([s_2.3891, s_4.2776] , [s_1.4641, s_2.8303])

Z ₃ ([s_4.4127, s_5.6950] , [s_1.1487, s_2.3050]) ([s_2.8143, s_4.8245] , [s_1.2746, s_2.3050]) ([s_2.6217, s_4.3199] , [s_1.1096, s_3.3254])

Z ₃ ([s_3.6644, s_5.1092] , [s_1.7366, s_2.7339]) ([s_3.5775, s_4.4475] , [s_1.5118, s_2.4580]) ([s_2.9624, s_4.2335] , [s_1.7366, s_2.5210])

	Z ¹	Z ²	Z ³
Z ₁	([s_4.7869, s_6.0452] , [s_1.1487, s_1.6438])	([s_3.1356, s_4.8327] , [s_1.0000, s_2.5508])	([s_2.7861, s_4.7567] , [s_1.1487, s_2.3246])
Z ₂	([s_3.4360, s_5.0020] , [s_1.8239, s_2.8714])	([s_2.3592, s_4.0486] , [s_1.2457, s_3.4146])	([s_2.3891, s_4.2776] , [s_1.4641, s_2.8303])
Z ₃	([s_4.4127, s_5.6950] , [s_1.1487, s_2.3050])	([s_2.8143, s_4.8245] , [s_1.2746, s_2.3050])	([s_2.6217, s_4.3199] , [s_1.1096, s_3.3254])
Z ₃	([s_3.6644, s_5.1092] , [s_1.7366, s_2.7339])	([s_3.5775, s_4.4475] , [s_1.5118, s_2.4580])	([s_2.9624, s_4.2335] , [s_1.7366, s_2.5210])

Step 3. Obtain the supports $S_{i}^{kt} = Sup (Z_{i}^{k}, Z_{i}^{t})$ according to Eq.(51): $S_{1}^{12} = S_{1}^{21} = 0.8772$ ,

$S_{1}^{13} = S_{1}^{31} = 0.8483, S_{1}^{23} = S_{1}^{32} = 0.9711$ ,

$S_{2}^{12} = S_{2}^{21} = 0.9026, S_{2}^{13} = S_{2}^{31} = 0.8892$ ,

$S_{2}^{23} = S_{2}^{32} = 0.9866, S_{3}^{12} = S_{3}^{21} = 0.8682$ ,

$S_{3}^{13} = S_{3}^{31} = 0.8818, S_{3}^{23} = S_{3}^{32} = 0.9864$ ,

$S_{4}^{12} = S_{4}^{21} = 0.9274, S_{4}^{13} = S_{4}^{31} = 0.8953$ ,

$S_{4}^{23} = S_{4}^{32} = 0.9679$

Step 4. Get the supports $T_{i}^{k} = T (Z_{i}^{k})$ and the weights $w_{i}^{k}$ according to Eq.(24) and Eq.(25): $T_{1}^{1} = \sum_{t = 1, t \neq 1}^{3} Sup (Z_{1}^{1}, Z_{1}^{t})$

$= Sup (Z_{1}^{1}, Z_{1}^{2}) + Sup (Z_{1}^{1}, Z_{1}^{3}) = 1.7255$

$T_{1}^{2} = 1.8483, T_{1}^{3} = 1.8194, T_{2}^{1} = 1.7918,$

$T_{2}^{2} = 1.8892, T_{2}^{3} = 1.8757, T_{3}^{1} = 1.7501,$

$T_{3}^{2} = 1.8547, T_{3}^{3} = 1.8682, T_{4}^{1} = 1.8227$ ,

$T_{4}^{2} = 1.8953, T_{4}^{3} = 1.8632$

$ω_{1}^{1} = \frac{λ_{1} (1 + T_{1}^{1})}{\sum_{k = 1}^{3} λ_{k} (1 + T_{1}^{k})} = 0.2356$

Similarly, we have

$ω_{1}^{2} = 0.5207, ω_{1}^{3} = 0.2437$ ,

$ω_{2}^{1} = 0.2370, ω_{2}^{2} = 0.5188, ω_{2}^{3} = 0.2441$ ,

$ω_{3}^{1} = 0.2359, ω_{3}^{2} = 0.5180, ω_{3}^{2} = 0.2461$ ,

$ω_{4}^{1} = 0.2390, ω_{4}^{2} = 0.5186, ω_{3}^{2} = 0.2424;$

Step 5. Get the collective perference valuse Z_i according to the G-GLIVIFCWPA operator in Eq.(45):

Z₁ = ([s_3.5131, s_5.1567] , [s_1.0687, s_2.2485]),

Z₂ = ([s_2.6424, s_4.3525] , [s_1.4184, s_3.1305]),

Z₃ = ([s_3.2033, s_4.9469] , [s_1.2020, s_2.5225]),

Z₄ = ([s_3.4572, s_4.5699] , [s_1.6161, s_2.5368])

Step 6. Compute the score values of every alternative on the basis of Definition 6, we have S (a₁) =5.3381, S (a₂) =4.6115, S (a₃) =5.1064, S (a₄) =4.9686.

The rank of alternatives is a₁ ≻ a₃ ≻ a₄ ≻ a₂, and so a₁ is the best alternative.

The ordering results of alternatives use other ECs proposed in present work are list in Table 4. From Table 4, we can see that there are slight differences among the scores of alternatives by different types of copulas, but the ranking results are exactly the same. This suggests that the use of different types of copulas has no obvious effect on the aggregation results.

Table 4

The ordering results of alternatives using other different Copulas

Type of Copula	Parameters	Score index of Ψ_i (i = 1, 2, 3, 4)	Ranking order
Gumbel	θ = 1, q = 1	5.3381, 4.6115, 5.1064, 4.9686	Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂
Clayton	θ = 1, q = 1	5.4976, 4.7272, 5.2069, 5.1958	Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂
Frank	θ = 1, q = 1	5.3034, 4.5598, 5.0709, 4.8063	Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂
Ali-Mikhail-Haq	θ = -1, q = 1	5.3035, 4.5768, 5.0802, 4.9185	Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂
Joe	θ = 1, q = 1	5.2701, 4.5769, 5.0448, 4.9546	Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂

6.2 Sensitivity analysis

In this subsection, the influence of parameter q, θ on the results will be analyzed. Without loss of generality, we select the Gumbel type copula to conduct an analysis with respect to different values of q and θ. The score value and the order relation are listed in Table 5, Table 6, Figure 2 and Figure 3.

Table 5
The influence of parameters θ on the rank of alternatives

Parameters (q = 1) Score index of Ψ_i (i = 1, 2, 3, 4) Ranking order

θ = 1 5.3381, 4.6115, 5.1064, 4.9686, Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂

θ = 1.5 5.4181, 4.6912, 5.1649, 5.0807 Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂

θ = 2 5.4958, 4.7640, 5.2213, 5.1849 Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂

θ = 3 5.6434, 4.8933, 5.3274, 5.3745 Ψ₁ ≻ Ψ₄ ≻ Ψ₃ ≻ Ψ₂

θ = 5 5.8900, 5.1051, 5.4990, 5.6944 Ψ₁ ≻ Ψ₄ ≻ Ψ₃ ≻ Ψ₂

Parameters (q = 1)	Score index of Ψ_i (i = 1, 2, 3, 4)	Ranking order
θ = 1	5.3381, 4.6115, 5.1064, 4.9686,	Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂
θ = 1.5	5.4181, 4.6912, 5.1649, 5.0807	Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂
θ = 2	5.4958, 4.7640, 5.2213, 5.1849	Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂
θ = 3	5.6434, 4.8933, 5.3274, 5.3745	Ψ₁ ≻ Ψ₄ ≻ Ψ₃ ≻ Ψ₂
θ = 5	5.8900, 5.1051, 5.4990, 5.6944	Ψ₁ ≻ Ψ₄ ≻ Ψ₃ ≻ Ψ₂

Table 6

The influence of parameters q on the rank of alternatives

Parameters (θ = 1)	Score index of Ψ_i (i = 1, 2, 3, 4)	Ranking order
q = 0.5	5.3087, 4.5747, 5.0842, 4.9256	Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂
q = 1	5.3381, 4.6115, 5.1064, 4.9686	Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂
q = 2	5.4011, 4.6872, 5.1552, 5.0548	Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂
q = 3	5.4657, 4.7604, 5.2078, 5.1387	Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂
q = 5	5.5901, 4.8935, 5.3158, 5.2965	Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂

Fig. 2

Scores of A_i (i = 1, 2, 3, 4) when θ = 1, q ∈ [0, 10].

Fig. 3

Scores of A_i (i = 1, 2, 3, 4) when q = 1, θ ∈ [0, 10].

From them, it is easy to derive the following conclusions: (1)the score value of alternatives are different by applying different parameters, but the ranking order of alternatives are almost the same; (2)although the final decision resultants of alternatives are slightly diverse, the optimal alternative is the same; (3)the parameter q can be regarded as a risk preference attitude of managers, i.e., the parameter q has a better control capability for the comprehensive evaluation value of alternatives;(4)the score value of all candidates increase monotonically with the increase of the parameter value q or θ.

Furthermore, we can see from Figure 2 and Figure3 that the order of Ψ₃ and Ψ₄ will be changed slightly under different parameter q or θ, but the best and worst candidates will not change.

In summary, we can derive that the sorting orders of alternatives are relatively stable with respect to different parameter q, θ. Accordingly, the proposed method can validly avoid interferences and acquire the best alternative.

6.3 Comparative analysing

In what follows, the proposed approach will be analyzed and compared with other existing methods.

Example 2. The data is the same as in example 1. As mentioned in the introduction, LIVIFNs is a more general form of LIFNs. Therefore, in order to compare with the existing approaches[26 , 43], we assume s_(a+b)/2 and s_(c+d)/2 is the given preferences of DMs. Also we consider λ = (0.243, 0.514, 0.243) be WV of DMs, and w = (0.4, 0.25, 0.2, 0.15) be WV of the attributes. The final score values and rankings results are displayed as in Table 7. From Table 7, we can get the same ranking order with the existing approaches. However, the existing approaches [26 , 38–40] can not handle LIVIFI. Although the existing approach[43] can handle LIVIFI, but it cannot provide a unique way in practical applications, so that DMs can choose different parameter values according to their own experience.

Table 7
Comparison with existing approaches for example 2

Methods Score index of Ψ_i (i = 1, 2, 3, 4) Ranking order

Chen et al[26] 2.5456, 0.9160, 2.0208, 4.9271 Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂

Zhang[38] 5.3920, 4.6849, 5.1104, 4.9271 Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂

LIU and Wang[39] 2.5259, 1.0904, 2.0827, 1.7157 Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂

Garg and Kumar[43] 5.4172, 4.7036, 5.1332, 4.9454 Ψ₄ ≻ Ψ₃ ≻ Ψ₂ ≻ Ψ₁

Liu and Liu[40] 6.5503, 6.3528, 6.5002, 6.4591 Ψ₄ ≻ Ψ₃ ≻ Ψ₂ ≻ Ψ₁

Methods	Score index of Ψ_i (i = 1, 2, 3, 4)	Ranking order
Chen et al[26]	2.5456, 0.9160, 2.0208, 4.9271	Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂
Zhang[38]	5.3920, 4.6849, 5.1104, 4.9271	Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂
LIU and Wang[39]	2.5259, 1.0904, 2.0827, 1.7157	Ψ₁ ≻ Ψ₃ ≻ Ψ₄ ≻ Ψ₂
Garg and Kumar[43]	5.4172, 4.7036, 5.1332, 4.9454	Ψ₄ ≻ Ψ₃ ≻ Ψ₂ ≻ Ψ₁
Liu and Liu[40]	6.5503, 6.3528, 6.5002, 6.4591	Ψ₄ ≻ Ψ₃ ≻ Ψ₂ ≻ Ψ₁

Example 3. This example is to select a new management information system. There are four alternatives(software system) A_i (i = 1, 2, 3, 4) to be considered. Following there are four attributes C₁, C₂, C₃, C₄ to be evaluated by three decision makers using LIVAIFNS, where C₁: The costs; C₂: The reliability of software development from outsourcing enterprise; C₃: the contribution to enterprise performance; C₄: The effort to transition to new system from the old systems. The decision matrices are shown in TableIII-V[48]. Let λ = (0.2, 0.5, 0.3) be WV of DMs, and w = (0.5, 0.3, 0.1, 0.1) be WV of the attributes. The data in example is expressed by interval-valued intuitionistic fuzzy number(IVIFN). The score values and rankings of alternatives are separately displayed as in Table 8. From it, we can draw a conclusion that the ranking orders are almost the same, and the the optimal selections are all A₄, so we can see the proposed approach is workable and efficient.

Table 8

Comparison with existing approaches for example 3

Methods	Parameters	Score index of A_i (i = 1, 2, 3, 4)	Ranking order
He et al[49]	λ = 2	0.0167, 0.0931, 0.0808, 0.2676	A₄ ≻ A₃ ≻ A₂ ≻ A₁
YU and WU[50]	p = 2, q = 2	0.0740, 0.1078, 0.2868, 0.3025	A₄ ≻ A₃ ≻ A₂ ≻ A₁
LIU[48]	p = 2, q = 2	0.0719, 0.1090, 0.2846, 0.3022	A₄ ≻ A₃ ≻ A₂ ≻ A₁
Gumbel type	θ = 1, q = 1	0.4640, 0.4996, 0.5220, 0.5396	A₄ ≻ A₃ ≻ A₂ ≻ A₁
Clayton type	θ = 1, q = 1	0.5138, 0.5547, 0.6401, 0.6757	A₄ ≻ A₃ ≻ A₂ ≻ A₁
Frank type	θ = 1, q = 1	0.4728, 0.5200, 0.5356, 0.5742	A₄ ≻ A₃ ≻ A₂ ≻ A₁
Ali-Mikhail-Haq type	θ = -1, q = 1	0.4884, 0.5146, 0.5701, 0.6033	A₄ ≻ A₃ ≻ A₂ ≻ A₁
Joe type	θ = 1, q = 1	0.4910, 0.4909, 0.5158, 0.6059	A₄ ≻ A₃ ≻ A₁ ≻ A₂

In this example, we changed the attribute value $r_{41}^{3}$ from ([0.4, 0.5] , [0.1, 0.2]) to ([0, 0] , [0.2, 0.2]), then the score values of A_i (i = 1, 2, 3, 4) by YU and WU[50] is (0.0740, 0.1078, 0.2868, 0.2800), so the ranking result is A₃ ≻ A₄ ≻ A₂ ≻ A₁. However, the ranking result of the methods proposed in this paper have not changed. Take Gumbel type as an example, the score value of A_i (i = 1, 2, 3, 4) is (0.4640, 0.4996, 0.5149, 0.5220). The proposed method in this paper considered the generalized weighted PA operator, it can relieve the influence of the irrational data provided by biased DMs. The comparison results verify the superiority of the proposed method. Compared with the methods proposed by and LIU[48], our method can handle linguistic interval-valued intuitionistic fuzzy number, which is a generalization of IVIFN.

In what follows, the proposed approach will be analyzed and compared with other existing method approaches.

(1) Chen et al.’s LIFWA operator [26], Zhang’s LIFWA operator [38] and Liu and Wang’s ILIFWA operator [39] are all based on LIFS. The operational rules are special form of ECs and ECCs. The proposed method will be degenerate to the above methods when t = 1, θ = 1, p = 1,and q = 0, and only consider s_(a+b)/2 and s_(c+d)/2. Therefore, the proposed method is more general.

(2) Compared with Tao’s method [55]. If t = 1, q = 1, and only consider s_(a+b)/2 and s_(c+d)/2, LIVIFCWPA operator in this paper reduce to IFCAA_ω. Therefore, compared with IFCAA_ω [55], LIVIFCWPA is the generalization of Tao’s method.

(3) Compared to the method of [45] and [43], the proposed method is more universal and flexible. It can be regarded five aggregation functions through assigning diverse copulas to it and flexible parameters θ, q can also be selected according to decision maker¡¯s attitude.

(4) In [46], Qin et al. combine the MM operator and the PA operator under the Archimedean T-norm and T-conorm operations(ATT). Compared with our method, Qin et al consider the interconnection of diverse attributes, but the designed method is based on ATT which is the special form of ECs and ECCs.

A detailed comparative analysis for aforementioned approaches are displayed in Table 9.

Table 9

The characteristic comparison of different AOs

AOs	eliminate extreme values	generalization	Flexibility	Deal with LIVIFNs
Chen et al. [26]	No	No	No	No
Zhang [38]	No	Yes	No	No
Liu and Wang [39]	Yes	No	No	No
Garg and Kumar [43]	No	Yes	No	Yes
Tao et al. [55]	No	No	Yes	Yes
Liu and Qin [45]	Yes	Yes	No	No
Qin et al. [46]	Yes	Yes	Yes	No
the proposed method	Yes	Yes	Yes	Yes

The merit of LIVIFCA operators are summarized as follows:

•(1) As far as the operational rules are concerned. On the one hand, the operational rules of Garg and Kumar [43] were built based on algebraic t-norm (TN_A) and t-conorm (TC_A). Copulas and co-copulas are classical examples of TNs and TCs. It is seen from Section 3 that TN_A and TC_A are the special cases of ECs and ECCs, which not only reflect the coherence between variables, but also prevent information losing in the process of integration. As Copulas and Co-copulas can only handle some situation on [0, 1], but it is invalid for some situations on [0, t] (t > 0). Therefore, we extend classical copulas and Co-copulas from [0, 1] to [0, t] to handle some DMPs with linguistic information. Hence, these AOs based on the ECs and ECCs are more diversified and also avoid information loss in decision making process. On the other hand, some different LIVIFS’ operational rules are obtained based on different cases of ECs and ECCs in our work, and different types of AOs have been proposed correspondingly. The main advantage of these AOs is that it can provide more choices for DMs.

•(2) In the matter of AOs. Garg and Kumar [43] introduced the LIVAIFWA_ω operator, LIVAIFOWA_ω, LIVAIFHA_ω operator to manage LIVIFI. The proposed LIVIFCWA operators are different families with different ECs and ECCs. The details have been studied with particulars in Section 4. What’s more, combined PA operator, we proposed generalized weighted power (GLIVIFCWPA)operator. Therefore, the proposed AOs are more flexible for DMs to make more choices in real DMPs.

•(3) Moreover, taking into account the irrational data given by biased decision makers in real DMPs, it is very interesting to integrate the Copula operations and PA operator for MAGDM under LIVIFS environment. The proposed operators not only make the decision making approach more flexible and robust in practical DMPs, but also reduce the impact of irrational data given by biased DMs.

7 Conclusions

Linguistic interval-valued intuitionistic fuzzy theory is a powerful tool to handle imprecise and uncertain information. In order to build the new approach for MAGDM under Linguistic interval-valued intuitionistic fuzzy environment, we propose a new version of copulas and co-copulas named ECs and ECCs, and present a LIVIFCWA operator, LIVIFCWPA operator and a GLIVIFCWPA operator respectively. Also some properties of the proposed AOs have been proved and five specific forms of AOs are obtained when EC and ECC take different types of ECs and ECCs. On this basis, a new MAGDM method is proposed and a detailed numerical example is given. Finally, the effectiveness, feasibility and superiorities of proposed method is proved by some comparative study. In future, we shall focus especially on the correlation between attributes, incomplete attribute information, as well as the large scale decision making algorithm based upon linguistic assessment theory and methodology.

Conflict of interest

The authors declare that they have no conflict of interest.

Footnotes

Acknowledgments

This work was supported in part by Sichuan Province Youth Science and Technology Innovation Team under Grant 2019JDTD0015, Application Basic Research Plan Project of Sichuan Province under Grant 2017JY0199; Scientific Research Project of Department of Education of Sichuan Province (No.17ZB0220, No.18ZA0273); The Application Basic Research Plan Project of Sichuan Province (No.2017JY0199); The Scientific Research Innovation Team of Neijiang Normal University (No.18TD008);Open fund of Data Recovery Key Lab of Sichuan Province(No.DRN19018)

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