Some spherical linguistic Muirhead mean operators with their application to multi-attribute decision making

Abstract

This paper aims to propose a new tool to express decision makers’ preference information in multi-attribute decision making (MADM) producers. By taking advantages of spherical fuzzy sets (SFSs) and linguistic variables (LVs), we give the definition of spherical linguistic sets (SLSs) and provide operations of spherical linguistic numbers (SLNs). Based on the proposed operations, we incorporate Muirhead mean (MM) into SLSs and introduce novel spherical linguistic aggregation operators. These proposed operators adsorb the inherent advantages of MM, i.e., taking the interrelationship among any numbers of aggregated inputs into account and producing flexible information fusion process. Furthermore, we apply the proposed method in MADM and present the main steps of a new method. In order to show its effectiveness, we use the method to solve an actual MADM problem. The advantages and superiorities of the proposed method are also studied.

Keywords

Spherical fuzzy set spherical linguistic set spherical linguistic muirhead mean multi-attribute decision making

1 Introduction

We always encounter decision-making problems in real-life and mostly, the decision-making problems refer to multi-attribute decision making (MADM) [1 –5]. For example, when we buy a car we usually evaluate all potential alternatives from multiple aspects, such as degree of comfort, price, brand reputation, overall performance, etc. In recent years, the MADM methods and related decision making techniques have been a very active research field, which has attracted many scholars’ interests [6 –15]. In the framework of MADM theory, there are four fundamental elements, i.e., candidate alternatives, multiple attributes, evaluation values and decision-making methods. Evidently, in advance of the procedure of determining the most optimal alternative, the possible alternatives and attributes that are evaluated have been confirmed already. Hence, the most important two parts are the evaluation values and decision-making methods. For evaluation values it is widely known that decision makers (DMs) can hardly get all information of alternatives within a limited time. Therefore, DMs would like to use fuzzy numbers rather than crisp numbers to express their evaluation values. With the development of fuzzy sets theories, DMs have more choices to select an appropriate tool to denote their preferences. Recently, Prof. Yager [16] provided a new methodology for dealing with complicated MADM problems, called Pythagorean fuzzy sets (PFSs). Since the introduction of PFSs, more and more scholars and scientists have shifted their attention to research on PFSs based MADM. Zhang and Xu [17] gave operations of Pythagorean fuzzy numbers (PFNs) as well we Pythagorean fuzzy TOPSIS method. Zhang [18] introduced the Pythagorean fuzzy QUAIFLEX approach based on a new comparison method for PFNs. Garg [19] extended PFSs to interval-valued PFSs and investigated their applications in MADM. Zhang [20] studied similarity measure for PFSs. Ren et al. [21] proposed a novel Pythagorean fuzzy MADM approach from the point of view of TODIM. Gou et al. [22] focused on continuous Pythagorean fuzzy information, investigated their important properties and introduced a methodology for integrating it. Garg [23] introduced new Pythagorean fuzzy correlation coefficients between PFSs and investigated their applications in pattern recognition and medical diagnosis. More scholars and scientists shifted their attention to aggregation operators for Pythagorean fuzzy information and some operators, such as Pythagorean fuzzy Bonferroni means [24, 25], Pythagorean fuzzy Maclaurin symmetric means [26], Pythagorean fuzzy Muirhead mean [27, 28], Pythagorean fuzzy power mean [29], and Pythagorean fuzzy point operators [30] have been developed. Some scholars devoted themselves to the investigation of Pythagorean fuzzy operations and a few operational rules of PFNs, such as Einstein operations [31 –33] symmetric operations [34], interaction operations [35], Hamacher operations [36], exponential operations [37], and Frank operations [38] were introduced.

In intuitionistic fuzzy sets (IFSs) and PFSs we have membership grades (MGs) and non-membership grades (NGs) and the indeterminacy grades (IGs) are a default once the MGS and NGs are determined. For example, in IFSs if the MG is 0.2 and NG is 0.3, then the IG is 1 - 0.2 - 0.3 = 0.5 and in PFSs the IG is $\sqrt{1 - 0 . 2^{2} - 0 . 3^{2}} = 0.93$ . However, in Coung’s [39, 40] opinion the IGs should be determined by DMs rather than a default. Hence, Coung [39, 40] proposed a new computational intelligence tool, i.e. PIcture Fuzzy Sets (PIFSs). In the framework, DMs are allowed to express not only the positive and negative information, but also the neutral ideas. Hence, PIFSs can describe more information and are more effective to deal with human opinion involving more answers of types: yes, abstain, no, and refusal, such as voting. To distinguish IFSs and PIFSs, Cuong called the three information functions the positive membership degree (PMD), the negative membership degree (ND), and the neutral membership degree (NMD). It is not difficult to find out that PIFSs express more information than IFSs and are more powerful to deal with complicated decision making systems. Since the appearance, PIFSs have gained extensive attention from scholars all over the world. Basically, recent researches on PIFs mainly focus on similarity measures [41 –43], outranking methods [44, 45] and picture fuzzy aggregation operators [46 –51].

Recently, motivated by PFSs Ashraf et al. [52] relaxed the restraint of PIFs and proposed the concept of spherical fuzzy set (SFS), whose constraint is that the sum of squares of PMD, ND, and NMD is less than or equal to 1. Evidently, SFSs take full advantages of both PFSs and PIFSs. Compared with PFSs, SFSs contains more decision information and are better to represent DMs’ opinions. Compared with PIFSs, SPFSs give DMs more freedom and consequently less information distortion is lead. Afterwards, scholars started to study on SFSs based on MADM methods and quite a few achievements have been reported [53, 54]. However, SFSs still have drawbacks as they only express DMs’ quantitative evaluation information and neglect their qualitative assessments. More and more scholars have started to know the high necessity to take DMs’ qualitative decision making information into account before determining the optimal choices [55 –57]. Therefore, this paper tries to extend the power SFSs theory by additionally taking DMs’ qualitative evaluation into consideration and propose the spherical linguistic sets (SLSs). In the framework of SLSs, DMs utilize LVs to express their evaluation information and additionally, they are also allowed to provide the PMDs, NDs, and NMDs of LVs, such that the sum of the square of three degrees is less than or equal to one. Hence, the proposed SLSs inherit the advantages and superiorities of LVs and SFSs, i.e. they not only describe DMs’ preference information quantificationally and qualitatively, but also give DMs enough freedom to express their evaluations. SLSs are parallel to picture fuzzy linguistic sets (PFLSs) [58], but are more powerful as they have a laxer constraint, which gives experts more freedom to express their preferences. In addition, SLSs are also stronger than Pythagorean linguistic sets (PLSs) [59] as they additionally captures the DMs’ neutral grades. When fusing spherical linguistic (SL) information, the SL aggregation operators are needed. Recently the good performance of the Muirhead mean (MM) [60] in capturing the interrelationship among any numbers of input variables has deeply impressed us [61 –63]. Therefore, we extend MM to SLSs and propose new spherical linguistic operators. Finally, we use the proposed operators to introduce a new MADM method.

For easy description, the following of this paper is organized as follows. Section 2 proposes the SLSs and related concepts. Section 3 describes the SL aggregation operators and their properties. Section 4 gives a new algorithm for MADM in which attributes are in the form of SL information. Section 5 shows the effectiveness of the proposed method.

2 Preliminaries

Concepts that will be used in the paper are discussed in this section.

2.1 Spherical fuzzy sets

Definition 1. [52] Let X be an ordinary fixed set, a spherical fuzzy set (SFS) A defined on X is given by $A = {(x, μ_{A} (x), η_{A} (x), v_{A} (x)) | x \in X}$ (1)

Where μ_A (x), η_A (x) and v_A (x) represent the positive membership degree, the neutral membership degree, and the negative membership degree respectively, satisfying the condition that μ_A (x) , η_A (x) , v_A (x) ∈ [0, 1] and 0 ≤ μ_A (x) ² + η_A (x) ² + v_A (x) ² ≤ 1. The refusaldegree of A is expressed as $π_{A} (x) = \sqrt{1 - μ_{A} (x)^{2} - η_{A} (x)^{2} - v_{A} (x)^{2}}$ . For convenience, (μ_A (x) , η_A (x) , v_A (x)) is called a spherical fuzzy number (SFN), which can be simple denoted by α = (μ, η, v).

It is easy to find out that the SFS only considers DMs’ quantitative evaluation information. Basically, DMs usually give their evaluations both quantitatively and qualitatively. Hence, we combine SFSs and linguistic variables (LVs) to more comprehensively describe DMs’ assessment information.

Definition 2. Let X be an ordinary fixed set, a spherical linguistic set (SLS) defined on X can be given as $A = {(s_{θ (x)}, μ_{A} (x), η_{A} (x), v_{A} (x)) | x \in X}$ (2)

Where $s_{θ (x)} \in \bar{S}$ , and μ_A (x), η_A (x), and v_A (x) represent the positive membership degree, the neutral membership degree, and the negative membership degree, satisfying the conditions that 0 ≤ μ_A (x) , η_A (x) , v_A (x) ≤1 and 0 ≤ μ_A (x) ² + η_A (x) ² + v_A (x) ² ≤ 1. Then for each x ∈ X, the refusal degree of x to s_θ(x) is $π_{A} (x) = \sqrt{1 - μ_{A} (x)^{2} - η_{A} (x)^{2} - v_{A} (x)^{2}}$ . For convenience, we call α = (s_θ(x), μ_A (x) , η_A (x) , v_A (x)) a spherical linguistic number (SLN), which can be denoted asα = (s_θ, μ, η, v).

Based on the operations of SFNs proposed by Ashraf [52], we proposed operational rules of SLNs.

Definition 3. Let α₁ = (s_{θ
₁}, μ₁, η₁, v₁), α₂ = (s_{θ
₂}, μ₂, η₂, v₂) and α = (s_θ, μ, η, v) be any three SLNs, and λ be positive real number, then

$α_{1} \oplus α_{2} = (s_{θ_{1} + θ_{2}}, \sqrt{μ_{1}^{2} + μ_{2}^{2} - μ_{1}^{2} μ_{2}^{2}},$ η₁η₂, v₁v₂ );

$α_{1} \otimes α_{2} = (s_{θ_{1} \times θ_{2}}, μ_{1} μ_{2}, \sqrt{η_{1}^{2} + η_{2}^{2} - η_{1}^{2} η_{2}^{2}},$ $\sqrt{v_{1}^{2} + v_{2}^{2} - v_{1}^{2} v_{2}^{2}})$ ;

$λ α = (s_{λ \times θ}, \sqrt{1 - {(1 - μ^{2})}^{λ}}, η^{λ}, v^{λ})$ ;

$α^{λ} = (s_{θ^{λ}}, μ^{λ}, \sqrt{1 - {(1 - η^{2})}^{λ}},$ $\sqrt{1 - {(1 - v^{2})}^{λ}})$ .

Definition 4. Let α = (s_θ, μ, η, v) be a SLN, then the sore is defined as follows $S (α) = \frac{1}{4} (μ^{2} - v^{2} + 2) \times s_{θ} = s_{\frac{1}{4} (μ^{2} - v^{2} + 2) \times θ}$ (3)

For any two SLNs α₁ = (s_{θ
₁}, μ₁, η₁, v₁) and α₂ = (s_{θ
₂}, μ₂, η₂, v₂), if S (α₁) > S (α₂), then α₁ > α₂ and if S (α₁) > S (α₂), then α₁ = α₂.

2.2 Muirhead mean and dual Muirhead mean

The MM is an aggregation operator introduced by Muirhead [60] for crisp numbers. This operator can capture the interrelationship among all the aggregated arguments.

Definition 5. [60] Let a_j (j = 1, 2, . . . , n) be a collection of crisp numbers and Q = (q₁, q₂, . . . , q_n) ∈ Rⁿ be a vector of parameter, then the MM operator is defined as ${MM}^{Q} (a_{1}, a_{2}, . . ., a_{n}) = {(\frac{1}{n!} \sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} a_{ϑ (j)}^{q_{j}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}$ (4)

Where ϑ (j) (j = 1, 2, . . . , n) is any permutation of (1, 2, . . . , n), and T_n is the collection of all permutations of (1, 2, . . . , n).

The dual form of MM is called the dual Muirhead mean (DMM) and its definition is given as follows.

Definition 6. [64] Let a_j (j = 1, 2, . . . , n) be a collection of crisp numbers and Q = (q₁, q₂, . . . , q_n) ∈ Rⁿ be a vector of parameter, then the DMM operator is defined as

$\begin{matrix} {DMM}^{Q} (a_{1}, a_{2}, . . ., a_{n}) \\ = \frac{1}{\sum_{j = 1}^{n} q_{j}} {(\prod_{ϑ \in T_{n}} \sum_{j = 1}^{n} (q_{j} a_{ϑ (j)}))}^{\frac{1}{n!}} \end{matrix}$ (5) where ϑ (j) (j = 1, 2, . . . , n) is any permutation of (1, 2, . . . , n), and T_n is the collection of all permutations of (1, 2, . . . , n).

3 Some spherical linguistic aggregation operators

3.1 The spherical linguistic Muirhead mean (SLMM) operator

Definition 7. Let α_j = (s_{θ
_j}, μ_j, η_j, v_j) be a collection of SLNs, and Q = (q₁, q₂, . . . , q_n) ∈ Rⁿ be a set of parameters, then the spherical linguistic Muirhead mean (SLMM) operator is defined as

$\begin{matrix} {SLMM}^{Q} (α_{1}, α_{2}, . . ., α_{n}) \\ = {(\frac{1}{n!} \sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} α_{ϑ (j)}^{q_{j}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}} \end{matrix}$ (6)

Where ϑ (j) (j = 1, 2, . . . , n) is any permutation of (1, 2, . . . , n), and T_n is the collection of all permutations of (1, 2, . . . , n).

Theorem 1. Let α_j = (s_{θ
_j}, μ_j, η_j, v_j) (j = 1, 2, . . . , n) be a collection of SLNs, Q = (q₁, q₂, . . . , q_n) ∈ Rⁿ be a vector of parameters, then the aggregated value by the SLMM operator is still a SLNs and ${SLMM}^{Q} (α_{1}, α_{2}, . . ., α_{n}) = (s_{{(\frac{1}{n!} \sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} θ_{ϑ (j)}^{q_{j}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}},$ ${(\sqrt{1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} μ_{ϑ (j)}^{2 q_{j}}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}},$ $\sqrt{1 - {(1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} {(1 - η_{ϑ (j)}^{2})}^{q_{j}}))}^{\frac{1}{n!}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}},$ $\sqrt{1 - {(1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} {(1 - v_{ϑ (j)}^{2})}^{q_{j}}))}^{\frac{1}{n!}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}})$ (7)

Proof. According to the operations of Definition 3, we can obtain $\begin{matrix} α_{ϑ (j)}^{q_{j}} = (s_{θ_{ϑ (j)}^{q_{j}}}, μ_{ϑ (j)}^{q_{j}}, \sqrt{1 - {(1 - η_{ϑ (j)}^{2})}^{q_{j}}}, \\ \sqrt{1 - {(1 - v_{ϑ (j)}^{2})}^{q_{j}}}), \end{matrix}$ and $\begin{matrix} \prod_{j = 1}^{n} α_{ϑ (j)}^{q_{j}} = (s_{\prod_{j = 1}^{n} θ_{ϑ (j)}^{q_{j}}}, \prod_{j = 1}^{n} μ_{ϑ (j)}^{q_{j}}, \\ \sqrt{1 - \prod_{j = 1}^{n} {(1 - η_{ϑ (j)}^{2})}^{q_{j}}}, \end{matrix}$ $\sqrt{1 - \prod_{j = 1}^{n} {(1 - v_{ϑ (j)}^{2})}^{q_{j}}})$

Then, $\begin{matrix} \sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} α_{ϑ (j)}^{q_{j}} = (s_{\sum_{ϑ \in T} \prod_{j = 1}^{n} θ_{ϑ (j)}^{q_{j}}}, \\ \sqrt{1 - \prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} μ_{ϑ (j)}^{2 q_{j}})}, \end{matrix}$

$\begin{matrix} \prod_{ϑ \in T_{n}} \sqrt{1 - \prod_{j = 1}^{n} {(1 - η_{ϑ (j)}^{2})}^{q_{j}}}, \\ \prod_{ϑ \in T_{n}} \sqrt{1 - \prod_{j = 1}^{n} {(1 - v_{ϑ (j)}^{2})}^{q_{j}}}) \end{matrix}$

Further, $\begin{matrix} \frac{1}{n!} \sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} α_{ϑ (j)}^{q_{j}} = (s_{\frac{1}{n!} \sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} θ_{ϑ (j)}^{q_{j}}}, \\ \sqrt{1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} μ_{ϑ (j)}^{2 q_{j}}))}^{\frac{1}{n!}}}, \\ {(\prod_{ϑ \in T_{n}} \sqrt{1 - \prod_{j = 1}^{n} {(1 - η_{ϑ (j)}^{2})}^{q_{j}}})}^{\frac{1}{n!}}, \\ {(\prod_{ϑ \in T_{n}} \sqrt{1 - \prod_{j = 1}^{n} {(1 - v_{ϑ (j)}^{2})}^{q_{j}}})}^{\frac{1}{n!}}) \end{matrix}$

Consequently, we have $\begin{matrix} {(\frac{1}{n!} \sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} α_{ϑ (j)}^{q_{j}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}} = (s_{{(\frac{1}{n!} \sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} θ_{ϑ (j)}^{q_{j}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}}, \\ {(\sqrt{1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} μ_{ϑ (j)}^{2 q_{j}}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}, \\ \sqrt{1 - {(1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} {(1 - η_{ϑ (j)}^{2})}^{q_{j}}))}^{\frac{1}{n!}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}}, \\ \sqrt{1 - {(1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} {(1 - ν_{ϑ (j)}^{2})}^{q_{j}}))}^{\frac{1}{n!}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}}) \end{matrix}$ which completes the proof of Theorem 1.□

In what follows, we will explore some properties of the SLMM operator.

Property 1 (Idempotency). Let α_j = (s_{θ
_j}, μ_j, η_j, v_j) (j = 1, 2, . . . , n) be a collection of SLNs, if all SLNs are equal, i.e., α_j = α = (s_{θ
_j}, μ_j, η_j, v_j) for all j, then

${SLMM}^{Q} (α_{1}, α_{2}, \dots, α_{n}) = α = (s_{θ}, μ, η, v)$ (8)Proof. Since α_j = α = (s_θ, μ, η, v), then we have $\begin{matrix} {SLMM}^{Q} (α_{1}, α_{2}, \dots, α_{n}) = (s_{{(\frac{1}{n!} \sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} θ_{ϑ (j)}^{q_{j}})}^{1 / - \sum_{j = 1}^{n} q_{j}}}, \\ {(\sqrt{1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} μ_{ϑ (j)}^{2 q_{j}}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}, \\ \sqrt{1 - {(1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} {(1 - η_{ϑ (j)}^{2})}^{q_{j}}))}^{\frac{1}{n!}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}}, \\ \sqrt{1 - {(1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} {(1 - v_{ϑ (j)}^{2})}^{q_{j}}))}^{\frac{1}{n!}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}}) \\ = (s_{{(\frac{1}{n!} \sum_{θ \in T_{n}} θ^{\sum_{j = 1}^{n} q_{j}})}^{1 / - \sum_{j = 1}^{n} q_{j}}}, \\ {(\sqrt{1 - {(\prod_{ϑ \in T_{n}} (1 - μ_{ϑ (j)}^{2 \sum_{j = 1}^{n} q_{j}}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}, \end{matrix}$

$\begin{matrix} \sqrt{1 - {(1 - {(\prod_{ϑ \in T_{n}} (1 - {(1 - η_{ϑ (j)}^{2})}^{\sum_{j = 1}^{n} q_{j}}))}^{\frac{1}{n!}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}}, \\ \sqrt{1 - {(1 - {(\prod_{ϑ \in T_{n}} (1 - {(1 - v_{ϑ (j)}^{2})}^{\sum_{j = 1}^{n} q_{j}}))}^{\frac{1}{n!}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}}) \end{matrix}$ $= (s_{{(θ^{\sum_{j = 1}^{n} q_{j}})}^{1 / \sum_{j = 1}^{n} q_{j}}}, {(\sqrt{1 - {({(1 - μ_{ϑ (j)}^{2 \sum_{j = 1}^{n} q_{j}})}^{n!})}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}, \sqrt{1 - {(1 - {({(1 - {(1 - η_{ϑ (j)}^{2})}^{\sum_{j = 1}^{n} q_{j}})}^{n!})}^{\frac{1}{n!}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}}, \sqrt{1 - {(1 - {({(1 - {(1 - v_{ϑ (j)}^{2})}^{\sum_{j = 1}^{n} q_{j}})}^{n!})}^{\frac{1}{n!}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}}) = (s_{θ}, μ, η, v)$

Property 2 (Monotonicity). Let α_j and β_j (j = 1, 2, …, n) be any two collections of SLNs. If α_j ≥ β_j for all j, then ${SLMM}^{Q} (α_{1}, α_{2}, \dots, α_{n}) \geq {SLMM}^{Q} (β_{1}, β_{2}, \dots, β_{n})$ (9) Proof. As α_j ≥ β_j holds for all j, then $α_{ϑ (j)}^{q_{j}} \geq β_{ϑ (j)}^{q_{j}},$ and $\prod_{j = 1}^{n} α_{ϑ (j)}^{q_{j}} \geq \prod_{j = 1}^{n} β_{ϑ (j)}^{q_{j}}$

Hence, $\sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} α_{ϑ (j)}^{q_{j}} \geq \sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} β_{ϑ (j)}^{q_{j}},$ and $\frac{1}{n!} \sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} α_{ϑ (j)}^{q_{j}} \geq \frac{1}{n!} \sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} β_{ϑ (j)}^{q_{j}}$

Therefore, ${(\frac{1}{n!} \sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} α_{ϑ (j)}^{q_{j}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}} \geq {(\frac{1}{n!} \sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} β_{ϑ (j)}^{q_{j}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}},$ which completes the proof.□

Property 3 (Boundedness). Let α_j (j = 1, 2, . . , n) be a collection of SLNs, if $α^{-} = (s_{θ_{j}}, \min (μ_{j}), \max (η_{j}), max (v_{j})),$ and $α^{+} = (s_{θ_{j}}, \max (μ_{j}), \min (η_{j}), \min (v_{j})),$

$α^{+} \geq {SLMM}^{Q} (α_{1}, α_{2}, \dots, α_{n}) \geq α^{+},$ (10)Proof. Based on Properties 1 and 2, $\begin{matrix} {SLMM}^{Q} (α_{1}, α_{2}, \dots, α_{n}) \leq \\ {SLMM}^{Q} (α^{+}, α^{+}, \dots, α^{+}) = α^{+}, \end{matrix}$ and $\begin{matrix} {SLMM}^{Q} (α_{1}, α_{2}, \dots, α_{n}) \geq \\ {SLMM}^{Q} (α^{-}, α^{-}, \dots, α^{-}) = α^{-} . \end{matrix}$

So, we can get α⁺ ≥ SLMM^Q (α₁, α₂, ⋯ , α_n) ≥ α^-.

The SLMM operator is proposed based on the MM operator. Hence, SLMM inherits the powerfulness and flexibility of MM. To better illustrate the flexibility of the SLMM operator, in the followings we discuss special cases of SLMM with respect to the parameter vector Q.□

Special case 1: when Q = (1, 0, ⋯ , 0), then the SLMM operator reduces to the spherical linguistic average (SLA) operator, i.e.

$\begin{matrix} {SLMM}^{(1, 0, \dots, 0)} (α_{1}, α_{2}, \dots, α_{n}) = (s_{\frac{1}{n} \sum_{j = 1}^{n} θ_{j}}, \\ \sqrt{1 - \prod_{j = 1}^{n} {(1 - μ_{j}^{2})}^{1 / - n}}, \prod_{j = 1}^{n} η_{j}^{1 / - n}, \prod_{j = 1}^{n} v_{j}^{1 / - n}) \end{matrix}$ (11)Special case 2: when Q = (1, 1, 0, 0, ⋯ , 0), then the SLMM operator reduces to the spherical linguistic Bonferroni mean (BM) operator, i.e. $\begin{matrix} {SLMM}^{(1, 1, 0, 0, \dots, 0)} (α_{1}, α_{2}, \dots, α_{n}) \\ = (s_{{(\frac{1}{n (n - 1)} {\prod_{i, j = 1}}_{i \neq j}^{n} (θ_{i} θ_{j}))}^{\frac{1}{2}}}, \\ {(\sqrt{1 - {({\prod_{i, j = 1}}_{i \neq j}^{n} (1 - μ_{i}^{2} μ_{j}^{2}))}^{\frac{1}{n (n - 1)}}})}^{\frac{1}{2}}, \\ \sqrt{1 - {(1 - {\prod_{i, j = 1}}_{i \neq j}^{n} {(1 - (1 - η_{i}^{2}) (1 - η_{j}^{2}))}^{\frac{1}{n (n - 1)}})}^{\frac{1}{2}}}, \\ \sqrt{1 - {(1 - {\prod_{i, j = 1}}_{i \neq j}^{n} {(1 - (1 - v_{i}^{2}) (1 - v_{j}^{2}))}^{\frac{1}{n (n - 1)}})}^{\frac{1}{2}}}) \end{matrix}$ (12)

Special case 3: when $Q = {\overset{︷}{(1, 1, \dots 1,}}^{k} {\overset{︷}{0, 0, \dots 0)}}^{n - k}$ , then the SLMM operator reduces to the spherical linguistic Maclaurin symmetric mean (MSM) operator, i.e. $\begin{matrix} {SLMM}^{{\overset{︷}{(1, 1, \dots 1,}}^{k} {\overset{︷}{0, 0, \dots 0)}}^{n - k}} (α_{1}, α_{2}, \dots, α_{n}) \end{matrix}$ $\begin{matrix} = (s_{{(\frac{1}{C_{n}^{k}} \prod_{1 \leq i_{1} < . . . < i_{k} \leq n} (\sum_{j = 1}^{k} θ_{i_{j}}))}^{1 / k}}, \\ {(\sqrt{1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} μ_{i_{j}}^{2}))}^{1 / C_{n}^{k}}})}^{1 / k}, \end{matrix}$

$\begin{matrix} \sqrt{1 - {(1 - {(\prod_{1 \leq i_{1} < . . . < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - η_{i_{j}}^{2})))}^{1 / C_{n}^{k}})}^{1 / k}}, \\ \sqrt{1 - {(1 - {(\prod_{1 \leq i_{1} < . . . < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - v_{i_{j}}^{2})))}^{1 / C_{n}^{k}})}^{1 / k}}) \end{matrix}$ (13) Special case 4: when Q = (1, 1, ⋯ , 1) or Q = (1/n, 1/n, ..., 1/n), then the SLMM operator reduces spherical linguistic geometric (SLG) operator, i.e.

$\begin{matrix} {SLMM}^{(1, 1, \dots, 1) or (1 / n, 1 / n, \dots, 1 / n)} (α_{1}, α_{2}, \dots, α_{n}) \\ = (s_{\prod_{j = 1}^{n} θ_{j}^{1 / n}}, \prod_{j = 1}^{n} μ_{j}^{1 / n}, \sqrt{1 - \prod_{j = 1}^{n} {(1 - η_{j}^{2})}^{1 / n}}, \\ \sqrt{1 - \prod_{j = 1}^{n} {(1 - v_{j}^{2})}^{1 / n}}) \end{matrix}$ (14)

3.2 The spherical linguistic weighted Muirhead mean (SLWMM) operator

Definition 8. Let α_j = (s_{θ
_j}, μ_j, η_j, v_j) (j = 1, 2, …, n) be a collection of SLNs and Q = (q₁, q₂, . . . , q_n) ∈ Rⁿ be a vector of parameter. Let w = (w₁, w₂, . . . , w_n) ^T be the weight vector, satisfying w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ . If

$\begin{matrix} {SLWMM}^{Q} (α_{1}, α_{2}, . . ., α_{n}) \\ = {(\frac{1}{n!} \sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} {({nw}_{ϑ (j)} α_{ϑ (j)})}^{q_{j}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}} \end{matrix}$ (15)

Then SLWMM^Q is the SLWMM operator, where ϑ (j) (j = 1, 2, . . . , n) is any permutation of (1, 2, . . . , n), and T_n is the collection of all permutations of (1, 2, . . . , n).

Theorem 2. Let α_j = (s_{θ
_j}, μ_j, η_j, v_j) (j = 1, 2,. . . , n) be a collection of SLNs and Q = (q₁, q₂, . . . , q_n) ∈ Rⁿ be a vector of parameter. Then, the aggregated value by using the SLWMM operator is still a SLN and

$\begin{matrix} {SLWMM}^{Q} (α_{1}, α_{2}, . . ., α_{n}) \\ = (s_{{(\frac{1}{n!} \sum_{ϑ \in T_{n}} \prod_{j = 1}^{n} {({nw}_{ϑ (j)} θ_{ϑ (j)})}^{q_{j}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}}, \\ {(\sqrt{1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} {(1 - {(1 - μ_{ϑ (j)}^{2 q_{j}})}^{{nw}_{ϑ (j)}})}^{q_{j}}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}, \\ \sqrt{1 - {(1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} {(1 - η_{ϑ (j)}^{2 {nw}_{ϑ (j)}})}^{q_{j}}))}^{\frac{1}{n!}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}}, \\ \sqrt{1 - {(1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} {(1 - v_{ϑ (j)}^{2 {nw}_{ϑ (j)}})}^{q_{j}}))}^{\frac{1}{n!}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}}) \end{matrix}$ (16)

The proof of Theorem 2 is similar to that of Theorem 1. Similarly, the SLWMM operator also has the properties of monotonicity and boundedness.

3.3 The spherical linguistic dual Muirhead mean (SLDMM) operator

Definition 9. Let α_j = (s_{θ
_j}, μ_j, η_j, v_j) (j = 1, 2, …, n) be a collection of SLNs and Q = (q₁, q₂, . . . , q_n) ∈ Rⁿ be a vector of parameter. Then the SLDMM operator is defined as

$\begin{matrix} {SLDMM}^{Q} (α_{1}, α_{2}, . . ., α_{n}) \\ = \frac{1}{\sum_{j = 1}^{n} q_{j}} {(\prod_{ϑ \in T_{n}} \sum_{j = 1}^{n} (q_{j} α_{ϑ (j)}))}^{\frac{1}{n!}} \end{matrix}$ (17)

Where ϑ (j) (j = 1, 2, . . . , n) is any permutation of (1, 2, . . . , n), and S_n is the collection of all permutations of (1, 2, . . . , n).

Theorem 4. Let α_j = (s_{θ
_j}, μ_j, η_j, v_j) (j = 1, 2, . . . , n) be a collection of SLNs and Q = (q₁, q₂, . . . , q_n) ∈ Rⁿ be a vector of parameter. Then, the aggregated value by using the SLDMM operator is also a SLN, and

$\begin{matrix} {SLDMM}^{Q} (α_{1}, α_{2}, . . ., α_{n}) \\ = (s_{\frac{1}{\sum_{j = 1}^{n} q_{j}} {(\prod_{ϑ \in T_{n}} \sum_{j = 1}^{n} (q_{j} θ_{ϑ (j)}))}^{1 / n!}}, \\ \sqrt{1 - {(1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} {(1 - μ_{ϑ (j)}^{2})}^{q_{j}}))}^{\frac{1}{n!}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}}, \\ {(\sqrt{1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} η_{ϑ (j)}^{2 q_{j}}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}, \\ {(\sqrt{1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} v_{ϑ (j)}^{2 q_{j}}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{j = 1}^{n} q_{j}}}) \end{matrix}$ (18)

The proof of Theorem 4 is similar to that of Theorem 1. In what follows, we will discuss some special cases of the SLDMM operator through changing the values of parameter vector Q.

Special case 1. when Q = (1, 0, ⋯ , 0), then the SLMM operator reduces to the SLG operator, which is shown as Equation (14)

Special case 2. when Q = (1, 1, 0, 0, ⋯ , 0), then the SLMM operator reduces to the spherical linguistic geometric BM operator, i.e.

$\begin{matrix} {SLDMM}^{(1, 1, 0, 0, \dots, 0)} (α_{1}, α_{2}, \dots, α_{n}) \\ = (s_{\frac{1}{2} {({\prod_{i, j = 1}}_{i \neq j}^{n} (θ_{i} + θ_{j}))}^{\frac{1}{n (n + 1)}}}, \end{matrix}$

$\begin{matrix} \sqrt{1 - {(1 - {\prod_{i, j = 1}}_{i \neq j}^{n} {(1 - (1 - μ_{i}^{2}) (1 - μ_{j}^{2}))}^{\frac{1}{n (n - 1)}})}^{\frac{1}{2}}}, \\ {(\sqrt{1 - {({\prod_{i, j = 1}}_{i \neq j}^{n} (1 - η_{i}^{2} η_{j}^{2}))}^{\frac{1}{n (n - 1)}}})}^{\frac{1}{2}}, \\ {(\sqrt{1 - {({\prod_{i, j = 1}}_{i \neq j}^{n} (1 - v_{i}^{2} v_{j}^{2}))}^{\frac{1}{n (n - 1)}}})}^{\frac{1}{2}}) \end{matrix}$ (19)

Special case 3. when $Q = {\overset{︷}{(1, 1, \dots 1,}}^{k} {\overset{︷}{0, 0, \dots 0)}}^{n - k}$ , then the SLMM operator reduces to the spherical linguistic dual MSM operator, i.e. $\begin{matrix} {SLDMM}^{{\overset{︷}{(1, 1, \dots 1,}}^{k} {\overset{︷}{0, 0, \dots 0)}}^{n - k}} (α_{1}, α_{2}, \dots, α_{n}) \\ = (s_{\frac{1}{k} {(\prod_{1 \leq i_{1} < . . . i_{k} \leq n} (\sum_{j = 1}^{k} θ_{i_{j}}))}^{\frac{1}{C_{n}^{k}}}}, \\ \sqrt{1 - {(1 - {(\prod_{1 \leq i_{1} < . . . < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - μ_{i_{j}}^{2})))}^{1 / C_{n}^{k}})}^{1 / k}}, \\ {(\sqrt{1 - {(\prod_{1 \leq i_{1} < . . . < i_{k} \leq n} (1 - \prod_{j = 1}^{k} η_{i_{j}}^{2}))}^{1 / C_{n}^{k}}})}^{1 / k}, \\ {(\sqrt{1 - {(\prod_{1 \leq i_{1} < . . . < i_{k} \leq n} (1 - \prod_{j = 1}^{k} v_{i_{j}}^{2}))}^{1 / C_{n}^{k}}})}^{1 / k}) \end{matrix}$ (20)

Special case 5. when Q = (1, 1, ⋯ , 1) or Q = (1/ - n, 1/ - n, ⋯ , 1/ - n), then the SLMM operator reduces SLA operator, which is shown as Equation (11).

3.4 The spherical linguistic weighted dual Muirhead mean (SLWDMM) operator

Definition 10. Let α_j = (s_{θ
_j}, μ_j, η_j, v_j) (j = 1, 2, …, n) be a collection of SLNs and Q = (q₁, q₂, . . . , q_n) ∈ Rⁿ be a vector of parameter. Let w = (w₁, w₂, . . . , w_n) ^T be the weight vector, satisfying w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ . If

$S L W D M M^{Q} (α_{1}, α_{2}, \dots, α_{n}) = \frac{1}{\sum_{j = 1}^{n} q j} {(\prod_{ϑ \in T_{n}} \sum_{j = 1}^{n} (q j^{α_{ϑ (j)}^{n w ϑ (j)}}))}^{\frac{1}{n!}}$ (21)

Then SLWDMM^Q is the SLWDMM operator, where ϑ (j) (j = 1, 2, . . . , n) is any permutation of (1, 2, . . . , n), and T_n is the collection of all permutations of (1, 2, . . . , n).

Theorem 5. Let α_j = (s_{θ
_j}, μ_j, η_j, v_j) (j = 1, 2,. . . , n) be a collection of SLNs and Q = (q₁, q₂, . . . , q_n) ∈ Rⁿ be a vector of parameter. Then, the aggregated value by using the SLWDMM operator is still a SLN and $\begin{array}{l} S L W D M M^{Q} (α_{1}, α_{2}, \dots, α_{n}) = (\begin{array}{l} s \\ \frac{1}{\sum_{j = 1}^{n} q j} {(\prod_{ϑ \in T_{n}} \sum_{j = 1}^{n} (q j^{θ_{ϑ (j)}^{n w ϑ (j)}}))}^{\frac{1}{n!}} \end{array}, \\ \sqrt{1 - (1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} {(1 - μ_{ϑ (j)}^{2 n w ϑ (j)})}^{a j}))}^{\frac{1}{n!}}) \frac{1}{\sum_{j = 1}^{n} q j}}, \\ {(\sqrt{1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} {(1 - {(1 - η_{ϑ (j)}^{2})}^{n w ϑ (j)})}^{a j}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{j = 1}^{n} q j}}, \\ {(\sqrt{1 - {(\prod_{ϑ \in T_{n}} (1 - \prod_{j = 1}^{n} {(1 - {(1 - υ_{ϑ (j)}^{2})}^{n w ϑ (j)})}^{a j}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{j = 1}^{n} q j}}) \end{array}$ (22)

The proof of Theorem 5 is similar to that of Theorem 1. In addition, SLWDMM has the properties of monotonicity and boundedness.

4 A novel approach to MADM based on the proposed operators

In this section, we present a new MADM method based on the proposed operators.

4.1 Description of a typical MADM problem with spherical linguistic information

A typical MADM problem with spherical linguistic information can be described as follows. Suppose there are m alternatives A = {A₁, A₂, . . . , A_m} with n attributes G = {G₁, G₂, . . . , G_n}. Weight vector of attributes is w = (w₁, w₂, . . . , w_n) ^T, satisfying 0 ≤ w_j ≤ 1 and $\sum_{j = 1}^{n} w_{j} = 1$ . For attribute G_j (j = 1, 2, . . . , n) of alternative A_i (i = 1, 2, . . . , m), decision makers are required to utilize a SLN α = (s_{θ
_ij}, μ_ij, η_ij, v_ij) to express their evaluation value. Hence, a decision matrix can be denoted by R = (α_ij) _m×n. In the followings, we present a method for solving such a MADM based on the proposed operators.

4.2 An algorithm to spherical linguistic MADM problems

Step 1. Normalize original decision matrix according to the following formula. $α_{ij} = {\begin{matrix} (s_{θ_{ij}}, μ_{ij}, η_{ij}, v_{ij}) G_{j} is benefit type \\ (s_{θ_{ij}}, v_{ij}, μ_{ij}, η_{ij}) G_{j} is cos t type \end{matrix}$ (23)Step 2. For alternative A_i (i = 1, 2, . . . , m), utilize the SLWMM operator $α_{i} = {SLWMM}^{Q} (α_{i 1}, α_{i 2}, . . ., α_{in})$ (24) or the SLWDMM operator $α_{i} = {SLWDMM}^{Q} (α_{i 1}, α_{i 2}, . . ., α_{in})$ (25) to compute the comprehensive evaluation value.

Step 3. According to Definition 4, compute the score values of overall evaluation values.

Step 4. Rank alternatives according to their score values and select the optimal one.

5 An application of the proposed method in investment project selection

In this section, we apply the proposed method in an investment selection problem to demonstrate the effectiveness of our proposed method. Now an investment company’s wants to invest its money to a project. In order to obtain a stable return, this company invites a set of experts to evaluate four possible projects from four aspects. Alternatives can be denoted as {A₁, A₂, A₃, A₄} and the attributes are {G₁, G₂, G₃, G₄}, wherein G₁ is the reputation of project, G₂ denotes the ability of risk tolerance, G₃ represents the socio-economic influence, and G₄ is the environmental friendliness. The weight vector of attributes is w = (0.32, 0.26, 0.18, 0.24) ^T. Experts are required to use SLNs to express their evaluation information. Hence, a spherical linguistic decision matrix is shown in Table 1. In the followings, we determine the best alternative on the basis of the proposed method.

Table 1
Spherical linguistic decision matrix

G ₁ G ₂ G ₃ G ₄

A ₁ (s₅, 0.2, 0.1, 0.7) (s₃, 0.2, 0.2, 0.5) (s₅, 0.1, 0.3, 0.5) (s₃, 0.2, 0.2, 0.5)

A ₂ (s₄, 0.2, 0.2, 0.6) (s₅, 0.4, 0.1, 0.5) (s₄, 0.2, 0.3, 0.5) (s₄, 0.4, 0.1, 0.5)

A ₃ (s₃, 0.2, 0.1, 0.7) (s₄, 0.1, 0.2, 0.5) (s₄, 0.1, 0.2, 0.6) (s₅, 0.3, 0.3, 0.3)

A ₄ (s₆, 0.3, 0.2, 0.5) (s₃, 0.1, 0.2, 0.7) (s₄, 0.2, 0.1, 0.6) (s₃, 0.1, 0.4, 0.5)

	G ₁	G ₂	G ₃	G ₄
A ₁	(s₅, 0.2, 0.1, 0.7)	(s₃, 0.2, 0.2, 0.5)	(s₅, 0.1, 0.3, 0.5)	(s₃, 0.2, 0.2, 0.5)
A ₂	(s₄, 0.2, 0.2, 0.6)	(s₅, 0.4, 0.1, 0.5)	(s₄, 0.2, 0.3, 0.5)	(s₄, 0.4, 0.1, 0.5)
A ₃	(s₃, 0.2, 0.1, 0.7)	(s₄, 0.1, 0.2, 0.5)	(s₄, 0.1, 0.2, 0.6)	(s₅, 0.3, 0.3, 0.3)
A ₄	(s₆, 0.3, 0.2, 0.5)	(s₃, 0.1, 0.2, 0.7)	(s₄, 0.2, 0.1, 0.6)	(s₃, 0.1, 0.4, 0.5)

5.1 Decision-making process

Step 1. As all attributes are benefit, the original spherical linguistic decision matrix does not need to be normalized.

Step 2. Utilize the SLWMM operator to calculate the overall evaluation values of alternatives. Without loss if generality, suppose Q = (1, 1, 1, 1) and we can get $\begin{matrix} α_{1} = (s_{3.7932}, 0.1664, 0.2319, 0.5755); \\ α_{2} = (s_{4.1424}, 0.2799, 0.2120, 0.5405); \\ α_{3} = (s_{3.8549}, 0.1549, 0.2319, 0.5695); \\ α_{4} = (s_{3.7547}, 0.1549, 0.2693, 0.5978) . \end{matrix}$ Step 3. Calculate the scores of the overall values and we can derive $\begin{matrix} S (α_{1}) = 1.6088, S (α_{2}) = 1.8498, \\ S (α_{3}) = 1.6380, S (α_{4}) = 1.5644 \end{matrix}$ Step 4. Rank the suppliers according to the rank of overall values. That is A₂ > A₃ > A₁ > A₄, and A₂ is the best alternatives.

In Step 2, if we utilize the SLWDMM operator (suppose Q = (1, 1, 1, 1)) to aggregate decision makers’ preference information for each alternative, we can get $\begin{matrix} α_{1} = (s_{4.1579}, 0.2014, 0.1842, 0.5384); \\ α_{2} = (s_{4.4306}, 0.3358, 0.1549, 0.5180); \\ α_{3} = (s_{4.1568}, 0.2120, 0.1842, 0.4959); \\ α_{4} = (s_{4.1570}, 0.2120, 0.1979, 0.5635); \end{matrix}$

Hence, the scores of alternatives are $\begin{matrix} S (α_{1}) = 1.8198, S (α_{2}) = 2.0430, \\ S (α_{3}) = 1.8695, S (α_{4}) = 1.7952 \end{matrix}$

Thus, the ranking order of the alternatives is A₂ > A₃ > A₁ > A₄, which means that A₂ is the best alternatives.

5.2 The influence of the parameters on the results

In this subsection, we analyze the impacts of the parameter vector Q on the scores and ranking orders. Obviously, the parameter vector Q is very important for the aggregation results by the SLWMM and SLWDMM operators. To better investigate how the parameter Q influents the aggregation results, we assign different values to Q in the SLWMM and SLWDMM operators, and present the results in Tables 2 and 3.

Table 2
Scores and ranking orders with different parameter vector Q in the SLWMM operator

Q Score function S (α_i) (i = 1, 2, 3, 4) Ranking orders

Q = (1, 0, 0, 0) S (α₁) = 2.3439 S (α₂) = 2.6358 S (α₃) = 2.4039 S (α₄) = 2.3195 A₂ ≻ A₃ ≻ A₁ ≻ A₄

Q = (1, 1, 0, 0) S (α₁) = 2.0298 S (α₂) = 2.2985 S (α₃) = 2.0557 S (α₄) = 1.9885 A₂ ≻ A₃ ≻ A₁ ≻ A₄

Q = (1, 1, 1, 0) S (α₁) = 1.6448 S (α₂) = 1.8773 S (α₃) = 1.6698 S (α₄) = 1.6025 A₂ ≻ A₃ ≻ A₁ ≻ A₄

Q = (1, 1, 1, 1) S (α₁) = 1.6088 S (α₂) = 1.8498 S (α₃) = 1.6380 S (α₄) = 1.5644 A₂ ≻ A₃ ≻ A₁ ≻ A₄

Q	Score function S (α_i) (i = 1, 2, 3, 4)	Ranking orders
Q = (1, 0, 0, 0)	S (α₁) = 2.3439 S (α₂) = 2.6358 S (α₃) = 2.4039 S (α₄) = 2.3195	A₂ ≻ A₃ ≻ A₁ ≻ A₄
Q = (1, 1, 0, 0)	S (α₁) = 2.0298 S (α₂) = 2.2985 S (α₃) = 2.0557 S (α₄) = 1.9885	A₂ ≻ A₃ ≻ A₁ ≻ A₄
Q = (1, 1, 1, 0)	S (α₁) = 1.6448 S (α₂) = 1.8773 S (α₃) = 1.6698 S (α₄) = 1.6025	A₂ ≻ A₃ ≻ A₁ ≻ A₄
Q = (1, 1, 1, 1)	S (α₁) = 1.6088 S (α₂) = 1.8498 S (α₃) = 1.6380 S (α₄) = 1.5644	A₂ ≻ A₃ ≻ A₁ ≻ A₄

Table 3

Scores and ranking orders with different parameter vector Q in the SLWDMM operator

Q	Score function S (α_i) (i = 1, 2, 3, 4)	Ranking orders
Q = (1, 0, 0, 0)	S (α₁) = 2.2946 S (α₂) = 2.6873 S (α₃) = 2.3498 S (α₄) = 2.2201	A₂ ≻ A₃ ≻ A₁ ≻ A₄
Q = (1, 1, 0, 0)	S (α₁) = 2.1034 S (α₂) = 2.3844 S (α₃) = 2.1385 S (α₄) = 2.0528	A₂ ≻ A₃ ≻ A₁ ≻ A₄
Q = (1, 1, 1, 0)	S (α₁) = 1.8612 S (α₂) = 2.1008 S (α₃) = 1.8793 S (α₄) = 1.8285	A₂ ≻ A₃ ≻ A₁ ≻ A₄
Q = (1, 1, 1, 1)	S (α₁) = 1.8198 S (α₂) = 2.0430 S (α₃) = 1.8695 S (α₄) = 1.7952	A₂ ≻ A₃ ≻ A₁ ≻ A₄

As seen from Tables 2 and 3, an apparent fact is that with different parameter vector Q in the SLWMM and SLWDMM operators, different scores can be obtained. However, it is not difficult to find out that the ranking orders of alternatives are the same with different parameter vector Q. This phenomena demonstrates the robustness of our proposed method. In addition, the interrelationship among more attributes is considered, the decrease of the scores will become. Therefore, the parameter vector Q can be viewed as decision makers’ appetite towards to risk. Generally, decision makers can choose the value of Q according tom actual need. In addition, as discussed above the main flexibility of proposed method is that it has the power to capture any numbers of attributes. For instance, when Q = (1, 1, 0, 0), then our proposed method play the same role as the Bonferroni mean does, i.e. they consider the interrelationship among any two attributes. When Q = (1, 1, 1, 0), the proposed operators can capture the interrelationship among multiple attributes, which is similar to the Maclaurin symmetric mean operator. When Q = (1, 1, 1, 1), then the proposed method reflect the interrelationship among all attributes, which is the most distinct characteristic of the proposed method.

5.3 Advantages of the proposed method

To better illustrate the advantages and superiorities of our proposed method, we conduct comparative analysis. Details can be found in the following subsections.

5.3.1 The ability of capturing the interrelationship among attributes

We utilize the method proposed by Liu and Zhang [58] based on the Archimedean picture fuzzy linguistic weighted arithmetic averaging (A-PFLWAA) operator and our proposed method based on the SLWMM operator to solve the above investment project selection problem and present the decision results in Table 4.

Table 4
Scores and ranking orders of alternatives by different MADM methods

Method Score function S (α_i) (i = 1, 2, 3, 4) Ranking orders

The method proposed by Liu and Zhang [58] based on the A-PFLWAA operator S (α₁) = 1.4565 S (α₂) = 1.7081 A₂ ≻ A₃ ≻ A₄ ≻ A₁

(suppose g (x) = - log x) S (α₃) = 1.5976 S (α₄) = 1.4674

The method based on the SLWMM operator presented in this paper S (α₁) = 1.6088 S (α₂) = 1.8498 A₂ ≻ A₃ ≻ A₁ ≻ A₄

(suppose Q = (1, 1, 1, 1)) S (α₃) = 1.6380 S (α₄) = 1.5644

Method	Score function S (α_i) (i = 1, 2, 3, 4)	Ranking orders
The method proposed by Liu and Zhang [58] based on the A-PFLWAA operator	S (α₁) = 1.4565 S (α₂) = 1.7081	A₂ ≻ A₃ ≻ A₄ ≻ A₁
(suppose g (x) = - log x)	S (α₃) = 1.5976 S (α₄) = 1.4674
The method based on the SLWMM operator presented in this paper	S (α₁) = 1.6088 S (α₂) = 1.8498	A₂ ≻ A₃ ≻ A₁ ≻ A₄
(suppose Q = (1, 1, 1, 1))	S (α₃) = 1.6380 S (α₄) = 1.5644

As seen from Table 4, the best alternatives obtained by the method introduced by Liu and Zhang [58] and our developed method are the same, which proves the effectiveness of the newly presented method. Nonetheless, the ranking order of alternatives derived by Liu and Zhang’s [58] method is slightly different with that gained by our method. This is because Liu and Zhang’s [58] method is based on the assumption that attributes are independent. However, this assumption is usually unreliable in real MADM problem. As a matter of fact, instead of independence there always exists weak or strong interrelationship between attributes. In the above MADM problem, obviously there is interrelationship among the attributes G₁ and G₂, i.e. the increase of the ability of risk tolerance leads to the increase of projects’ reputation. Our proposed method is based on the SLWMM operator, which is capable to capture such kind of interrelationship between attributes, making it more suitable to deal with real decision making problems. Hence, our method is more powerful, suitable and flexible than Liu and Zhang’s [58] MADM method.

5.3.2 The greater freedom that provides for DMs

The constraint of PFLSs is that the sum of PDM, ND, and NMD of a LV should be less than or equal to one, i.e. μ + η + v ≤ 1. However, this restriction cannot be always strictly satisfied in practical MADM problems. Hence, PFLSs cannot comprehensively express attribute values proposed by DMs. To better demonstrate this situation, we replace the evaluation value of G₁ of alternative A₂ with (s₄, 0.4, 0.4, 0.6). Afterwards, we can compare the proposed method with that developed by Liu and Zhang [58]. The decision results are listed in Table 5.

Table 5
Scores and ranking orders of alternatives by different MADM methods based on revised decision matrix

Method Score function S (α_i) (i = 1, 2, 3, 4) Ranking orders

The method proposed by Liu and Zhang [58] based on the A-PFLWAA operator Cannot be calculated None

The method based on the SLWMM operator presented in this paper S (α₁) = 1.6088 S (α₂) = 1.8758 A₂ ≻ A₃ ≻ A₁ ≻ A₄

(suppose Q = (1, 1, 1, 1)) S (α₃) = 1.6380 S (α₄) = 1.5644

Method	Score function S (α_i) (i = 1, 2, 3, 4)	Ranking orders
The method proposed by Liu and Zhang [58] based on the A-PFLWAA operator	Cannot be calculated	None
The method based on the SLWMM operator presented in this paper	S (α₁) = 1.6088 S (α₂) = 1.8758	A₂ ≻ A₃ ≻ A₁ ≻ A₄
(suppose Q = (1, 1, 1, 1))	S (α₃) = 1.6380 S (α₄) = 1.5644

As seen in Table 5, the method introduced by Liu and Zhang [58] cannot deal with the revised decision matrix. This is due to the rigorous restriction and narrow scope of application of PFLSs. While our method can effectively deal with cases wherein μ + η + v ≥ 1 and μ² + η² + v² ≤ 1. Hence, our method is more flexible and can handle more complicated MADM problems and results in less information loss.

5.3.3 The capability of considering the NMD

Additionally, our proposed method is also more powerful than the method proposed by Peng and Yang [59]. This is because the Pythagorean fuzzy linguistic sets do not consider the degrees that DMs cannot express their information value. In other words, the Pythagorean fuzzy linguistic sets only consider PMDs, NDs while neglect the NMDs of LVs. The proposed SLSs not only consider the PMDs and NDs but also take the NMDs into account. Hence, SLS is more general and flexible than Pythagorean fuzzy linguistic set and Pythagorean fuzzy linguistic set can be regarded as a special case of SLS.

To sum up, the advantages of the proposed method are: (1) It considers the interrelationship among any numbers of attributes; (2) It takes into account not only DMs’ PMDs and NDs but also their NMDs, which can comprehensively describe DMs’ preference information; (3) It permits the square sum of the three information functions to be less than or equal to one, giving DMs more freedom to express their evaluation values and results in less information distortion. Hence, our method is more powerful and suitable to deal with real MADM problems. To better illustrate the advantages and superiorities of our proposed method, we list the main characteristics of some existing methods in Table 6.

Table 6
Characteristics of different methods

Method Whether consider Whether consider Whether allows the sum Whether considers the

the PMD and ND the NMD of information functions interrelationship

to be greater than one among attributes

The method proposed by Liu and Zhang [58] Yes Yes No No

The method presented by Peng and Yang [59] Yes No No No

The proposed method in this paper Yes Yes Yes No

Method	Whether consider	Whether consider	Whether allows the sum	Whether considers the
The method proposed by Liu and Zhang [58]	Yes	Yes	No	No
The method presented by Peng and Yang [59]	Yes	No	No	No
The proposed method in this paper	Yes	Yes	Yes	No

6 Conclusions

The paper proposed the SLSs by combing SFSs with LVs, which are more effective to deal with both DMs’ quantitative and qualitative evaluation information. Afterwards, we proposed a set of SL aggregation operators to deal with the interrelationship among SLNs. To make our proposed method more convenient to use, we gave an algorithm to solve MADM problems based on our proposed aggregation operators. An application in an investment selection problem has shown that our proposed method has good effectiveness. Given the good of performance of SLSs in representing DMs’ evaluation values, in the future we shall investigate more aggregation operators for SLNs.

Footnotes

Acknowledgments

This research is supported by National Natural Science Foundation of China (71532002) and a major project of the National Social Science Foundation of China (18ZDA086).

References

Li ,

R.T.

Zhang ,

Wang ,

X.P.

Shang and

K.Y.

Bai , A novel approach to multi-attribute group decision-making with q-rung picture linguistic information, Symmetry 10 (2018), 172.

K.Y.

Bai ,

X.M.

Zhu ,

Wang and

R.T.

Zhang , Some partitioned Maclaurin symmetric mean based on q-rung orthopair fuzzy information for dealing with multi-attribute group decision making, Symmetry 10 (2018), 383.

Wang ,

R.T.

Zhang ,

X.M.

Zhu ,

Y.P.

Xing and

Borut , Some hesitant fuzzy linguistic Muirhead means with their application to multiattribute group decision-making, Complexity 2018 (2018), 16. Article ID 5087851.

Garg , Hesitant Pythagorean fuzzy Maclaurin symmetric mean operators and its applications to multiattribute decision-making process, International Journal of Intelligent Systems 34 (2018), 601–626.

Garg and

Arora , Maclaurin symmetric mean aggregation operators based on t-norm operations for the dual hesitant fuzzy soft set, Journal of Ambient Intelligence and Humanized Computing (2019), 1–36.

S.P.

Wan and

D.F.

Li , Fuzzy LINMAP approach to heterogeneous MADM considering comparisons of alternatives with hesitation degrees, Omega: The International Journal of management Science 41 (2013), 925–940.

D.F.

Li and

S.P.

Wan , A fuzzy inhomogenous multi-attribute group decision making approach to solve outsourcing provider selection problems, Knowledge-Based Systems 67 (2014), 71–89.

G.F.

Yu ,

Fei and

D.F.

Li , A compromise-typed variable weight decision method for hybrid multi-attribute decision making, IEEE Transactions on Fuzzy Systems 27 (2019), 861–872.

G.F.

Yu ,

D.F.

Li and

Fei , A novel method for heterogeneous multi-attribute group decision making with preference deviation, Computers & Industrial Engineering 124 (2018), 58–64.

10.

Lopez-Ortega and

Castro-Espinoza , Fuzzy similarity metrics and their application to consensus reaching in group decision making, Journal of Intelligent and Fuzzy Systems 36 (2019), 3095–3104.

11.

Ullah ,

Garg ,

Mahmood ,

Jan and

Ali , Correlation coefficients for T-spherical fuzzy sets and their applications in clustering and multi-attribute decision making, Soft Computing. DOI: 10.1007/s00500-019-03993-6

12.

Garg ,

Munir ,

Ullah ,

Mahmood and

Jan , Algorithm for T-Spherical fuzzy multi-attribute decision making based on improved interactive aggregation operators, Symmetry 10 (2018), 670.

13.

D.F.

Li and

S.P.

Wan , Fuzzy heterogeneous multiattribute decision making method for outsourcing provider selection, Expert Systems with Applications 41 (2014), 3047–3059.

14.

D.F.

Li ,

G.H.

Chen and

Z.G.

Huang , Linear programming method for multiattribute group decision making using IF sets, Information Sciences 180 (2010), 1591–1609.

15.

Garg and

Kumar , Linguistic interval-valued Atanassov intuitionistic fuzzy sets and their applications to group decision-making problems, IEEE Transactions on Fuzzy Systems (2019), 1–1.

16.

R.R.

Yager , Pythagorean membership grades in multi-criteria decision making, IEEE Transactions on Fuzzy Systems 22 (2014), 958–965.

17.

X.L.

Zhang and

Z.S.

Xu , Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets, International Journal of Intelligent Systems 29 (2014), 1061–1078.

18.

X.L.

Zhang , Multicriteria Pythagorean fuzzy decision analysis: A hierarchical QUALIFLEX approach with the closeness index-based ranking methods, Information Sciences 330 (2016), 104–124.

19.

Garg , A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem, Journal of Intelligent & Fuzzy Systems 31 (2016), 529–540.

20.

X.L.

Zhang , A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making, International Journal of Intelligent Systems 31 (2016), 593–611.

21.

P.J.

Ren ,

Z.S.

Xu and

X.J.

Gou , Pythagorean fuzzy TODIM approach to multi-criteria decision making, Applied Soft Computing 42 (2016), 246–259.

22.

X.J.

Gou ,

Z.S.

Xu and

P.J.

Ren , The properties of continuous Pythagorean fuzzy information, International Journal of Intelligent Systems 31 (2016), 401–424.

23.

Garg , A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decision-making processes, International Journal of Intelligent Systems 31 (2016), 1234–1252.

24.

X.M.

Zhu ,

K.Y.

Bai ,

Wang ,

R.T.

Zhang and

Y.P.

Xing , Pythagorean fuzzy interaction power partitioned Bonferroni means with applications to multi-attribute group decision making, Journal of Intelligent & Fuzzy Systems 36 (2019), 3423–3438.

25.

R.T.

Zhang ,

Wang ,

X.M.

Zhu ,

M.M.

Xia and

Yu , Some generalized Pythagorean fuzzy Bonferroni mean aggregation operators with their application to multiattribute group decision-making, Complexity 2017 (2017), 1–16.

26.

G.W.

Wei and

Lu , Pythagorean fuzzy Maclaurin symmetric mean operators in multiple attribute decision making, International Journal of Intelligent Systems 33 (2018), 1043–1070.

27.

Xu ,

X.P.

Shang and

Wang , Pythagorean fuzzy interaction Muirhead means with their application to multi-attribute group decision-making, Information 9 (2018), 157.

28.

Li ,

R.T.

Zhang ,

Wang , X.,

Zhu and

Y.P.

Xing , Pythagorean fuzzy power Muirhead mean operators with their application to multi-attribute decision making, Journal of Intelligent and Fuzzy Systems 35 (2018), 2035–2050.

29.

G.W.

Wei and

Lu , Pythagorean fuzzy power aggregation operators in multiple attribute decision making, International Journal of Intelligent Systems 33 (2018), 169–186.

30.

Biswas and

Sarkar , Pythagorean fuzzy multicriteria group decision making through similarity measure based on point operators, International Journal of Intelligent Systems 33 (2018), 1731–1744.

31.

Garg , A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making, International Journal of Intelligent Systems 31 (2016), 886–920.

32.

Rahman ,

Abdullah ,

Ahmed and

Ullah , Pythagorean fuzzy Einstein weighted geometric aggregation operator and their application to multiple attribute group decision making, Journal of Intelligent & Fuzzy Systems 33 (2017), 635–647.

33.

Garg , Generalized Pythagorean fuzzy geometric aggregation operators using Einstein t-norm and t-conorm for multicriteria decision-making process, International Journal of Intelligent Systems 32 (2017), 597–630.

34.

Z.M.

Ma and

Z.S.

Xu , Symmetric Pythagorean fuzzy weighted geometric/averaging operators and their application in multicriteria decision-making problems, International Journal of Intelligent Systems 31 (2016), 1198–1219.

35.

Wei , Pythagorean fuzzy interaction aggregation operators and their application to multiple attribute decision making, Journal of Intelligent & Fuzzy Systems 33 (2017), 2119–2132.

36.

Gao , Pythagorean fuzzy Hamacher prioritized aggregation operators in multiple attribute decision making, Journal of Intelligent & Fuzzy Systems 35 (2018), 2229–2245.

37.

Garg , New exponential operational laws and their aggregation operators for interval-valued Pythagorean fuzzy multicriteria decision-making, International Journal of Intelligent Systems 33 (2018), 653–683.

38.

Y.P.

Xing ,

R.T.

Zhang ,

Wang and

X.M.

Zhu , Some new Pythagorean fuzzy Choquet-Frank aggregation operators for multi-attribute decision making, International Journal of Intelligent Systems 33 (2018), 2189–2215.

39.

B.C.

Cuong , Picture fuzzy sets -first results. part 1, seminar neuro-fuzzy systems with applications. Tech. rep., Instiute of Mathematics, Hanoi, 2013.

40.

B.C.

Cuong , Picture fuzzy sets -first results. part 2, seminar neuro-fuzzy systems with applications. Tech. rep., Instiute of Mathematics, Hanoi, 2013.

41.

G.W.

Wei , Some similarity measures for picture fuzzy sets and their applications, Iranian Journal of Fuzzy Systems 15 (2018), 77–89.

42.

G.W.

Wei and

Gao , The generalized Dice similarity measures for picture fuzzy sets and their applications, Informatica 29 (2018), 107–124.

43.

G.W.

Wei , Some cosine similarity measures for picture fuzzy sets and their applications to strategic decision making, Informatica 28 (2017), 547–564.

44.

G.W.

Wei ,

F.E.

Alsaadi ,

Hayat and

Alsaedi , Projection models for multiple attribute decision making with picture fuzzy information, International Journal of Machine Learning and Cybernetics 9 (2018), 713–719.

45.

G.W.

Wei , TODIM method for picture fuzzy multiple attribute decision making, Informatica 29 (2018), 555–566.

46.

G.W.

Wei , Picture fuzzy aggregation operators and their application to multiple attribute decision making, Journal of Intelligent & Fuzzy Systems 33 (2017), 713–724.

47.

G.W.

Wei , Picture fuzzy Hamacher aggregation operators and their application to multiple attribute decision making, Fundamenta Informaticae 157 (2018), 271–320.

48.

Jana ,

Senapati ,

Pal and

R.R.

Yager , Picture fuzzy Dombi aggregation operators: Application to MADM process, Applied Soft Computing 74 (2019), 99–109.

49.

Xu ,

X.P.

Shang ,

Wang ,

R.T.

Zhang ,

W.Z.

Li and

Y.P.

Xing , A method to multi-attribute decision making with picture fuzzy information based on Muirhead mean, Journal of Intelligent & Fuzzy Systems (2018) (Preprint), 1–17.

50.

H.R.

Zhang ,

R.T.

Zhang ,

H.Q.

Huang and

Wang , Some picture fuzzy dombi heronian mean operators with their application to multi-attribute decision-making, Symmetry 10 (2018), 593.

51.

Garg , Some picture fuzzy aggregation operators and their applications to multicriteria decision-making, Arabian Journal for Science and Engineering 42 (2017), 5275–5290.

52.

Ashraf ,

Abdullah ,

Mahmood ,

Ghani and

F.T.

Mahmood , Spherical fuzzy sets and their applications in multi-attribute decision making problems, Journal of Intelligent & Fuzzy Systems (Preprint), 1–16.

53.

F.K.

Gündoğdu and

Kahraman , Spherical fuzzy sets and spherical fuzzy TOPSIS method, Journal of Intelligent & Fuzzy Systems 36 (2019), 337–352.

54.

Ashraf and

Abdullah , Spherical aggregation operators and their application in multiattribute group decision-making, International Journal of Intelligent Systems 34 (2019), 493–523.

55.

Li ,

R.T.

Zhang ,

Wang and

X.P.

Shang , Some q-rung orthopair linguistic Heronian mean operators with their application to multi-attribute group decision making, Archives of Control Sciences 28 (2018), 551–583.

56.

Li ,

R.T.

Zhang ,

Wang ,

X.P.

Shang and

K.Y.

Bai , A novel approach to multi-attribute group decision-making with q-rung picture linguistic information, Symmetry 10 (2018), 172.

57.

Wang ,

R.T.

Zhang ,

Li ,

X.M.

Zhu and

X.P.

Shang , A novel approach to multi-attribute group decision making based on q-rung orthopair uncertain linguistic information, Journal of Intelligent and Fuzzy Systems. DOI: 10.3390/sym10050172

58.

P.D.

Liu and

X.H.

Zhang , A novel picture fuzzy linguistic aggregation operator and its application to group decision-making, Cognitive Computation 10 (2018), 242–259.

59.

X.D.

Peng and

Yang , Multiple attribute group decision making methods based on Pythagorean fuzzy linguistic set, Computer and Engineering Application 52 (2016), 50–54.

60.

R.F.

Muirhead , Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters, Proceedings of the Edinburgh Mathematical Society 21 (1902), 144–162.

61.

Garg , Multi-criteria decision-making method based on prioritized Muirhead mean aggregation operator under neutrosophic set environment, Symmetry 10 (2018), 280.

62.

Xu ,

X.P.

Shang ,

Wang ,

H.M.

Zhao ,

R.T.

Zhang and

K.Y.

Bai , Some interval-valued q-rung dual hesitant fuzzy Muirhead mean operators with their application to multi-attribute decision-making, IEEE Access 7 (2019), 2169–3536.

63.

Wang ,

R.T.

Zhang ,

X.M.

Zhu ,

Zhou ,

Z.P.

Shang and

W.Z.

Li , Some q-rung orthopair fuzzy Muirhead means with their application to multi-attribute group decision making, Journal of Intelligent and Fuzzy Systems 36 (2019), 1599–1614.

64.

Liu and

D.F.

Li , Some Muirhead mean operators for intuitionistic fuzzy numbers and their applications to group decision making, Plos One 12 (2017), Article ID: e0168767.