New extension of TOPSIS method based on Pythagorean hesitant fuzzy sets with incomplete weight information

Abstract

Pythagorean Hesitant fuzzy set (PHFS) which permits the membership degree and non-membership degree of an element to a set represented by several possible values is deliberated as a powerful tool to express uncertain information in the process of multi-attribute decision making (MADM) problems. In this paper, we propose a novel approach based on TOPSIS method and the maximizing deviation method for solving MADM problems where the evaluation information provided by the decision makers (DMs) is expressed in form of Pythagorean hesitant fuzzy numbers and the information about attribute weights is incomplete. To determine the attribute weight we develop an optimization model based on maximizing deviation method. Finally we provide a practical decision-making problem to demonstrate the implementation process of the proposed method.

Keywords

Pythagorean hesitant fuzzy set maximizing deviation method TOPSIS method

1. Introduction

The researches which are keen on contributing to socially relevant research either through pure or applied streams are looking forward to solve real life problems. Multi-attribute decision making (MADM) which addresses the problem of making a finest choice that has the highest degree of satisfaction from a set of alternatives that are characterized in terms of their attributes, is a susual task in human activities and these kinds of MADM problems arise in many real-world situations. It is a major component of decision science, whose theory has been widely applied in the fields of economy [19, 27], management [12, 30], engineering [10, 11], etc. Many approaches have been proposed to handle the MADM problems, such as TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) [13], ELECTRE [28], and PROMETHEE [18]. In classical MADM, the assessments of alternatives are surely known [6, 31]. In addition, for the uncertainty of the MADM problems, the DMs are difficult to provide the precise assessments for alternatives. To solve this issue, fuzzy set theory [43] has been applied to MADM [4, 20], which provides a crucial means of describing the complex information. However, the fuzzy set theory is still confronted with some limitations when decision makers (DMs) intend to deal with some uncertain information induced from several sources of vagueness, the attributes involved in decision making problems are not always expressed in real numbers, and some are better suited to be denoted by fuzzy values, such as interval values (IVs) [5, 46], linguistic variables (LVs) [7, 21], intuitionistic fuzzy values (IFVs) [1, 37], and hesitant fuzzy elements (HFEs) [32, 33], Dual hesitant fuzzy elements (DHFEs) [48], Pythagorean fuzzy values (PFVs) [40, 41], just to mention a few. Many methods for MADM, such as the TOPSIS method [14, 39], the maximizing deviation method [35], have been extended to take different types of attribute values into account, such as interval values, linguistic variables, intuitionistic fuzzy values, Pythagorean fuzzy values, Dual hesitant fuzzy elements [48], intuitionistics hesitant fuzzy values. All of the above methods, however, have not yet been accommodated to fit the Pythagorean hesitant fuzzy information provided by the decision makers (DMs).

Among the various extension forms of fuzzy set theory, intuitionistic fuzzy set (IFS), due to Atanassov is generally assumed as an intuitively straightforward extension of fuzzy set theory [1, 2]. The IFS is characterized by a membership degree and a nonmembership degree meaning the degree of one alternative satisfies a certain criterion and dissatisfies the same criterion respectively. Moreover the membership degree and a nonmembership degree in IFS satisfy the condition that the sum of membership degree and a nonmembership degree is equal or less than 1. Based on the theory of IFS as a brand-new extension of fuzzy set, Yager proposed the concept of Pythagorean fuzzy set (PFS) [40], which has been proved useful to deal with uncertain information in decision-making procedures. Yager [42] introduced a series of aggregation operations based on Pythagorean fuzzy information for MADM problems. Yager and Abbasov [39] presented the Pythagorean membership grades and the relationship between the Pythagorean membership grades and the complex numbers. Zhang and Xu [45] extended the technique for order preference by similarity to an ideal solution (TOPSIS) method to MADM and some operation rules, the score function, and the distance measure were discussed. Peng and Yong [25] and Gou et al. [9] proposed the subtraction and division operations of Pythagorean fuzzy numbers (PFNS), some characteristics of the Pythagorean fuzzy aggregation operators were also discussed. Zhang [47] analyzed the continuity and derivability of Pythagorean fuzzy information and extended PFS to interval-valued Pythagorean fuzzy sets (IVPFSs). Peng and Yong [24] developed interval-valued Pythagorean fuzzy weighted average operator, interval-valued Pythagorean fuzzy weighted geometric operator, interval valued Pythagorean fuzzy point operators, interval-valued Pythagorean fuzzy point weighted averaging operators, and proposed an interval-valued Pythagorean fuzzy ELECTRE (ELimination Et Choix Traduisant la REalit’e) method to solve the decision-making problem. Liang and Zhang [17] developed a maximizing deviation method to solve decision-making problems under interval-valued Pythagorean fuzzy circumstances. Zeng [44] developed a Pythagorean fuzzy multi-attribute group decision making (MAGDM) method based on probabilistic information and the ordered weighted averaging (OWA) approach. Garg [8] proposed an improved accuracy function for the ranking order of interval-valued Pythagorean fuzzy sets (IVPFSs). Khan et al. [52], introduced the TOPSIS method for MADM problems based on Choquet integral with interval-valued Pythagorean fuzzy setting. In [51] Khan and Abdullah proposed an interval-valued Pythagorean fuzzy gray relational analysis (GRA) method for MADM problems.

Hesitant fuzzy sets (HFSs), which are another extension of fuzzy sets, are highly useful in handling situations where people are hesitant in providing their preferences with regard to objects in a decision-making process and have provided a theory for solving MADM problems in certain situations. HFS was first introduced by Torra and Narukawa [32] and Torra [33], and permits the membership degree of an element to be a set of several possible values between 0 and 1. To date, HFSs have been the subject of a great deal of research. For example Xu and Zhang, [38] developed a novel approach based on TOPSIS and the maximizing deviation method for solving MADM problems, in which the evaluation information provided by the decision maker is expressed in hesitant fuzzy elements and the information about attribute weights is incomplete.

In [26], Qian et al. generalized the notion of hesitant fuzzy set (HFS) with intuitionistic fuzzy sets (IFSs) and referred to them as generalized hesitant fuzzy set (GHFS), which, in essence, extended the element of HFSs from a real number to intuitionistic fuzzy numbers (IFNs). Zhu et al. in [28] developed the concept of dual hesitant fuzzy set (DHFS) and also discussed their basic operations and properties. In [34] the authors proposed a variety of distance measures for dual hesitant fuzzy sets. The authors investigated the connections of the aforementioned distance measures and further developed a number of dual hesitant ordered weighted distance measures. They also developed a TOPSIS approach based on proposed distance measures for the weapon selection problem. Peng et al. [22] introduced a MADM approach with hesitant interval-valued intuitionistic fuzzy sets (HIVIFSs) which are an extension of dual IVHFSs. However dual HFSs are defined in terms of sets of values as opposed to precise numbers for the membership degrees and non-membership degrees of IFSs. In [23] the authors apply the concept of Intuitionistic hesitant fuzzy set (IHFS) to group decision making (GDM) problems using fuzzy cross-entropy. Beg and Rashid [3] introduced the concept of intuitionistic hesitant fuzzy sets (IHFS) by merging the concept of IFS and HFS. It helps to manage those situations of uncertainty in which some values are possible as membership values of element as well as non-membership values of the same element. They introduced the concept of distance between IHFS. Based on this distance they developed fuzzy TOPSIS for IHFS. However there may be a situation where the decision maker may provide the degree of membership and nonmembership of a particular attribute in such a way that their sum is greater than 1. To overcome this shortcoming Khan et al. [29] introduced the concept of Pythagorean hesitant fuzzy set as a generalization of the concept of intuitionistic hesitant fuzzy set under the restriction that the square sum of its membership degrees is less than or equal to 1. The authors discussed some basic operational laws and developed some aggregation operators under Pythagorean hesitant fuzzy environments.

Due the inspiration and motivation of the above discussion in this paper a novel decision making method based on the TOPSIS will be developed to deal effectively with the MADM problems with Pythagorean hesitant fuzzy information. To do so the remainder of this paper is organized as follows: In Section 2 we review briefly some basic definitions. In Section 3 we develop a Pythagorean hesitant fuzzy TOPSIS approach to solve the MADM problems with PHFSs. In Section 4 we provide a practical decision-making problem to demonstrate the implementation process of the proposed method and to conduct a comparison analysis. Section 5 presents our conclusions.

2. Basic concepts

In this section, we introduce some basic definition and properties.

Definition 2.1. [41] Let X be a fixed set. Then a Pythagorean fuzzy set (PFS), P in X can be defined as follows: $P = {〈 x, h_{P} (x), h_{P}^{'} (x) 〉 | x \in X},$ (1) where h _P (x) and $h_{P}^{'} (x)$ are mappings from X to [0, 1], such that 0 ≤ h _P (x) ≤1, $0 \leq h_{P}^{'} (x) \leq 1$ and also $0 \leq h_{P}^{2} (x) + h_{P}^{' 2} (x) \leq 1,$ for all x ∈ X, here h _P (x) and h′ (x) denotes the membership degree and nonmembership degree of element x ∈ X to set $P$ , respectively. Let $π_{P} (x) = \sqrt{1 - h_{P}^{2} (x) - h_{P}^{' 2} (x)}$ . Then it is commonly called the Pythagorean fuzzy index of element x ∈ X to set $P$ , representing the degree of indeterminacy of x to P. Also, 0 ≤ π _P (x) ≤1, for every x ∈ $X$ . We denote the Pythagorean fuzzy number (PFN) by $\hat{h} = 〈 Λ_{\hat{h}}, Γ_{\hat{h}} 〉 .$

Definition 2.2. [33] Let X be fixed set. Then a hesitant fuzzy set(HFS) on X is a function from X to a subset of [0, 1]. To understand hesitant fuzzy set we write this mathematically as follows: $H = {〈 x, h_{H} (x) 〉 | x \in X}$ (2) where h _H (x) denotes the set of some values belonging to [0, 1], that is the possible membership degree of the element x ∈ X to the set H. For convenience we denote a hesitant fuzzy number (HFN) by h = h _H (x) and HFN the set of all HFNs.

Definition 2.3. [29] Let X be a fixed set. A Pythagorean hesitant fuzzy set abbreviated as PHFS P _H in X is an object with the following notion. $P_{H} = {〈 x, Λ_{P_{H}} (x), Γ_{P_{H}} (x) | x \in X 〉}$ (3) where Λ _{P
_H} (x) and Γ _{P
_H} (x) are mappings from X to [0, 1], denoting a possible degree of membership and non-membership degree of element x ∈ X in P _H respectively, and for each element x ∈ X, ∀ h _{P
_H} (x) ∈ Λ _{P
_H} (x), $\exists {h^{'}}_{P H} (x) \in Γ_{P H} (x)$ such that $0 \leq h_{P H}^{2} (x) + {h^{'}}_{P H}^{2} (x) \leq 1$ , and ∀ $h_{P_{H}}^{'} (x) \in Γ_{P_{H}} (x)$ , ∃h _{P
_H} (x) ∈ Λ _{P
_H} (x) such that $0 \leq h_{P H}^{2} (x) + {h^{'}}_{P H}^{2} (x) \leq 1$

Moreover, PHFS(X) denotes the set of all elements of PHFSs. If X has only one element 〈x, Λ _{P
_H} (x), Γ _{P
_H} (x)〉 is said to be Pythagorean hesitant fuzzy number and is denoted by $\hat{h} = 〈 Λ_{\hat{h}}, Γ_{\hat{h}} 〉$ for convenience. We denote the set of all PHFNs by PHFNS. For all x ∈ X if Λ _{P
_H} (x) and Γ _{P
_H} (x) have only element. Then the PHFSs become a PFSs. If the non-membership degree is {0 }, then PHFSs become a HFSs. For any PHFS P _H = {〈x, Λ _{P
_H} (x), Γ _{P
_H} (x) |x ∈ X〉} and for all x ∈ X,

$\prod_{P H} (x) = \cup_{h_{P H}} \in Λ_{P H}, {h^{'}}_{P H} \in Γ_{P H} \sqrt{1 - h_{P H}^{2} - {h^{'}}_{P H}^{2}}$ is said to be the degree of indetrminacy of x to P _H, where $1 - h_{P H}^{2} - {h^{'}}_{P H}^{2} \geq 0$ .

Definition 2.4. Let ${\hat{h}}_{i} = (Λ_{{\hat{h}}_{i}}, Γ_{{\hat{h}}_{i}}) (i = 1, 2)$ be two PHFNs, a nature quasi-ordering on the PHFNs is defined as follows: ${\hat{h}}_{1} \geq {\hat{h}}_{2}$ if and only if $Λ_{{\hat{h}}_{1}} \geq Λ_{{\hat{h}}_{2}}$ and $Γ_{{\hat{h}}_{1}} \leq Γ_{{\hat{h}}_{2}} .$

Definition 2.5. [29] Let $\hat{h} = (Λ_{\hat{h}}, Γ_{\hat{h}})$ be a PHFN. Then we define the score function of $\hat{h}$ as follows: $S (\hat{h}) = (\frac{1}{l_{h_{\hat{h}} \in Λ_{\hat{h}}}} \sum_{h_{\hat{h}} \in Λ_{\hat{h}}} h_{\hat{h}})^{2} - (\frac{1}{l_{h_{\hat{h}}^{'} \in Γ_{\hat{h}}}} \sum_{h_{\hat{h}}^{'} \in Γ_{\hat{h}}} h_{\hat{h}}^{'})^{2}$ (4) where $S (\hat{h}) \in [- 1, 1]$ , $l_{h_{\hat{h}}}$ denotes the number of elements in $Λ_{\hat{h}}$ and $l_{h_{\hat{h}}^{'}}$ denotes the number of elements in $Γ_{\hat{h}}$ .

Definition 2.6. [29] Let, $\hat{h} = 〈 Λ_{\hat{h}}, Γ_{\hat{h}} 〉$ be a PHFN. Then the deviation degree of $\hat{h}$ is denoted by $\bar{σ} (\hat{h})$ and can be defined as follows: $\begin{matrix} \bar{σ} (\hat{h}) = {(\frac{1}{l_{h_{\hat{h}} \in Λ_{\hat{h}}}} \sum_{h_{\hat{h}} \in Λ_{\hat{h}}} h_{\hat{h}} - S (\hat{h}))}^{2} \\ + {(\frac{1}{l_{h_{\hat{h}}^{'} \in Γ_{\hat{h}}}} \sum_{h_{\hat{h}}^{'} \in Γ_{\hat{h}}} h_{\hat{h}}^{'} - S (\hat{h}))}^{2} \end{matrix}$ (5)

Here we can see that $S (\hat{h})$ is just the mean value in statistics, and $\bar{σ} (\hat{h})$ is just the standard variance, which reflects the deviation degree between all values in the PHFN $\hat{h}$ and their mean value. Inspired by this idea, based on the score $S (\hat{h})$ and the deviation degree $\bar{σ} (\hat{h})$ , we can comparison and rank, two PHFNs in the following as:

Definition 2.7. [29] Let ${\hat{h}}_{1}$ and ${\hat{h}}_{2}$ be two PHFNs, $S ({\hat{h}}_{1})$ be the score of ${\hat{h}}_{1}$ , $S ({\hat{h}}_{2})$ be the score of ${\hat{h}}_{2}$ , and $\bar{σ} ({\hat{h}}_{1})$ be the deviation degree of ${\hat{h}}_{1}$ , $\bar{σ} ({\hat{h}}_{2})$ be the deviation degree of ${\hat{h}}_{2}$ . Then

If $S ({\hat{h}}_{1}) < S ({\hat{h}}_{2})$ , then ${\hat{h}}_{1} < {\hat{h}}_{2}$ .

If $S ({\hat{h}}_{1}) > S ({\hat{h}}_{2})$ , then ${\hat{h}}_{1} > {\hat{h}}_{2}$ .

If $S ({\hat{h}}_{1}) = S ({\hat{h}}_{2})$ , then ${\hat{h}}_{1} \sim {\hat{h}}_{2}$ .

If $\bar{σ} ({\hat{h}}_{1}) < \bar{σ} ({\hat{h}}_{2})$ , then ${\hat{h}}_{1} < {\hat{h}}_{2}$ .

If $\bar{σ} ({\hat{h}}_{1}) > \bar{σ} ({\hat{h}}_{2})$ , then ${\hat{h}}_{1} > {\hat{h}}_{2}$ .

If $\bar{σ} ({\hat{h}}_{1}) = \bar{σ} ({\hat{h}}_{2})$ , then ${\hat{h}}_{1} \sim {\hat{h}}_{2}$ .

3. Pythagorean hesitant fuzzy TOPSIS for MAGDM based on maximizing deviation method

In this In this section, we introduce the concept of distance measure and similarity measure for PHFSs. We give example for the proposed distance. Subsequently, the extension of TOPSIS method, in which the proposed distance measure is used to calculate separation measures, to Pythagorean hesitant fuzzy set (PHFS) environment is demonstrated to solve multi-criteria group decision making (MCGDM) problems using optimal criteria weights determined with linear programming model based on the concept of maximizing relative closeness coefficient. This section puts forward a framework for determining attribute weights and the ranking orders for all the alternatives with incomplete weight information under hesitant fuzzy environment.

3.1. Distance measure between Pythagorean hesitant fuzzy numbers

In this section we propose the distance measure between Pythagorean hesitant fuzzy numbers.

Definition 3.1.1. Let ${\hat{h}}_{1}, {\hat{h}}_{2}$ and ${\hat{h}}_{3}$ be three PHFSs in X. A distance measure $D ({\hat{h}}_{1}, {\hat{h}}_{2})$ is a mapping D : PHFS × PHFS → [0, 1], satisfies the following properties: $\begin{matrix} (d_{1}) 0 \leq D ({\hat{h}}_{1}, {\hat{h}}_{2}) \leq 1; \\ (d_{2}) D ({\hat{h}}_{1}, {\hat{h}}_{2}) = D ({\hat{h}}_{2}, {\hat{h}}_{1}), \\ (d_{3}) D ({\hat{h}}_{1}, {\hat{h}}_{2}) = 0 if and only if {\hat{h}}_{1} = {\hat{h}}_{2} . \\ (d_{4}) If {\hat{h}}_{1} \subseteq {\hat{h}}_{2} \subseteq {\hat{h}}_{3}, thenD ({\hat{h}}_{1}, {\hat{h}}_{2}) \leq D ({\hat{h}}_{1}, {\hat{h}}_{3}) \\ and D_{1} ({\hat{h}}_{2}, {\hat{h}}_{3}) \leq D_{1} ({\hat{h}}_{1}, {\hat{h}}_{3}) \end{matrix}$

Definition 3.1.2. Let ${\hat{h}}_{1}$ and ${\hat{h}}_{2}$ be two PHFNs on a set X ={ x ₁, x ₂, …, x _n }. Then we define the distance measure between ${\hat{h}}_{1}$ and ${\hat{h}}_{2}$ as follows: $D ({\hat{h}}_{1}, {\hat{h}}_{2}) = \frac{1}{2} (\begin{array}{l} \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(h_{{\hat{h}}_{1}}^{_{σ (i)}} (x_{i}))}^{2} - {(h_{{\hat{h}}_{2}}^{_{σ (i)}} (x_{i}))}^{2} | + \\ \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(h_{{\hat{h}}_{1}}^{'_{σ (i)}} (x_{i}))}^{2} - {(h_{{\hat{h}}_{2}}^{'_{σ (i)}} (x_{i}))}^{2} | + \\ \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(π_{{\hat{h}}_{1}}^{_{σ (i)}} (x_{i}))}^{2} - {(π_{{\hat{h}}_{2}}^{_{σ (i)}} (x_{i}))}^{2} | \end{array}) .$

where, $h_{{\hat{h}}_{1}}^{_{σ (k)}} (x_{i})$ , $h_{{\hat{h}}_{1}}^{'_{σ (k)}}$ , $h_{{\hat{h}}_{2}}^{_{σ (k)}}$ , $h_{{\hat{h}}_{2}}^{'_{σ (k)}}$ , be the i^th largest value in ${\hat{h}}_{1}$ and ${\hat{h}}_{2}$ respectively

Since the number of elements for different PHFNs could be different, we can make them equivalent through adding elements to the PHFN that has a less number of elements. We can add the smallest element in terms of pessimistic principle while the opposite case will be adopted in optimistic principle.

Example 3.1.3. Let ${\hat{h}}_{1} = 〈 {0.5, 0.6}, {0.5, 0.7, 0.8} 〉$ and ${\hat{h}}_{2} = 〈 {0.4, 0.5, 0.6}, {0.7, 0.8} 〉$ be two PHFNs. Then

D ({\hat{h}}_{1}, {\hat{h}}_{2}) = \frac{1}{2} (\begin{array}{l} \frac{1}{3} (| {0.5}^{2} - {0.4}^{2} | + | {0.5}^{2} - {0.5}^{2} | + | {0.6}^{2} - {0.6}^{2} |) + \\ \frac{1}{3} (| {0.5}^{2} - {0.7}^{2} | + | {0.7}^{2} - {0.5}^{2} | + | {0.8}^{2} - {0.8}^{2} |) + \\ \frac{1}{6} (\begin{array}{l} | {0.7071}^{2} - {0.5916}^{2} | + | {0.5099}^{2} - {0.4472}^{2} | + \\ | {0.3317}^{2} - {0.5099}^{2} | + | {0.6245}^{2} - {0.3317}^{2} | + \\ | {0.3873}^{2} - {0.3873}^{2} | + | {0.00}^{2} - {0.00}^{2} | \end{array}) \end{array})

D ({\hat{h}}_{1}, {\hat{h}}_{2}) = 0 .1483

Where,

$π_{{\hat{h}}_{1}} = {0 . 7071, 0 . 5099, 0 . 3317, 0 . 6245, 0 . 3873, 0}$

and

$π_{{\hat{h}}_{2}} = {0 . 5916, 0 . 4472, 0 . 5099, 0 . 3317, 0 . 3873, 0}$

Theorem 3.1.4. $D ({\hat{h}}_{1}, {\hat{h}}_{2})$ is the distance between two PHFSs ${\hat{h}}_{1}$ and ${\hat{h}}_{2}$ in X.

Proof. Proof is omitted.

3.2. Description of the Problem

A MADM problem can be expressed as a decision matrix whose elements indicate the evaluation information of all alternatives with respect to an attribute. We construct a Pythagorean hesitant fuzzy decision matrix, whose elements are PHFNs, which are given not only the information that the alternative X _i satisfies the attributes A _j, but also the information that the alternative X _i does not satisfies the attributes A _j may initiate from a doubt between a few different values.

Suppose that there are m alternatives X = {X ₁, X ₂, …, X _m} and n attributes A = {A ₁ A ₂, …, A _n} to be evaluated. To evaluate the performance of the alternative X _i under the attributes A _j, the decision maker is required to provide not only the information that the alternative X _i satisfies the attributes A _j, but also the information that the alternative X _i does not satisfy the attributes A _j. These two part information can be expressed by Λ _ij and Γ _ij which denote the degrees that the alternative X _i satisfy the attribute A _j and does not satisfy the attribute A _j, then the performance of the alternative X _i under the attribute A _j can be expressed by an PHFN ${\hat{h}}_{ij} = 〈 Λ_{ij}, Γ_{ij} 〉$ with the condition that for all h _ij ∈ Λ _ij, ∃ $h_{ij}^{'} \in Γ_{ij}$ such that 0≤ (h _ij) ² + ${(h_{ij}^{'})}^{2} \leq 1$ , and for all h _ij ∈ Γ _ij, ∃ $h_{ij}^{'} \in Λ_{ij}$ such that $0 \leq {(h_{ij})}^{2} + {(h_{ij}^{'})}^{2} \leq 1,$ i = 1, 2, …, n, j = 1, 2, …, m and k = 1, 2, … t. The Pythagorean hesitant fuzzy decision matrix H, can be written as:

$H = [\begin{matrix} {\hat{h}}_{11} & {\hat{h}}_{12} & \dots & {\hat{h}}_{1 n} \\ {\hat{h}}_{21} & {\hat{h}}_{22} & \dots & {\hat{h}}_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\hat{h}}_{m 1} & {\hat{h}}_{m 2} & \dots & {\hat{h}}_{mn} \end{matrix}]$ (6)

Considering that the attributes have different importance degrees, the weight vector of all the attributes, given by the DMs, is defined by w = (w ₁, w ₂, …, w _n) ^T, 1 ≥ w _j ≥ 0, j = 1, 2, …, n, $\sum_{j = 1}^{n} w_{j} = 1$ , and w _j is the importance degree of each attribute. Due to the complexity and uncertainty of practical decision making problems and the inherent subjective nature of human thinking, the information about attribute weights is usually incomplete. For convenience, let Ω be a set of the known weight information [15, 16], where Ω can be constructed by the following forms, for i ≠ j:

A weak ranking: {w _i ≥ w _j};

A strict ranking: {w _i - w _j ≥ λ _i (>0)};

A ranking with multiples: {w _i ≥ λ _i w _j}, 0 ≤ λ _i ≤ 1;

An interval form: {λ _i ≤ w _i ≤ λ _i + δ _i}, 0 ≤ λ _i ≤ λ _i + δ _i ≤ 1;

A ranking of differences: {w _i - w _j ≥ w _k - w _l}, for j ≠ k ≠ l.

3.3. Determining the optimal weights by the maximizing deviation method

Since in MADM the estimation of the attribute weights plays an vital role. Therefore in [35], Wang proposed a maximizing deviation method to determine the attribute weights for solving MADM problems with numerical information. Wang [35] suggested that for a MADM problem, the attribute with a larger deviation value among alternatives should be assigned a larger weight, while the attribute with a small deviation value among alternatives should be signed a smaller weight. So from the viewpoint of ranking the alternatives, if one attribute has similar attribute values across alternatives, it should be assigned a small weight; otherwise, the attribute which makes larger deviations should be evaluated a bigger weight, in spite of the degree of its own importance. Particularly, if all existing alternatives score equally corresponding to a given attribute, then such an attribute will be judged unimportant by most of DMs. According to Wang [35] the zero weight should be assigned to the corresponding attribute. Thus, we here construct an optimization model based on the maximizing deviation method to determine the optimal relative weights of attributes under Pythagorean hesitant fuzzy environment. For the attribute A _j ∈ A, the deviation of the alternative X _i to all the other alternatives can be expressed as:

$\begin{matrix} D_{ij} (w) = \sum_{i = 1}^{m} d ({\hat{h}}_{ij}, {\hat{h}}_{kj}) w_{j}, i = 1, 2, \dots, m, \\ k = 1, 2, \dots, m, j = 1, 2, \dots, n \end{matrix}$ (7) where $d ({\hat{h}}_{i j}, {\hat{h}}_{k j}) = \frac{1}{2} (\begin{array}{l} \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(h_{{\hat{h}}_{i j}}^{_{σ (k)}})}^{2} - {(h_{{\hat{h}}_{k j}}^{_{σ (k)}})}^{2} | + \\ \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(h_{{\hat{h}}_{i j}}^{'_{σ (k)}})}^{2} - {(h_{{\hat{h}}_{k j}}^{'_{σ (k)}})}^{2} | + \\ \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(π_{{\hat{h}}_{i j}}^{_{σ (k)}})}^{2} - {(π_{{\hat{h}}_{k j}}^{_{σ (k)}})}^{2} | \end{array}) .$ denote the Pythagorean hesitant fuzzy distance between Pythagorean hesitant fuzzy numbers ${\hat{h}}_{ij}$ and ${\hat{h}}_{kj}$ Defined in Equation (7).

Let $\begin{array}{l} D_{j} (w) = \sum_{i = 1}^{m} D_{i j} (w) \\ = \sum_{i = 1}^{m} \sum_{k = 1}^{m} w_{j} [\frac{1}{2} (\begin{array}{l} \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(h_{{\hat{h}}_{i j}}^{_{σ (i)}})}^{2} - {(h_{{\hat{h}}_{k j}}^{_{σ (i)}})}^{2} | + \\ \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(h_{{\hat{h}}_{i j}}^{'_{σ (i)}})}^{2} - {(h_{{\hat{h}}_{k j}}^{'_{σ (i)}})}^{2} | + \\ \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(π_{{\hat{h}}_{i j}}^{_{σ (i)}})}^{2} - {(π_{{\hat{h}}_{k j}}^{_{σ (i)}})}^{2} | \end{array})], \end{array}$ where j = 1, 2, …, n, then D _j (w) represents the deviation value of all alternatives to other alternatives for the attribute A _j ∈ A.

Based on the above analysis, we can construct a non-linear programming model to select the weight vector w which maximizes all deviation values for all the attributes, as follows: $M - 1 {\begin{matrix} Max D (w) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} w_{j} d ({\hat{h}}_{ij}, {\hat{h}}_{kj}) \\ s . t . w_{j} \geq 0, j = 1, 2, \dots, n, \sum_{j = 1}^{n} w_{j} = 1 \end{matrix}$ (8)

To solve the above model, we let $L (w, ζ) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} d ({\hat{h}}_{ij}, {\hat{h}}_{kj}) + \frac{ζ}{2} (\sum_{j = 1}^{n} w_{j}^{2} - 1)$ (9) which indicates the Lagrange function of the constrained optimization problem (M-1), where ζ is a real number, denoting the Lagrange multiplier variable. The partial derivatives of $L$ are computed as:

$\frac{\partial L}{\partial w_{j}} = \sum_{i = 1}^{m} \sum_{k = 1}^{m} d ({\hat{h}}_{ij}, {\hat{h}}_{kj}) + ζ \sum_{j = 1}^{n} w_{j} = 0$ (10)

$\frac{\partial L}{\partial ζ} = \frac{1}{2} (\sum_{j = 1}^{n} w_{j}^{2} - 1) = 0$ (11)

It follows from Equation (14) that $w_{j} = \frac{- \sum_{i = 1}^{m} \sum_{k = 1}^{m} d ({\hat{h}}_{ij}, {\hat{h}}_{kj})}{ζ}, j = 1, 2, \dots, n$ (12)

Putting Equation (16) into Equation (15), we have $ζ = - \sqrt{\sum_{j = 1}^{n} {(\sum_{i = 1}^{m} \sum_{k = 1}^{m} d ({\hat{h}}_{ij}, {\hat{h}}_{kj}))}^{2}}$ (13) obviously, ζ < 0, $\sum_{i = 1}^{m} \sum_{k = 1}^{m} d ({\hat{h}}_{ij}, {\hat{h}}_{kj})$ means the sum of deviations of all the alternatives with respect to the jth attribute and $\sqrt{\sum_{j = 1}^{n} {(\sum_{i = 1}^{m} \sum_{k = 1}^{m} d ({\hat{h}}_{ij}, {\hat{h}}_{kj}))}^{2}}$ means the sum of deviations of all the alternatives with respect to all the attributes.

Then combining Equation (16) and Equation (17), we can get $w_{j} = \frac{\sum_{i = 1}^{m} \sum_{k = 1}^{m} d ({\hat{h}}_{ij}, {\hat{h}}_{kj})}{\sqrt{\sum_{j = 1}^{n} {(\sum_{i = 1}^{m} \sum_{k = 1}^{m} d ({\hat{h}}_{ij}, {\hat{h}}_{kj}))}^{2}}}$ (14)

By normalizing w _j (j = 1, 2, …, n), we make their sum into a unit, and get $w_{j} = \frac{\sum_{i = 1}^{m} \sum_{k = 1}^{m} (\begin{array}{l} \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(h_{{\hat{h}}_{i j}}^{_{σ (k)}})}^{2} - {(h_{{\hat{h}}_{k j}}^{_{σ (k)}})}^{2} | + \\ \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(h_{{\hat{h}}_{i j}}^{'_{σ (k)}})}^{2} - {(h_{{\hat{h}}_{k j}}^{'_{σ (k)}})}^{2} | + \\ \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(π_{{\hat{h}}_{i j}}^{_{σ (k)}})}^{2} - {(π_{{\hat{h}}_{k j}}^{_{σ (k)}})}^{2} | \end{array})}{\sum_{j = 1}^{n} (\sum_{i = 1}^{m} \sum_{k = 1}^{m} (\begin{array}{l} \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(h_{{\hat{h}}_{i j}}^{_{σ (k)}})}^{2} - {(h_{{\hat{h}}_{k j}}^{_{σ (k)}})}^{2} | + \\ \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(h_{{\hat{h}}_{i j}}^{'_{σ (k)}})}^{2} - {(h_{{\hat{h}}_{k j}}^{'_{σ (k)}})}^{2} | + \\ \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(π_{{\hat{h}}_{i j}}^{_{σ (k)}})}^{2} - {(π_{{\hat{h}}_{k j}}^{_{σ (k)}})}^{2} | \end{array}))}$

However, there are actual situations that the information about the weight vector is not completely unknown but partially known. For these cases, based on the set of the known weight information Ω, we construct the following constrained optimization model: M-2

${\begin{matrix} Max D (w) = \frac{1}{2} \sum_{j = 1}^{n} w_{j} (\sum_{i = 1}^{m} \sum_{k = 1}^{m} (\begin{matrix} \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(h_{{\hat{h}}_{i j}}^{_{σ (i)}})}^{2} - {(h_{{\hat{h}}_{k j}}^{_{σ (i)}})}^{2} | + \\ \begin{array}{l} \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(h_{{\hat{h}}_{i j}}^{'_{σ (i)}})}^{2} - {(h_{{\hat{h}}_{k j}}^{'_{σ (i)}})}^{2} | + \\ \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(π_{{\hat{h}}_{i j}}^{_{σ (i)}})}^{2} - {(π_{{\hat{h}}_{k j}}^{_{σ (i)}})}^{2} | \end{array} \end{matrix})) \\ s .t . w_{j} \geq 0, j = 1, 2, …, n, \sum_{j = 1}^{n} w_{j} = 1 \end{matrix}$ where Ω is also a set of constraint conditions that the weight value w _j should satisfy according to the requirements in real situations. The model (M-2) is a linear programming model that can be executed using the LINGO 11.0 mathematics software package. By solving this model, we get the optimal solution w = (w ₁, w ₂, …, w _n) ^T, which can be used as the weight vector of attributes.

3.4. Approaches to MADM with Pythagorean hesitant fuzzy information

In the process of Pythagorean hesitant fuzzy information aggregation, it produces the loss of too much information due to the complexity of the aggregation process of Pythagorean hesitant fuzzy aggregation operators, which implies a deficiency of precision in the final results. Therefore, in order to overcome this disadvantage, we have extended the TOPSIS method to take Pythagorean hesitant fuzzy information into account and used the distance measures of PHFNs to obtain the final ranking of the alternatives. TOPSIS is a kind of method to solve MADM problems, which aims at choosing the alternative with the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS), and is widely used for deal with the ranking problems in real situations. Under Pythagorean hesitant fuzzy environment, the Pythagorean hesitant fuzzy PIS, denoted by p ⁺, and the Pythagorean hesitant fuzzy NIS, denoted by p ^-, can be defined as follows: $p^{+} = {{\hat{h}}_{1}^{+}, {\hat{h}}_{2}^{+}, \dots, {\hat{h}}_{m}^{+}}^{T}$ (15)

where

${\hat{h}}_{i}^{+} = \cup_{j = 1}^{n} {\hat{h}}_{ij} = {\begin{matrix} \cup_{h_{{\hat{h}}_{i 1}} \in Λ_{{\hat{h}}_{i 1}}, h_{{\hat{h}}_{i 2}} \in Λ_{{\hat{h}}_{i 2}}, \dots, h_{{\hat{h}}_{in}} \in Λ_{{\hat{h}}_{in}}} \\ max {h_{{\hat{h}}_{i 1}}, h_{{\hat{h}}_{i 2}}, \dots, h_{{\hat{h}}_{in}}}, \\ \cup_{h_{{\hat{h}}_{i 1}}^{'} \in Γ_{{\hat{h}}_{i 1}}, h_{{\hat{h}}_{i 2}}^{'} \in Γ_{{\hat{h}}_{i 2}}, \dots, h_{{\hat{h}}_{in}}^{'} \in Γ_{{\hat{h}}_{in}}} \\ min {h_{{\hat{h}}_{i 1}}^{'}, h_{{\hat{h}}_{i 2}}^{'}, \dots, h_{{\hat{h}}_{in}}^{'}} | i = 1, 2, \dots, m . \end{matrix}}^{T}$ $p^{-} = {{\hat{h}}_{1}^{-}, {\hat{h}}_{2}^{-}, \dots, {\hat{h}}_{m}^{-}}$ (16) where

${\hat{h}}_{i}^{-} = \cap_{i = 1}^{n} {\hat{h}}_{ij} = {\begin{matrix} \cup_{h_{{\hat{h}}_{i 1}} \in Λ_{{\hat{h}}_{i 1}}, h_{{\hat{h}}_{i 2}} \in Λ_{{\hat{h}}_{i 2}}, \dots, h_{{\hat{h}}_{in}} \in Λ_{{\hat{h}}_{in}}} \\ min {h_{{\hat{h}}_{i 1}}, h_{{\hat{h}}_{i 2}}, \dots, h_{{\hat{h}}_{in}}}, \\ \cup_{h_{{\hat{h}}_{i 1}}^{'} \in Γ_{{\hat{h}}_{i 1}}, h_{{\hat{h}}_{i 2}}^{'} \in Γ_{{\hat{h}}_{i 2}}, \dots, h_{{\hat{h}}_{in}}^{'} \in Γ_{{\hat{h}}_{in}}} \\ max {h_{{\hat{h}}_{i 1}}^{'}, h_{{\hat{h}}_{i 2}}^{'}, \dots, h_{{\hat{h}}_{in}}^{'}} | i = 1, 2, \dots, m . \end{matrix}}^{T}$

The separation between alternatives can be measured by Equation (7). The separation measures, d ⁺ and d ^-, of each alternative from the Pythagorean hesitant fuzzy PIS p ⁺ and the Pythagorean hesitant fuzzy NIS p ^-, respectively, are derived from $\begin{array}{l} d^{+} = \sum_{j = 1}^{n} d ({\hat{h}}_{i j}, {\hat{h}}_{j}^{+}) \\ = \frac{1}{2} \sum_{j = 1}^{n} w_{j} (\begin{matrix} \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(h_{{\hat{h}}_{i j}}^{_{σ (k)}})}^{2} - {[{(h_{{\hat{h}}_{j}}^{_{σ (k)}})}^{+}]}^{2} | + \\ \begin{array}{l} \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(h_{{\hat{h}}_{i j}}^{'_{σ (k)}})}^{2} - {[{(h_{{\hat{h}}_{j}}^{'_{σ (k)}})}^{+}]}^{2} | + \\ \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(π_{{\hat{h}}_{i j}}^{_{σ (k)}})}^{2} - {[{(π_{{\hat{h}}_{j}}^{_{σ (k)}})}^{+}]}^{2} | \end{array} \end{matrix}) \end{array}$ $i = 1, 2, …, m$

$\begin{array}{l} d^{-} = \sum_{j = 1}^{n} d ({\hat{h}}_{i j}, {\hat{h}}_{j}^{-}) \\ = \frac{1}{2} \sum_{j = 1}^{n} w_{j} (\begin{matrix} \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(h_{{\hat{h}}_{i j}}^{_{σ (k)}})}^{2} - {[{(h_{{\hat{h}}_{j}}^{_{σ (k)}})}^{-}]}^{2} | + \\ \begin{array}{l} \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(h_{{\hat{h}}_{i j}}^{'_{σ (k)}})}^{2} - {[{(h_{{\hat{h}}_{j}}^{'_{σ (k)}})}^{-}]}^{2} | + \\ \frac{1}{l_{x_{i}}} \sum_{i = 1}^{l_{x_{i}}} | {(π_{{\hat{h}}_{i j}}^{_{σ (k)}})}^{2} - {[{(π_{{\hat{h}}_{j}}^{_{σ (k)}})}^{-}]}^{2} | \end{array} \end{matrix}) \end{array}$

The relative closeness coefficient of an alternative X _i with respect to the Pythagorean hesitant fuzzy PIS p ⁺ is defined as:

$C_{i} = \frac{d_{i}^{-}}{d_{i}^{+} - d_{i}^{-}}$ (17) where 0 ≤ C _i ≤ 1, i = 1, 2, …, m. Obviously, an alternative X _i is closer to the Pythagorean hesitant fuzzy PIS p ⁺ and farther from the Pythagorean hesitant fuzzy NIS p ^- as C _i approaches 1. Therefore, according to the closeness coefficient C _i, we can determine the ranking orders of all alternatives and select the best one from a set of feasible alternatives.

Based on the above models, we shall develop a practical approach for solving MADM problems, in which the information about attribute weights is incompletely known or completely unknown, and the attribute values take the form of Pythagorean hesitant fuzzy information.

Step 1. For a MADM problem, we construct the decision matrix $H = [{\hat{h}}_{ij}]_{m \times n}$ , where all the arguments ${\hat{h}}_{ij}$ (i = 1, 2, …, m; j = 1, 2, …, n) are PHFNs, given by the DMs, for the alternative X _i ∈ X with respect to the attribute A _j ∈ A.

Step 2. If the information about the attribute weights is completely unknown, then we can obtain the attribute weights by using Equation (19); if the information about the attribute weights is partly known, then we solve the model (M-2) to obtain the attribute weights.

Step 3. Utilize Equation (20) and Equation (21) to determine the corresponding Pythagorean hesitant fuzzy PIS p ⁺ and the Pythagorean hesitant fuzzy NIS p ^-.

Step 4. Utilize Equation (22) and Equation (23) to calculate the separation measures $d_{i}^{+}$ and $d_{i}^{-}$ of each alternative X _i from the Pythagorean hesitant fuzzy PIS p ⁺ and the Pythagorean hesitant fuzzy NIS p ^-, respectively.

Step 5. Utilize Equation (24) to calculate the relative closeness coefficient C _i of each alternative X _i to the Pythagorean hesitant fuzzy PIS p ⁺.

Step 6. Rank the alternatives X _i (i = 1, 2, …, m) according to the relative closeness coefficients C _i (i = 1, 2, …, m) to the Pythagorean hesitant fuzzy PIS p ⁺ and then select the most desirable one(s).

4. Illustrative example

In this section, we consider a Pythagorean hesitant fuzzy multi-attribute decision making problem that concerns the evaluation of the service quality among domestic airlines (adapted from the literature 49, 50) to illustrate the application of the proposed method and conduct a comparison analysis with the references [38, 45].

Owing to the development of high-speed railroad, the domestic airline marketing has faced a stronger challenge in Taiwan, People’s Republic of China. More and more airlines have attempted to attract customers by reducing price. Unfortunately, they soon found that there was a no-win situation and only service quality is the critical and fundamental element to survive in this highly competitive domestic market. To improve the service quality of domestic airline, the civil aviation administration of Taiwan (CAAT) wants to know which airline is the best in Taiwan and then calls for the others to learn from it. So the CAAT constructs a committee to investigate the four major domestic airlines, which are

X₁: UNI Air

X₂: Transasia

X₃: Mandarin

X₄: Daily Air

X₅: Tigerair Taiwan

According to the following four major attributes:

A₁: Booking and ticketing service,

A₂: check-in and boarding process,

A₃: cabin service,

A₄: responsiveness.

In order to avoid influence each other, the decision makers are required to evaluate the five possible emerging technology enterprises X _i (i = 1, 2, 3, 4, 5) under the above four attributes in anonymity, and Pythagorean hesitant fuzzy decision matrix $C = [{\hat{h}}_{ij}]_{5 \times 4}$ is constructed as shown in Table 1, where ${\hat{h}}_{ij} = 〈 Λ_{ij}, Γ_{ij} 〉 (i = 1, 2, 3,$ 4, 5, j = 1, 2, 3, 4) are in the form of PHFNs.

Table 1
Pythagorean hesitant fuzzy decision matrix

A ₁ A ₂ A ₃ A ₄

X ₁ 〈{0.3, 0.4}, {0.8, 0.9} 〉〈 {0.4, 0.5, 0.6}, {0.7, 0.8} 〉〈{0.7, 0.8, 0.9}, {0.3, 0.4} 〉〈{0.7, 0.8}, {0.5, 0.6} 〉

X ₂ 〈{0.6, 0.7}, {0.5, 0.6, 0.7〉〈{0.5, 0.8 {0.6, 0.7〉〈{0.5, 0.6, 0.7}, {0.7, 0.8} 〉〈{0.3, 0.4, 0.5}, {0.6, 0.7, 0.8} 〉

X ₃ 〈{0.8, 0.9}, {0.3, 0.4} 〉〈{0.6, 0.7}, {0.6, 0.8} 〉〈{0.3, 0.4}, {0.8, 0.9} 〉〈{0.4, 0.6, 0.7}, {0.7, 0.8} 〉

X ₄ 〈{0.4, 0.5}, {0.7, 0.8} 〉〈{0.8, 0.9}, {0.3, 0.4} 〉〈{0.6, 0.7}, {0.5, 0.6, 0.7} 〉〈{0.2, 0.3, 0.4}, {0.8, 0.9} 〉

X ₅ 〈{0.7, 0.8}, {0.5, 0.6} 〉〈{0.3, 0.4, 0.6}, {0.8} 〉〈{0.4, 0.5, 0.6}, {0.7, 0.8} 〉〈{0.5, 0.6, 0.7}, {0.6, 0.7, 0.8} 〉

	A ₁	A ₂	A ₃	A ₄
X ₁	〈{0.3, 0.4}, {0.8, 0.9} 〉	〈 {0.4, 0.5, 0.6}, {0.7, 0.8} 〉	〈{0.7, 0.8, 0.9}, {0.3, 0.4} 〉	〈{0.7, 0.8}, {0.5, 0.6} 〉
X ₂	〈{0.6, 0.7}, {0.5, 0.6, 0.7〉	〈{0.5, 0.8 {0.6, 0.7〉	〈{0.5, 0.6, 0.7}, {0.7, 0.8} 〉	〈{0.3, 0.4, 0.5}, {0.6, 0.7, 0.8} 〉
X ₃	〈{0.8, 0.9}, {0.3, 0.4} 〉	〈{0.6, 0.7}, {0.6, 0.8} 〉	〈{0.3, 0.4}, {0.8, 0.9} 〉	〈{0.4, 0.6, 0.7}, {0.7, 0.8} 〉
X ₄	〈{0.4, 0.5}, {0.7, 0.8} 〉	〈{0.8, 0.9}, {0.3, 0.4} 〉	〈{0.6, 0.7}, {0.5, 0.6, 0.7} 〉	〈{0.2, 0.3, 0.4}, {0.8, 0.9} 〉
X ₅	〈{0.7, 0.8}, {0.5, 0.6} 〉	〈{0.3, 0.4, 0.6}, {0.8} 〉	〈{0.4, 0.5, 0.6}, {0.7, 0.8} 〉	〈{0.5, 0.6, 0.7}, {0.6, 0.7, 0.8} 〉

Obviously the numbers of values in different PHFNs of PHFSs are different. In order to more accurately calculate the distance between two PHFSs, we should extend the shorter one until both of them have the same length when we compare them. According to the regulations mentioned above, we consider that the DM is pessimistic in according to Definition 12, and change the Pythagorean hesitant fuzzy data by adding the minimal values as listed in Table 2.

Table 2

Pessimist Pythagorean hesitant fuzzy decision matrix

	A ₁	A ₂	A ₃	A ₄
X ₁	〈{0.3, 0.3, 0.4}, {0.8, 0.8, 0.9} 〉	〈{0.4, 0.5, 0.6}, {0.7, 0.7, 0.8} 〉	〈{0.7, 0.8, 0.9}, {0.3, 0.3, 0.4} 〉	〈{0.7, 0.7, 0.8}, {0.5, 0.5, 0.6} 〉
X ₂	〈{0.6, 0.6, 0.7}, {0.5, 0.6, 0.7} 〉	〈{0.5, 0.5, 0.8}, {0.6, 0.6, 0.7} 〉	〈{0.5, 0.6, 0.7}, {0.7, 0.7, 0.8} 〉	〈{0.3, 0.4, 0.5}, {0.6, 0.7, 0.8} 〉
X ₃	〈{0.8, 0.8, 0.9}, {0.3, 0.3, 0.4} 〉	〈{0.6, 0.6, 0.7}, {0.6, 0.6, 0.8} 〉	〈{0.3, 0.3, 0.4}, {0.8, 0.8, 0.9} 〉	〈{0.4, 0.6, 0.7}, {0.7, 0.7, 0.8} 〉
X ₄	〈{0.4, 0.4, 0.5}, {0.7, 0.7, 0.8} 〉	〈{0.8, 0.8, 0.9}, {0.3, 0.3, 0.4} 〉	〈{0.6, 0.6, 0.7}, {0.5, 0.6, 0.7} 〉	〈{0.2, 0.3, 0.4}, {0.8, 0.8, 0.9} 〉
X ₅	〈{0.7, 0.7, 0.8}, {0.5, 0.5, 0.6} 〉	〈{0.3, 0.4, 0.6}, {0.8, 0.8, 0.8} 〉	〈{0.4, 0.5, 0.6}, {0.7, 0.7, 0.8} 〉	〈{0.5, 0.6, 0.7}, {0.6, 0.7, 0.8} 〉

Then, we utilize the approach developed to get the most desirable alternative (s), which involves the following two cases:

Case 1. Assume that the information about the attribute weights is completely unknown; we get the most desirable alternative(s) according to the following steps:

Step 1. Utilize the Equation (19) to get the optimal weight vector:

$w = (0.3434, 0.2272, 0.2444, 0.1851)^{T}$

Step 2. Utilize Equations (20, 21) to determine the Pythagorean hesitant fuzzy PIS p ⁺ and the Pythagorean hesitant fuzzy NIS p ^-, respectively: $\begin{matrix} p^{+} = {〈 {0.8, 0.8, 0.9}, {0.3, 0.3, 0.4} 〉, 〈 {0.8, 0.8, 0.9}, \\ {0.3, 0.3, 0.4} 〉, 〈 {0.7, 0.8, 0.9}, {0.3, 0.3, 0.4} 〉 \\ 〈 {0.7, 0.7, 0.8}, {0.5, 0.5, 0.6} 〉} \\ p^{-} = {〈 {0.3, 0.3, 0.4}, {0.8, 0.8, 0.9} 〉, 〈 {0.3, 0.4, 0.6}, \\ {0.8, 0.8, 0.8} 〉, 〈 {0.3, 0.3, 0.4}, {0.8, 0.8, 0.9} 〉, \\ 〈 {0.2, 0.3, 0.4}, {0.8, 0.8, 0.9} 〉} \end{matrix}$

Step 3. Utilize Equations (22, 23) to calculate the separation measures $d_{i}^{+}$ and $d_{i}^{-}$ of each alternative X _i from the Pythagorean hesitant fuzzy PIS p ⁺ and the Pythagorean hesitant fuzzy NIS p ^-, respectively. $\begin{matrix} d_{1}^{+} = 0.3354, d_{2}^{+} = 0.3398, \\ d_{3}^{+} = 0.2873, d_{4}^{+} = 0.3366, \\ d_{5}^{+} = 0.3445 \end{matrix}$ $\begin{matrix} d_{1}^{-} = 0.2329, d_{2}^{-} = 0.2285, \\ d_{3}^{-} = 0.2810, d_{4}^{-} = 0.2317, \\ d_{5}^{-} = 0.2238 \end{matrix}$

Step 4. Utilize Equation (24) to calculate the relative closeness C _i of each alternative X _i to the Pythagorean hesitant fuzzy PIS p ⁺: $\begin{matrix} C_{1} = & 0.4098, C_{2} = & 0.4021, \\ C_{3} = & 0.4944, C_{4} = & 0.4077, \\ C_{5} = & 0.3939 \end{matrix}$

Step 5. Rank the alternatives X _i (i = 1, 2, 3, 4, 5), according to the relative closeness coefficients C _i (i = 1, 2, 3, 4, 5): $X_{3} > X_{1} > X_{4} > X_{2} > X_{5}$

Thus the most desirable alternative is X ₃.

Case 2. The information about the attribute weights is partly known and the known weight information is given as follows: $Ω = {\begin{matrix} 0.3 \leq w_{1} \leq 0.35, 0.18 \leq w_{2} \leq 0.22, \\ 0.2 \leq w_{3} \leq 0.25, 0.15 \leq w_{4} \leq 0.2, \sum_{j = 1}^{n} w_{i} = 1 \end{matrix}}$

Step 1. Utilize the model (M-2) to construct the single-objective model as follows: ${\begin{matrix} Max D (w) & = & 7.1750 w_{1} + 4.7467 w_{2} \\ + 5.10667 w_{3} + 3.8667 w_{4} \\ s . t . w \in Ω, w_{j} \geq 0, j = 1, 2, 3, 4 \end{matrix}$

By solving this model, we get the optimal weight vector w = (0.4, 0.2, 0.25, 0.15)

Step 2. Utilize Equation (20) and Equation (21) to determine the Pythagorean hesitant fuzzy PIS p ⁺ and the Pythagorean hesitant fuzzy NIS p ^-, respectively: $\begin{matrix} p^{+} = {〈 {0.8, 0.8, 0.9}, {0.3, 0.3, 0.4} 〉, 〈 {0.8, 0.8, 0.9}, \\ {0.3, 0.3, 0.4} 〉, 〈 {0.7, 0.8, 0.9}, {0.3, 0.3, 0.4} 〉, \\ 〈 {0.7, 0.7, 0.8}, {0.5, 0.5, 0.6} 〉 \\ p^{-} = {〈 {0.3, 0.3, 0.4}, {0.8, 0.8, 0.9} 〉, 〈 {0.3, 0.4, 0.6}, \\ {0.8, 0.8, 0.8} 〉, 〈 {0.3, 0.3, 0.4}, {0.8, 0.8, 0.9} 〉, \\ 〈 {0.2, 0.3, 0.4}, {0.8, 0.8, 0.9} 〉} \end{matrix}$

Step 3. Utilize Equations (22, 23) to calculate the separation measures $d_{i}^{+}$ and $d_{i}^{-}$ of each alternative X _i from the Pythagorean hesitant fuzzy PIS p ⁺ and the Pythagorean hesitant fuzzy NIS p ^-, respectively. $\begin{matrix} d_{1}^{+} = 0.3355, d_{2}^{+} = 0.3392, \\ d_{3}^{+} = 0.2863, d_{4}^{+} = 0.3382, \\ d_{5}^{+} = 0.3425 \end{matrix}$ $\begin{matrix} d_{1}^{-} = 0.2333, d_{2}^{-} = 0.2295, \\ d_{3}^{-} = 0.2824, d_{4}^{-} = 0.2305, \\ d_{5}^{-} = 0.2262 \end{matrix}$

Step 4. Utilize Equation (24) to calculate the relative closeness C _i of each alternative X _i to the Pythagorean hesitant fuzzy PIS p ⁺: $\begin{matrix} C_{1} = & 0.4102, C_{2} = & 0.4036, \\ C_{3} = & 0.4966, C_{4} = & 0.4054, \\ C_{5} = & 0.3978 \end{matrix}$

Step 5. Rank the alternatives X _i (i = 1, 2, 3, 4, 5), according to the relative closeness coefficients C _i (i = 1, 2, 3, 4, 5): $X_{3} > X_{1} > X_{4} > X_{2} > X_{5}$

Thus the most desirable alternative is X ₃.

5. Comparison analysis

In order to verify the validity and effectiveness of the proposed approach, a comparative study is conducted using the methods of PFNs Yager et al. [41] and HFNs Torra et al. [33] which are special cases of PHFNs, to the same illustrative example.

5.1. Comparison analysis with Pythagorean fuzzy approach

PFNs can be considered as a special case of PHFNs when there is only one element in membership and non-membership degree. For comparison, the PHNs can be transformed to PFNs by calculating the average value of the membership and non-membership degrees. After transformation the Pythagorean information can be shown in Table 3 and weight of attributes are consider as: w = (0.3434, 0.2272, 0.2444, 0.1851) ^T.

Table 3
Pythagorean fuzzy decision matrix

A ₁ A ₂ A ₃ A ₄

X ₁ 〈0.35, 0.85〉〈0.50, 0.75〉〈0.80, 0.35〉〈0.75, 0.55〉

X ₂ 〈0.65, 0.60〉〈0.65, 0.65〉〈0.60, 0.75〉〈0.40, 0.70〉

X ₃ 〈0.85, 0.35〉〈0.65, 0.70〉〈0.35, 0.85〉〈0.57, 0.75〉

X ₄ 〈0.45, 0.75〉〈0.85, 0.35〉〈0.65, 0.60〉〈0.30, 0.85〉

X ₅ 〈0.75, 0.55〉〈0.43, 0.80〉〈0.50, 0.75〉〈0.60, 0.70〉

	A ₁	A ₂	A ₃	A ₄
X ₁	〈0.35, 0.85〉	〈0.50, 0.75〉	〈0.80, 0.35〉	〈0.75, 0.55〉
X ₂	〈0.65, 0.60〉	〈0.65, 0.65〉	〈0.60, 0.75〉	〈0.40, 0.70〉
X ₃	〈0.85, 0.35〉	〈0.65, 0.70〉	〈0.35, 0.85〉	〈0.57, 0.75〉
X ₄	〈0.45, 0.75〉	〈0.85, 0.35〉	〈0.65, 0.60〉	〈0.30, 0.85〉
X ₅	〈0.75, 0.55〉	〈0.43, 0.80〉	〈0.50, 0.75〉	〈0.60, 0.70〉

Now we calculate the comprehensive evaluation values using the Pythagorean fuzzy TOPSIS developed by Zhang and Xu [45], The Pythagorean fuzzy PIS p ⁺ and Pythagorean fuzzy NIS p ^- are: $p^{+} = {\begin{matrix} 〈 0.85, 0.35 〉, 〈 0.85, 0.35 〉, \\ 〈 0.85, 0.35 〉, 〈 0.75, 0.55 〉 \end{matrix}}$ $p^{-} = {\begin{matrix} 〈 0.35, 0.85 〉, 〈 0.43, 0.80 〉, \\ 〈 0.35, 0.85 〉, 〈 0.30, 0.85 〉 \end{matrix}}$

The separation measures $d_{i}^{+}$ and $d_{i}^{-}$ of each alternative X _i from the Pythagorean fuzzy PIS p ⁺ and the Pythagorean fuzzy NIS p ^-, respectively are: $\begin{matrix} d_{1}^{+} = 0.3104, d_{2}^{+} = 0.3032, \\ d_{3}^{+} = 0.2579, d_{4}^{+} = 0.3052, \\ d_{5}^{+} = 0.3144 \end{matrix}$ $\begin{matrix} d_{1}^{-} = 0.2357, d_{2}^{-} = 0.2429, \\ d_{3}^{-} = 0.2882, d_{4}^{-} = 0.2409, \\ d_{5}^{-} = 0.2317 \end{matrix}$

The relative closeness C _i of each alternative X _i to the Pythagorean fuzzy PIS p ⁺ are: $\begin{matrix} C_{1} = 0.4316, C_{2} = 0.4448, \\ C_{3} = 0.5277, C_{4} = 0.4411, \\ C_{5} = 0.4242, \end{matrix}$

Rank the alternatives X _i (i = 1, 2, 3, 4, 5), according to the relative closeness coefficients C _i (i = 1, 2, 3, 4, 5): $X_{3} > X_{2} > X_{4} > X_{1} > X_{5}$

Thus the most desirable alternative is X ₃.

There is a difference in the alternatives in our approach X ₁ > X ₄ and X ₄ > X ₂ while in Zhang and Xu [45], X ₄ > X ₁ and X ₂ > X ₄ These shows that the ranking order of these pairs are just the inverse. The main reason is that the our approach we consider the Pythagorean hesitant fuzzy information which is represented by several possible values in the membership degree and non-membership degree. while if adopting the Zhang and Xu [45], method, it needs to transform PHFNs into PFNs, which gives rise to a difference in the accuracy of data, it will have an effect on the final decision results. Thus it is not hard to see that our approach has some desirable advantages over the Zhang and Xu [45], method as follows:

Our approach, by extending the TOPSIS method to take into account the Pythagorean hesitant fuzzy assessments which are well-suited to handle the ambiguity and impreciseness inherent in MADM problems, does not need to transform PHFNs into PFNs but directly deals with these problems, and thus obtains better final decision results.

In particular, when we meet some situations where the information is represented by several possible values, our approach shows its great superiority in handling those decision making problems with Pythagorean hesitant fuzzy information. Thus PHFSs are more flexible than PFSs because they consider the situations where decision makers would like to use several possible values to express the membership and non-membership degrees.

5.2. Comparison analysis with hesitant fuzzy approach

HFNs can be considered as a special case of PHFNs when decision makers only consider membership degrees in evaluation. For comparison, the PHFNs can be transformed to HFNs by remaining only the membership degrees, and the hesitant fuzzy information can be represented in Table 4 and weight of attributes are consider as: w = (0.3434, 0.2272, 0.2444, 0.1851) ^T.

Table 4
Hesitant fuzzy decision matrix

A ₁ A ₂ A ₃ A ₄

X ₁ {0.3,0.3,0.4} {0.4,0.5,0.6} {0.7,0.8,0.9} {0.7,0.7,0.8}

X ₂ {0.6,0.6,0.7} {0.5,0.5,0.8} {0.5,0.6,0.7} {0.3,0.4,0.5}

X ₃ {0.8,0.8,0.9} {0.6,0.6,0.7} {0.3,0.3,0.4} {0.4,0.6,0.7}

X ₄ {0.4,0.4,0.5} {0.8,0.8,0.9} {0.6,0.6,0.7} {0.2,0.3,0.4}

X ₅ {0.7,0.7,0.8} {0.3,0.4,0.6} {0.4,0.5,0.6} {0.5,0.6,0.7}

	A ₁	A ₂	A ₃	A ₄
X ₁	{0.3,0.3,0.4}	{0.4,0.5,0.6}	{0.7,0.8,0.9}	{0.7,0.7,0.8}
X ₂	{0.6,0.6,0.7}	{0.5,0.5,0.8}	{0.5,0.6,0.7}	{0.3,0.4,0.5}
X ₃	{0.8,0.8,0.9}	{0.6,0.6,0.7}	{0.3,0.3,0.4}	{0.4,0.6,0.7}
X ₄	{0.4,0.4,0.5}	{0.8,0.8,0.9}	{0.6,0.6,0.7}	{0.2,0.3,0.4}
X ₅	{0.7,0.7,0.8}	{0.3,0.4,0.6}	{0.4,0.5,0.6}	{0.5,0.6,0.7}

Now we calculate the comprehensive evaluation values using the hesitant fuzzy TOPSIS proposed by Xu and Zhang [38]. The hesitant fuzzy PIS h ⁺ and hesitant fuzzy NIS h ^- are: $\begin{matrix} h^{+} = {\begin{matrix} {0.8, 0.8, 0.9}, {0.8, 0.8, 0.9}, \\ {0.7, 0.8, 0.9}, {0.7, 0.7, 0.8} \end{matrix}} \\ h^{-} = {\begin{matrix} {0.3, 0.3, 0.4}, {0.3, 0.4, 0.6}, \\ {0.3, 0.3, 0.4}, {0.2, 0.3, 0.4} \end{matrix}} \end{matrix}$

The separation measures $d_{i}^{+}$ and $d_{i}^{-}$ of each alternative X _i from the hesitant fuzzy PIS p ⁺ and the hesitant fuzzy NIS p ^-, respectively are: $\begin{matrix} d_{1}^{+} = 0.2483, d_{2}^{+} = 0.3080, \\ d_{3}^{+} = 0.2657, d_{4}^{+} = 0.3500, \\ d_{5}^{+} = 0.2747 . \end{matrix}$ $\begin{matrix} d_{1}^{-} = 0.3297, d_{2}^{-} = 0.2700, \\ d_{3}^{-} = 0.3123, d_{4}^{-} = 0.2280, \\ d_{5}^{-} = 0.3033 . \end{matrix}$

The relative closeness C _i of each alternative X _i to the Pythagorean fuzzy PIS p ⁺ are: $\begin{matrix} C_{1} = 0.5704, C_{2} = 0.4671, \\ C_{3} = 0.5404, C_{4} = 0.3945, \\ C_{5} = 0.5248 \end{matrix}$

Rank the alternatives X _i (i = 1, 2, 3, 4, 5), according to the relative closeness coefficients C _i (i = 1, 2, 3, 4, 5):

$X_{1} > X_{3} > X_{5} > X_{2} > X_{4}$

Thus the most desirable alternative is X ₁. X ₃ > X ₁ > X ₄ > X ₂ > X ₅

Obviously, the ranking order of the alternatives obtained by the Xu and Zhang [38] method is different from that obtained by the approach proposed in this paper. The differences are the ranking orders between X ₁ and X ₃, and between X ₂, X ₄ and X ₅, i.e., in our approach X ₃ > X ₁ and X ₄ > X ₂ > X ₅ while in the Xu and Zhang [38] method X ₁ > X ₃ and X ₅ > X ₂ > X ₄. Which shows that the ranking orders of these pairs of alternatives are just converse. The main reasons is that, HFNs only consider the membership degrees of an element and ignore the non-membership degrees, which may result in information distortion and loss. which gives rise to a difference in the accuracy of data, it will have an effect on the final decision results. Thus it is not hard to see that our approach has some desirable advantages over the Xu and Zhang [38], method as follows:

Our approach, by extending the TOPSIS method to take into account the Pythagorean hesitant fuzzy information’s which are well-suited to handle the ambiguity and impreciseness inherent in MADM problems, does not need to transform PHFNs into HFNs but directly deals with these problems, and thus obtains better final decision results.

In particular, when we meet some situations where the information is represented by several possible values in membership degree and non-membership degree, our approach shows its great superiority in handling those decision making problems with Pythagorean hesitant fuzzy information.

6. Conclusion

Since many real-worlds MADM problems take place in a complex environment and usually adhere to imprecise data and uncertainty. The PHFS is suitable for dealing with the vagueness of a DM’s judgments over alternatives with respect to attributes. In this paper, based on the idea that the attribute with a larger deviation value among alternatives should be evaluated with a larger weight, we have first developed a method called the maximizing deviation method to determine the optimal relative weights of attributes under Pythagorean hesitant fuzzy environment. An important advantage of the proposed method is its ability to relieve the influence of subjectivity of the DMs and at the same time remain the original decision information sufficiently. Then we have proposed a novel approach on the basis of TOPSIS to solve MADM problems with Pythagorean hesitant fuzzy information. The approach is based on the relative closeness of each alternative to determine the ranking order of all alternatives, which avoids producing the loss of too much information in the process of information aggregation. Finally, the effectiveness and applicability of the proposed method has been illustrated with an example. Apparently, our approach is straightforward and has less loss of information, and can be applied easily to other managerial decision making problems under Pythagorean hesitant fuzzy environment.

In future, we will introduce the concept TODIM methods under Pythagorean hesitant fuzzy environment. Further we will define Pythagorean hesitant fuzzy linguistic sets and will propose a MADM based TOPSIS and TODIM methods under Pythagorean hesitant fuzzy linguistic environment.

References

K.T.

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

K.T.

Atanassov , Intuitionistic fuzzy logics as tools for evaluation of data mining processes, Knowl-Based Syst 80 (2015), 122–130.

Beg and

Rashid , Group decision making using intuitionistic hesitant fuzzy sets, International Journal of Fuzzzy Logic and Intelligent Systems 14(3) (2014), 181–187.

R.E.

Bellman and

L.A.

Zadeh , Decision-making in a fuzzy environment, Management Science 17 (1970), 141–161.

Q.W.

Cao and

Wu , The extended COWG operators and their application to multiple attributive group decision making problems with interval numbers, Applied Mathematical Modelling 35 (2011), 2075–2086.

J.S.

Dyer ,

P.C.

Fishburn ,

R.E.

Steuer ,

Wallenius and

Zionts , Multiple criteria decision making, multiattribute utility theory: The next ten years, Management Science 38 (1992), 645–654.

Z.P.

Fan and

Feng , A multiple attributes decision making method using individual and collaborative attribute data in a fuzzy environment, Information Sciences 179 (2009), 3603–3618.

Garg , A novel improved accuracy function for interval valued pythagorean fuzzy sets and its applications in the decision-making process, Int J Intell Syst (2017), 1–14.

Gou ,

Xu and

Ren , The properties of continuous Pythagorean fuzzy information, Int J Intell Syst 31 (5) (2016), 401–424.

10.

Giirbiiz and

Y.E.

Albayrak , An engineering approach to human resources performance evaluation: Hybrid MCDM application with interactions, Applied Soft Computing 21 (2014), 365–375.

11.

Hashemkhani Zolfani ,

Maknoon and

E.K.

Zavadskas , Multiple nash equilibriums and evaluation of strategies. New application of MCDM methods, Journal of Business Economics and Management 16(2) (2015), 290–306.

12.

Y.H.

Hung ,

S.C.T.

Chou and

G.H.

Tzeng , Knowledge man-agement adoption and assessment for SMEs by a novel MCDM approach, Decision Support Systems 51 (2011), 270–291.

13.

C.L.

Hwang and

K.S.

Yoon , Multiple attribute decision methods and applications, Germany: Springer, Berlin, 1981.

14.

G.R.

Jahanshahloo ,

F.H.

Lotfi and

Izadikhah , An algo-rithmic method to extend TOPSIS for decision-making problems with interval data, Applied Mathematics and Computation 175 (2006), 1375–1384.

15.

S.H.

Kim ,

S.H.

Choi and

Kim , An interactive procedure for multiple attribute group decision making with incomplete information: Range-based approach, European Journal of Operational Research 118 (1999), 139–152.

16.

Liang ,

Zhang and

Liu , The Maximizing Deviation Method Based on Interval-Valued Pythagorean Fuzzy Weighted Aggregating Operator for Multiple Criteria Group Decision Analysis, Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume (2015), Article ID 746572, p. 15.

17.

Liang ,

X.L.

Zhang and

M.F.

Liu , The maximizing devi-ation method based on interval-valued Pythagorean fuzzy weighted aggregating operator for multiple criteria group decision analysis, Discrete Dyn Nat Soc (2015), 1–15. 10.1155/2015/746572

18.

Mareschal ,

J.P.

Brans and

Vincke , PROMETHEE: A new family of outranking methods in multicriteria analysis, ULB, Universite Libre de Bruxelles, 1984.

19.

Mulliner ,

Malys and

Maliene , Comparative analysis of MCDM methods for the assessment of sustainable housing affordability, Omega (2015). in press.

20.

Nakamura , Preference relations on a set of fuzzy utilities as a basis for decision making, Fuzzy Sets and Systems 20 (1986), 147–162.

21.

R.O.

Parreiras ,

P.Y.

Ekel ,

J.S.C.

Martini and

R.M.

Palhares , A flexible consensus scheme for multicriteria group decision making under linguistic assessments, Information Sciences 180 (2010), 1075–1089.

22.

J.J.

Peng ,

J.Q.

Wang ,

Wang and

X.H.

Chen , Multi-criteria decision-making approach with hesitant interval valued intuitionistic fuzzy set, The Scientific World Journal (2014), 22. Article ID 868515.

23.

J.J.

Peng ,

J.Q.

Wang ,

X.H.

Wu ,

H.Y.

Zhang and

X.H.

Chen , The fuzzy cross-entropy for intuitionistic hesitant fuzzy sets and their application in multi-criteria decision-making, International Journal of Systems Science (2014).

24.

X.D.

Peng and

Yang , Fundamental properties of interval-valued Pythagorean fuzzy aggregation operators, Int J Intell Syst 31(5) (2015), 444–487.

25.

X.D.

Peng and

Yang , Some results for Pythagorean fuzzy sets, Int J Intell Syst 30(11) (2015), 1133–1160.

26.

Qian ,

G.H.

Wang and

X.Q.

Feng , Generalized hesitant fuzzy sets and their application in decision support system, Knowledge-Based Systems 37 (2013), 357–365.

27.

Rabbani ,

Zamani ,

Yazdani-Chamzini , et al., Proposing a new integrated model based on sustainabil- ity balanced scorecard (SBSC) and MCDM approaches by using linguistic variables for the performance evaluation of oil producing companies, Expert Systems with Applications 41(16) (2014), 7316–7327.

28.

Roy and

Bertier , La methode ELECTRE II: Une methode de classement en prédence decriteres multiples, 1971.

29.

Sajjad

Ali Khan, S.

Abdullah, A. Ali and

Sadiqui , Pythagorean hesitant fuzzy sets and their application to group decision making, Journal of Intelligent and Fuzzy Systems 33 (2017), 3971–3985. DOI: 10.3233/JIFS-17811

30.

Sakhuja ,

Jain and

Dweiri , Application of an integrated MCDM approach in selecting outsourcing strategies in hotel industry, International Journal of Logistics Systems and Management 20(3) (2015), 304–324.

31.

T.J.

Stewart , A critical survey on the status of multiple criteria decision making theory and practice, Omega 20 (1992), 569–586.

32.

Torra and

Narukawa , On hesitant fuzzy sets and decision, in: The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 2009, pp. 1378–1382.

33.

Torra , Hesitant fuzzy sets, International Journal of Intelligent Systems 25 (2010), 529–539.

34.

Wang ,

Xu and

Ni , Distance and similarity measures of dual hesitant fuzzy sets with their applications to multiple attribute decision making, 2014 IEEE International Conference on Progress in Informatics and Computing, Shanghai, 2014, pp. 88–92.

35.

Y.M.

Wang , Using the method of maximizing deviations to make decision for multi-indices, System Engineering and Electronics 7 (1998), 24–26.

36.

Z.S.

Xu , A deviation-based approach to intuitionistic fuzzy multiple attribute group decision making, Group Decision and Negotiation 19 (2010), 57–76.

37.

Z.S.

Xu , Intuitionistic preference relations and their application in group decision making, Information Sciences 177 (2007), 2363–2379.

38.

Xu and

Zhang , Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information, Knowledge-Based Systems 52 (2013), 53–64.

39.

Z.L.

Yue , An extended TOPSIS for determining weights of decision makers with interval numbers, Knowledge-Based Systems 24 (2011), 146–153.

40.

R.R.

Yager and

A.M.

Abbasov , Pythagorean membership grades, complex numbers, and decision making, Int J Intell Syst 28(5) (2013), 436–452.

41.

R.R.

Yager , Pythagorean fuzzy subsets, In Proc Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, Canada, 2013, pp. 57–61.

42.

R.R.

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

43.

L.A.

Zadeh , Fuzzy sets, Information and Control 8 (1965), 338–356.

44.

Zeng , Pythagorean fuzzy multi-attribute group decision making with probabilistic information and OWA approach, Int J Intell Syst (2017), 1–15.

45.

X.L.

Zhang and

Z.S.

Xu , Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets, Int J Intell Syst 29(12) (2014), 1061–1078.

46.

Zhang and

P.D.

Liu , Method for multiple attribute decision-making under risk with interval numbers, Inter-national Journal of Fuzzy Systems 12 (2010), 237–242.

47.

X.L.

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

48.

X.U.

Zhu ,

Z.S.

Xu and

M.M.

Xia , Dual hesitant fuzzy sets, Journal of Applied-Mathematics (2012), 13. Article ID 879629.

49.

H.C.

Liao and

Z.S.

Xu , A VIKOR-based method for hesitant fuzzy multi-criteria decision making, Fuzzy Optim Decis Mak 12 (2013), 373–392.

50.

X.L.

Zhang and

Z.S.

Xu , The TODIM analysis approach based on novel measured functions under hesitant fuzzy environment, Knowl-Based Syst 61 (2014), 48–58.

51.

Sajjad

Ali Khan, S.

Abdullah, M.Y. Ali ,

Hussain and

Farooq , Extension of TOPSIS method base on Choquet integral under interval-valued Pythagorean fuzzy environ-ment, Journal of Intelligent & Fuzzy Systems 34 (2018), 267–282. DOI: 10.3233/JIFS-171164

52.

Sajjad Ali Khan and

Abdullah , Interval-valued Pythagorean fuzzy GRA method for multiple attribute decision making with incomplete weight information, IntJIntell Syst (2018), 1–28. 10.1002/int.21992.