Pythagorean Uncertain Linguistic Variable Hamy Mean Operator and Its Application to Multi-attribute Group Decision Making

Ⅰ. INTRODUCTION

The multi-attribute group decision making (MAGDM) problem relates to the selection of the best alternative(s) under multiple attribute values given by several experts, and has been widely applied in the fields of supplier selection [1], [2], transportation engineering [3], [4], risk assessment [5]-[7], etc. [8], [9]. In the multi-attribute decision making (MADM) problem, linguistic variables are very convenient to describe evaluation intention. For example, the decision makers can use "Very low", "Low", "Fair", "High", and "Very high" to evaluate the investment amount. With the increasing complexity and uncertainty of decision making environment, uncertain linguistic is used to describe the evaluation information, which can significantly express the decision makers' intension. For example, decision makers give the evaluation value of investment amount between "Fair" and "High", which means it is superior to "Fair" but inferior to "High". Consequently, research of uncertain linguistic variables in MAGDM is significantly valuable.

Intuitionistic fuzzy set (IFS) was first proposed by Atanassov [10] to express uncertainty, which involves membership degree and non-membership degree, and attracted many researchers' attentions [11]-[14]. However, IFS has its limitation that the sum of membership degree and non-membership degree must be equal to or less than 1, which is not able to fully express the ideas of the decision makers. Therefore, Yager extended the IFS and proposed the Pythagorean fuzzy set (PFS) [15], which can overcome the constraints of IFS and provide more flexibility than IFS for handling uncertain information. For example, the degree of membership and non-membership given by a decision maker are 0.8 and 0.4, respectively. Obviously, it can not be handled by IFS, but it can be expressed by PFSs. Therefore, PFS received more attentions, recently [16]-[22]. Zhang and Xu [16] defined the operation rules of PFS, and extended the TOPSIS method with PFSs. By combining PFSs with hesitant fuzzy sets (HFSs), Liang and Xu [17] proposed a new concept of hesitant Pythagorean fuzzy sets (HPFSs) and extended the TOPSIS method with HPFSs. Zhang [18] extended PFSs to interval-valued Pythagorean fuzzy sets (IVPFSs) and introduced the basic operations of interval-valued Pythagorean fuzzy numbers (IVPFNs). In addition, they developed a closeness index-based Pythagorean fuzzy QUALIFLEX method to deal with hierarchical multi-criteria decision making (MCDM) problems. Wang et al. [19] proposed a new bi-directional projection model to handle uncertain linguistic variables. Compared with projection model, the bi-directional projection model can effectively get the ranking order even when alternatives distribute on the perpendicular bisector of ideal alternatives. Ren et al. [20] introduced an extended TODIM method to select the governor of Asian Infrastructure Investment Bank with Pythagorean fuzzy information. A new Pythagorean Fuzzy Stochastic MCDM method was proposed by Peng and Dai [21] based on prospect theory and regret theory. Xue et al. [22] defined the concept of entropy of PFSs and extended the LINMAP method. Combining Pythagorean fuzzy sets with linguistic variables, Peng and Yang [23] proposed Pythagorean fuzzy linguistic sets (PFLSs) and defined the operation rules of PFLSs. Since uncertain linguistic variable is more convenient to express uncertain information than linguistic variable, Liu et al. [24] proposed the definition of Pythagorean fuzzy uncertain linguistic sets (PFULSs) based on PFLSs and uncertain linguistic variable, and extended VIKOR method with the PFULSs.

VIKOR method was initially proposed by Opricovic [25], which can obtain a set of compromising solutions when the criterion conflicts with each other. The compromising solution provides a balance between the maximum group utility value and the minimum individual regret value. Recently, VIKOR method has been widely used to deal with different types of MADM problems. Wu et al. [26] proposed an extended VIKOR method under the uncertain linguistic environment to solve the supplier selection problem of nuclear power industry in China. To select the best green supplier development program, Awasthi and Kannan [27] presented a fuzzy NGT-VIKOR method. Chen [28] developed a new VIKOR-based method, and applied it to solve service quality and Internet stock performance evaluation problems. Traditional VIKOR method can effectively express the decision makers' behavior, but, it can not describe the interaction relationship of each other.

For MAGDM problems, information aggregation operator is a common method being used to aggregate multi-dimensional individual information into a comprehensive value [29]-[34] However, some information aggregation operators, such as OWA operator [34], do not take the interaction relationship into account. In fact, attributes interact with each other in general.

As being discussed previously, it is natural for decision makers to express the evaluation information with linguistic terms in realistic MAGDM problems. On the face of complexity and uncertainty in real decision making problems, Pythagorean uncertain linguistic variable has been proved to be a powerful and effective tool to enhance the expression ability of uncertain information.

To overcome the drawbacks listed in VIKOR method and OWA operator, we propose a new approach for MAGDM problems with Pythagorean uncertain linguistic variables. Hamy mean [35] (HM) operator, a useful tool to cope with the interaction relationship between the aggregated arguments, is introduced for information aggregation of experts' evaluations. As a result, we extend the HM operator to the fields of Pythagorean uncertain linguistic variables, and propose a new powerful information aggregation operator, named as Pythagorean uncertain linguistic variable Hamy mean (PULVHM) operator. Furthermore, inspired by the score function of Pythagorean fuzzy number (PFN), we develop a new score function of Pythagorean uncertain linguistic variables (PULVs), named as PULVSF, which is used to transform the PULVs into crisp numbers. Up to now, on the basis of the proposed PULVHM operator and PULVSF, we integrate the VIKOR method in the final decision making stage to obtain the ranking order of all the optional alternatives.

The rest of this paper is organized as follows. Section Ⅱ presents some basic definitions of linguistic variable, Pythagorean fuzzy set, and Hamy mean operator. In Section Ⅲ, we propose the concept of PULVHM operator. A new score function of PULFSs is provided in Section Ⅳ and the MAGDM procedures are listed in Section Ⅴ. In Section Ⅵ, the effectiveness of the proposed method is demonstrated by a practical MAGDM problem, comparative analysis and sensitivity analysis are conducted subsequently. Finally, Section Ⅶ comes to some conclusions.

Ⅱ. PRELIMINARIES

In this section, some basic conceptions of UVL, PFS, and HM operator are given and some properties about ULV, PFS, and HM operator are introduced.

A Uncertain Linguistic Variable (ULV)

Definition 1 [36], [37]: Let $S=\{s_i|i=0, 1, \ldots, 2z\}$ be a linguistic discrete term set, where denotes an evaluation value of linguistic variable. We call $s=[\underline{s}_\alpha, \bar{s}_\beta]$ the uncertain linguistic variable (ULV), where $\underline{s}_\alpha, \bar{s}_\beta\in S$ and $0\leq\alpha\leq\beta$, $\underline{s}_\alpha, \bar{s}_\beta$ denote the lower bound and the upper bound, respectively.

To preserve all the given information, Xu [38] extended the discrete term set to a continuous linguistic term set $\tilde{S}=\{s_i|s_1 < s_i\leq s_{2z}, i\in[1, 2z]\}$.

Let $s_1=[\underline{s}_{\alpha_1}, \bar{s}_{\beta_1}]$ and $s_2=[\underline{s}_{\alpha_2}, \bar{s}_{\beta_2}]$ be two arbitrary ULVs, then we have [38].

1) $s_1\oplus s_2=[\underline{s}_{\alpha_1}, \bar{s}_{\beta_1}]\oplus[\underline{s}_{\alpha_2}, \bar{s}_{\beta_2}]=[\underline{s}_{\alpha_1+\alpha_2}, \bar{s}_{\beta_1+\beta_2}]$

2) $s_1\otimes s_2=[\underline{s}_{\alpha_1}, \bar{s}_{\beta_1}]\otimes[\underline{s}_{\alpha_2}, \bar{s}_{\beta_2}]=[\underline{s}_{\alpha_1\times\alpha_2}, \bar{s}_{\beta_1\times\beta_2}]$

3) $\lambda_s=\lambda[\underline{s}_\alpha, \bar{s}_{\beta}]=[\underline{s}_{\lambda\cdot\alpha}, \overline{s}_{\lambda\cdot\beta}], \lambda\geq0$

4) $(s)^{\lambda}=[\underline{s}_\alpha, \bar{s}_{\beta}]^{\lambda}=[\underline{s}_{(\alpha)^{\lambda}}, \bar{s}_{(\beta)^{\lambda}}], \lambda\geq0$.

B Pythagorean Fuzzy Sets (PFSs)

Definition 2 [16]: Let $X$ be a fixed set. A PFS in $X$ takes the form of:

where function $u_p(x): X\rightarrow[0, 1]$ and $v_p(x): X\rightarrow[0, 1]$ denote membership function and non-membership function of $x\in X$, respectively $u_p^2(x)+v_p^2(x)\leq 1$, and $\pi_{p} \left(x \right)=\sqrt{1-u_{p}^{2} \left(x \right)-v_{p}^{2} \left(x \right)}$ denotes the hesitate degree of $x\in X$.

For the sake of simplicity, Zhang and Xu [16] called $P(u_P(x), v_P(x))$ as PFN denoted by $\beta=P(u_{\beta}, v_{\beta})$, where $u_p(x), v_p(x)\in[0, 1]$, $\pi_{p} \left(x \right)=\sqrt{1-u_{p}^{2} \left(x \right)-v_{p}^{2} \left(x \right)}$ and $u_p^2(x)+v_p^2(x)\leq 1$.

Let $\beta_1=P(u_{\beta_1}, v_{\beta_1})$ and $\beta_2=P(u_{\beta_2}, v_{\beta_2})$ be two PFNs, then the basic operation rules are summarized as follows [16]:

1) $\beta_{1} \oplus \beta_{2} =P\left({\sqrt{u_{\beta_{1} }^{2} +u_{\beta_{2} }^{2} -u_{\beta_{1} }^{2} u_{\beta_{2} }^{2} }, \nu_{\beta _{1} } \nu_{\beta_{2} } } \right)$

2) $\beta_{1} \otimes \beta_{2} =P\left({u_{\beta_{1} } u_{\beta_{2} }, \sqrt{\nu_{\beta_{1} }^{2} +\nu_{\beta_{2} }^{2} -\nu_{\beta_{1} }^{2} \nu_{\beta_{2} }^{2} }} \right)$

3) $k{\kern 1pt}\beta =P\left({\sqrt{1-\left({1-u_{\beta }^{2} } \right)^{k}}, \left({\nu_{\beta } } \right)^{k}} \right), {\kern 1pt}k\geq 0$

4) $\beta^{k} =P\left({\left({u_{\beta } } \right)^{k}, \sqrt{1-\left({1-\nu_{\beta }^{2} } \right)^{k}}} \right), {\kern 1pt}k\geq 0$.

According to the definition of PFNs, we can know that the subspace of PFNs is bigger than IFNs, as is shown in Fig. 1.

Figure  1.  The space relationship of PFNs and IFNs memberships functions

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Definition 3 [24]: Let $X$ be a fixed set $\tilde{P}=\{\langle x_i|([s_{\alpha_i}, s_{\beta_i}], P(u_p(x_i), v_p(x_i)))\rangle|x_i\in X\}$. denotes the PULVs on $X$, where function $u_p(x):X\rightarrow[0, 1]$ and $v_p(x):X\rightarrow[0, 1]$ denote membership function and non-membership function of $x\in X$, respectively, with $u_p^2(x)+v_p^2(x)\leq 1$.

Let $\tilde{\alpha_1}=\langle[s_{\alpha_1}, s_{\beta_1}], P(u_p(x_1), v_p(x_1))\rangle$ and $\tilde{\alpha_2}=\langle[s_{\alpha_2}, s_{\beta_2}], P(u_p(x_2), v_p(x_2))\rangle$ be two PULVs, we can obtain the operation rules based on that of ULVs and PFNs as follows:

1)

2)

3)

4)

Theorem 1: For any two PULVs $\dot{\alpha_1}=\langle[s_{\alpha_1}, s_{\beta_1}], $ $P(u_p(x_1), v_p(x_1))\rangle$ and $\dot{\alpha_2}=\langle[s_{\alpha_2}, s_{\beta_2}], P(u_p(x_2), v_p(x_2))\rangle$, the calculation rules meet the following properties:

1) $\tilde{\alpha}_1\oplus\tilde{\alpha}_2=\tilde{\alpha}_2\oplus\tilde{\alpha}_1$

2) $\tilde{\alpha}_1\otimes\tilde{\alpha}_2=\tilde{\alpha}_2\otimes\tilde{\alpha}_1$

3) $\gamma(\tilde{\alpha}_1\oplus\tilde{\alpha}_2)=\gamma\tilde{\alpha}_1\oplus\tilde{\alpha}_2, \gamma\geq0$

4) $(\tilde{\alpha})^{\gamma_1+\gamma_2}=\tilde{\alpha}^{\gamma_1}\otimes\tilde{\alpha}^{\gamma_2}, \gamma_1, \gamma_2\geq0$

5) $\gamma_1\tilde{\alpha}\oplus\gamma_2\tilde{\alpha}=(\gamma_1+\gamma_2)\tilde{\alpha}, \gamma_1, \gamma_2\geq0$

6) $\tilde{\alpha}_1^{\gamma}\otimes\tilde{\alpha}_2^{\gamma}=(\tilde{\alpha}_1\otimes\tilde{\alpha}_2)^{\gamma}, \gamma\geq0$

Proof:

C Hamy Mean (HM) Operator

Definition 4 [35]: Let $h_j(j=1, 2, \ldots, n)$ be a collection of nonnegative real numbers. $HM^{(p)}$ is called the Hamy mean(HM)operator, which is defined as follows:

where $p(p=1, 2, \ldots, n)$ is the parameter of HM operator, and $(k_1, k_2, \ldots, k_p)$ is p-tuple full combination of $(1, 2, \ldots, n)$, and $\left({{ $$\begin{array}{*{20}c} n \hfill \\ p \hfill \\ \end{array}$$ }} \right)=C_n^p=\dfrac{n!}{p!(n-p)!}$ is the binomial coefficient.

Obviously, the HM operator meets the following properties:

1) $HM^{(P)}(0, 0, \ldots, 0)=0$, if $\delta_k=0(k=1, 2, \ldots, n)$

2) $HM^{(P)}(\delta, \delta, \ldots, \delta)=\delta$, if $\delta_k=\delta(k=1, 2, \ldots, n)$

3) $HM^{(P)}(\delta_1, \delta_2, \ldots, \delta_k)\leq HM^{(P)}=(\beta_1, \beta_2, \ldots, \beta_k)$, if $\delta_k\leq\beta_k(k=1, 2, \ldots, n)$

4) ${\text{min}}_k(\delta_k)\leq HM^{(P)}(\delta_1, \delta_2, \ldots, \delta_k)\leq {\text{max}}_k(\delta_k)$.

Ⅲ. PYTHAGOREAN UNCERTAIN LINGUISTIC VARIABLE HAMY MEAN (PULVHM) OPERATOR

Definition 5: The collection of PULVs is denoted by$\widetilde{P}=\left\{ {\left\langle {x_{i} |\left({\left[ {s_{\alpha _{i} }, s_{\beta_{i} } } \right], P\left({u_{\widetilde{P}} \left({x_{i} } \right), v_{\widetilde{P}} \left({x_{i} } \right)} \right)} \right)} \right\rangle |x_{i} \in X} \right\}{\kern 1pt}$, $i=1, 2, \ldots, n$. Then PULVHM operator is defined as follows:

where $\varphi(\varphi=1, 2, \ldots, n)$ is the parameter of PULVHM operator, $(i_1, i_2, \ldots, i_{\varphi})$ is $\varphi$-tuple full combination of $(1, 2, \ldots, n)$, and $\left({{ $$\begin{array}{*{20}c} n \hfill \\ \phi \hfill \\ \end{array}$$ }} \right)=C_n^{\varphi}=\frac{n!}{\varphi!(n-\varphi)!}$ is the binomial coefficient and satisfies $1\leq i_1 < \ldots < i_{\varphi}\leq n$.

Theorem 2: The collection of PULVs is denoted as $\widetilde{P}=\left\{ {\left\langle {x_{i} |\left({\left[ {s_{\alpha_{i} }, s_{\beta_{i} } } \right], P\left({u_{\widetilde{P}} \left({x_{i} } \right), v_{\widetilde{P}} \left({x_{i} } \right)} \right)} \right)} \right\rangle |x_{i} \in X} \right\}{\kern 1pt}$, $i=1, 2, \ldots, n$. Based on the operation rules of PULVs, we can know that the result of PULVHM is still a PULV, consequently, the PULVHM operator can also be expressed as:

Proof:

According to Theorem 1, we can get

Then,

Therefore,

Next, we will prove that the aggregation result of multiple PULVs by PULVHM operator is still a PULV. We already know that $0\leq u_{\tilde{p}}(x_{i_j})\leq 1$, $0\leq v_{\tilde{p}}(x_{i_j})\leq 1$ and $(u_{\tilde{p}}(x_{i_j}))^2+(v_{\tilde{p}}(x_{i_j}))^2\leq 1$, then,

Example 1: There are three PULVs:

$\tilde{p}_1=\langle[s_4, s_5], P(0.8, 0.3)\rangle$, $\tilde{p}_2=\langle[s_3, s_4], P(0.7, 0.5)\rangle$ and $\tilde{p}_3=\langle[s_6, s_7], P(0.9, 0.2)\rangle$, $\varphi=2$ is the parameter of PULVHM operator. Then, we can get:

Therefore,

Theorem 3: The PULVHM operator meets the following properties:

1) $PULVHM^{\left(\phi \right)}\left({0, 0, \ldots, } \right)=0$

2) $PULVHM^{\left(\phi \right)}\left({\widetilde{q}, \widetilde{q}, \ldots, \widetilde{q}} \right)=\widetilde{q}$

3) $PULVHM^{\left(\phi \right)}\left({\widetilde{q_{1} }, \widetilde{q_{2} }, \ldots, \widetilde{q_{k} }} \right)\leq PULVHM^{\left(\phi \right)}(\widetilde{p_{1} }, \widetilde{p_{2} }, $ $\ldots, \widetilde{p_{k} })$, if $\widetilde{q_{k} }\leq \widetilde{p_{k} }\left({k=1, 2, \ldots, n} \right)$

4) ${\min} \left({\widetilde{q_{k} }} \right)\leq PULVHM^{\left(\phi \right)}\left({\widetilde{q_{1} }, \widetilde{q_{2} }, \ldots, \widetilde{q_{n} }} \right)\leq {\max }\left({\widetilde{q_{k} }} \right), k=1, 2, \ldots, n$

Proof:

1) The result of property 1 can be easily drawn out from the basic mathematical operation.

2) For any PULVs $q=\langle[s_{\alpha_i}, s_{\beta_1}], P[u_{\tilde{p}}(x_i), v_{\tilde{p}}(x_i)]\rangle$,

3) Let $\tilde{q}_i=\langle[s_{\alpha_i}, s_{\beta_i}], P[u_q(x_i), v_q(x_i)]\rangle$ and $\tilde{p}_i=\langle[s_{\gamma_i}, s_{\gamma_i}], P[u_{\tilde{p}}(x_i), v_{\tilde{p}}(x_i)]\rangle$ be two groups of PULVs, and $\tilde{q}_i\leq \tilde{p}_i(i=1, 2, \ldots, n)$.

For any $i$, it can be seen that $\alpha_i\leq\gamma_i$, $\beta_i\leq\lambda_i$, $u_{\tilde{q}}(x_i)\leq u_{\tilde{p}}(x_i)$ and $v_{\tilde{q}}(x_i)\leq v_{\tilde{p}}(x_i)$.

Then, we have $\prod_{j=1}^{\varphi}\alpha_{i_j}\leq\prod_{j=1}^{\varphi}\gamma_{i_j}$, $\prod_{j=1}^{\varphi}\beta_{i_j}\leq\prod_{j=1}^{\varphi}\lambda_{i_j}$

Similarly, we can get

From the calculation above, we can know that

Similarly, we can obtain

Accordingly, if, then $PULVHM^{\left(\phi \right)}\left({\widetilde{q_{1} }, \widetilde{q_{2} }, \ldots, \widetilde{q_{k} }} \right)\leq$ $ PULVHM^{\left(\phi \right)}\left({\widetilde{p_{1} }, \widetilde{p_{2} }, \ldots, \widetilde{p_{k} }} \right)$

4) For arbitrary PULV, we have:

and

It is easy to get that $s_{\min \left({\alpha_{i} } \right)} \leq s_{\alpha_{i} } \leq s_{\max \left({\alpha_{i} } \right)} $, and $\min(u_{\tilde{p}}(x_i))\leq u_{\tilde{p}}(x_i)\leq \max(u_{\tilde{p}}(x_i))$, $\min(v_{\tilde{p}}(x_i))\leq v{\tilde{p}}(x_i)\leq \max(v_{\tilde{p}}(x_i))$. Based on the properties (2) and (3) in Theorem 3, we can obtain that: $PULVHM^{\left(\phi \right)}\left({\widetilde{q_{1} }, \widetilde{q_{2} }, \ldots, \widetilde{q_{k} }} \right)\geq PULVHM^{\left(\phi \right)}\left({\widetilde{q}^{-}, \widetilde{q}^{-}, \ldots, \widetilde{q}^{-}} \right)=\widetilde{q}^{-}$$PULVHM^{\left(\phi \right)}\left({\widetilde{q_{1} }, \widetilde{q_{2} }, \ldots, \widetilde{q_{k} }} \right)\leq PULVHM^{\left(\phi \right)}\Big(\widetilde{q}^{+}, \widetilde{q}^{+}, $ $\ldots, \widetilde{q}^{+} \Big)=\widetilde{q}^{+}$Therefore, ${\min }\left({\widetilde{q_{k} }} \right)\leq PULVHM^{\left(\phi \right)}\Big(\widetilde{q_{1} }, \widetilde{q_{2} }, $ $\ldots, \widetilde{q_{n} } \Big)\leq {\max }\left({\widetilde{q_{k} }} \right), k=1, 2, \ldots, n$

Ⅳ. PYTHAGOREAN UNCERTAIN LINGUISTIC VARIABLE SCORE FUNCTION (PULVSF)

Zhang proposed the score function [16] and accuracy function [39] to compare the values of PFNs, the definitions are as follows:

Definition 6 [16]: For arbitrary PFN $\beta=P(u_{\beta}, v_{\beta})$, the score function of $\beta$ is:

Definition 7 [39]: For arbitrary PFN $\beta=P(u_{\beta}, v_{\beta})$, the accuracy function of $\beta$ is:

For two PFNs $\beta_{1} =P\left({u_{\beta_{1} }, \nu_{\beta_{1} } } \right)$ and $\beta_{2} =P\left({u_{\beta_{2} }, \nu_{\beta_{2} } } \right)$, the comparison laws of the two PFNs are defined as follows:

1) $score(\beta_1)>score(\beta_2)$, which means $\beta_1$ is bigger than $\beta_2$, expressed by $\beta_1>\beta_2$.

2) $score(\beta_1)>score(\beta_2)$, which means $\beta_1$ is smaller than $\beta_2$, expressed by $\beta_1 < \beta_2$.

3) $score(\beta_1)\!\!=\!\! score(\beta_2)$, then $\left\{\!\!\! $$\begin{array}{l} H(\beta_1)\! < \!H(\beta_2)\!\Rightarrow\!\beta_1\! < \!\beta_2\\ H(\beta_1)\!=\!H(\beta_2)\!\Rightarrow\!\beta_1\!:\!\beta_2\\ H(\beta_1)\!>\!H(\beta_2)\!\Rightarrow\!\beta_1\!>\!\beta_2\end{array}$$ \right.$.

Based on the score function of Pythagorean fuzzy number [16], a new Pythagorean fuzzy number score function (PFNSF) is proposed, which considers the decision makers' hesitation degree.

Definition 8: For a PFN $\beta=P(u_{\beta}, v_{\beta})$, the new score function of $\beta$ is:

Theorem 4: For PFNSF $score(\beta)$, it satisfies the properties as follows.

1) Monotonicity. $score(\beta)$ is a monotonous increasing function of membership degree $\mu_{\beta}$ and a monotonous decreasing function of non-membership degree $\nu_{\beta}$.

2) Boundedness. The maximum of $score(\beta)$ is 1, while the minimum of $score(\beta)$ is -1. That is to say, $score(\beta)\in[-1, 1]$.

3) Comparision rules. For two PFNs $\beta_{1} =P\left({u_{\beta_{1} }, \nu_{\beta_{1} } } \right)$ and $\beta_{2} =P\left({u_{\beta_{2} }, \nu_{\beta_{2} } } \right)$, the comparison rules of the PFNs are:

Proof:

From $\pi_{\beta } {\rm =}\sqrt{1-u_{\beta }^{2} -v_{\beta }^{2} }$, we can obtain $score\left(\beta \right)=2u_{\beta }^{2} +\frac{1-u_{\beta }^{2} -v_{\beta }^{2} }{1+v_{\beta }^{2} }-1$

Obviously, $\frac{\partial score\left(\beta \right)}{\partial u_{\beta } }>0$. Then, we can know that is a monotonous increasing function of membership degree.

Obviously, $\frac{\partial score\left(\beta \right)}{\partial v_{\beta } } < 0$. Then, we can know that $score(\beta)$ is a monotonous decreasing function of non-membership degree $\nu_{\beta}$.

2) According to monotonicity, $score(\beta)$ is a monotonous increasing function of membership degree $\mu_{\beta}$ and a monotonous decreasing function of non-membership degree $\nu_{\beta}$. Therefore, the maximum of $score(\beta)$ is 1 if $\mu_{\beta}=1, \nu_{\beta}=0$ and only if and the minimum $score(\beta)$ of is -1 if and only if $\mu_{\beta}=0, \nu_{\beta}=1$.

Example 2: For $\beta_1=(0.8, 0.4)$ and $\beta_2=(0.7, 0.1)$. According to definition 6 and definition 7, we can get $score(\beta_1)=0.48$ and $score(\beta_2)=0.48$, i.e., $score(\beta_1)=score(\beta_2)$. We need to further compute the accuracy functions of the two PFNs. $H(\beta_1)=0.8$ and $H(\beta_2)=0.5$, therefore, $\beta_1>\beta_2$.

According to definition 8, $score(\beta_1)=0.45$, $score(\beta_2)=0.48$, and we can directly obtain the comparision result of $\beta_1 < \beta_2$ from $score(\beta_1) < score(\beta_2)$.

Remark 1: The proposed score function has the following advantages:

1) Compared with the score function proposed by [16], we can gain the final comparision result just in one step without another accuracy function.

2) The hesitation degree of decision makers can be taken into account.

Next, We further extend the PFNSF to deal with Pythagorean uncertain linguistic variables and propose a new score function of Pythagorean uncertain linguistic variables, abbreviated as PULVSF for the sake of convenience.

Definition 9:

For any PULV $\tilde{\alpha}=\langle[s_{\alpha_i}, s_{\beta_1}], P(u_p(x_i), v_p(x_i))\rangle$, the score function of $\tilde{\alpha}$ is:

where $I(s_{(\alpha+\beta)/2})=(\alpha+\beta)/2$.

Theorem 5: For PULVSF $score(\tilde{\alpha})$, it holds some properties as follows:

1) Monotonicity. $score(\tilde{\alpha})$ is a monotonous increasing function of membership degree $\mu_{\beta}$ and is a monotonous decreasing function of non-membership degree $\nu_{\beta}$.

2) Comparision rules. For two PULVs $\tilde{\alpha_1}=\langle[s_{\alpha_1}, s_{\beta_1}], P(u_p(x_1), v_p(x_1))\rangle$ and $\tilde{\alpha_2}=\langle[s_{\alpha_2}, s_{\beta_2}], P(u_p(x_2), v_p(x_2))\rangle$, the comparison rules of the PULVs are as follows:

a) If $score(\tilde{\alpha_1})>score(\tilde{\alpha_2})$, then $\tilde{\alpha_1}>\tilde{\alpha_2}$.

b) If $score(\tilde{\alpha_1}) < score(\tilde{\alpha_2})$, then $\tilde{\alpha_1} < \tilde{\alpha_2}$.

c) If $score(\tilde{\alpha_1})=score(\tilde{\alpha_2})$, then $\tilde{\alpha_1}:\tilde{\alpha_2}$.

Ⅴ. MAGDM METHOD BASED ON PULVHM OPERATOR

Based on the PULVHM operator derived from Section Ⅲ and the PULVSF presented in Section IV, we propose a new MAGDM approach. The framework of the proposed method is depicted in Fig. 2. The proposed method consists of three stages. Firstly, construct the group decision making problem with finite m alternatives (marked as $A$), n attributes (marked as $C$) and k experts (marked as $E$). Secondly, aggregate all the experts' evaluation information by the PULVHM operator, and further transform the PULVs into crisp numbers based on PULVSF. Finally, obtain the ranking order of the alternative(s) by VIKOR method and select the best alternative(s).

Figure  2.  Diagram of the proposed method

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Step 1: Decision information input. Define the set of $C=\{c_1, c_2, \ldots, c_n\}$ attributes and weighting vector of the attributes $w=\{w_1, w_2, \ldots, w_n\}$, which meets $0\leq w_i\leq 1(i=1, 2, \ldots, n)$, $\sum_{i=1}^nw_i=1$, a series of alternatives $A=\{a_1, a_2, \ldots, a_m\}$ and decision makers group $E=\{e_1, e_2, \ldots, e_k\}$. In addition, the attribute values of each alternative are given by PULVs.

Step 2: Each decision maker provides his/her individual evaluation value with respect to each attribute, then we can get the decision matrix of each expert.

Step 3: Normalize the decision matrix. For PULVs $\tilde{p}_{ij}=\langle[s_{\alpha_{ij}}, s_{\beta_{ij}}], P(u_{\tilde{p}}(x_{ij}), v_{\tilde{p}}(x_{ij}))\rangle$.

For beneficial attributes, $\tilde{p}_{ij}=\tilde{p}_{ij}=\langle[s_{\alpha_{ij}}, s_{\beta_{ij}}], P(u_{\tilde{p}}(x_{ij})$, $v_{\tilde{p}}(x_{ij}))\rangle$. For cost attributes,

where $(\alpha_{ij})^{-1}=l+1-\alpha_{ij}$, $(\beta_{ij})^{-1}=l+1-\beta_{ij}$ and $l$ is the number of language terms.

Step 4: Based on PULVHM operator, we can get group decision matrix (GDM, as shown in Table Ⅰ), a result of the aggregation with respect to each decision maker's evaluation value.

Table  Ⅰ.  GROUP DECISION MATRIX

	$c_1$	$c_2$	$\ldots$	$c_n$
$a_1$	$\tilde{p}_{11}$	$\tilde{p}_{12}$	$\ldots$	$\tilde{p}_{1n}$
$a_2$	$\tilde{p}_{21}$	$\tilde{p}_{22}$	$\ldots$	$\tilde{p}_{2n}$
$\ldots$	$\ldots$	$\ldots$	$\ldots$	$\ldots$
$a_m$	$\tilde{p}_{m1}$	$\tilde{p}_{m2}$	$\ldots$	$\tilde{p}_{mn}$

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Step 5: Transform the GDM into score function matrix (as shown in Table Ⅱ) based on definition 9.

Table  Ⅱ.  SCORE FUNCTION MATRIX

	$c_1$	$c_2$	$\ldots$	$c_n$
$a_1$	$G_{11}$	$G_{12}$	$\ldots$	$G_{1n}$
$a_2$	$G_{21}$	$G_{22}$	$\ldots$	$G_{2n}$
$\ldots$	$\ldots$	$\ldots$	$\ldots$	$\ldots$
$a_m$	$G_{m1}$	$G_{m2}$	$\ldots$	$G_{mn}$

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Step 6: Determine the positive ideal solution and negative ideal solution, respectively.

Positive ideal solution: $G^+\!\!=\!\!\{\!{\text{max}} G_{i1}, {\text{max}} G_{i2}, ..., {\text{max}} G_{in}\!\}$;

Negative ideal solution: $G^-\!\!=\!\!\{\!\text{max} G_{i1}, \text{max} G_{i2}, ..., \text{max} G_{in}\!\}$

Step 7: Obtain group utility $S_i$, individual regret $R_i$ and compromise value $Q_i$ via the following equations.

where $S^+=\max S_i$, $S^-=\min S_i$, $R^+=\max R_i$, $R^-=\min R_i$. Besides, $\theta$ is the weight for the strategy of the maximum group utility and $1-\theta$ is the weight of individual regret. When $\theta>0.5$, decision makers tend to make decision by the strategy of maximum group utility; when $\theta=0.5$, by consensus, group utility and individual regret are of the same importance; and when $\theta < 0.5$, by the strategy of individual regret.

Step 8: Ranking the alternatives in an ascending order by the values of $S_i$, $R_i$ and $Q_i$. The results are three ranking lists.

Step 9: Based on the value of $Q_i$, we can get $a^{(1)}, a^{(2)}, \ldots, a^{(m)}$, and alternative $a^{(1)}$ is named as the best solution, if it meets the following conditions.

Condition 1: $Q(a^{(2)})-Q(a^{(1)})\geq\frac{1}{m-1}$;

Condition 2: The alternative $a^{(1)}$ must be the best ranking either by the value of $S_i$ or $R_i$.

If the conditions can not be satisfied simultaneously, we can still get compromise solution by the following situations:

If Condition 2 is not satisfied, then alternatives $a^{(1)}$ and $a^{(2)}$ are both the compromise solutions;

If Condition 1 is not satisfied, the alternatives $a^{(1)}$, $a^{(2)}$, $\ldots$, $a^{(d)}$ are all the compromise solutions, where $a^{(d)}$ meets $Q(a^{(d)})-Q(a^{(1)}) < \frac{1}{m-1}$

Ⅵ. NUMERICAL STUDY

In this section, a numerical example is used to illustrate the effectiveness of the proposed method.

The risk investment project evaluation decision is based on the information obtained by the risk investment company, using scientific means and methods to analyze and evaluate the quality of the project carefully, and decide whether to invest the project in light of some factors, such as its potential investment income and risk. An investment company plans to invest one of the following industries: $a_1$: high-technology industry; $a_2$ tourism industry; $a_3$ financial industry; $a_4$ manufacturing industry. And after synthetical consideration, the main factors are taken into account: $c_1$ investment rewards; $c_2$ investment risks; $c_3$ investment capitals, $c_4$ market prospects. Besides, three experts in this area provide the consultant service for the company as shown in Table Ⅲ-Ⅴ. $E=\{e_1, e_2, e_3\}$ denotes the experts group. Attributes set is $C=\{c_1, c_2, c_3, c_4\}$, and $w=\{0.4, 0.3, 0.2, 0.1\}$ is the weight vector of the corresponding attributes.

Table  Ⅲ.  DECISION MATRIX OF EXPERT 1

	$c_1$	$c_2$	$c_3$	$c_4$
$a_1$	$\langle[s_5, s_6], P(0.7, 0.3)\rangle$	$\langle[s_6, s_7], P(0.8, 0.3)\rangle$	$\langle[s_6, s_7], P(0.9, 0.2)\rangle$	$\langle[s_4, s_6], P(0.6, 0.5)\rangle$
$a_2$	$\langle[s_4, s_5], P(0.8, 0.3)\rangle$	$\langle[s_3, s_4], P(0.7, 0.5)\rangle$	$\langle[s_3, s_5], P(0.5, 0.5)\rangle$	$\langle[s_2, s_3], P(0.7, 0.4)\rangle$
$a_3$	$\langle[s_6, s_7], P(0.6, 0.5)\rangle$	$\langle[s_6, s_7], P(0.7, 0.2)\rangle$	$\langle[s_4, s_6], P(0.7, 0.4)\rangle$	$\langle[s_5, s_6], P(0.8, 0.3)\rangle$
$a_4$	$\langle[s_3, s_4], P(0.7, 0.4)\rangle$	$\langle[s_1, s_2], P(0.8, 0.2)\rangle$	$\langle[s_2, s_4], P(0.6, 0.3)\rangle$	$\langle[s_2, s_3], P(0.7, 0.4)\rangle$

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Table  Ⅳ.  DECISION MATRIX OF EXPERT 2

	$c_1$	$c_2$	$c_3$	$c_4$
$a_1$	$\langle[s_6, s_7], P(0.8, 0.2)\rangle$	$\langle[s_4, s_5], P(0.7, 0.4)\rangle$	$\langle[s_5, s_7], P(0.6, 0.5)\rangle$	$\langle[s_6, s_7], P(0.9, 0.2)\rangle$
$a_2$	$\langle[s_5, s_6], P(0.7, 0.4)\rangle$	$\langle[s_4, s_5], P(0.6, 0.3)\rangle$	$\langle[s_5, s_6], P(0.8, 0.2)\rangle$	$\langle[s_5, s_7], P(0.6, 0.4)\rangle$
$a_3$	$\langle[s_5, s_7], P(0.8, 0.3)\rangle$	$\langle[s_5, s_6], P(0.8, 0.4)\rangle$	$\langle[s_5, s_6], P(0.7, 0.4)\rangle$	$\langle[s_6, s_7], P(0.8, 0.3)\rangle$
$a_4$	$\langle[s_4, s_6], P(0.6, 0.5)\rangle$	$\langle[s_3, s_4], P(0.7, 0.4)\rangle$	$\langle[s_4, s_5], P(0.7, 0.4)\rangle$	$\langle[s_4, s_5], P(0.8, 0.3)\rangle$

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DownLoad: CSV

Table  Ⅴ.  DECISION MATRIX OF EXPERT 3

	$c_1$	$c_2$	$c_3$	$c_4$
$a_1$	$\langle[s_4, s_6], P(0.7, 0.4)\rangle$	$\langle[s_5, s_6], P(0.8, 0.4)\rangle$	$\langle[s_5, s_7], P(0.6, 0.4)\rangle$	$\langle[s_5, s_7], P(0.8, 0.3)\rangle$
$a_2$	$\langle[s_4, s_5], P(0.8, 0.4)\rangle$	$\langle[s_5, s_6], P(0.7, 0.5)\rangle$	$\langle[s_4, s_6], P(0.8, 0.4)\rangle$	$\langle[s_4, s_5], P(0.7, 0.3)\rangle$
$a_3$	$\langle[s_5, s_6], P(0.7, 0.3)\rangle$	$\langle[s_6, s_7], P(0.8, 0.3)\rangle$	$\langle[s_4, s_6], P(0.6, 0.5)\rangle$	$\langle[s_4, s_6], P(0.7, 0.4)\rangle$
$a_4$	$\langle[s_4, s_5], P(0.7, 0.4)\rangle$	$\langle[s_3, s_4], P(0.7, 0.3)\rangle$	$\langle[s_4, s_5], P(0.6, 0.3)\rangle$	$\langle[s_3, s_4], P(0.8, 0.4)\rangle$

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DownLoad: CSV

A Investment Project Selections

Step 1: Standardized decision matrix. $c_1$, $c_4$ are beneficial attributes, whereas, $c_2$, $c_3$ are cost attributes. And standard expert decision matrices are as shown in Table Ⅵ-Ⅷ.

Table  Ⅵ.  STANDARD DECISION MATRIX OF EXPERT 1

	$c_1$	$c_2$	$c_3$	$c_4$
$a_1$	$\langle[s_5, s_6], P(0.7, 0.3)\rangle$	$\langle[s_1, s_2], P(0.3, 0.8)\rangle$	$\langle[s_1, s_2], P(0.2, 0.9)\rangle$	$\langle[s_4, s_6], P(0.6, 0.5)\rangle$
$a_2$	$\langle[s_4, s_5], P(0.8, 0.3)\rangle$	$\langle[s_4, s_5], P(0.5, 0.7)\rangle$	$\langle[s_4, s_6], P(0.5, 0.5)\rangle$	$\langle[s_2, s_3], P(0.7, 0.4)\rangle$
$a_3$	$\langle[s_6, s_7], P(0.6, 0.5)\rangle$	$\langle[s_1, s_2], P(0.2, 0.7)\rangle$	$\langle[s_3, s_5], P(0.4, 0.7)\rangle$	$\langle[s_5, s_6], P(0.8, 0.3)\rangle$
$a_4$	$\langle[s_3, s_4], P(0.7, 0.4)\rangle$	$\langle[s_6, s_7], P(0.2, 0.8)\rangle$	$\langle[s_4, s_6], P(0.3, 0.6)\rangle$	$\langle[s_2, s_3], P(0.7, 0.4)\rangle$

 | Show Table

DownLoad: CSV

Table  Ⅶ.  STANDARD DECISION MATRIX OF EXPERT 2

	$c_1$	$c_2$	$c_3$	$c_4$
$a_1$	$\langle[s_6, s_7], P(0.8, 0.2)\rangle$	$\langle[s_3, s_4], P(0.4, 0.7)\rangle$	$\langle[s_1, s_3], P(0.5, 0.6)\rangle$	$\langle[s_6, s_7], P(0.9, 0.2)\rangle$
$a_2$	$\langle[s_5, s_6], P(0.7, 0.4)\rangle$	$\langle[s_3, s_4], P(0.3, 0.6)\rangle$	$\langle[s_2, s_3], P(0.2, 0.8)\rangle$	$\langle[s_5, s_7], P(0.6, 0.4)\rangle$
$a_3$	$\langle[s_5, s_7], P(0.8, 0.3)\rangle$	$\langle[s_2, s_3], P(0.4, 0.8)\rangle$	$\langle[s_2, s_3], P(0.4, 0.7)\rangle$	$\langle[s_6, s_7], P(0.8, 0.3)\rangle$
$a_4$	$\langle[s_4, s_6], P(0.6, 0.5)\rangle$	$\langle[s_4, s_5], P(0.4, 0.7)\rangle$	$\langle[s_3, s_4], P(0.4, 0.7)\rangle$	$\langle[s_4, s_5], P(0.8, 0.3)\rangle$

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DownLoad: CSV

Table  Ⅷ.  STANDARD DECISION MATRIX OF EXPERT 3

	$c_1$	$c_2$	$c_3$	$c_4$
$a_1$	$\langle[s_4, s_6], P(0.7, 0.4)\rangle$	$\langle[s_2, s_3], P(0.4, 0.8)\rangle$	$\langle[s_1, s_3], P(0.4, 0.6)\rangle$	$\langle[s_5, s_7], P(0.8, 0.3)\rangle$
$a_2$	$\langle[s_4, s_5], P(0.8, 0.4)\rangle$	$\langle[s_2, s_3], P(0.5, 0.7)\rangle$	$\langle[s_2, s_4], P(0.4, 0.8)\rangle$	$\langle[s_4, s_5], P(0.7, 0.3)\rangle$
$a_3$	$\langle[s_5, s_6], P(0.7, 0.3)\rangle$	$\langle[s_1, s_2], P(0.3, 0.8)\rangle$	$\langle[s_2, s_4], P(0.5, 0.6)\rangle$	$\langle[s_4, s_6], P(0.7, 0.4)\rangle$
$a_4$	$\langle[s_4, s_5], P(0.7, 0.4)\rangle$	$\langle[s_4, s_5], P(0.3, 0.7)\rangle$	$\langle[s_2, s_3], P(0.3, 0.6)\rangle$	$\langle[s_3, s_4], P(0.8, 0.4)\rangle$

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Step 2: By PULVHM operator, we can get group decision matrix as shown in Table Ⅸ.

Table  Ⅸ.  GROUP DECISION MATRIX

	$c_1$	$c_2$	$c_3$	$c_4$
$a_1$	$\langle[s_{4.13}, s_{6.32}], P(0.73, 0.30)\rangle$	$\langle[s_{1.87}, s_{2.10}], P(0.37, 0.77)\rangle$	$\langle[s_1, s_{2.63}], P(0.83, 0.23)\rangle$	$\langle[s_{4.94}, s_{6.65}], P(0.77, 0.34)\rangle$
$a_2$	$\langle[s_{4.31}, s_{5.32}], P(0.76, 0.36)\rangle$	$\langle[s_{2.91}, s_{3.93}], P(0.21, 0.91)\rangle$	$\langle[s_1, s_{2.63}], P(0.83, 0.23)\rangle$	$\langle[s_{3.48}, s_{4.79}], P(0.79, 0.36)\rangle$
$a_3$	$\langle[s_{5.31}, s_{6.65}], P(0.69, 0.08)\rangle$	$\langle[s_{1.27}, s_{2.29}], P(0.29, 0.77)\rangle$	$\langle[s_{2.29}, s_{3.93}], P(0.43, 0.66)\rangle$	$\langle[s_{4.94}, s_{6.32}], P(0.76, 0.33)\rangle$
$a_4$	$\langle[s_{3.64}, s_{4.94}], P(0.66, 0.43)\rangle$	$\langle[s_{4.59}, s_{5.61}], P(0.29, 0.77)\rangle$	$\langle[s_{2.91}, s_{4.20}], P(0.33, 0.63)\rangle$	$\langle[s_{2.91}, s_{3.93}], P(0.76, 0.40)\rangle$

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Step 3: Transform the GDM into a score function matrix as shown in Table Ⅹ by the proposed score function.

Table  Ⅹ.  SCORE FUNCTION MATRIX

	$c_1$	$c_2$	$c_3$	$c_4$
$a_1$	2.2	-1.1	1.1	2.6
$a_2$	2.1	-2.9	-1.8	1.9
$a_3$	2.9	-1.1	-1.2	2.5
$a_4$	0.8	-3.2	-1.5	1.3

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Step 4: Determine the positive ideal solution and negative ideal solution, respectively.

Positive ideal solution: $G^+=\{2.86, -1.11, 1.14, 2.57\}$;

Negative ideal solution: $G^-=\{0.84, -3.19, -1.75, 1.34\}$

Step 5: Calculate group utility $S_i$, individual regret $R_i$ and compromise value $Q_i$, wherein, $\theta=$ 0.5.

Step 6: Get the best alternative(s) in an ascending order by the values of $S_i$, $R_i$ and $Q_i$. We have $Q_1 \prec Q_3 \prec Q_2 \prec Q_4$, $S_1 \prec S_3 \prec S_2 \prec S_4$ and $R_1 \prec R_3 \prec R_2 \prec R_4$, besides, $Q^{(2)}-Q^{(1)}=0.051 < \frac{1}{m-1}=0.33$.

Therefore, based on the conditions mentioned in Step 9 of Section V, we can get the final ranking order of all alternatives, which is $a_1\approx a_3 \succ a_2 \succ a_4$, that is to say, $a_1$ and $a_3$ are the best alternatives a period.

B Sensitive Analysis

To analyze the influence of different values of parameter $\theta$ on the final ranking order, we apply the proposed method to the MAGDA problem of venture capital. The values of parameter $\theta$ vary from 0 to 1 increasing by 0.1 and the sensitivity analysis results are given in Table Ⅺ and Fig. 3.

Table  Ⅺ.  THE RESULTS WITH DIFFERENT θ VALUES

	$c_1$	$c_2$	$c_3$	$c_4$	ranking order
$\theta =$ 0	$Q_1 =$ 0	$Q_2 =$ 0.44	$Q_3 =$ 0.07	$Q_4=$ 0.1	$a_1\approx a_3 \succ a_2 \succ a_4$
$\theta =$ 0.1	$Q_1 =$ 0	$Q_2 =$ 0.46	$Q_3 =$ 0.07	$Q_4 =$ 1	$a_1\approx a_3 \succ a_2 \succ a_4$
$\theta=$ 0.2	$Q_1 =$ 0	$Q_2 =$ 0.48	$Q_3 =$ 0.06	$Q_4 =$ 1	$a_1\approx a_3 \succ a_2 \succ a_4$
$\theta =$ 0.3	$Q_1 =$ 0	$Q_2 =$ 0.5	$Q_3 =$ 0.06	$Q_4 =$ 1	$a_1\approx a_3 \succ a_2 \succ a_4$
$\theta =$ 0.4	$Q_1 =$ 0	$Q_2 =$ 0.52	$Q_3 =$ 0.06	$Q_4 =$ 1	$a_1\approx a_3 \succ a_2 \succ a_4$
$\theta =$ 0.5	$Q_1 =$ 0	$Q_2 =$ 0.54	$Q_3 =$ 0.05	$Q_4 =$ 1	$a_1\approx a_3 \succ a_2 \succ a_4$
$\theta =$ 0.6	$Q_1 =$ 0	$Q_2 =$ 0.56	$Q_3 =$ 0.05	$Q_4 =$ 1	$a_1\approx a_3 \succ a_2 \succ a_4$
$\theta =$ 0.7	$Q_1 =$ 0	$Q_2 =$ 0.58	$Q_3 =$ 0.04	$Q_4 =$ 1	$a_1\approx a_3 \succ a_2 \succ a_4$
$\theta =$ 0.8	$Q_1 =$ 0	$Q_2 =$ 0.6	$Q_3 =$ 0.04	$Q_4 =$ 1	$a_1\approx a_3 \succ a_2 \succ a_4$
$\theta =$ 0.9	$Q_1 =$ 0	$Q_2 =$ 0.62	$Q_3 =$ 0.03	$Q_4 =$ 1	$a_1\approx a_3 \succ a_2 \succ a_4$
$\theta =$ 1	$Q_1 =$ 0	$Q_2 =$ 0.63	$Q_3 =$ 0.03	$Q_4 =$ 1	$a_1\approx a_3 \succ a_2 \succ a_4$

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Figure  3.  The results with different $\theta$ values

DownLoad: Full-Size Img PowerPoint

Based on the sensitivity analysis, we can see the ranking order of alternatives is stable with different values of $\theta$. Therefore, the proposed method can effectively reduce noise interference and obtain stable optimal alternative(s).

C Comparison Analysis

To verify the validity of our method, another two approaches are adopted for this numerical example, named as VIKOR with PULWAA method [24] and score function method. The results of comparision analysis are shown in Table Ⅻ.

Table  Ⅻ.  COMPARISON ANALYSIS WITH OTHER METHODS

Methods	Values	Order of alternatives
VIKOR with PULWAA[24]	$Q_1=0, Q_2=1, Q_3=$0.19, $Q_4$=0.84	$a_1\approx a_3 \succ a_4 \succ a_2$
Score function method	$Score(A_1)=1.01, Score(A_2)=-0.22, $
	$Score(A_3)=0.82, Score(A_4)=-0.79, $	$a_1\approx a_3 \succ a_2 \succ a_4$
Our method	$Q_1 = $ 0, $Q_2 = $ 0.53, $Q_3 = $ 0.05, $Q_4 = $ 1	$a_1\approx a_3 \succ a_2 \succ a_4$

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DownLoad: CSV

As it can be observed that the result of our method is slightly different from the results of Liu's method and the score function method. Take the second and the fourth alternatives for an example, the ranking order by using Liu's method is $a_4 \succ a_2$, whereas the result of our method is $a_4 \succ a_2$. The reason is that our method takes the interaction relationship between the aggregated arguments into account during the computing process. On the other hand, as for the the ranking order of $a_1, a_3$, a distinguished ranking result is achieved with the score function method. However, the same results are obtained by our method and Liu's method, i.e. $a_1\approx a_3$, which are both compromising solutions. The kind of difference arises from the feature of VIKOR method, which can obtain a set of compromising solutions when the criterion conflicts with each other.

Ⅶ. CONCLUSIONS

It is a natural and convenient way for decision makers to give an evaluation in the form of linguistic terms, and it is widely used in realistic decision making applications. This study proposed a new PULVHM operator by combining the merits of Pythagorean uncertain linguistic variable and Hamy mean operator. The definition and some useful properties of PULVHM operator were put forward and proved mathematically. The effectiveness for the integration of multiple PULVs was proved and an illustrative example was given in detail. The group decision matrix was obtained by using the PULVHM operator for integrating the evaluation information given by the decision makers. At the same time, a new score function of Pythagorean uncertain linguistic variables, named as PULVSF, was developed. Some properties, comparison rules, and illustrative example of PULVSF were given out and discussed. Based on the proposed PULVHM operator and PULVSF, a new MAGDM approach was presented integrating with the VIKOR method. A numerical study about the investment project selection problem was performed. The applicability and superiority of the proposed method were verified by sensibility analysis and comparison analysis with another two methods.