Incomplete pythagorean fuzzy preference relation for subway station safety management during COVID-19 pandemic

Abstract

Completing the Pythagorean fuzzy preference relations (PFPRs) based on additive consistency may exceed the defined domain. Therefore, we develop a group decision-making (GDM) method with incomplete PFPRs. Firstly, sufficient conditions for the expressibility of estimated preference values in PFPRs based on additive consistency are presented. Next, the correction algorithm is developed to correct the inexpressible elements in incomplete PFPRs. Then, a GDM method based on incomplete PFPRs is proposed to determine the objective weights of decision-makers. Finally, an example of subway station safety management during COVID-19 is selected to illustrate the applicability of the developed GDM method. The results show that the developed GDM method effectively identifies the crucial risk factor in subway station safety management and has better performance in terms of computational time complexity than the multiplicative consistency method.

1. Introduction

The group decision-making (GDM) method is to obtain a more objective, reasonable, and comprehensive decision result (Lotfi et al., 2022, Çetinkaya et al., 2022, Khalilpourazari et al., 2019). Multiple decision-makers are organized to participate in the decision, and the decision information provided by all decision-makers is integrated in a suitable way to obtain the most suitable decision result (Tikidji-Hamburyan et al., 2020, Kropat et al., 2020a, Kropat et al., 2020b). Because of the advantages such as rationality and comprehensiveness of GDM results, GDM methods have been widely used in industrial design, supply chain management, operations research, and risk evaluation (Maghsoodi et al., 2019, Banaeian et al., 2018, Islam et al., 2019, Zhang et al., 2021, He et al., 2021). However, in a complex decision environment, it may occur that the evaluation attributes cannot be determined, or the decision-maker is unable to perform the evaluation due to too many attributes, their own knowledge structure, or limited ability level (Kropat et al., 2016, Kropat and Weber, 2018, Tirkolaee and Aydin, 2022, Kalantari et al., 2022, Goli et al., 2021). Thus, the decision-makers choose a very effective and simple method to compare alternatives two by two to determine the preference between alternatives, constructing a judgment matrix that reflects the preference relations (PRs) (Orlovsky, 1978, Grobelny et al., 2021). Currently, PRs can generally be described using multiplicative PRs (MPRs) (Herrera et al., 2001), fuzzy PRs (FPRs) (Orlovsky, 1978), linguistic PRs (LPRs) (Dong et al., 2008, Elibal and Özceylan, 2022, Zhang et al., 2022), interval-valued FPRs (IVFPRs) (Barrenechea et al., 2014) and intuitionistic FPRs (IFPRs) (Xu, 2007, Zhang et al., 2022). The IFPRs are the preferred methods to describe uncertainty or non-judgment (Yang et al., 2019, Liu et al., 2020). It deals with both uncertainty and ambiguity of decision makers' preferences, shows imprecise judgments, and expresses affirmation and negation with the help of definitions of membership and non-membership degrees.

Pythagorean fuzzy sets (PFSs) are the extension of intuitionistic fuzzy sets (IFSs), continuing the advantages of IFSs containing positive, neural, and negative information (Ghosh et al., 2022, Ghosh et al., 2022, Jana and Roy, 2022, Mondal and Roy, 2022). The value range consisting of membership and non-membership values is extended from $\mu +\nu ⩽1$ to ${\mu }^2+{\nu }^2⩽1$. The information of PFSs has 1.57 times more than that of IFSs. The PRs have been combined with Pythagorean fuzzy environment and developed the concept of Pythagorean fuzzy preference relations (PFPRs) (Zhang et al., 2019, Zhang et al., 2020, Mandal and Ranadive, 2019, Wu et al., 2021). Zhang et al. (2019) regarded PFPRs as Pythagorean fuzzy numbers (PFNs) and developed a Pythagorean fuzzy PROMETHEE method to solve the GDM problem. Mandal and Ranadive (2019) reached a group consensus degree greater than a given threshold by updating the individual PFPRs and ranked the alternatives by calculating the arithmetic average values. Zhang et al. (2020) developed an adjustment method to correct the PFPRs that do not meet consistency requirements. Wu et al. (2021) defined the multiplicative consistency of PFPRs, developed a decision support system based on PFPRs, and applied it to crucial risk factors affecting finance management. When using PFPRs for decision-making, decision-makers need to construct the corresponding judgment matrix by providing the preference degree of one alternative over others through a two-by-two comparison of the alternatives.

However, due to limited time, lack of knowledge, and the complex decision environment, it is sometimes difficult for decision-makers to make complete judgments, especially for higher-order PRs with more alternatives. As a result, incomplete PFPRs with missing preference values are prone to occur. Thus, a large number of studies have been devoted to incomplete FPRs (Xu et al., 2011, Liao et al., 2014a, Liao et al., 2014b, Wang and Li, 2016, Chen and Xu, 2019, Chu et al., 2020), especially on how to estimate the missing preference values to make incomplete FPRs complete and consistent. Wang and Li (2016) made acceptable incomplete IFPRs complete based on quadratic programming models. Xu et al. (2011) introduced the concept of multiplicative consistent IFPRs (MCIFPRs) and proposed two algorithms for estimating missing preference values. Liao et al., 2014a, Liao et al., 2014b introduced the concept of multiplicative transferability of IFPRs and constructed complete PRs. Based on additive consistency, Chen and Xu (2019) corrected the inexpressible missing preference values of incomplete IFPRs. Chu et al. (2020) developed a complementary method for the fuzzy judgment matrix using trust relationships and social networks and designed a large-scale group fuzzy judgment matrix decision method applied to product recommendation.

Additive and multiplicative consistency play a crucial role in estimating unknown or missing values in incomplete FPRs (Tang et al., 2019). However, most existing studies have used multiplicative consistency for incomplete FPRs (Song et al., 2021, Liu et al., 2019). According to the characteristics of additive and multiplicative operations, an additive consistency method is much simpler in estimating the missing preference values than the multiplicative consistency approach. Moreover, using additive consistency to complete missing values in incomplete FPRs can significantly reduce the time complexity of the computation. However, there are few studies on incomplete PFPRs based on additive consistency. Because it has certain limitations, the preference values estimated using additive consistency often exceed the defined domain and lose their practical significance (Zhang et al., 2018, Guo et al., 2020).

Thus, we develop a GDM method based on additive consistent incomplete PFPRs. Firstly, two important theorems are presented to describe sufficient conditions for the expressibility of estimated preference values based on additive consistency after defining the concept of additive consistent PFPRs (ACPFPRs). Second, when the sufficient conditions are not satisfied, the algorithm to correct the original incomplete PFPR is developed so that the corrected PRs satisfy the expressible sufficient conditions. Then, a GDM method based on incomplete PFPRs is developed, and the objective weights of decision-makers are determined. Finally, an example of subway station safety management is selected to illustrate the applicability of the developed GDM method through the decision-making process and comparative analysis.

The rest of this paper is organized as follows. Some concepts of FPRs, IVFPRs, IFPRs, PFSs, and PFPRs are introduced in Section 2. Then, the sufficient conditions for incomplete PFPRs to be expressible are presented in Section 3. For some inexpressible PFPRs, the correction algorithm is provided to correct the missing preference values in Section 4. Next, a GDM method with incomplete PFPRs is developed in Section 5. To illustrate the effectiveness of the developed GDM method, an example of subway station safety management is used in Section 6. Finally, some conclusions are made in Section 7.

2. Preliminaries

2.1. Some concepts of FPRs, IFPRs, and IVFPRs

In practical decision problems, decision-makers are sometimes unable to determine the attribute values of each alternative due to too many evaluation attributes or time constraints. Still, they can compare two alternatives and determine the preference degrees of one alternative over another. The PRs provide a useful tool to construct a judgment matrix that can express the decision maker's preference information. Thus, Orlovsky (1978) defined the concept of FPRs.

Definition 1

(Orlovsky, 1978) Let $F={\left(f_{\mathit{ij}}\right)}_{n\times n}$ be an FPR on the attribute set $X=\left\{x_1,x_2,\dots ,x_n\right\}$, then.

$$f_{\mathit{ij}}+f_{\mathit{ji}}=1,f_{\mathit{ii}}=0.5,f_{\mathit{ij}}⩾0\left(i,j=1,2,\dots ,n\right)$$

where $f_{\mathit{ij}}$ denotes the preference degree of $x_i$ over$x_j$.

Remark 1

The larger the value$f_{\mathit{ij}}$, the stronger the preference for$x_i$over$x_j$. $f_{\mathit{ij}}>0.5$indicates$x_i$is better than$x_j$; $f_{\mathit{ij}}<0.5$means$x_j$is better than$x_i$; $f_{\mathit{ij}}=0.5$means$x_i$is equivalence with$x_j$. $f_{\mathit{ij}}=1$indicates that$x_i$is better than$x_j$, while$f_{\mathit{ij}}=0$explains that$x_j$is definitely than$x_i$.

Next, Tanino (1984) provided the definitions of additive and multiplicative consistent FPRs, respectively.

Definition 2

(Tanino, 1984) Let $F={\left(f_{\mathit{ij}}\right)}_{n\times n}$ be an FPR, then.

(1) if $F$ is an additive consistent FPR (ACFPR), then $F$ satisfies additive transitivity, i.e., $f_{\mathit{ij}}=f_{\mathit{ik}}-f_{\mathit{jk}}+0.5$;

(2) if $F$ is a multiplicative consistent FPR (MCFPR), then $F$ satisfies multiplicative transitivity, i.e., $f_{\mathit{ij}}{f}_{\mathit{jk}}{f}_{\mathit{ki}}=f_{\mathit{ik}}{f}_{\mathit{kj}}{f}_{\mathit{ji}}$.

Based on the fact that IFSs can more intuitively describe the good features of positive, negative, and hesitant in the decision-making process, Xu (2007) expressed the PRs in terms of intuitionistic fuzzy numbers (IFNs) and defined the IFPRs as follows.

Definition 3

(Xu, 2007) Let $Q={\left(q_{\mathit{ij}}\right)}_{n\times n}$ be an IFPR on $X=\left\{x_1,x_2,\dots ,x_n\right\}$, where $q_{\mathit{ij}}=\left({\mu }_{\mathit{ij}},{\nu }_{\mathit{ij}}\right)$, then.

$$0⩽{\mu }_{\mathit{ij}}+{\nu }_{\mathit{ij}}⩽1,{\mu }_{\mathit{ij}}={\nu }_{\mathit{ji}},{\nu }_{\mathit{ij}}={\mu }_{\mathit{ji}},{\mu }_{\mathit{ii}}={\nu }_{\mathit{ii}}=0.5,\forall i,j=1,2,\dots ,n$$

where ${\mu }_{\mathit{ij}}$ denotes the certain degree that $x_i$ is better than $x_j$ and ${\nu }_{\mathit{ij}}$ denotes the certain degree that $x_j$ is better than $x_i$. ${\pi }_{\mathit{ij}}=1-{\mu }_{\mathit{ij}}-{\nu }_{\mathit{ij}}$ is the hesitation degree of $q_{\mathit{ij}}$. $q_{\mathit{ij}}=\left({\mu }_{\mathit{ij}},{\nu }_{\mathit{ij}}\right)$ is an IFN.

The decision-maker needs to compare $n$ alternatives by $n\left(n-1\right)/2$ times to obtain the IFPR. However, the decision-maker may not complete the comparison of all alternatives due to time constraints and the complexity of the problem, resulting in some preference values being absent or missing, resulting in an incomplete IFPR.

Definition 4

(Xu, 2007) Let $Q={\left(q_{\mathit{ij}}\right)}_{n\times n}$ be an IFPR on $X=\left\{x_1,x_2,\dots ,x_n\right\}$, where $q_{\mathit{ij}}=\left({\mu }_{\mathit{ij}},{\nu }_{\mathit{ij}}\right)$. If $Q$ is an incomplete IFPR, some preference values in $Q$ are missing, and the remaining preference values that exist still satisfy the condition that.

$$0⩽{\mu }_{\mathit{ij}}+{\nu }_{\mathit{ij}}⩽1,{\mu }_{\mathit{ij}}={\nu }_{\mathit{ji}},{\nu }_{\mathit{ij}}={\mu }_{\mathit{ji}},{\mu }_{\mathit{ii}}={\nu }_{\mathit{ii}}=0.5$$

When making a decision based on an incomplete IFPR, the missing values should be filled in first. The incomplete IFPR is acceptable if all the missing values can be obtained using consistency. Otherwise, it is unacceptable (Chen and Xu, 2019). To fill in the missing values in the IFPRs, Xu and Cai (2009) defined the MCIFPRs.

Definition 5

(Xu and Cai, 2009) Let $Q={\left(q_{\mathit{ij}}\right)}_{n\times n}$ be an IFPR, where $0⩽{\mu }_{\mathit{ij}}+{\nu }_{\mathit{ij}}⩽1,{\mu }_{\mathit{ij}}={\nu }_{\mathit{ji}},{\nu }_{\mathit{ij}}={\mu }_{\mathit{ji}},{\mu }_{\mathit{ii}}={\nu }_{\mathit{ii}}=0.5,\forall i,j=1,2,\dots ,n$. If $Q$ satisfies multiplicative consistency, then.

${\mu }_{\mathit{ij}}=\left\{\begin{array}{c}0,if\left({\mu }_{\mathit{ik}},{\mu }_{\mathit{kj}}\right)=\left(0,1\right)or\left(1,0\right)\\ \frac{{\mu }_{\mathit{ik}}{\mu }_{\mathit{kj}}}{{\mu }_{\mathit{ik}}{\mu }_{\mathit{kj}}+\left(1-{\mu }_{\mathit{ik}}\right)\left(1-{\mu }_{\mathit{kj}}\right)},otherwise\end{array}\right)$, and

$${\nu }_{\mathit{ij}}=\left\{\begin{array}{c}0,if\left({\nu }_{\mathit{ik}},{\nu }_{\mathit{kj}}\right)=\left(0,1\right)or\left(1,0\right)\\ \frac{{\nu }_{\mathit{ik}}{\nu }_{\mathit{kj}}}{{\nu }_{\mathit{ik}}{\nu }_{\mathit{kj}}+\left(1-{\nu }_{\mathit{ik}}\right)\left(1-{\nu }_{\mathit{kj}}\right)},otherwise\end{array}\right)$$

The concepts of IVFPRs and additive consistent IVFPRs are first introduced to define additive consistent IFPRs (ACIFPRs).

Definition 6

(Khalid and Beg, 2016) Let $R={\left(r_{\mathit{ij}}\right)}_{n\times n}$ be an IVFPR on $X=\left\{X_1,X_2,\dots ,X_n\right\}$, where $r_{\mathit{ij}}=\left[r_{\mathit{ij}}^{-},r_{\mathit{ij}}^{+}\right]$, then.

$$0⩽r_{\mathit{ij}}^{-}⩽r_{\mathit{ij}}^{+}⩽1,r_{\mathit{ij}}^{-}+r_{\mathit{ji}}^{+}=r_{\mathit{ij}}^{+}+r_{\mathit{ji}}^{-}=1,r_{\mathit{ii}}^{-}=r_{\mathit{ii}}^{+}=0.5\left(i,j=1,2,\dots ,n\right)$$

where $r_{\mathit{ij}}$ denotes the interval-valued preference degree that $x_i$ is better than $x_j$. Here, $r_{\mathit{ij}}^{-}$ and $r_{\mathit{ij}}^{+}$ denote the lower and upper bounds of $r_{\mathit{ij}}$, respectively.

Definition 7

(Khalid and Beg, 2016) Let $R={\left(r_{\mathit{ij}}\right)}_{n\times n}$ be an IVFPR on $X=\left\{X_1,X_2,\dots ,X_n\right\}$,$r_{\mathit{ij}}=\left[r_{\mathit{ij}}^{-},r_{\mathit{ij}}^{+}\right]$, where $r_{\mathit{ij}}^{-}$ and $r_{\mathit{ij}}^{+}$ denote the lower and upper bounds of $r_{\mathit{ij}}$, respectively. If $R$ satisfies additive consistency, then.

$$r_{\mathit{ij}}^{-}=r_{\mathit{ik}}^{-}+r_{\mathit{kj}}^{-}-0.5,r_{\mathit{ij}}^{+}=r_{\mathit{ik}}^{+}+r_{\mathit{kj}}^{+}-0.5$$

2.2. Some concepts of PFS and PFPRs

Definition 8

(Yager, 2013) Let $A=\left\{〈x,{\rho }_A\left(x\right),{\sigma }_A\left(x\right)〉\left|x\in X\right)\right\}$ be a PFS on the attribute set $X$, where the membership degree ${\rho }_A$ and the non-membership degree ${\sigma }_A$ satisfy the condition that.

$${\rho }_A:X\to \left[0,1\right],{\sigma }_A:X\to \left[0,1\right],0⩽{\rho }_A^2+{\sigma }_A^2⩽1$$ (1)

Furthermore, ${\tau }_A\left(x\right)=\sqrt{1-{\sigma }_A^2\left(x\right)-{\rho }_A^2\left(x\right)}$ is the hesitancy degree of the element $x$ affiliated to the set $A$. Obviously, $0⩽{\tau }_A\left(x\right)⩽1$. Assume that $\phi =\left\{〈x_i,{\rho }_{\phi }\left(x_i\right),{\sigma }_A\left(x_i\right)〉\right\}$ be an element of the PFS $A$, then $\phi$ is called a PFN. The PFN $\phi$ can be expressed as ${\phi }_i=\left({\rho }_{{\phi }_i},{\sigma }_{{\phi }_i}\right)$. For the convenience of calculation, the PFN $\phi$ is simplified as $\phi =\left({\rho }_{\phi },{\sigma }_{\phi }\right)$, which satisfies ${\rho }_{\phi }^2+{\sigma }_{\phi }^2⩽1$ and ${\rho }_{\phi },{\sigma }_{\phi }\in \left[0,1\right]$.

To compare the magnitude of PFNs, Zhang and Xu (2014) defined the score and accuracy functions of PFNs $\phi =\left({\rho }_{\phi },{\sigma }_{\phi }\right)$ as $s\left(\phi \right)={\rho }_{\phi }^2-{\sigma }_{\phi }^2$ and $h\left(\phi \right)={\rho }_{\phi }^2+{\sigma }_{\phi }^2$, respectively, and provided the following comparison rules.

Definition 9

(Zhang and Xu, 2014) Let ${\phi }_1$ and ${\phi }_2$ be two PFNs, then.

(1) If $s\left({\phi }_1\right)>s\left({\phi }_2\right)$, then ${\phi }_1$ is better than ${\phi }_2$, denoted as ${\phi }_1\succ {\phi }_2$.

(2) If $s\left({\phi }_1\right)=s\left({\phi }_2\right)$, then.

(a) if $h\left({\phi }_1\right)=h\left({\phi }_2\right)$, then ${\phi }_1$ and ${\phi }_2$ are equivalent, denoted as ${\phi }_1\sim {\phi }_2$.

(b) if $h\left({\phi }_1\right)>h\left({\phi }_2\right)$, then ${\phi }_1$ is better than ${\phi }_2$, denoted as ${\phi }_1\succ {\phi }_2$.

There is a close connection between PFPRs and IVFPRs (Mandal and Ranadive, 2019). The PFPRs can be transformed into IVFPRs by $r_{\mathit{ij}}^{-}={\rho }_{\mathit{ij}}^2,r_{\mathit{ij}}^{+}=1-{\sigma }_{\mathit{ij}}^2\left(i,j=1,2,\dots ,n\right)$. Inspired by the definition of additive consistent IVFPR, assume that ${\rho }_{\mathit{ij}}^2=r_{\mathit{ij}}^{-},{\sigma }_{\mathit{ij}}^2=1-r_{\mathit{ij}}^{+}\left(i,j=1,2,\dots ,n\right)$, the concept of ACPFPRs can be obtained.

Definition 10

(Wu et al., 2021) Let $P={\left(p_{\mathit{ij}}\right)}_{n\times n}$ be a PFPR on $X=\left\{x_1,x_2,\dots ,x_n\right\}$, where $p_{\mathit{ij}}=\left({\rho }_{\mathit{ij}},{\sigma }_{\mathit{ij}}\right)$, then.

$$0⩽{\rho }_{\mathit{ij}}^2+{\sigma }_{\mathit{ij}}^2⩽1,{\rho }_{\mathit{ij}}={\sigma }_{\mathit{ji}},{\sigma }_{\mathit{ij}}={\rho }_{\mathit{ji}},{\rho }_{\mathit{ii}}={\sigma }_{\mathit{ii}}=\sqrt{2}/2,i,j=1,2,\dots ,n$$

where ${\rho }_{\mathit{ij}}$ is the preference value of $x_i$ over $x_j$ and ${\sigma }_{\mathit{ij}}$ is the preference value of $x_j$ over $x_i$, ${\tau }_{\mathit{ij}}=\sqrt{1-{\rho }_{\mathit{ij}}^2-{\sigma }_{\mathit{ij}}^2}$ is the hesitant degree of $p_{\mathit{ij}}$, where $p_{\mathit{ij}}=\left({\rho }_{\mathit{ij}},{\sigma }_{\mathit{ij}}\right)$ is a PFN.

The comparison between IFPR and PFPR is presented in Fig. 1 . An element $q_{\mathit{ij}}$ of the IFPR $Q={\left(q_{\mathit{ij}}\right)}_{n\times n}={\left({\mu }_{\mathit{ij}},{\nu }_{\mathit{ij}}\right)}_{n\times n}$ is represented by a point in the region formed by the constraint $0⩽{\mu }_{\mathit{ij}}+{\nu }_{\mathit{ij}}⩽1$, and an element $p_{\mathit{ij}}$ of the PFPR $P={\left(p_{\mathit{ij}}\right)}_{n\times n}={\left({\rho }_{\mathit{ij}},{\sigma }_{\mathit{ij}}\right)}_{n\times n}$ is represented by a point in the region formed by the constraint $0⩽{\rho }_{\mathit{ij}}^2+{\sigma }_{\mathit{ij}}^2⩽1$. If the IFPR and PFPR are reciprocal, then $\left({\mu }_{\mathit{ij}},{\nu }_{\mathit{ij}}\right)=\left({\mu }_{\mathit{ji}},{\nu }_{\mathit{ji}}\right)$ and $\left({\rho }_{\mathit{ij}},{\sigma }_{\mathit{ij}}\right)=\left({\rho }_{\mathit{ji}},{\sigma }_{\mathit{ji}}\right)$. Thus, the element $\left({\mu }_{\mathit{ji}},{\nu }_{\mathit{ji}}\right)$ is the mirror image of $\left({\mu }_{\mathit{ij}},{\nu }_{\mathit{ij}}\right)$ with respect to line ${\mu }_q={\nu }_q$, and the element $\left({\rho }_{\mathit{ji}},{\sigma }_{\mathit{ji}}\right)$ is the mirror image of $\left({\rho }_{\mathit{ij}},{\sigma }_{\mathit{ij}}\right)$ with respect to line ${\rho }_p={\sigma }_p$.

Definition 11

(Wu et al., 2021) Let $P={\left(p_{\mathit{ij}}\right)}_{n\times n}$ be a PFPR, where $p_{\mathit{ij}}=\left({\rho }_{\mathit{ij}},{\sigma }_{\mathit{ij}}\right)$. If $P$ satisfies additive consistency, then.

$${\rho }_{\mathit{ij}}^2={\rho }_{\mathit{ik}}^2+{\rho }_{\mathit{kj}}^2-0.5,{\sigma }_{\mathit{ij}}^2={\sigma }_{\mathit{ik}}^2+{\sigma }_{\mathit{kj}}^2-0.5$$ (2)

Fig. 1.

Comparison between IFPR and PFPR.

Additive consistency is one of the simplest and most effective tools for estimating missing preferences, but its drawback is that the estimated preference values often exceed the defined domain (Liao et al., 2014a, Liao et al., 2014b). An estimated preference value is expressible based on the consistency condition if the estimated value does not exceed the defined domain. Otherwise, it is inexpressible (Khalid and Awais, 2014). The expressible and inexpressible preference values in incomplete PFPRs are defined as follows:

Definition 12

Let$P={\left(p_{\mathit{ij}}\right)}_{n\times n}$be an incomplete PFPR on$X=\left\{x_1,x_2,\dots ,x_n\right\}$. If the estimated preference value by Eq. (2)belongs to the interval [0, 1], then the preference value is expressible. Otherwise, it is inexpressible.

Example 1

An incomplete PFPR is as follows:

$$\left[\begin{array}{cccc}\left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.7280,0.6708\right)& \left(-,-\right)& \left(-,-\right)\\ \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.7810,0.6083\right)& \left(-,-\right)\\ \left(-,-\right)& \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(-,-\right)\\ \left(-,-\right)& \left(-,-\right)& \left(0.5099,0.8485\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)\end{array}\right]$$

By estimating the missing values from Eq.(2), the following complete PFPR is obtained:

$$\left[\begin{array}{cccc}\left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.7280,0.6708\right)& \left(0.8000,0.5657\right)& \left(0.9274,0.2828\right)\\ \left(0.6708,0.7280\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.7810,0.6083\right)& \left(0.9000,0.3317\right)\\ \left(0.5657,0.8000\right)& \left(0.6083,0.7810\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.8485,0.5099\right)\\ \left(0.2828,0.9274\right)& \left(0.3317,0.9000\right)& \left(0.5099,0.8485\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)\end{array}\right]$$

Example 2

A 4 × 4 incomplete PFPR is as follows:

$$\left[\begin{array}{cccc}\left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)\\ \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(-,-\right)& \left(-,-\right)\\ \left(0.8832,0.4123\right)& \left(0.3162,0.7746\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.8944,0.3162\right)\\ \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)\end{array}\right]$$

From Eq. (2), the complete PFPR is:

$$\left[\begin{array}{cccc}\left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(-,0.9381\right)& \left(0.4123,0.8832\right)& \left(0.6856,0.6164\right)\\ \left(0.9381,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.7746,0.3162\right)& \left(0.9487,-\right)\\ \left(0.8832,0.4123\right)& \left(0.3162,0.7746\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.8944,0.3162\right)\\ \left(0.6164,0.6856\right)& \left(-,0.9487\right)& \left(0.3162,0.8944\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)\end{array}\right]$$

It can be found that when using Eq. (2) to solve the incomplete PFPR in Example 1 and Example 2, respectively, different situations arise. The former estimated preferences are all expressible, while the latter estimated preferences $q_{12}=\left(-,0.9381\right)$,$q_{21}=\left(0.9381,-\right)$, $q_{24}=\left(0.9487,-\right)$, $q_{42}=\left(-,0.9487\right)$ are all inexpressible.

The ACPFPRs in Definition 11 will be used to deal with the GDM problem with incomplete PFPRs.

3. Sufficient conditions for incomplete PFPRs to be expressible

In practical decision-making, it is usually unrealistic for decision-makers to provide complete preference information. Thus, scholars have investigated many methods for estimating missing preference values based on multiplicative consistency and additive consistency (Xu et al., 2011, Xu and Cai, 2015, Liao et al., 2014a, Liao et al., 2014b, Wang and Li, 2016). Compared with multiplicative consistency, the estimation of missing preference values based on additive consistency greatly reduces the computation time complexity. However, it has the limitation that the estimated preference values tend to exceed the defined domain, as in the case of Example 2. Two important theorems are given below. Based on the ACPFPRs in Definition 11, these two theorems discuss the conditions under which the estimated preference values are expressible. Also, a corollary on the inexpressibility of the estimated preference values is obtained.

Theorem 1

Let$P={\left(p_{\mathit{ij}}\right)}_{n\times n}$be an incomplete PFPR on$X=\left\{x_1,x_2,\dots ,x_n\right\}$, with$n$known preference values$p_{\mathit{ks}}=\left({\rho }_{\mathit{ks}},{\sigma }_{\mathit{ks}}\right)$, where$k$is fixed and$s=1,2,\dots ,n$. Let$\epsilon ={\rho }_{\mathit{ki}}=\min \left\{{\rho }_{\mathit{ks}}:s=1,2,\dots ,n\right\}$, if$\epsilon ⩾\sqrt{0.5}$, then all the estimated preference values by Eq.(2)are expressible.

Proof. Because $\epsilon ={\rho }_{\mathit{ki}}=\min \left\{{\rho }_{\mathit{ks}}:s=1,2,\dots ,n\right\}⩾\sqrt{0.5}$, then

$$0.5⩽{\rho }_{\mathit{ki}}^2⩽{\rho }_{\mathit{ks}}^2,s=1,2,\dots ,n$$ (3)

Furthermore, by the definition of PFPRs we have ${\rho }_{\mathit{ks}}^2+{\sigma }_{\mathit{ks}}^2⩽1$, then

$${\sigma }_{\mathit{ks}}^2⩽1-{\rho }_{\mathit{ks}}^2⩽1-{\rho }_{\mathit{ki}}^2⩽0.5$$ (4)

From the definition of PFPRs, we have ${\rho }_{\mathit{sk}}={\sigma }_{\mathit{ks}}$. Then, according to Definition 11, it is easy to obtain that

$${\rho }_{\mathit{st}}^2={\rho }_{\mathit{sk}}^2+{\rho }_{\mathit{kt}}^2-0.5={\sigma }_{\mathit{ks}}^2+{\sigma }_{\mathit{kt}}^2-0.5,\text{where}s,t=1,2,\dots ,n,s\ne k$$ (5)

From Eqs. (3) - (5), we have

$$0⩽{\sigma }_{\mathit{ks}}^2⩽{\sigma }_{\mathit{ks}}^2+{\rho }_{\mathit{kt}}^2-0.5⩽{\rho }_{\mathit{kt}}^2⩽1$$ (6)

Thus, $0⩽{\rho }_{\mathit{st}}⩽1$.

Similarly, $0⩽{\sigma }_{\mathit{st}}⩽1$.

In summary, the estimated preference value is expressible, i.e., Theorem 1 is proved.

Example 3

An incomplete PFPR is known as follows:

$$\left[\begin{array}{cccc}\left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)\\ \left(0.8944,0.3162\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.7746,0.4472\right)& \left(0.7071,0.6325\right)\\ \left(-,-\right)& \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(-,-\right)\\ \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)\end{array}\right]$$

Obviously, $\epsilon =\sqrt{0.5}$. The PFPR in Example 3 satisfies the conditions in Theorem 1, so the missing value can be estimated using Eq. (2), and the estimated preference value is expressible. As a result, the complete PFPR is obtained as follows:

$$\left[\begin{array}{cccc}\left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.3162,0.8944\right)& \left(0.4472,0.7071\right)& \left(0.3162,0.8367\right)\\ \left(0.8944,0.3162\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.7746,0.4472\right)& \left(0.7071,0.6325\right)\\ \left(0.7071,0.4472\right)& \left(0.4472,0.7746\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.4472,0.7071\right)\\ \left(0.8367,0.3162\right)& \left(0.6325,0.7071\right)& \left(0.7071,0.4472\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)\end{array}\right]$$

Theorem 1 shows that if the membership value of each PFPR is not less than $\sqrt{0.5}$, then all the missing preference values are estimated. However, these estimated preference values are still expressible. However, it is not very realistic to assume that some alternatives are preferred over others. In Example 2, when completing the PRs based on ACPFPRs in Definition 11, there is a situation where the estimated values are not expressible. It can be found that $\epsilon =0.3162<\sqrt{0.5}$ in incomplete PRs do not satisfy the conditions in Theorem 1. Then, can we infer that if $\epsilon <\sqrt{0.5}$ in the incomplete PFPR, the estimated missing preference value is inexpressible? The concept of critical missing preference for incomplete PFPRs was first introduced to investigate the problem, inspired by incomplete FPRs (Khalid and Awais, 2014).

Definition 13

Let$P={\left(p_{\mathit{ij}}\right)}_{n\times n}$be an incomplete PFPR on$X=\left\{x_1,x_2,\dots ,x_n\right\}$with$n$known preference values$p_{\mathit{ks}}=\left({\rho }_{\mathit{ks}},{\sigma }_{\mathit{ks}}\right)$, where$k$is fixed and$s=1,2,\dots ,n$. If${\rho }_{\mathit{ki}}=\min \left\{{\rho }_{\mathit{ks}}:s=1,2,\dots ,n\right\}$, ${\sigma }_{\mathit{kj}}=\min \left\{{\sigma }_{\mathit{ks}}:s=1,2,\dots ,n\right\}$, then$p_{\mathit{ij}}=\left({\rho }_{\mathit{ij}},{\sigma }_{\mathit{ij}}\right)$is called the critical missing preference.

The key missing preference will play a crucial role in completing the incomplete PFPR because it determines the expressibility of the other estimated missing preferences, as detailed in Lemma 1.

Lemma 1

Let$P={\left(p_{\mathit{ij}}\right)}_{n\times n}$be an incomplete PFPR with$n$known preference values$p_{\mathit{ks}}=\left({\rho }_{\mathit{ks}},{\sigma }_{\mathit{ks}}\right)$, where$k$is fixed and$s=1,2,\dots ,n$. If the key missing preference is expressible, then the other missing preferences can be estimated by Eq.(2), and the estimated values are expressible.

Proof. From Definition 13, we get ${\rho }_{\mathit{ki}}=\min \left\{{\rho }_{\mathit{ks}}:s=1,2,\dots ,n\right\}$ and ${\sigma }_{\mathit{kj}}=\min \left\{{\sigma }_{\mathit{ks}}:s=1,2,\dots ,n\right\}$, then

$${\rho }_{\mathit{ki}}⩽{\rho }_{\mathit{ks}},{\sigma }_{\mathit{kj}}⩽{\sigma }_{\mathit{ks}},s=1,2,\dots ,n$$ (7)

It is known that $p_{\mathit{ij}}=\left({\rho }_{\mathit{ij}},{\sigma }_{\mathit{ij}}\right)$ is expressible, i.e., $0⩽{\rho }_{\mathit{ij}}⩽1$ and $0⩽{\sigma }_{\mathit{ij}}⩽1$.

From Definition 11, we have.

${\rho }_{\mathit{ij}}^2={\rho }_{\mathit{ik}}^2+{\rho }_{\mathit{kj}}^2-0.5={\sigma }_{\mathit{ki}}^2+{\rho }_{\mathit{kj}}^2-0.5⩾0$, and

$${\sigma }_{\mathit{ij}}^2={\sigma }_{\mathit{ik}}^2+{\sigma }_{\mathit{kj}}^2-0.5={\rho }_{\mathit{ki}}^2+{\sigma }_{\mathit{kj}}^2-0.5⩾0$$

where $s,t=1,2,\dots ,n,s\ne k$. The estimated missing preference values are obtained as follows:

$${\rho }_{\mathit{st}}^2={\rho }_{\mathit{sk}}^2+{\rho }_{\mathit{kt}}^2-0.5={\sigma }_{\mathit{ks}}^2+{\rho }_{\mathit{kt}}^2-0.5⩾{\sigma }_{\mathit{kj}}^2+{\rho }_{\mathit{ki}}^2-0.5⩾0$$

$${\sigma }_{\mathit{st}}^2={\sigma }_{\mathit{sk}}^2+{\sigma }_{\mathit{kt}}^2-0.5={\rho }_{\mathit{ks}}^2+{\sigma }_{\mathit{kt}}^2-0.5⩾{\rho }_{\mathit{ki}}^2+{\sigma }_{\mathit{kj}}^2-0.5⩾0$$

$${\rho }_{\mathit{st}}^2+{\sigma }_{\mathit{st}}^2={\sigma }_{\mathit{ks}}^2+{\rho }_{\mathit{kt}}^2-0.5+{\rho }_{\mathit{ks}}^2+{\sigma }_{\mathit{kt}}^2-0.5⩽2-1=1$$

In summary, the estimated preference values $p_{\mathit{st}}=\left({\rho }_{\mathit{st}},{\sigma }_{\mathit{st}}\right)$ are expressible.

Theorem 2

Let$P={\left(p_{\mathit{ij}}\right)}_{n\times n}$be an incomplete PFPR on$X=\left\{x_1,x_2,\dots ,x_n\right\}$with$n$known preference values$p_{\mathit{ks}}=\left({\rho }_{\mathit{ks}},{\sigma }_{\mathit{ks}}\right)$, where$k$is fixed and$s=1,2,\dots ,n$. Assume that$\epsilon ={\rho }_{\mathit{ki}}=\min \left\{{\rho }_{\mathit{ks}}:s=1,2,\dots ,n\right\}$，$\delta ={\sigma }_{\mathit{kj}}=\min \left\{{\sigma }_{\mathit{ks}}:s=1,2,\dots ,n\right\}$. If$\epsilon <\sqrt{0.5}$and.

$$0.5⩽{\epsilon }^2+{\delta }^2$$ (8)

then the missing preference value can be estimated by Eq. (2), and the estimated value is expressible.

Proof. From Definition 13, $p_{\mathit{ij}}=\left({\rho }_{\mathit{ij}},{\sigma }_{\mathit{ij}}\right)$ is a key missing preference value.

It is known that $\epsilon ={\rho }_{\mathit{ki}}<\sqrt{0.5}$, ${\epsilon }^2+{\delta }^2={\rho }_{\mathit{ki}}^2+{\sigma }_{\mathit{kj}}^2⩾0.5$, $p_{\mathit{ij}}=\left({\rho }_{\mathit{ij}},{\sigma }_{\mathit{ij}}\right)$ estimated by Eq. (2) satisfies

$${\rho }_{\mathit{ij}}^2={\rho }_{\mathit{ik}}^2+{\rho }_{\mathit{kj}}^2-0.5={\sigma }_{\mathit{ki}}^2+{\rho }_{\mathit{kj}}^2-0.5⩾{\rho }_{\mathit{ki}}^2+{\sigma }_{\mathit{kj}}^2-0.5⩾0$$

${\sigma }_{\mathit{ij}}^2={\sigma }_{\mathit{ik}}^2+{\sigma }_{\mathit{kj}}^2-0.5={\rho }_{\mathit{ki}}^2+{\sigma }_{\mathit{kj}}^2-0.5⩾0$, and

$${\rho }_{\mathit{ij}}^2+{\sigma }_{\mathit{ij}}^2={\sigma }_{\mathit{ki}}^2+{\rho }_{\mathit{kj}}^2-0.5+{\rho }_{\mathit{ki}}^2+{\sigma }_{\mathit{kj}}^2-0.5⩽1$$

Therefore, the estimated critical missing preference value $p_{\mathit{ij}}=\left({\rho }_{\mathit{ij}},{\sigma }_{\mathit{ij}}\right)$ is expressible.

According to Lemma 1, it can be concluded that the estimated missing preference value by Eq. (2) is expressible.

Theorem 2 is proved.

Corollary 1

Let$P={\left(p_{\mathit{ij}}\right)}_{n\times n}$be an incomplete PFPR with$n$known preference values$p_{\mathit{ks}}=\left({\rho }_{\mathit{ks}},{\sigma }_{\mathit{ks}}\right)$, where$k$is fixed and$s=1,2,\dots ,n$. Assume that$\epsilon ={\rho }_{\mathit{ki}}=\min \left\{{\rho }_{\mathit{ks}}:s=1,2,\dots ,n\right\}$and$\delta ={\sigma }_{\mathit{kj}}=\min \left\{{\sigma }_{\mathit{ks}}:s=1,2,\dots ,n\right\}$. If${\delta }^2+{\epsilon }^2<0.5$, then at least one of the missing preference values estimated by Eq.(2)is inexpressible.

Proof. From Eq. (2), we have ${\rho }_{\mathit{ji}}^2={\rho }_{\mathit{jk}}^2+{\rho }_{\mathit{ki}}^2-0.5$.

From the definition of PFPR, ${\rho }_{\mathit{jk}}={\sigma }_{\mathit{kj}}=\delta$.

From the condition ${\delta }^2+{\epsilon }^2<0.5$, then

$${\rho }_{\mathit{ji}}^2={\rho }_{\mathit{jk}}^2+{\rho }_{\mathit{ki}}^2-0.5={\delta }^2+{\epsilon }^2-0.5<0$$

That is ${\rho }_{\mathit{ji}}\notin \left[0,1\right]$. In other words, $p_{\mathit{ji}}=\left({\rho }_{\mathit{ji}},{\sigma }_{\mathit{ji}}\right)$ is inexpressible.

Therefore, Corollary 1 is proven.

Example 4

A 4 × 4 incomplete PFPR is known as follows:

$$\left[\begin{array}{cccc}\left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.3873,0.8367\right)& \left(0.4472,0.7746\right)& \left(0.5916,0.6325\right)\\ \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(-,-\right)& \left(-,-\right)\\ \left(-,-\right)& \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(-,-\right)\\ \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)\end{array}\right]$$

In the PFPR of Example 4, it can be found that $\epsilon ={\rho }_{12}=0.3873$ and $\delta ={\sigma }_{14}=0.6325$, which implies $\epsilon <\sqrt{2}/2$ and ${\delta }^2+{\epsilon }^2=0.55>0.5$, exactly satisfies the conditions in Theorem 2. Therefore, when using the ACPFPRs in Definition 11, the missing preference values are estimated, and the estimated preference value is expressible. Thus, the above PFPR after completion is as follows:

$$\left[\begin{array}{cccc}\left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.3873,0.8367\right)& \left(0.4472,0.7746\right)& \left(0.5916,0.6325\right)\\ \left(0.8367,0.3873\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.6325,0.5000\right)& \left(0.7416,0.2236\right)\\ \left(0.7746,0.4472\right)& \left(0.5000,0.6325\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.6708,0.3162\right)\\ \left(0.6325,0.5916\right)& \left(0.2236,0.7416\right)& \left(0.3162,0.6708\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)\end{array}\right]$$

4. Correction algorithm for inexpressible PFPRs

Suppose an incomplete PFPR satisfies the conditions in Theorem 1 or Theorem 2. All the missing values can be filled by the ACPFPRs in Definition 11, and the estimated values are all expressible. Otherwise, like the incomplete PFPR in Example 2, the preference estimated based on additive consistency may be meaningless beyond the definition domain. Thus, the membership or non-membership values of the incomplete PFPR are corrected. This section proposes a correction algorithm to handle the incomplete PFPRs in two different cases.

In the PFPR $P={\left(p_{\mathit{ij}}\right)}_{n\times n}$, $n$ preference values $p_{i^{\prime }j}$ are known, where $i^{\prime }$ is given, and $j\in \left\{1,2,\dots ,n\right\}$. Without loss of generality, take $i^{\prime }=1$. In this section, we consider a set of membership values and a set of non-membership values in the PFPRs.

Let $A=\left\{{\rho }_{12},{\rho }_{13},\dots ,{\rho }_{1n}\right\}$, $B=\left\{{\sigma }_{12},{\sigma }_{13},\dots ,{\sigma }_{1n}\right\}$ and $\inf \left(A\right)<\sqrt{0.5}$. Furthermore,

$${\gamma }_{\rho }=\left\{{\rho }_{1k}:{\rho }_{1k}\text{satisfyTheorem}2\right\}\subset A$$

$${\gamma }_{\sigma }=\left\{{\sigma }_{1k}:{\sigma }_{1k}\text{satisfyTheorem}2\right\}\subset B$$

Algorithm I

Case (I): If$\left|{\gamma }_{\sigma }\right|⩾\left[\frac{n-1}{2}\right]$, let$\delta =\sqrt{0.5-{\left(\inf \left(A\right)\right)}^2}$.

Step 1: Take $\inf \left(B\right)={\delta }_1$.

If ${\delta }_1\notin {\gamma }_{\sigma }$, the membership degree of the PFPR is denoted as $M\left({\delta }_1\right)$, then the preference value is modified as

$$\left(M\left({\delta }_1\right),{\delta }_1\right)=\left\{\begin{array}{c}\left(M\left({\delta }_1\right),{\delta }_1\right),\text{if}M\left({\delta }_1\right)⩽\sqrt{1-{\delta }^2}\\ \left(\sqrt{1-{\delta }^2-{\left(\underset{j}{\min }\left\{\left|{\tau }_{\mathit{ij}}\right|\right\}\right)}^2},\delta \right),ifM\left({\delta }_1\right)>\sqrt{1-{\delta }^2}\end{array}\right)$$ (9)

Otherwise, no correction is needed.

Step 2: Take $\inf \left(B/{\delta }_1\right)={\delta }_2$.

If ${\delta }_2\notin {\gamma }_{\sigma }$, then the correction is the same as Step 1, i.e.,

$$\left(M\left({\delta }_2\right),{\delta }_2\right)=\left\{\begin{array}{c}\left(M\left({\delta }_2\right),{\delta }_2\right),\text{if}M\left({\delta }_2\right)⩽\sqrt{1-{\delta }^2}\\ \left(\sqrt{1-{\delta }^2-{\left(\underset{j}{\min }\left\{\left|{\tau }_{\mathit{ij}}\right|\right\}\right)}^2},\delta \right),\text{if}M\left({\delta }_2\right)>\sqrt{1-{\delta }^2}\end{array}\right)$$ (10)

Otherwise, no correction is required.

…….

Step$\left|B\right|-\left|{\gamma }_{\sigma }\right|$: take $\inf \left(B/{\delta }_{\left|B\right|-\left|{\gamma }_{\sigma }\right|-1}\right)={\delta }_{\left|B\right|-\left|{\gamma }_{\sigma }\right|}$.

If ${\delta }_{\left|B\right|-\left|{\gamma }_{\sigma }\right|}\notin {\gamma }_{\sigma }$, then the reference value is modified as

$$\left(M\left({\delta }_{\left|B\right|-\left|{\gamma }_{\sigma }\right|}\right),{\delta }_{\left|B\right|-\left|{\gamma }_{\sigma }\right|}\right)=\left\{\begin{array}{c}\left(M\left({\delta }_{\left|B\right|-\left|{\gamma }_{\sigma }\right|}\right),{\delta }_{\left|B\right|-\left|{\gamma }_{\sigma }\right|}\right),\text{if}M\left({\delta }_{\left|B\right|-\left|{\gamma }_{\sigma }\right|}\right)⩽\sqrt{1-{\delta }^2}\\ \left(\sqrt{1-{\delta }^2-{\left(\underset{j}{\min }\left\{\left|{\tau }_{\mathit{ij}}\right|\right\}\right)}^2},\delta \right),\text{if}M\left({\delta }_{\left|B\right|-\left|{\gamma }_{\sigma }\right|}\right)>\sqrt{1-{\delta }^2}\end{array}\right)$$ (11)

Otherwise, no correction is required.

Case (II): if $\left|{\gamma }_{\rho }\right|⩾\left[\frac{n-1}{2}\right]$, let $\epsilon =\sqrt{{\left(\sup \left(A\right)\right)}^2-0.5}$.

Step 1: take $\inf \left(A\right)={\epsilon }_1$.

If ${\epsilon }_1\in {\gamma }_{\rho }$, the non-membership value of the PFPR is $N\left({\epsilon }_1\right)$, then the corresponding correction is as follows:

$$\left({\epsilon }_1,N\left({\epsilon }_1\right)\right)=\left\{\begin{array}{c}\left(\epsilon ,N\left({\epsilon }_1\right)\right),\text{if}N\left({\epsilon }_1\right)⩽\sqrt{1-{\epsilon }^2}\\ \left(\epsilon ,\sqrt{1-{\epsilon }^2-{\left(\underset{j}{\min }\left\{\left|{\tau }_{\mathit{ij}}\right|\right\}\right)}^2}\right),\text{if}N\left({\epsilon }_1\right)>\sqrt{1-{\epsilon }^2}\end{array}\right)$$ (12)

Otherwise, no correction is needed.

Step 2: take $\inf \left(A/{\epsilon }_1\right)={\epsilon }_2$.

If ${\epsilon }_2\in {\gamma }_{\rho }$, then

$$\left({\epsilon }_2,N\left({\epsilon }_2\right)\right)=\left\{\begin{array}{c}\left(\epsilon ,N\left({\epsilon }_2\right)\right),\text{if}N\left({\epsilon }_2\right)⩽\sqrt{1-{\epsilon }^2}\\ \left(\epsilon ,\sqrt{1-{\epsilon }^2-{\left(\underset{j}{\min }\left\{\left|{\tau }_{\mathit{ij}}\right|\right\}\right)}^2}\right),\text{if}N\left({\epsilon }_2\right)>\sqrt{1-{\epsilon }^2}\end{array}\right)$$ (13)

Otherwise, no correction is needed.

…….

Step$\left|{\gamma }_{\rho }\right|$: take $\inf \left(A/{\epsilon }_{\left|{\gamma }_{\rho }\right|-1}\right)={\epsilon }_{\left|{\gamma }_{\rho }\right|}$.

If ${\epsilon }_{\left|{\gamma }_{\rho }\right|}\in {\gamma }_{\rho }$, then

$$\left({\epsilon }_{{\gamma }_{\rho }},N\left({\epsilon }_{{\gamma }_{\rho }}\right)\right)=\left\{\begin{array}{c}\left(\epsilon ,N\left({\epsilon }_{{\gamma }_{\rho }}\right)\right),\text{if}N\left({\epsilon }_{{\gamma }_{\rho }}\right)⩽\sqrt{1-{\epsilon }^2}\\ \left(\epsilon ,\sqrt{1-{\epsilon }^2-{\left(\underset{j}{\min }\left\{\left|{\tau }_{\mathit{ij}}\right|\right\}\right)}^2}\right),\text{if}N\left({\epsilon }_{{\gamma }_{\rho }}\right)>\sqrt{1-{\epsilon }^2}\end{array}\right)$$ (14)

Otherwise, no correction is required.

Example 5

A 7 × 7 incomplete PFPR is known as follows.

$$\left[\begin{array}{ccccccc}\left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.7746,0.5477\right)& \left(0.8246,0.2828\right)& \left(0.5568,0.7746\right)& \left(0.4472,0.8367\right)& \left(0.5477,0.7416\right)& \left(0.9592,0.1414\right)\\ \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)\\ \left(-,-\right)& \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)\\ \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)\\ \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(-,-\right)& \left(-,-\right)\\ \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(-,-\right)\\ \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)\end{array}\right]$$

Clearly, $\epsilon =0.4472$, $\delta =0.1414$ and ${\epsilon }^2+{\delta }^2=0.22<0.5$, which satisfy the conditions in Corollary 1. At this point, several estimated preference values are inexpressible if the missing preferences are estimated using Eq. (2). Therefore, some preference values in the PFPR of Example 5 need to be corrected.

From the PFPR, then

$$A=\left\{0.7746,0.8246,0.5568,0.4472,0.5477,0.9592\right\}$$

$$B=\left\{0.5477,0.2828,0.7746,0.8367,0.7416,0.1414\right\}$$

$${\gamma }_{\rho }=\left\{0.7746,0.8246,0.5568,0.4472,0.5477,0.9592\right\}$$

$${\gamma }_{\sigma }=\left\{0.5477,0.7746,0.8367,0.7416\right\}$$

Since $\left|{\gamma }_{\sigma }\right|=4⩾\left[6/2\right]$，then $\delta =\sqrt{0.5-{\left(\inf \left(A\right)\right)}^2}=0.5477$, the preference value is corrected according to Case (I) in Algorithm I.

Step 1: take $\inf \left(B\right)={\sigma }_{17}=0.1414={\delta }_1$,

then ${\delta }_1\notin {\gamma }_{\sigma }$ and ${\rho }_{17}=0.9592>\sqrt{1-0.{5477}^2}=0.8367$.

Next, $\underset{j}{\min }\left\{\left|{\tau }_{\mathit{ij}}\right|\right\}=\min \left\{0.3162,0.4899,0.3000,0.3162,0.3873,0.2450\right\}=0.2450$.

The revised reference value by Eq.(9) is

$$\left({\rho }_{17},{\sigma }_{17}\right)=\left(\sqrt{1-0.{5477}^2-0.{2450}^2},0.5477\right)=\left(0.8000,0.5477\right)$$

Step 2: take $\inf \left(B/{\delta }_1\right)=0.08={\delta }_2$,

then ${\delta }_2\notin {\gamma }_{\sigma }$ and ${\rho }_{13}=0.8246<\sqrt{1-0.{5477}^2}=0.8367$.

From Eq. (10), then $\left({\rho }_{13},{\sigma }_{13}\right)=\left(0.8246,0.5477\right)$.

Since $0.3\in {\gamma }_{\sigma }$, there is no need to revise.

It is found that the key missing preference value $p_{57}=\left(0.9165,0\right)$ is expressible, and the modified PRs satisfy the conditions in Theorem 2.

Therefore, the missing preferences are estimated by Eq. (2), and the complete PFPR is obtained as follows:

$$\left[\begin{array}{ccccccc}\left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.7746,0.5477\right)& \left(0.8246,0.5477\right)& \left(0.5568,0.7746\right)& \left(0.4472,0.8367\right)& \left(0.5477,0.7416\right)& \left(0.8000,0.5477\right)\\ \left(0.5477,0.7746\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.6928,0.6325\right)& \left(0.3162,0.8367\right)& \left(0,0.8944\right)& \left(0.3162,0.8062\right)& \left(0.6633,0.6325\right)\\ \left(0.5477,0.8246\right)& \left(0.6325,0.6928\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.3317,0.8832\right)& \left(0,0.9381\right)& \left(0.3162,0.8544\right)& \left(0.6633,0.6928\right)\\ \left(0.7746,0.5568\right)& \left(0.8367,0.3162\right)& \left(0.8832,0.3317\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.5477,0.7141\right)& \left(0.6325,0.6000\right)& \left(0.8602,0.3317\right)\\ \left(0.8367,0.4472\right)& \left(0.8944,0\right)& \left(0.9381,0\right)& \left(0.7141,0.5477\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.7071,0.3873\right)& \left(0.9165,0\right)\\ \left(0.7416,0.5477\right)& \left(0.8062,0.3162\right)& \left(0.8544,0.3162\right)& \left(0.6000,0.6325\right)& \left(0.3873,0.7071\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.8307,0.3162\right)\\ \left(0.5477,0.8000\right)& \left(0.6325,0.6633\right)& \left(0.6928,0.6633\right)& \left(0.3317,0.8602\right)& \left(0,0.9165\right)& \left(0.3162,0.8307\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)\end{array}\right]$$

It can be seen that all values are expressible. Obviously, the obtained complete PFPRs are also additive consistent.

Example 6

A 5 × 5 incomplete PFPR is shown as follows:

$$\left[\begin{array}{ccccc}\left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.3162,0.8944\right)& \left(0.8367,0.5477\right)& \left(0.8944,0.3162\right)& \left(0.8367,0.4472\right)\\ \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)\\ \left(-,-\right)& \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(-,-\right)& \left(-,-\right)\\ \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(-,-\right)\\ \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)\end{array}\right]$$

If we directly use Eq. (2) to estimate the missing preference values, we can obtain complete PFPR as follows:

$$\left[\begin{array}{ccccc}\left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.3162,0.8944\right)& \left(0.8367,0.5477\right)& \left(0.8944,0.3162\right)& \left(0.8367,0.4472\right)\\ \left(0.8944,0.3162\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(1,-\right)& \left(1.0488,-\right)& \left(1,-\right)\\ \left(0.5477,0.8367\right)& \left(-,1\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.7746,0.5477\right)& \left(0.7071,0.6325\right)\\ \left(0.3162,0.8944\right)& \left(-,1.0488\right)& \left(0.5477,0.7746\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.5477,0.7071\right)\\ \left(0.4472,0.8367\right)& \left(-,1\right)& \left(0.6325,0.7071\right)& \left(0.7071,0.5477\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)\end{array}\right]$$

Clearly, some of the estimated missing preferences are inexpressible.

In fact, ${\epsilon }^2+{\delta }^2=0.{3162}^2+0.{3162}^2=0.2<0.5$, which is consistent with the condition in Corollary 1. Then

$$A=\left\{0.3162,0.8367,0.8944,0.8367\right\}B=\left\{0.8944,0.5477,0.3162,0.4472\right\}{\gamma }_{\rho }=\left\{0.3162\right\}{\gamma }_{\sigma }=\left\{0.8944\right\}$$

Clearly, $\left|{\gamma }_{\rho }\right|=1<\left[4/2\right]=2$ and $\epsilon =\sqrt{{\left(\sup \left(A\right)\right)}^2-0.5}=\sqrt{0.{8944}^2-0.5}=0.5477$.

Therefore, the following result is obtained by Case (II) in Algorithm I.

Step 1: take $inf\left(A\right)={\epsilon }_1={\rho }_{12}=0.3162$.

Then, ${\epsilon }_1\in {\gamma }_{\rho }$,${\sigma }_{12}=0.8944>\sqrt{1-{\epsilon }^2}=\sqrt{1-0.{5477}^2}=0.8367$. Meanwhile,$\underset{j}{\min }\left\{\left|{\tau }_{\mathit{ij}}\right|\right\}=\min \left\{0.3162,0,0.3162,0.3162\right\}=0$, then the revised preference value is$\left({\rho }_{12},{\sigma }_{12}\right)=\left(0.5477,\sqrt{1-0.{5477}^2-0}\right)=\left(0.5477,0.8367\right)$.

The correction in Case (II) is over.

Next, $A^{\prime }=\left\{0.5477,0.8367,0.8944,0.8367\right\}$,$B^{\prime }=\left\{0.8367,0.5477,0.3162,0.4472\right\}$, then ${\epsilon }^{\prime }=0.5477<\sqrt{2}/2$,${\left({\epsilon }^{\prime }\right)}^2+{\left({\delta }^{\prime }\right)}^2=0.3+0.1<0.5$.

Thus, ${\gamma }_{\rho }=\left\{0.5477,0.8367,0.8944,0.8367\right\}$ and ${\gamma }_{\sigma }=\left\{0.8367,0.5477,0.4472\right\}$, then$\left|{\gamma }_{\sigma }\right|=3>\left[4/2\right]=2$.

$\delta =\sqrt{0.5-{\left(\inf \left(A^{\prime }\right)\right)}^2}=0.4472$. Case I in Algorithm (I) is adopted.

Step 2: take $\inf \left(A^{\prime }\right)={\delta }_1={\sigma }_{14}=0.3162$.

Since ${\delta }_1\notin {\gamma }_{\sigma }$,${\rho }_{14}=0.8944⩽\sqrt{1-{\delta }^2}=\sqrt{1-0.{4472}^2}=0.8944$.

The corrected preference value by Eq.(9) is $\left({\rho }_{14},{\sigma }_{14}\right)=\left(0.8944,0.4472\right)$.

After the preference values are corrected, the complete ACPFPR is then obtained as follows:

$$\left[\begin{array}{ccccc}\left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.5477,0.8367\right)& \left(0.8367,0.5477\right)& \left(0.8944,0.4472\right)& \left(0.8367,0.4472\right)\\ \left(0.8367,0.5477\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.9487,0.3162\right)& \left(1,0\right)& \left(0.9487,0\right)\\ \left(0.5477,0.8367\right)& \left(0.3162,0.9487\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.7746,0.6325\right)& \left(0.7071,0.6325\right)\\ \left(0.4472,0.8944\right)& \left(0,1\right)& \left(0.6325,0.7746\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)& \left(0.6325,0.7071\right)\\ \left(0.4472,0.8367\right)& \left(0,0.9487\right)& \left(0.6325,0.7071\right)& \left(0.7071,0.6325\right)& \left(\sqrt{2}/2,\sqrt{2}/2\right)\end{array}\right]$$

5. GDM method with incomplete PFPRs

Decision-makers differ in terms of knowledge level, area of expertise, and professional competence and thus play different roles in the group decision-making process. Assigning appropriate weights to decision-makers is crucial in the group decision-making problem. Inspired by IFPRs (Chu et al., 2016), a GDM method with incomplete PFPRs is developed to determine the weights of decision-makers based on the consensus degree between individuals and groups.

Let $D=\left\{d_1,d_2,\dots ,d_m\right\}$ be a set of decision-makers and $\omega ={\left\{{\omega }_1,{\omega }_2,\dots ,{\omega }_m\right\}}^T$ be the weight vector of decision-makers, where${\sum }_{k=1}^m{\omega }_k=1,{\omega }_k⩾0,k=1,2,\dots ,m$. The evaluation matrix given by the decision-maker $d_k$ is a PFPR, denoted as $P^k={\left(p_{\mathit{ij}}^k\right)}_{n\times n}$, where $p_{\mathit{ij}}^k=\left({\rho }_{\mathit{ij}}^k,{\sigma }_{\mathit{ij}}^k\right)$.

Wu and Xu (2012) defined the deviation measure of two PRs. Subsequently, Chu et al. (2016) extended it to IFPRs. Then, the divergence measure of two PFPRs is defined as follows:

Definition 14

Let$P^1={\left(p_{\mathit{ij}}^1\right)}_{n\times n}=\left({\rho }_{\mathit{ij}}^1,{\sigma }_{\mathit{ij}}^1\right)$and$P^2={\left(p_{\mathit{ij}}^2\right)}_{n\times n}=\left({\rho }_{\mathit{ij}}^2,{\sigma }_{\mathit{ij}}^2\right)$be two PFPRs. The deviation between$P^1$and$P^2$is defined as follows:

$$d\left(P^1,P^2\right)=\frac{2}{n\left(n-1\right)}\sum _{i<j}0.5\left(\left|{\left({\rho }_{\mathit{ij}}^1\right)}^2-{\left({\rho }_{\mathit{ij}}^2\right)}^2\right|+\left|{\left({\sigma }_{\mathit{ij}}^1\right)}^2-{\left({\sigma }_{\mathit{ij}}^2\right)}^2\right|\right)$$ (15)

Remark 2

From the definition of PFPRs, ${\sigma }_{\mathit{ij}}^1={\rho }_{\mathit{ji}}^1,{\sigma }_{\mathit{ij}}^2={\rho }_{\mathit{ji}}^2,{\rho }_{\mathit{ii}}^1={\rho }_{\mathit{ii}}^2=0.5$. Therefore, the deviation defined by Eq.(15)is simplified as.

$$d\left(P^1,P^2\right)=\frac{2}{n\left(n-1\right)}\sum _{i,j}0.5\left(\left|{\left({\rho }_{\mathit{ij}}^1\right)}^2-{\left({\rho }_{\mathit{ij}}^2\right)}^2\right|\right)$$

Definition 15

Let$\left\{P^1,P^2,\dots ,P^m\right\}$be$m$PFPRs, and a group PFPR is defined as follows:

$$P={\left(p_{\mathit{ij}}\right)}_{n\times n}$$ (16)

$$p_{\mathit{ij}}=\left({\rho }_{\mathit{ij}},{\sigma }_{\mathit{ij}}\right)=\left(\sqrt{\sum _{k=1}^m{\omega }_k{\left({\rho }_{\mathit{ij}}^k\right)}^2},\sqrt{\sum _{k=1}^m{\omega }_k{\left({\sigma }_{\mathit{ij}}^k\right)}^2}\right),{\omega }_k>0\left(k=1,2,\dots ,n\right)$$

Based on the additive consistency in Definition 11 and the modified algorithm for inexpressible PFPRs, a new GDM method for incomplete PFPRs is proposed.

Algorithm II

Step 1:Complement the incomplete PFPRs. If the conditions inTheorem 1orTheorem 2are satisfied, the missing preference values can be estimated directly by Eq.(2). Otherwise, the missing preference values are estimated by Eq.(2)after modifying the preference values usingAlgorithm I.

Step 2: Determine the weight ${\omega }_k$ of the decision-maker $d_k$.

Step 2.1: Calculate the mean of the PFPRs. Assume that the mean of all decision matrices is the ideal decision matrix. Then, the average PFPR is

$$P^{\ast }={\left(p_{\mathit{ij}}^{\ast }\right)}_{n\times n}$$

where

$$p_{\mathit{ij}}^{\ast }=\left({\rho }_{\mathit{ij}}^{\ast },{\sigma }_{\mathit{ij}}^{\ast }\right)=\left(\sqrt{\sum _{k=1}^m\frac{1}{m}{\left({\rho }_{\mathit{ij}}^k\right)}^2},\sqrt{\sum _{k=1}^m\frac{1}{m}{\left({\sigma }_{\mathit{ij}}^k\right)}^2}\right)$$ (17)

Step 2.2: Calculate the consensus degree $MCI\left(P_l\right)$ of the decision-maker $d_l\left(l=1,2,\dots ,m\right)$, which measures the deviation of the ideal PFPR from the individual PFPRs. The consensus degree of the average PFPR is defined as

$$MCI\left(P_l\right)=1-d\left(P^l,P^{\ast }\right)=1-\frac{2}{n\left(n-1\right)}\sum _{i,j}0.5\left(\left|{\left({\rho }_{\mathit{ij}}^l\right)}^2-{\left({\rho }_{\mathit{ij}}^{\ast }\right)}^2\right|\right),l=1,2,\dots ,m$$ (18)

Step 2.3: Obtain the weight ${\omega }_k$ of the decision-maker $d_k$. Obviously, the larger the value of $MCI\left(P^l\right)$, the closer the corresponding decision-maker is to the ideal decision. That is, the greater the consensus degree between $P^l$ and $P^{\ast }$, the more important the decision-maker $d_k$ is. Therefore, the weight of the decision-maker $d_k$ is determined by the following equation:

$${\omega }_k=\frac{MCI\left(P^k\right)}{{\sum }_{l=1}^{m}MCI\left(P^l\right)}$$ (19)

Step 3: Construct group PFPRs. In the GDM process, after obtaining the weight vector $\omega ={\left\{{\omega }_1,{\omega }_2,\dots ,{\omega }_m\right\}}^T$ of decision-makers, all individual decision matrices $P^k$ are assembled into a group decision matrix $P$ by Eq. (16).

Step 4: obtain the preferences of the alternative $x_i\left(i=1,2,\dots ,n\right)$ as follows:

$${\rho }_i=\sqrt{\frac{1}{n}\sum _{j=1}^n{\left({\rho }_{\mathit{ij}}\right)}^2},{\sigma }_i=\sqrt{\frac{1}{n}\sum _{j=1}^n{\left({\sigma }_{\mathit{ij}}\right)}^2}$$ (20)

Step 5: Calculate the score or accuracy and rank the alternatives according to Definition 9.

6. An example of subway station safety management during the COVID-19 pandemic

In this section, an example of subway station safety management during the COVID-19 pandemic is illustrated to verify the effectiveness of the developed GDM method.

6.1. Decision-making process and results

To find out the crucial risk factor in subway station safety management during COVID-19, a group of three decision-makers formed a panel of experts: subway operation engineer $d_1$, designer $d_2$, and professor $d_3$. The weight vector of decision-makers is unknown.

The experts compared five risk factors: station ventilation $x_1$, disinfection $x_2$, passenger flow$x_3$, temperature measurement $x_4$ and staff training $x_5$ (Xing et al., 2022, Xing et al., 2023). The detailed information is shown in Table 1 .

Table 1.

Five risk factors and their explanations.

Risk factors	Explanations
Ventilation $x_1$	The ventilation time of metro stations and trains is extended, and according to the requirement of not less than 22 h’ ventilation per day, the ventilation mode of stations is adjusted to full fresh air operation. The fresh air wells and filters are cleaned and disinfected, and the wind speed of the air outlets is tested regularly to deliver sufficient fresh air on time.
Disinfection $x_2$	The disinfection frequency is no less than twice a day for railings, handrails, seats, and other parts of subway stations with low passenger flow and no less than four times a day for subway stations with high passenger flow, such as those near hospitals, transportation hubs, and tourist attractions.
Passenger flow $x_3$	A batch approach is adopted to the station to reduce the accumulation of passengers when there is a large passenger flow outside the station. Inside the station, adding guidance staff and posting guidance signs are used to reduce passengers' time in the public area.
Temperature measurement $x_4$	Passengers whose temperature exceeds 37.3 ℃ through multiple measurements are notified to the hospital for treatment. A non-contact thermal imaging thermometer is used for subway stations with large passenger flow. The manual handheld, the non-contact thermometer is recommended for smaller passenger flow subway stations because of cost.
Staff training $x_5$	The front-line staff is trained to improve their emergency handling ability, grasp timely and accurate knowledge about wearing protective equipment, temperature detection, and epidemic prevention, and report abnormalities immediately to ensure that in case of an epidemic, they can do a good job of on-site handling under the requirements of the plan and effectively guard the first line of defense of passenger security.

The evaluation matrix $P={\left(p_{\mathit{ij}}^k\right)}_{5\times 5}\left(k=1,2,3\right)$ is the three incomplete PFPRs given by the three experts as follows. For example, $p_{41}^1=\left(\sqrt{0.1},\sqrt{0.9}\right)$ is the Pythagorean fuzzification of approval and disapproval degrees of expert $d_1$, $\sqrt{0.1}$ is the preference value of $x_4$ over $x_1$ and $\sqrt{0.9}$ is the preference value of $x_1$ over $x_4$.

$$P^1=\left[\begin{array}{ccccc}\left(\sqrt{0.5},\sqrt{0.5}\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)\\ \left(-,-\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)\\ \left(-,-\right)& \left(-,-\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(-,-\right)& \left(-,-\right)\\ \left(\sqrt{0.1},\sqrt{0.9}\right)& \left(\sqrt{0.3},\sqrt{0.7}\right)& \left(\sqrt{0.3},\sqrt{0.6}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)\\ \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)\end{array}\right]$$

$$P^2=\left[\begin{array}{ccccc}\left(\sqrt{0.5},\sqrt{0.5}\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)\\ \left(-,-\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)\\ \left(-,-\right)& \left(-,-\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(-,-\right)& \left(-,-\right)\\ \left(\sqrt{0.1},\sqrt{0.9}\right)& \left(\sqrt{0.3},\sqrt{0.7}\right)& \left(\sqrt{0.3},\sqrt{0.6}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)\\ \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)\end{array}\right]$$

$$P^3=\left[\begin{array}{ccccc}\left(\sqrt{0.5},\sqrt{0.5}\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)\\ \left(-,-\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)\\ \left(-,-\right)& \left(-,-\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(-,-\right)& \left(-,-\right)\\ \left(\sqrt{0.7},\sqrt{0.2}\right)& \left(\sqrt{0.1},\sqrt{0.8}\right)& \left(\sqrt{0.7},\sqrt{0.3}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.8},\sqrt{0.1}\right)\\ \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(-,-\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)\end{array}\right]$$

Step 1: The incomplete PFPR $P^1$ satisfy the conditions in Theorem 2, then the missing values can be estimated directly by Eq. (2) as follows:

$$P^1=\left[\begin{array}{ccccc}\left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.7},\sqrt{0.3}\right)& \left(\sqrt{0.7},\sqrt{0.2}\right)& \left(\sqrt{0.9},\sqrt{0.1}\right)& \left(\sqrt{0.9},\sqrt{0.1}\right)\\ \left(\sqrt{0.3},\sqrt{0.7}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.5},\sqrt{0.4}\right)& \left(\sqrt{0.7},\sqrt{0.3}\right)& \left(\sqrt{0.7},\sqrt{0.3}\right)\\ \left(\sqrt{0.2},\sqrt{0.7}\right)& \left(\sqrt{0.4},\sqrt{0.5}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.6},\sqrt{0.3}\right)& \left(\sqrt{0.6},\sqrt{0.3}\right)\\ \left(\sqrt{0.1},\sqrt{0.9}\right)& \left(\sqrt{0.3},\sqrt{0.7}\right)& \left(\sqrt{0.3},\sqrt{0.6}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)\\ \left(\sqrt{0.1},\sqrt{0.9}\right)& \left(\sqrt{0.3},\sqrt{0.7}\right)& \left(\sqrt{0.3},\sqrt{0.6}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)\end{array}\right]$$

However, $P^2$ and $P^3$ do not satisfy the conditions in Theorem 1 or Theorem 2. According to Algorithm I, the following complete PFPR can be obtained.

$$P^2=\left[\begin{array}{ccccc}\left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.8},\sqrt{0.2}\right)& \left(\sqrt{0.3},\sqrt{0.5}\right)& \left(\sqrt{0.7},\sqrt{0.3}\right)& \left(\sqrt{0.6},\sqrt{0.3}\right)\\ \left(\sqrt{0.2},\sqrt{0.8}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(0,\sqrt{0.8}\right)& \left(\sqrt{0.4},\sqrt{0.6}\right)& \left(\sqrt{0.3},\sqrt{0.6}\right)\\ \left(\sqrt{0.5},\sqrt{0.3}\right)& \left(\sqrt{0.8},0\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.7},\sqrt{0.1}\right)& \left(\sqrt{0.6},\sqrt{0.3}\right)\\ \left(\sqrt{0.3},\sqrt{0.7}\right)& \left(\sqrt{0.6},\sqrt{0.4}\right)& \left(\sqrt{0.1},\sqrt{0.7}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.4},\sqrt{0.5}\right)\\ \left(\sqrt{0.3},\sqrt{0.6}\right)& \left(\sqrt{0.6},\sqrt{0.3}\right)& \left(\sqrt{0.3},\sqrt{0.6}\right)& \left(\sqrt{0.5},\sqrt{0.4}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)\end{array}\right]$$

$$P^3=\left[\begin{array}{ccccc}\left(\sqrt{0.5},\sqrt{0.5}\right)& \left(0,\sqrt{0.9}\right)& \left(\sqrt{0.4},\sqrt{0.5}\right)& \left(\sqrt{0.2},\sqrt{0.7}\right)& \left(\sqrt{0.5},\sqrt{0.4}\right)\\ \left(\sqrt{0.9},0\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.9},\sqrt{0.1}\right)& \left(\sqrt{0.7},\sqrt{0.3}\right)& \left(1,0\right)\\ \left(\sqrt{0.5},\sqrt{0.4}\right)& \left(\sqrt{0.1},\sqrt{0.9}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.3},\sqrt{0.7}\right)& \left(\sqrt{0.6},\sqrt{0.4}\right)\\ \left(\sqrt{0.7},\sqrt{0.2}\right)& \left(\sqrt{0.3},\sqrt{0.7}\right)& \left(\sqrt{0.7},\sqrt{0.3}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.8},\sqrt{0.2}\right)\\ \left(\sqrt{0.4},\sqrt{0.5}\right)& \left(0,1\right)& \left(\sqrt{0.4},\sqrt{0.6}\right)& \left(\sqrt{0.2},\sqrt{0.8}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)\end{array}\right]$$

Step 2: obtain the weight of the decision-maker by Eq.(19).

Step 2.1: calculate the mean of the PFPRs by Eq.(17) as follows.

$$P^{\ast }=\left[\begin{array}{ccccc}\left(0.7071,0.7071\right)& \left(0.7071,0.6831\right)& \left(0.6831,0.6325\right)& \left(0.7746,0.6055\right)& \left(0.8165,0.5164\right)\\ \left(0.6831,0.7071\right)& \left(0.7071,0.7071\right)& \left(0.6831,0.6583\right)& \left(0.7746,0.6325\right)& \left(0.8165,0.5477\right)\\ \left(0.6325,0.6831\right)& \left(0.6583,0.6831\right)& \left(0.7071,0.7071\right)& \left(0.7303,0.6055\right)& \left(0.7746,0.5774\right)\\ \left(0.6055,0.7746\right)& \left(0.6325,0.7746\right)& \left(0.6055,0.7303\right)& \left(0.7071,0.7071\right)& \left(0.7528,0.6325\right)\\ \left(0.5164,0.8165\right)& \left(0.5477,0.8165\right)& \left(0.5774,0.7746\right)& \left(0.6325,0.7528\right)& \left(0.7071,0.7071\right)\end{array}\right]$$

Step 2.2: Calculate the consensus degree $MCI\left(P_l\right)$ of decision-maker $d_l\left(l=1,2,3\right)$ by Eq. (18).

$$MCI\left(P_1\right)=0.88,MCI\left(P_2\right)=0.81,MCI\left(P_3\right)=0.76$$

Step 2.3: Obtain the weight vector $\omega ={\left\{0.3587,0.3315,0.3098\right\}}^T$ of decision-makers by Eq. (19).

Step 3: Construct the group PFPR matrix $P={\left(p_{\mathit{ij}}\right)}_{5\times 5}$ according to Eq. (16) as follows.

$$P=\left[\begin{array}{ccccc}\left(0.7071,0.7071\right)& \left(0.7185,0.6728\right)& \left(0.6888,0.6264\right)& \left(0.7854,0.5934\right)& \left(0.8226,0.5092\right)\\ \left(0.6728,0.7185\right)& \left(0.7071,0.7071\right)& \left(0.6769,0.6631\right)& \left(0.7749,0.6320\right)& \left(0.8126,0.5536\right)\\ \left(0.6264,0.6888\right)& \left(0.6631,0.6769\right)& \left(0.7071,0.7071\right)& \left(0.7350,0.5980\right)& \left(0.7746,0.5753\right)\\ \left(0.5934,0.7854\right)& \left(0.6320,0.7749\right)& \left(0.5980,0.7350\right)& \left(0.7071,0.7071\right)& \left(0.7482,0.6380\right)\\ \left(0.5092,0.8226\right)& \left(0.5536,0.8126\right)& \left(0.5753,0.7746\right)& \left(0.6380,0.7482\right)& \left(0.7071,0.7071\right)\end{array}\right]$$

Step 4: fuse all the preference values in $P={\left(p_{\mathit{ij}}\right)}_{5\times 5}$ into the total preference value $p_i=\left({\rho }_i,{\sigma }_i\right)$, respectively.

Step 5: calculate the scores $p_i=\left({\rho }_i,{\sigma }_i\right)$ and rank the risk factors.

The scores are

$$s\left(p_1\right)=0.1655,s\left(p_2\right)=0.1020,s\left(p_3\right)=0.0702,s\left(p_4\right)=-0.0991,s\left(p_5\right)=-0.2386$$

Therefore, the ranking of the risk factors is $x_1\succ x_2\succ x_3\succ x_4\succ x_5$, i.e., $x_1$ is the crucial risk factor.

6.2. Comparative analysis

(1) GDM method with crisp consistency.

The developed GDM method based on PFPRs is compared with the crisp consistency method (Sarkar and Biswas, 2021). The crisp PRs adopt the membership values of the complete PFPRs with non-Pythagorean fuzzification and are as follows:

$P^1=\left[\begin{array}{ccccc}0.5& 0.7& 0.7& 0.9& 0.9\\ 0.3& 0.5& 0.5& 0.7& 0.7\\ 0.2& 0.4& 0.5& 0.6& 0.6\\ 0.1& 0.3& 0.3& 0.5& 0.5\\ 0.1& 0.3& 0.3& 0.5& 0.5\end{array}\right]$, $P^2=\left[\begin{array}{ccccc}0.5& 0.8& 0.3& 0.7& 0.6\\ 0.2& 0.5& 0& 0.4& 0.3\\ 0.5& 0.8& 0.5& 0.7& 0.6\\ 0.3& 0.6& 0.1& 0.5& 0.4\\ 0.3& 0.6& 0.3& 0.5& 0.5\end{array}\right]$, and

$$P^3=\left[\begin{array}{ccccc}0.5& 0& 0.4& 0.2& 0.5\\ 0.9& 0.5& 0.9& 0.7& 1\\ 0.5& 0.1& 0.5& 0.3& 0.6\\ 0.7& 0.3& 0.7& 0.5& 0.8\\ 0.4& 0& 0.4& 0.2& 0.5\end{array}\right]$$

Then, the mean of the three crisp PRs is

$$P^{\ast }=\left[\begin{array}{ccccc}0.5& 0.5& 0.4667& 0.6& 0.6667\\ 0.4667& 0.5& 0.4667& 0.6& 0.6667\\ 0.4& 0.4333& 0.5& 0.5333& 0.6\\ 0.3667& 0.4& 0.3667& 0.5& 0.5667\\ 0.2667& 0.3& 0.3333& 0.4& 0.5\end{array}\right]$$

Next, the deviation-based weight is $\omega ={\left\{0.3599,0.3297,0.3104\right\}}^T$.

The group crisp PR is

$$P=\left[\begin{array}{ccccc}0.5& 0.5157& 0.4750& 0.6168& 0.6769\\ 0.4533& 0.5& 0.4593& 0.6011& 0.6613\\ 0.3920& 0.4387& 0.5& 0.5398& 0.6000\\ 0.3522& 0.3989& 0.3582& 0.5& 0.5602\\ 0.2591& 0.3058& 0.3310& 0.4069& 0.5\end{array}\right]$$

The score are

$$s\left(x_1\right)=0.5569,s\left(x_2\right)=0.5350,s\left(x_3\right)=0.4941,s\left(x_4\right)=0.4339,s\left(x_5\right)=0.3605$$

Therefore, the ranking of the risk factors is the same as the developed GDM method, i.e.,$x_1\succ x_2\succ x_3\succ x_4\succ x_5$. However, the uncertainty of decision-making information provided by decision-making experts can not be fully expressed in the crisp PRs.

(2) GDM method with multiplicative consistency.

It is compared with the multiplicative consistency method (Wu et al., 2021) to illustrate the advantages of the developed GDM method. First, the missing elements are estimated by multiplicative consistency (Wu et al., 2021), and the PFPRs are aggregated and ranked, which is calculated as follows:

Step 1: Estimate the missing preference values in $P={\left(p_{\mathit{ij}}^k\right)}_{5\times 5}\left(k=1,2,3\right)$ by Eq.(21).

$${\rho }_{\mathit{ij}}=\sqrt{\frac{{\rho }_{\mathit{ik}}^2{\rho }_{\mathit{kj}}^2}{{\rho }_{\mathit{ik}}^2{\rho }_{\mathit{kj}}^2+\left(1-{\rho }_{\mathit{ik}}^2\right)\left(1-{\rho }_{\mathit{kj}}^2\right)}},{\sigma }_{\mathit{ij}}=\sqrt{\frac{{\sigma }_{\mathit{ik}}^2{\sigma }_{\mathit{kj}}^2}{{\sigma }_{\mathit{ik}}^2{\sigma }_{\mathit{kj}}^2+\left(1-{\sigma }_{\mathit{ik}}^2\right)\left(1-{\sigma }_{\mathit{kj}}^2\right)}}$$ (21)

Three complete PFPRs $P={\left(p_{\mathit{ij}}^k\right)}_{5\times 5}\left(k=1,2,3\right)$ are obtained as follows:

$$P^1=\left[\begin{array}{ccccc}\left(\sqrt{0.5},\sqrt{0.5}\right)& \left(0.8911,0.4537\right)& \left(0.8911,0.3780\right)& \left(\sqrt{0.9},\sqrt{0.1}\right)& \left(\sqrt{0.9},\sqrt{0.1}\right)\\ \left(0.4537,0.8911\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.5},0.6255\right)& \left(\sqrt{0.7},\sqrt{0.3}\right)& \left(\sqrt{0.7},\sqrt{0.3}\right)\\ \left(0.3780,0.8911\right)& \left(0.6255,\sqrt{0.5}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.6},\sqrt{0.3}\right)& \left(\sqrt{0.6},\sqrt{0.3}\right)\\ \left(\sqrt{0.1},\sqrt{0.9}\right)& \left(\sqrt{0.3},\sqrt{0.7}\right)& \left(\sqrt{0.3},\sqrt{0.6}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)\\ \left(\sqrt{0.1},\sqrt{0.9}\right)& \left(\sqrt{0.3},\sqrt{0.7}\right)& \left(\sqrt{0.3},\sqrt{0.6}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)\end{array}\right]$$

$$P^2=\left[\begin{array}{ccccc}\left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.8},\sqrt{0.1}\right)& \left(\sqrt{0.3},\sqrt{0.5}\right)& \left(\sqrt{0.7},\sqrt{0.3}\right)& \left(\sqrt{0.6},\sqrt{0.3}\right)\\ \left(\sqrt{0.1},\sqrt{0.8}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(0.2132,\sqrt{0.8}\right)& \left(0.4537,0.7947\right)& \left(0.3780,0.7947\right)\\ \left(\sqrt{0.5},\sqrt{0.3}\right)& \left(\sqrt{0.8},0.2132\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.7},0.3939\right)& \left(\sqrt{0.6},0.3939\right)\\ \left(\sqrt{0.3},\sqrt{0.7}\right)& \left(0.7947,0.4537\right)& \left(0.3939,\sqrt{0.7}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(0.6255,\sqrt{0.5}\right)\\ \left(\sqrt{0.3},\sqrt{0.6}\right)& \left(0.7947,0.3780\right)& \left(0.3939,\sqrt{0.6}\right)& \left(\sqrt{0.5},0.6255\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)\end{array}\right]$$

$$P^3=\left[\begin{array}{ccccc}\left(\sqrt{0.5},\sqrt{0.5}\right)& \left(0.1644,0.9504\right)& \left(0.6070,\sqrt{0.5}\right)& \left(\sqrt{0.2},\sqrt{0.7}\right)& \left(\sqrt{0.5},0.4537\right)\\ \left(0.9504,0.1644\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(0.9504,0.2132\right)& \left(\sqrt{0.8},\sqrt{0.1}\right)& \left(0.9701,0.1104\right)\\ \left(\sqrt{0.5},0.6070\right)& \left(0.2132,0.9504\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.3},\sqrt{0.7}\right)& \left(0.7947,0.4537\right)\\ \left(\sqrt{0.7},\sqrt{0.2}\right)& \left(\sqrt{0.1},\sqrt{0.8}\right)& \left(\sqrt{0.7},\sqrt{0.3}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)& \left(\sqrt{0.8},\sqrt{0.1}\right)\\ \left(0.4537,\sqrt{0.5}\right)& \left(0.1104,0.9701\right)& \left(0.4537,0.7947\right)& \left(\sqrt{0.1},\sqrt{0.8}\right)& \left(\sqrt{0.5},\sqrt{0.5}\right)\end{array}\right]$$

Step 2: the weights of the experts are subjectively given. Here, let the expert's weight vector be $\omega ={\left\{0.5,0.2,0.3\right\}}^T$. The group PFPR $P={\left(p_{\mathit{ij}}\right)}_{5\times 5}$ can be obtained by Eq. (16) as follows:

$$P=\left[\begin{array}{ccccc}\left(0.7071,0.7071\right)& \left(0.7518,0.6276\right)& \left(0.7534,0.5669\right)& \left(0.8062,0.5657\right)& \left(0.8485,0.4144\right)\\ \left(0.6276,0.7518\right)& \left(0.7071,0.7071\right)& \left(0.7281,0.6077\right)& \left(0.7945,0.5535\right)& \left(0.8130,0.5219\right)\\ \left(0.5669,0.7534\right)& \left(0.6077,0.7281\right)& \left(0.7071,0.7071\right)& \left(0.7280,0.6253\right)& \left(0.7807,0.4927\right)\\ \left(0.5657,0.8062\right)& \left(0.5535,0.7945\right)& \left(0.6253,0.7280\right)& \left(0.7071,0.7071\right)& \left(0.7538,0.6164\right)\\ \left(0.4144,0.8485\right)& \left(0.5291,0.8130\right)& \left(0.4927,0.7807\right)& \left(0.6164,0.7538\right)& \left(0.7071,0.7071\right)\end{array}\right]$$

Step 3: Similarly, using Eq. (20), the preference values in $P={\left(p_{\mathit{ij}}\right)}_{5\times 5}$ are assembled into total PFPR values $p_i=\left({\rho }_i,{\sigma }_i\right)$.

The scores are

$$s\left(p_1\right)=0.2591,s\left(p_2\right)=0.1391,s\left(p_3\right)=0.0197,s\left(p_4\right)=-0.1211,s\left(p_5\right)=-0.2968$$

It indicates that the risk factors are ranked as $x_1\succ x_2\succ x_3\succ x_4\succ x_5$. Therefore, the crucial risk factor remains $x_1$.

Next, the ranking results obtained from these two methods are compared and analyzed, and the comparison is shown in Fig. 2 . The ranking results of the developed method are identical to those of the multiplicative consistency method (Wu et al., 2021): $x_1$ is ranked first, followed by $x_2$, $x_3$, $x_4$, and finally $x_5$. It can also be observed that the rankings given by$d_1$, $d_2$ and $d_3$ do not change regardless of whether the developed method or multiplicative consistency method (Wu et al., 2021) is used. It can be obtained that:

Fig. 2.

Ranking results of the risk factors.

(1) In the decision-making process, the complete PFPRs obtained with the developed method can retain the characteristics of the original PFPRs. In estimating the missing preference values, the developed method ensures as little correction as possible to prevent the loss of useful information. The multiplicative consistency method (Wu et al., 2021) makes full use of multiplicative transferability to estimate missing preferences without modifying any values, preserving the original information to some extent. The results obtained by the developed method are consistent with those in the multiplicative consistency method (Wu et al., 2021), which indicates that the modified PRs do not change the characteristics of the decision matrix, and the estimated information is consistent with the original view of the decision-maker.

(2) Additive and multiplicative consistency are effective tools for estimating the missing values of incomplete FPRs. However, the existing studies assume these FPRs are multiplicative consistent because the missing or unknown preference values estimated by additive consistency sometimes exceed the definition domain, making the results invalid and insignificant. We define the concept of ACPFPRs; if the preference values obtained based on additive consistency are inexpressible, several values in the original PRs need to be corrected, and a correction algorithm is given; finally, additive consistency is then used to complement the corrected incomplete PFPRs. The developed method provides a new way of thinking to solve the decision problem of incomplete FPRs.

(3) A relatively objective decision maker's weight vector can be obtained directly from PFPRs by the developed method. The weight vectors of decision-makers are given directly in the existing methods (Wu et al., 2021). Thus, the weights of decision-makers are relatively subjective in the developed method.

7. Conclusions

PFPRs play a very important role in the decision-making process, and they are more realistic to express the PFPRs among alternatives flexibly without scoring all alternatives under the corresponding attributes. However, decision-makers can sometimes not determine preference values due to time constraints and lack of knowledge, which inevitably produces an incomplete PFPR. Additive consistency and multiplicative consistency are widely used to estimate the missing preferences in PRs. Estimating missing preferences using additive consistency sometimes results in obtaining estimates outside the definition domain and losing their practical meaning. Thus, we develop a new approach to solve the GDM problem for incomplete PFPRs. The main contributions of the developed method are threefold. Firstly, an ACPFPR is defined using the connection between PFPRs and IVFPRs. Secondly, two theorems are given that present sufficient conditions for the missing values to be expressible based on additive consistent estimates; if the incomplete PFPR satisfies these sufficient conditions, the missing values can be estimated using additive consistency, and the estimated preferences are expressible. Obviously, the obtained complete PFPRs are also additive consistent. Then, a correction algorithm is proposed if the incomplete PFPR does not satisfy these sufficient conditions; the corrected incomplete PFPR can be directly used to obtain the estimated preference values that are expressible by using additive consistency. Thirdly, based on the consensus degree, the weight of decision-makers in the GDM process can be directly determined by the PFPRs. Finally, the validity of the developed GDM method is verified by an example of subway station safety management. The results show that the developed GDM method effectively identifies the crucial risk factor in subway station safety management and has better performance in terms of computational time complexity than the multiplicative consistency method.

CRediT authorship contribution statement

Zhenyu Zhang: Conceptualization, Methodology. Huirong Zhang: Writing – review & editing, Supervision. Lixin Zhou: Visualization, Writing – original draft. Yong Qin: Writing – review & editing, Project administration. Limin Jia: Conceptualization, Investigation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

No data was used for the research described in the article.