Modified GRA method for probabilistic hesitant fuzzy MAGDM and application to network health performance evaluation of radial distribution system

Abstract

Radial distribution system is an important link connecting power supply and users, and its power supply reliability is directly related to users. Radial distribution network reconfiguration can transform the network structure by changing the switching state of the distribution network lines, and achieve the goals of reducing network operational losses, improving power quality, and power supply reliability while meeting various constraints such as radial operation, power supply and demand balance, capacity, and voltage. Radial distribution systems have the characteristics of multiple components and complex structures. How to quickly and accurately evaluate the health performance of radial distribution systems and find an optimal solution for network reconfiguration are important issues in distribution network analysis. The network health performance evaluation of radial distribution system is classical multiple attributes group decision making (MAGDM). The probabilistic hesitancy fuzzy sets (PHFSs) are used as a tool for characterizing uncertain information during the network health performance evaluation of radial distribution system. In this paper, we extend the classical grey relational analysis (GRA) method to the probabilistic hesitancy fuzzy MAGDM with unknown weight information. Firstly, the basic concept, comparative formula and Hamming distance of PHFSs are briefly introduced. Then, the definition of the score values is employed to compute the attribute weights based on the information entropy method. Then, probabilistic hesitancy fuzzy GRA (PHF-GRA) method is built for MAGDM under PHFSs. Finally, a practical case study for network health performance evaluation of radial distribution system is designed to validate the proposed method and some comparative studies are also designed to verify the applicability.

Keywords

Multiple attributes group decision making (MAGDM)probabilistic hesitant fuzzy sets grey relational analysis method (GRA)information entropy network health performance evaluation

1 Introduction

With the development of society, the complex and changing information environment is making it difficult to make accurate decisions. Therefore, scientific decision-making extensively uses scientific and technological methods, and combines qualitative and quantitative analysis to ensure the scientific and accuracy of decision-making [1, 2]. With a view of development of MAGDM, a large number of decision-making methods have been proposed based on different research principles and directions [3–5]. The more important ones, for instance, like VIKOR method [6], MABAC method [7], CODAS method [8], MOORA method [9], and TODIM method [10] etc. Due to the different calculation principles, these methods have their own advantages and disadvantages. Among them, the classical GRA method as the part of the grey system theory (GST) [11]. Zhang, Wei and Chen [12] extended GRA method in spherical fuzzy sets, and applied it in the emergency applies supplier selection. Liang, Kobina and Quan [13] constructed the developed GRA method based on geometric Bonferroni mean operators in probabilistic linguistic sets. The main principle of traditional GRA method is selecting the target solution as reference solution, and measure the relation between the reference solution and compared solution. However, the evaluation of the negative ideal point (NIP) has the limitation of ignoring the subjective factors that influence DMs’ behavior in reality. The GRA is then extended based on CPT to avoid the limitation of original GRA method and its extension has been applied in intuitionistic fuzzy sets [14], Spherical fuzzy sets [12], 2-tuple linguistic neutrosophic sets [15].

In addition, the decision-making process in reality cannot avoid the loss of information and ambiguity. Fuzzy decision environment develops from the simplest real number environment to the first generation of fuzzy sets (FS) Zadeh [16] concepts in 1970. Then the development of FS such as the intuitionistic fuzzy set(IFS) [17], hesitant fuzzy set (HFS) [18] depict the complex environment with the memberships whose values are between 0 and 1. While in real environment, DMs are always indecisive between different alternatives with the uncertain information. To make up for the weaknesses, the general form of HFS is then proposed as probabilistic hesitant fuzzy sets (PHFSs) [19] with different probabilities of different memberships. After the proposition of PHFS, the basic operating formulas such as the PHF geometric operators [19] and new normalization process Li, Chen, Niu and Wang [20] improving the probability of different multiplication operations, and some developed aggregation operators are then proposed by Zhang, Xu and He [21]. Meanwhile, the score function, the deviation and the rule of comparison [22], and the PHF weighted averaging geometric operators. Besides Xu and Zhou [19] proposed the PHF weighted averaging geometric operators to process PHFE information. And the improved PHFS was introduced by Zhang, Xu and He [23] to allow more room for hesitation and the integrations can be calculated by the developed operators.

The network health performance evaluation of radial distribution system is classical MAGDM. The PHFSs are used as a tool for characterizing uncertain information during the network health performance evaluation of radial distribution system. In this paper, we extend the classical GRA method [24] to the probabilistic hesitancy fuzzy MAGDM with unknown weight information. Firstly, the basic concept, comparative formula and Hamming distance of PHFSs are briefly introduced. Then, the definition of the score values is employed to compute the attribute weights based on the information entropy method. Then, the PHF-GRA method is built for MAGDM under PHFSs. Finally, a practical case study for network health performance evaluation of radial distribution system is designed to validate the proposed method and some comparative studies are also designed to verify the applicability.

In order to do so, the layout of this paper is as follows. Section 2 gives the basic introduction of PHFS. In Section 3, the model of PHF-GRA is then applied to MAGDM with information entropy weight. A numerical example for network health performance evaluation of radial distribution system is given in Section 4. In the last section, we drawn a conclusion and list the expect of future work.

2 Basic knowledge

We give a simple introduction about PHFSs.

2.1 PHFS

In 2017, the PHFSs is proposed by Xu and Zhou [19] with different probabilities of different memberships based on the HFSs.

Definition 1 [19]. Let y be a fixed collection, a PHFSs on y is defined as Equation (1):

$J = {< y, h_{y} (p_{y}) > | y \in Y}$ (1) where h_y (p_y) is the collection of values ranging from 0 to 1, denoting the possible membership degree of the element y ∈ Y to the set J, h_y is called a hesitant fuzzy values, p_y is a set of probabilities associated with k_y.

For convenience, h (p) is called a probabilistic hesitant fuzzy element (PHFE), and the set of all PHFEs can be depicted as h (p) ={ h^φ (p^φ) | φ = 1, 2, ⋯ , # h (p) }, where p^φ is the probability of the membership degree h^φ, # h (p) is the number of all different membership degrees, and $\leftarrow \sum_{φ = 1}^{# h (p)} p^{φ} = 1$ .

Definition 2 [19]. $h_{1} (p) = {h_{1}^{φ} (p_{1}^{φ}) | φ = 1, 2, \dots, # h_{1} (p)}$ and $h_{2} (p) = {h_{2}^{φ} (p_{2}^{φ}) | φ = 1, 2, \dots, # h_{2} (p)}$ be two PHFSs, # h₁ (p) and # h₂ (p) are the given length of h₁ (p) and h₂ (p). If # h₁ (p) > # h₂ (p), then add # h₁ (p) - # h₂ (p) hesitant fuzzy values to h₂ (p). The added hesitant fuzzy values are the defined smallest hesitant fuzzy values in h₂ (p) and the given probabilities are zero.

Definition 3 [19]. For the given $h_{1} (p) = {h_{1}^{φ} (p_{1}^{φ}) | φ = 1, 2, \dots, # h_{1} (p)}$ , the expected values EV (h₁ (p)) and deviation degree DD (h₁ (p)) is defined:

$EV (h_{1} (p)) = \sum_{φ = 1}^{# h_{1} (p)} (h_{1}^{φ} \times p_{1}^{φ})$ (2)

$DD (h_{1} (p)) = \sqrt{\sum_{φ = 1}^{# h_{1} (p)} {(h_{1}^{φ} \times p_{1}^{φ} - EV (h_{1} (p)))}^{2}}$ (3)

By Equations (2)–(3), the order between two given PHFE is built: (1) if EV (h₁ (p)) > EV (h₂ (p)), h₁ (p) > h₂ (p); (2) if EV (h₁ (p)) = EV (h₂ (p)), if DD (h₁ (p)) = DD (h₂ (p)), then h₁ (p) = h₂ (p); if DD (h₁ (p)) < DD (h₂ (p)), then, h₁ (p) > h₂ (p).

Definition 4 [19]. Let $h_{1} (p) = {h_{1}^{φ} (p_{1}^{φ}) | φ = 1, 2, \dots, # h_{1} (p)}$ and $h_{2} (p) = {h_{2}^{φ} (p_{2}^{φ}) | φ = 1, 2, \dots, # h_{2} (p)}$ be two PLTSs with # h₁ (p) = # h₂ (p) = # h (p), then Hamming distance $ED (L L_{1} (\tilde{p}), L L_{2} (\tilde{p}))$ is designed:\hrule \hrule

$d (h_{1} (p), h_{2} (p)) = \frac{\sum_{φ = 1}^{# h (p)} | h_{1}^{φ} \times p_{1}^{φ} - h_{2}^{φ} \times p_{2}^{φ} |}{# h (p)}$ (4)

3 PHF-GRA method for MAGDM with PHF information

This section aims to introduce PHF-GRA method to solve the MAGDM problems, where the decision-making information are represented by PHFSs. The listed mathematical notations are employed to denote the MAGDM issues under PHFSs. Let AL ={ AL₁, AL₂, ⋯ , AL_m } be chosen alternatives, and attributes set AT ={ AT₁, AT₂, ⋯ , AT_n } with weight vector aw = (aw₁, aw₂, ⋯ , aw_n), where aw_j ∈ [0, 1], j = 1, 2, ⋯ , n, $\sum_{j = 1}^{n} a w_{j} = 1$ . Suppose that there aren attribute and their values are evaluated and denoted as PHFSs information $h_{ij} (p) = {h_{ij}^{φ} (p_{ij}^{φ}) | φ = 1, 2, \dots, # h_{ij} (p)} (i = 1, 2, \dots, m, j = 1, 2, \dots, n)$ .

Then, PHF-GRA method is designed to solve the MAGDM issues with PHFSs and entropy weight. The detailed algorithms are given subsequently:

Step 1. Build the PHF-matrix H = [h_ij (p)] _m×n:

$\begin{matrix} \begin{matrix} A T_{1} & A T_{2} & \dots & A T_{n} \end{matrix} \\ H = {[h_{ij} (p)]}_{m \times n} = \begin{matrix} A L_{1} \\ A L_{2} \\ ⋮ \\ A L_{m} \end{matrix} [\begin{matrix} h_{11} (p) & h_{12} (p) & \dots & h_{1 n} (p) \\ h_{21} (p) & h_{22} (p) & \dots & h_{2 n} (p) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ h_{m 1} (p) & h_{m 2} (p) & \dots & h_{mn} (p) \end{matrix}] \end{matrix}$ (5)

$\begin{matrix} H = {[h_{ij} (p)]}_{m \times n} \\ = {h_{ij}^{φ} (p_{ij}^{φ}) | φ = 1, 2, \dots, # h_{ij} (p)} \\ (i = 1, 2, \dots, m, j = 1, 2, \dots, n) \end{matrix}$ (6)

Step 2. Standardize the matrix H = [h_ij (p)] _m×n into NH = [nh_ij (p)] _m×n.

${\begin{matrix} n h_{ij} (p) = h_{ij}^{φ} (p_{ij}^{φ}) & attribute \\ n h_{ij} (p) = {1 - h_{ij}^{φ} (p_{ij}^{φ})} & negative attribute \end{matrix}$ (7)

Step 3. Compute the attributes weight by using information entropy.

Entropy [25] is a conventional theory to derive weight. Firstly, the normalized PHF-matrix nnh_ij (p) is obtained:

$nn h_{ij} (p) = \frac{EV (n h_{ij} (p))}{\sum_{i = 1}^{m} EV (n h_{ij} (p))},$ (8) \hrule \hrule

Then, the Shannon entropy SE = (SE₁, SE₂, ⋯ , SE_n) is derived by Equation (8):

$S E_{j} = - \frac{1}{ln m} \sum_{i = 1}^{m} nn h_{ij} (p) ln nn h_{ij} (p)$ (9) and nnh_ij (p) ln nnh_ij (p) = 0 if nnh_ij (p) = 0.

Then, the weights aw = (aw₁, aw₂, ⋯ , aw_n) is derived:

$a w_{j} = \frac{1 - S E_{j}}{\sum_{j = 1}^{n} (1 - S E_{j})}, j = 1, 2, \dots, n .$ (10)

Step 4. Define the PHFPIS and PHFNIS:

$PHFPIS = (PHFPI S_{1}, PHFPI S_{2}, \dots, PHFPI S_{n})$ (11)

$\begin{matrix} PHFNIS \\ = (PHFNI S_{1}, PHFNI S_{2}, \dots, PHFNI S_{n}) \end{matrix}$ (12) where

$\begin{matrix} PHFPI S_{j} \\ = {h_{j}^{φ +} (p_{j}^{φ +}) | φ = 1, 2, \dots, # n h_{ij} (p)} \end{matrix}$ (13)

$\begin{matrix} PHFNI S_{j} \\ = {h_{j}^{φ -} (p_{j}^{φ -}) | φ = 1, 2, \dots, # n h_{ij} (p)} \end{matrix}$ (14)

$EV (PHFPI S_{j}) = {max_{i} EV (n h_{ij} (p))}$ (15)

$EV (PHFNI S_{j}) = {min_{i} EV (n h_{ij} (p))}$ (16)

Step 5. compute the grey relational coefficient of all alternatives from PHFPIS and PHFNIS.

$\begin{matrix} PHFPIS (ξ_{ij}) = \frac{min_{1 ⩽ i ⩽ m} HD (n h_{ij} (p), PHFPI S_{j}) + ρ max_{1 ⩽ i ⩽ m} HD (n h_{ij} (p), PHFPI S_{j})}{HD (n h_{ij} (p), PHFPI S_{j}) + ρ max_{1 ⩽ i ⩽ m} HD (n h_{ij} (p), PHFPI S_{j})} \\ i = 1, 2, \dots, m, j \in 1, 2, \dots, n \end{matrix}$ (17)

$\begin{matrix} PHFNIS (ξ_{ij}) = \frac{min_{1 ⩽ i ⩽ m} HD (n h_{ij} (p), PHFNI S_{j}) + ρ max_{1 ⩽ i ⩽ m} HD (n h_{ij} (p), PHFNI S_{j})}{HD (n h_{ij} (p), PHFNI S_{j}) + ρ max_{1 ⩽ i ⩽ m} HD (n h_{ij} (p), PHFNI S_{j})} \\ i = 1, 2, \dots, m, j \in 1, 2, \dots, n . \end{matrix}$ (18)

Suppose that the identification coefficient is ρ = 0.5.

$\begin{matrix} HD (n h_{ij} (p), PLPI S_{j}) \\ = \frac{\sum_{φ = 1}^{# h (p)} | h_{ij}^{φ} \times p_{ij}^{φ} - h_{j}^{φ +} \times p_{j}^{φ +} |}{# n h_{ij} (p} \end{matrix}$ (19)

$\begin{matrix} HD (n h_{ij} (p), PLNI S_{j}) \\ = \frac{\sum_{φ = 1}^{# h (p)} | h_{ij}^{φ} \times p_{ij}^{φ} - h_{j}^{φ -} \times p_{j}^{φ -} |}{# n h_{ij} (p} \end{matrix}$ (20)Step 6. Calculating the degree of grey relational coefficient of all alternatives from PHFPIS and PHFNIS:

$\begin{matrix} PHFPIS (ξ_{i}) \\ = \sum_{j = 1}^{n} a w_{j} PHFPIS (ξ_{ij}), i = 1, 2, \dots, m, \end{matrix}$ (21)

$\begin{matrix} PHFNIS (ξ_{i}) \\ = \sum_{j = 1}^{n} a w_{j} PHFNIS (ξ_{ij}), i = 1, 2, \dots, m, \end{matrix}$ (22)

The fundamental idea of GRA method is that the optimal alternative is supposed to possess the “largest degree of grey relational coefficient” from PHFPIS and “smallest degree of grey relational coefficient” from PHFNIS. Obviously, the larger PHFPIS (ξ_i) along with smallerPHFNIS (ξ_i), the better alternative AL_i is.

Step 7. Derive the probabilistic linguistic relative relational degree (PLRRD) of all possible alternatives from PHFPIS.

$\begin{matrix} PHFRRD (ξ_{i}) \\ = \frac{PHFPIS (ξ_{i})}{PHFPIS (ξ_{i}) + PHFNIS (ξ_{i})}, i = 1, 2, \dots, m \end{matrix}$ (23)

Step 8. According to PHFRRD (ξ_i), the sorting order of all possible alternatives can be obtained. If any alternative has the largest PHFRRD (ξ_i), then, it is optimal choice.

4 Case study and comparative analysis

4.1 Case study

The safety of equipment in power system is of great significance to the security of power grid [26]. Based on equipment condition evaluation, a condition based maintenance system has been gradually formed, that is, from the perspective of power system health diagnosis [27], to reduce the risk of system hidden dangers, the maintenance scheme is optimized according to the real-time health status of equipment. Compared with the conventional reliability analysis based on failure rate data, the health diagnosis theory has greatly improved in real-time and accuracy. For this reason, EA Company of the United Kingdom proposed a risk prevention management system based on the assessment of power grid assets, namely CBRM (condition based risk management) evaluation system [28–33]. Among them, the health index (HI), as the evaluation core, can be quantified to represent the real-time health status of the equipment. In order to ensure the reliable and continuous power supply of the power generation and transmission network at the upstream of the power system, on the one hand, N-1 or N-k criteria [34, 35] are used as the standard for planning and design; on the other hand, equipment status information provided by advanced status monitoring and diagnosis technology is used to analyze the health status of equipment during the operation stage, and maintenance is required before the failure occurs [36]. However, the connection mode of power distribution system is relatively simple, mainly radial. It contains a large number of components, and the relative cost is low, limited by the cost constraints of monitoring and testing data is relatively scarce. From the perspective of the overall operation performance of the system, Ao and Wang [37] proposed the concept of power grid condition based maintenance to meet the requirements of power distribution system condition based maintenance. Therefore, the maintenance of the power distribution system is mostly based on the connection relationship between the equipment, and the unit is the interconnected equipment group in the regional network. Zhang, Tan and He [38] screened a series of indicators from the perspective of the overall operation of the power grid, and constructed a set of quantitative reference systems to measure the health status of complex distribution networks. However, the evaluation indicators can only indirectly reflect the health status of power grid equipment groups. The network health performance evaluation of radial distribution system is classical MAGDM. The model of PHF-GRA is then applied to a numerical example for network health performance evaluation of radial distribution system. Six evaluation principles are given to evaluate the five radial distribution system AL_i (i = 1, 2, 3, 4, 5) as follows: AT₁ is cost; AT₂ is product; AT₃ is the management of radial distribution system; AT₄ is network health idea of enterprise; AT₅ is service capability; AT₆ is enterprise qualification.

Then, the model of PHF-GRA is then applied to a numerical example for network health performance evaluation of radial distribution system.

Step 1. Build the PHF-matrix H = [h_ij (p)] _5×4 (Table 1) with three experts’ evaluation information:

Table 1
Decision matrix H = [h_ij (p)] _5×4

Alternative AT₁ AT₂

AL₁ {0.8 (0.7), 0.3 (0.3)} {0.4 (1)}

AL₂ {0.3 (0.2), 0.2 (0.5), 0.1 (0.3)} {0.2 (0.2), 0.3 (0.5), 0.1 (0.3)}

AL₃ {0.1 (0.2), 0.2 (0.5), 0.3 (0.3)} {0.5 (0.5), 0.1 (0.5)}

AL₄ {0.6 (0.5), 0.5 (0.5)} {0.7 (0.7), 0.5 (0.3)}

AL₅ {0.3 (1)} {0.3 (0.5), 0.2 (0.5)}

Alternative AT₃ AT₄

AL₁ {0.3 (0.5), 0.1 (0.2), 0.2 (0.3)} {0.2 (0.2), 0.3 (0.3), 0.4 (0.5)}

AL₂ {0.1 (0.2), 0.2 (0.5), 0.4 (0.3)} {0.3 (0.7), 0.2 (0.3)}

AL₃ {0.4 (0.5), 0.2 (0.5)} {0.2 (0.5), 0.3 (0.5)}

AL₄ {0.7 (1)} {0.6 (0.7), 0.7 (0.3)}

AL₅ {0.3 (0.5), 0.2 (0.2), 0.1 (0.3)} {0.3 (0.2), 0.4 (0.3), 0.2 (0.5)}

Alternative AT₅ AT₆

AL₁ {0.4 (0.2), 0.3 (0.3), 0.1 (0.5)} {0.2 (0.7), 0.3 (0.3)}

AL₂ {0.2 (0.5), 0.3 (0.5)} {0.3 (0.5), 0.2 (0.5)}

AL₃ {0.1 (0.7), 0.3 (0.3} {0.4 (1)}

AL₄ {0.5 (0.2), 0.6 (0.8)} {0.6 (0.2), 0.7 (0.8)}

AL₅ {0.2 (0.5), 0.3 (0.2), 0.1 (0.3)} {0.3 (0.2), 0.2 (0.5), 0.4 (0.3)}

Alternative	AT₁	AT₂
AL₁	{0.8 (0.7), 0.3 (0.3)}	{0.4 (1)}
AL₂	{0.3 (0.2), 0.2 (0.5), 0.1 (0.3)}	{0.2 (0.2), 0.3 (0.5), 0.1 (0.3)}
AL₃	{0.1 (0.2), 0.2 (0.5), 0.3 (0.3)}	{0.5 (0.5), 0.1 (0.5)}
AL₄	{0.6 (0.5), 0.5 (0.5)}	{0.7 (0.7), 0.5 (0.3)}
AL₅	{0.3 (1)}	{0.3 (0.5), 0.2 (0.5)}
Alternative	AT₃	AT₄
AL₁	{0.3 (0.5), 0.1 (0.2), 0.2 (0.3)}	{0.2 (0.2), 0.3 (0.3), 0.4 (0.5)}
AL₂	{0.1 (0.2), 0.2 (0.5), 0.4 (0.3)}	{0.3 (0.7), 0.2 (0.3)}
AL₃	{0.4 (0.5), 0.2 (0.5)}	{0.2 (0.5), 0.3 (0.5)}
AL₄	{0.7 (1)}	{0.6 (0.7), 0.7 (0.3)}
AL₅	{0.3 (0.5), 0.2 (0.2), 0.1 (0.3)}	{0.3 (0.2), 0.4 (0.3), 0.2 (0.5)}
Alternative	AT₅	AT₆
AL₁	{0.4 (0.2), 0.3 (0.3), 0.1 (0.5)}	{0.2 (0.7), 0.3 (0.3)}
AL₂	{0.2 (0.5), 0.3 (0.5)}	{0.3 (0.5), 0.2 (0.5)}
AL₃	{0.1 (0.7), 0.3 (0.3}	{0.4 (1)}
AL₄	{0.5 (0.2), 0.6 (0.8)}	{0.6 (0.2), 0.7 (0.8)}
AL₅	{0.2 (0.5), 0.3 (0.2), 0.1 (0.3)}	{0.3 (0.2), 0.2 (0.5), 0.4 (0.3)}

Step 2. Standardize the matrix H = [h_ij (p)] _5×4 into NH = [nh_ij (p)] _5×4 (Table 2).

Table 2

Decision matrix NH = [nh_ij (p)] _5×4

Alternative	AT₁	AT₂
AL₁	{0.7 (0.3), 0.7 (0.0), 0.2 (0.7)}	{0.4 (1.0). 0.4 (0.0) 0.4 (0.0)},
AL₂	{0.9 (0.3), 0.8 (0.5), 0.7 (0.2)}	{0.2 (0.2), 0.3 (0.5), 0.1 (0.3)}
AL₃	{0.9 (0.2), 0.8 (0.5), 0.7 (0.3)}	{0.5 (0.5), 0.1 (0.5), 0.1 (0.0)}
AL₄	{0.5 (0.5), 0.5 (0.0), 0.4 (0.5)}	{0.7 (0.7), 0.5 (0.3), 0.5 (0.0)}
AL₅	{0.7 (1.0), 0.7 (0.0) 0.7 (0.0)}	{0.3 (0.5), 0.2 (0.5), 0.2 (0.0)}
Alternative	AT₃	AT₄
AL₁	{0.3 (0.5), 0.1 (0.2), 0.2 (0.3)}	{0.2 (0.2), 0.3 (0.3), 0.4 (0.5)}
AL₂	{0.1 (0.2), 0.2 (0.5), 0.4 (0.3)}	{0.3 (0.7), 0.2 (0.3), 0.2 (0.0)}
AL₃	{0.4 (0.5), 0.2 (0.5), 0.2 (0.0)}	{0.2 (0.5), 0.3 (0.5), 0.3 (0.0)}
AL₄	{0.7 (1.0), 0.7 (0.0) 0.7 (0.0)}	{0.7 (0.3), 0.6 (0.7), 0.6 (0.0)}
AL₅	{0.3 (0.5), 0.2 (0.2), 0.1 (0.3)}	{0.3 (0.2), 0.4 (0.3), 0.2 (0.5)}
Alternative	AT₅	AT₆
AL₁	{0.4 (0.2), 0.3 (0.3), 0.1 (0.5)}	{0.3 (0.3), 0.2 (0.7), 0.2 (0.0)}
AL₂	{0.3 (0.5), 0.2 (0.5), 0.2 (0.0)}	{0.3 (0.5), 0.2 (0.5), 0.2 (0.0)}
AL₃	{0.3 (0.3), 0.1 (0.7), 0.1 (0.0)}	{0.4 (1.0), 0.4 (0.0), 0.4 (0.0)}
AL₄	{0.6 (0.8), 0.5 (0.2), 0.5 (0.0)}	{0.7 (0.8), 0.6 (0.2), 0.6 (0.0)}
AL₅	{0.2 (0.5), 0.3 (0.2), 0.1 (0.3)}	{0.3 (0.2), 0.2 (0.5), 0.4 (0.3)}

Step 3. Compute the weight values: aw₁ = 0.1973, aw₂ = 0.4482, aw₃ = 0.1548, aw₄ = 0.1997.

Step 4. Determine the PHFPIS and PHFNIS (Table 3):

Table 3

The PHFPIS and PHFNIS

Alternative	AT₁	AT₂
PHFPIS	{0.9 (0.3), 0.8 (0.5), 0.7 (0.2)}	{0.7 (0.7), 0.5 (0.3), 0.5 (0.0)}
PHFNIS	{0.5 (0.5), 0.5 (0.0), 0.4 (0.5)}	{0.2 (0.2), 0.3 (0.5), 0.1 (0.3)}
Alternative	AT₃	AT₄
PHFPIS	{0.7 (1.0), 0.7 (0.0) 0.7 (0.0)}	{0.7 (0.3), 0.6 (0.7), 0.6 (0.0)}
PHFNIS	{0.1 (0.2), 0.2 (0.5), 0.4 (0.3)}	{0.3 (0.2), 0.4 (0.3), 0.2 (0.5)}
Alternative	AT₅	AT₆
PHFPIS	{0.6 (0.8), 0.5 (0.2), 0.5 (0.0)}	{0.7 (0.8), 0.6 (0.2), 0.6 (0.0)}
PHFNIS	{0.3 (0.3), 0.1 (0.7), 0.1 (0.0)}	{0.4 (1.0), 0.4 (0.0), 0.4 (0.0)}

Step 5. compute the grey relational coefficient of all alternatives from PHFPIS and PHFNIS (Tables 4, 5).

Table 4

The grey relational coefficient from PHFPIS

Alternatives	AT₁	AT₂	AT₃
AL₁	0.3379	0.9213	0.5284
AL₂	0.4829	0.8508	0.4829
AL₃	0.4567	0.4223	0.8965
AL₄	0.4829	0.6517	0.9804
AL₅	0.9078	0.4971	0.4567
Alternatives	AT₂	AT₃	AT₄
AL₁	0.9954	0.5192	0.4602
AL₂	0.8416	0.4737	0.3747
AL₃	0.4131	0.8873	0.6425
AL₄	0.6425	0.9954	0.9954
AL₅	0.4879	0.4475	0.4879

Table 5

The grey relational coefficient from PHFNIS

Alternatives	AT₁	AT₂	AT₃
AL₁	0.3658	0.3658	0.6621
AL₂	0.5510	0.4121	0.8287
AL₃	0.9954	0.4954	0.3525
AL₄	0.3525	0.3800	0.3287
AL₅	0.3287	0.9954	0.9954
Alternatives	AT₁	AT₂	AT₃
AL₁	0.3547	0.3547	0.6510
AL₂	0.5399	0.4010	0.8176
AL₃	0.9843	0.4843	0.3414
AL₄	0.3414	0.3689	0.3176
AL₅	0.3176	0.9843	0.9843

Step 6. Obtain the degree of grey relational coefficient of all alternatives (Table 6):

Table 6

PHFPIS (ξ_i) and PHFNIS (ξ_i) of all possible alternatives

Alternatives	PHFPIS (ξ_i)	PHFNIS (ξ_i)
AL₁	0.5399	0.5932
AL₂	0.5287	0.5588
AL₃	0.6054	0.6095
AL₄	0.7645	0.3581
AL₅	0.6566	0.7437

Step 7. Calculating the PHFRRD of each alternative from PHFPIS (Table 7).

Table 7

PHFRRD of each alternative from PHFPIS

Alternatives	AL₁	AL₂	AL₃	AL₄	AL₅
PHFRRD (ξ_i)	0.4809	0.4906	0.5029	0.6871	0.4733

Step 9. According to the PHFRRD (ξ_i) (i = 1, 2, 3, 4, 5), we can sort all the radial distribution systems. Obviously, the order is: AL₄ > AL₃ > AL₂ > AL₁ > AL₅ and the optimal radial distribution system isAL₄.

4.2 Comparative analysis

In this section, we use the same data to solve the problem using other methods to verify the validity to prove this method’s applicability. Thus, the PHF-GRA is compared with PHF-TODIM [39], PHFWA operator [19] and PHFWG operator [19]. The order is listed in Table 8.

Table 8
The rank of different models

Method The rank The best solution

PHF-TODIM [39] AL₄ > AL₃ > AL₂ > AL₁ > AL₅ AL ₄

PHFWA operator [19] AL₄ > AL₃ > AL₂ > AL₁ > AL₅ AL ₄

PHFWG operator [19] AL₄ > AL₃ > AL₁ > AL₂ > AL₅ AL ₄

PHF-GRA AL₄ > AL₃ > AL₂ > AL₁ > AL₅ AL ₄

Method	The rank	The best solution
PHF-TODIM [39]	AL₄ > AL₃ > AL₂ > AL₁ > AL₅	AL ₄
PHFWA operator [19]	AL₄ > AL₃ > AL₂ > AL₁ > AL₅	AL ₄
PHFWG operator [19]	AL₄ > AL₃ > AL₁ > AL₂ > AL₅	AL ₄
PHF-GRA	AL₄ > AL₃ > AL₂ > AL₁ > AL₅	AL ₄

From the above detailed analysis, it could be seen that these four methods have the same optimal choice. This verifies the PHF-GRA method is reasonable and effective.

5 Conclusions

The power distribution system is located at the end of the power system and directly connected to users. It is an important link for the entire power system, including power generation, transmission and distribution, to connect with users, supply and distribute electric energy to users. Due to the simultaneous characteristics of power generation, supply, and utilization in power production, once the equipment of the distribution system fails or undergoes maintenance, it will simultaneously cause the interruption of the entire system’s power supply to users. Therefore, the health performance of distribution systems is actually a centralized reflection of the overall power system structure and operational characteristics. According to statistics, more than 80% of user power outages are caused by faults in the distribution system, which has the greatest impact on user power supply reliability. Research on the health performance of radiant power distribution systems is an important measure to ensure the quality of power supply in power systems and improve the modernization level of the power industry. It plays an important guiding role in improving and improving the production technology and management level of the power industry, improving economic and social benefits, and carrying out the construction and transformation of urban power networks. Through the health performance evaluation of radiation type distribution systems, on the one hand, it is possible to understand the reliability distribution level of the entire radiation type distribution system, find out the weak links of the radiation type distribution network, point out the direction of network transformation, and provide reference opinions for urban distribution network transformation; On the other hand, the economic benefits of different enhancement measures can be calculated quantitatively, thereby maximizing the reliability of the system with limited funds. In this paper, we extend the classical GRA method to the probabilistic hesitancy fuzzy MAGDM with unknown weight information. Firstly, the basic concept, comparative formula and Hamming distance of PHFSs are briefly introduced. Then, the definition of the score values is employed to compute the attribute weights based on the information entropy method. Then, PHF-GRA method is built for MAGDM under PHFSs. Finally, a practical case study for network health performance evaluation of radial distribution system is designed to validate the proposed method and some comparative studies are also designed to verify the applicability.

Footnotes

Acknowledgments

This work was supported by scientific research fund of Tangshan Normal University, research on the reconfiguration strategy of Beijing Tianjin Hebei distribution network in the age of artificial intelligence(Project No. 2022C52), science and technology research project of colleges and universities in Hebei province, research on reconfiguration strategy of Beijing Tianjin Hebei distribution network based on genetic algorithm (Project No. ZC2022079) and Science and Technology Plan Project of Tangshan Science and Technology Bureau Tangshan Foundation Innovation Team of Digital Media Security (Project No. 21130212D).

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Modified GRA method for probabilistic hesitant fuzzy MAGDM and application to network health performance evaluation of radial distribution system

Abstract

Keywords

1 Introduction

2 Basic knowledge

2.1 PHFS

4.1 Case study

Table 8 The rank of different models Method The rank The best solution PHF-TODIM [39] AL4 > AL3 > AL2 > AL1 > AL5 AL 4 PHFWA operator [19] AL4 > AL3 > AL2 > AL1 > AL5 AL 4 PHFWG operator [19] AL4 > AL3 > AL1 > AL2 > AL5 AL 4 PHF-GRA AL4 > AL3 > AL2 > AL1 > AL5 AL 4

Footnotes

Acknowledgments

References