Research on node importance evaluation of complex products based on three-parameter interval grey number grey relational model

Abstract

In the process of product development, the identification and evaluation of important nodes is of great significance for the effective control of complex product engineering change. In order to identify and evaluate important nodes accurately, this paper proposes a method to evaluate the importance of complex product nodes. Firstly, an engineering change expression method based on multi-stage complex network is proposed. Then, the evaluation index system of important nodes of complex products is constructed. A three parameter grey relational model based on subjective and objective weights is proposed to identify and evaluate the important nodes of complex products. Finally, an example of a large permanent magnet synchronous centrifugal compressor is analyzed. The example shows that the top nodes are node 4, 1, 7, 9 and 24. Compared with other experiments, the proposed method can effectively and reasonably evaluate the node importance of complex products.

Keywords

Complex product node importance evaluation three-parameter interval grey number grey relational model

1 Introduction

In the era of demand diversification of product customization, complex product design process inevitably faces the fundamental problem of design change. Design change is the driving force for enterprise products to seek innovation in order to meet the new needs of customers. However, in the process of complex product design, due to the relationship between components, the change of local parts will lead to the change of other parts. And the corresponding design tasks will also cause chain reaction because of the change [1]. Coupled with the complex relationship between tasks, the impact of change may spread further, or even avalanche spread, which eventually makes the complex product design more difficult. If the control is not proper, it may lead to design task delay, cost increase, and even reduce the quality of product design. At present, most of the industrial and academic fields implement the control of complex product design change based on the important nodes in the complex product development network [2]. Therefore, how to effectively evaluate important nodes and make corresponding control measures is of great practical value for controlling complex product design changes, preventing avalanche propagation, minimizing market response time and development cost.

The identification of important nodes has received considerable attention from researchers and practitioners. The traditional methods of evaluating the importance of nodes are based on the centrality of degree, betweenness and compactness. Most researchers use graph theory and other methods to carry out static analysis and propose algorithms to identify hub nodes [3 –5]. However, these methods generally only consider the local attributes of nodes in CPD network, and do not consider the global attributes and location attributes of nodes in the network. Moreover, the algorithm for identifying hub nodes is complex and the amount of computation is large. And there are few studies on the evaluation of important nodes related to complex product engineering change in the existing literature. However, the research on important nodes of complex products is of great significance for controlling node changes. In order to solve this problem, this paper considers the multi-stage production process and focuses on the establishment of comprehensive indicators, and establishes an index system to comprehensively evaluate the important nodes of complex products. In addition, due to the complexity of parts and disciplines, the heterogeneity of knowledge, and the difficulty of data acquisition, there is great opacity in the evaluation process of important nodes. Therefore, this paper improves the three-parameter interval greyrelational model to study the parity of important nodes.

The remainder of this paper is organized as follows. Section 2 reviews some of the existing research results related to this paper. Section 3 presents the problem description. The construction of evaluation index system of node importance is shown in Section 4. In Section 5,we introduce the improved evaluation method. In Section 6,we present the node importance evaluation results of the photovoltaic direct-drive variable-frequency centrifugal chiller unit. Finally, some conclusions from this study are presented in Section 7.

2 Literature review

There are many methods to identify important nodes in complex networks, which originate from graph theory and graph based data mining. [6] Proposed PageRank algorithm based on random walk model, but its results are sometimes not very accurateIn order to effectively distinguish the importance of more than two nodes in the same k-core, [7] proposed a node importance ranking algorithm based on the number of neighbor cores. The importance of neighbor nodes is identified by neighbor core number, and the sum of k-core values of all neighbor nodes is defined as neighbor core number, but how to determine the range of neighbor nodes is not pointed out. [8] constructed the network model of coastal urban agglomeration, and designed the basic correlation tree algorithm to identify the important nodes and key links of coastal urban agglomeration. [9] introduced the concept of edge weight and influence coefficient, designs an IKs algorithm, and analyzes its recognition effect in Zachary network and real microblog network. [10] proposed an important node identification method based on neighborhood first asynchronous h-operation in dynamic networks. [11] proposed an important node network security evaluation index based on node degree and nearest neighbor information without prior knowledge of network organization. [12] proposed an important node recognition method based on local volume dimension. [5] proposed an important node discovery algorithm in complex networks based on local community aggregation and recognition.

In the existing research of important node identification, the calculation is more complex, and some of them can not reflect the comprehensive evaluation attributes of complex products.Therefore, this paper attempts to build a comprehensive evaluation index system considering network topology attributes and complex product attributes. And the node importance is evaluated and analyzed based on the three parameter interval grey number model which is more suitable for its product characteristics.

Many scholars have studied three-parameter interval grey number decision model. For the multi-attribute decision-making problem, [13] further expanded the greytarget decision model and gave a weighting method suitable for the ternary interval comprehensive target distance decision model. [14] proposed a decision-making method based on prospect theory for the multi-criteria decision-making problem where probability and criterion value are both three-parameter interval grey numbers. [15] proposed a dynamic multi-attribute decision-making method based on prospect theory for the dynamic multi-attribute decision-making problem where the attribute value of scheme is a three-parameter interval grey number. [16] proposed a three-parameter interval grey number group decision method based on prospect theory, taking into account the influence of decision makers’ satisfaction range of indexes and risk attitudes on group decision-making. [17] proposed an attribute reduction method based on theta grey dominance relation for incomplete information system whose attribute evaluation value is three-parameter interval grey number. [18] proposed a multi-attribute decision-making method based on cosine similarity to solve the multi-attribute decision-making problem that the evaluation information is three-parameter interval grey number and the grey information set is not fully utilized. [19] proposed a grey target decision-making method considering the value information of the grey number in the three-parameter interval and the risk attitude of the decision maker.

There are also some researches on the index weight of grey relational model. [20] calculated the weight of attributes by improved entropy method, calculated the relative closeness of different schemes by TOPSIS method and grey relation algorithm, and sorted the alternatives by the relative closeness. [21] and [22] respectively introduced entropy weight method into grey relation analysis to study the risk evaluation of tunnel, the representative volume evaluation of concrete and the comprehensive analysis of the influencing factors of gas outburst. Based on entropy TOPSIS grey relational, [23] and [24] respectively studied the path selection of the evaluation of the opening level of coastal cities in China and the evaluation of the implementation effect of TCM standards. [25] studied the grey relational decision model based on AHP and DEA. Based on the sensitivity and grey relational degree, [26] studied the variable weight evaluation method of distribution network operation mode. For the determination of the weight of three parameter interval grey number index, according to different decision-making schemes, most scholars consider the weight of the index one sidedly, without considering the limitations of the method. In fact, every scheme is different in different situations. So the index weights of different schemes should be different. Therefore, this article combines the advantages of subjective and objective to comprehensively empower it and apply it to the evaluation of complex product node importance.

The existing research has done a lot of research on the identification of important nodes in complex networks, but little attention has been paid to the important nodes in complex product changes. In the engineering change of complex products, the interaction and conflict between important nodes are the fundamental reasons for avalanche propagation of complex product design changes. The evaluation of important nodes plays an important role in the effective control of engineering change.Therefore, this paper evaluates the importance of complex product nodes. Based on multi-stage complex network model, a multi-stage network model of complex product design process production is constructed. Some evaluation indexes based on complex product attributes and network attributes are constructed to evaluate important nodes. Finally, due to the difficulty of obtaining engineering change data of complex products, the data quality is poor and the fuzziness is large. In this paper, three-parameter grey relational model based on BWM and Gini coefficient method is used to evaluate and rank the important nodes in engineering change. The framework is shown in the Fig. 1. (The picture in step 1 is drawn with a large permanent magnet synchronous centrifugal compressor as an example).

Fig. 1

Framework of the proposed complex product node importance evaluation method.

3 Problem description

Engineering change occurs in product design, process, manufacturing and other stages. When these changes occur, it is necessary to respond to the changes at any time to achieve real-time change design response, rather than follow a fixed process of production. All processes will be affected when the changes occur. It is very vital about how to effectively collect, organize and manage scattered product engineering change knowledge to ensure the integrity of product assembly structure. As an important part of the re-generation of product design schemes after product changes. At the same time,it is critical to timely feed back the engineering change information of complex products at different stages to the design department. Therefore,this paper constructs a complex network of multi-process expression from the aspects of design, process and manufacturing process.

In the complex product design stage, the structure and function are determined according to the market demand, which is the key to providing the guidelines for the subsequent production and manufacturing. Therefore, the product design knowledge includes those specific parts and components, as well as a series of knowledge including the structure, function and behavior of the product. There are three sources of parts knowledge in product design knowledge:one of them is the function knowledge from the transformation of customer requirements, the other one is selected knowledge from the basic parts base according to the requirements of the design structure, the last one is the redesign knowledge according to the subsequent changes. This knowledge network based on process must be able to accurately record the comprehensive information of each part for complex product. Knowledge network of complex product manufacturing stage is associated with the assembly material attribute information, supplier and quota information. In addition to material attribute information, self-made parts should also be associated with material quota, man hour quota, work center, tools, accessories, equipment and other information. The multi-process complex network construction process of complex product is shown in Fig. 2.

Fig. 2

Multi-stage complex network of complex products.

The single stage network is represented as: G_k = (V, E_K, W_K) V = (V_i, i = 1, 2, . . . N). If there are connecting edges between parts knowledge, the $e_{k, j}^{k} = 1$ , else, $e_{k, j}^{k} = 0$ . When calculating the multi-stage network, the connected edges and edge weights of its indexes are added and processed. For the same connected edge, the weight value is $w_{i, j} = \sum_{α}^{3} w_{i, j}^{α}$ .

4 Determination of evaluation index system

Through a large number of studies, researchers found that a large number of real networks generally have short average shortest path and high clustering coefficient, showing the characteristics of “small world”. Therefore, in the paper, considering the realistic significance of the model, the clustering coefficient is selected to measure the importance of nodes. Erdos number is selected to represent the shortest path of the node. At the same time, considering the definition of betweenness (the number of times a node is passed by all other shortest paths), if the betweenness of a node is larger, it means that the node is the necessary point for more shortest paths, then the node will be more important in the whole network. Therefore, it is very convincing that the betweenness can reflect the importance of nodes in the network. In addition, complex products have their own basic attributes to judge. This paper selects the node value which can reflect the important information of the node to evaluate. Therefore, this paper selects the modified clustering coefficient, Erdos number, betweenness, and parts’ own attribute index: parts value to comprehensively measure the importance of the constructed network node. Details of these indexes are as follows:

Clustering coefficient: It refers to the parameters used to measure the clustering of network nodes. Intuitively speaking, clustering coefficient describes the possibility that a node’s friends are still friends. The higher the clustering coefficient of a node, the more likely it is that its friend is still a friend, and the more important it is in the whole network circle.Generally, assuming that a node i in an undirected network has k adjacent nodes, then the ratio of the actual number of edges to the possible number of edges among the k nodes is called the clustering coefficient: $C_{i} = \frac{E_{i}}{k (k - 1) / 2} * k$ . Where, k is the degree of node i, which is equivalent to counting the number of friends of node i, and E_i is the actual number of edges between k nodes connected with node i. Suppose there are n nodes in a complex network, then the Erdos number of each node is defined as follows: $D_{i} = \frac{1}{N - 1} \sum_{i, j \neq i} d_{ij}$ . Where, d_ij is the length of the shortest path from node i to node j.

Erdos: Assuming that the current node is Erdos, and the Erdos number of other nodes is the length of the shortest path from the current node to other nodes, then the Erdos number of the current node is the length of the average path from the current node to other nodes.

Node betweenness: the number of paths that all shortest paths in the network pass through this node. The betweennes B_i of verte i is defined as: $B_{i} = \sum_{j, k \in V} \frac{n_{jk} (i)}{n_{jk}}$ . Where, n_jk (i) is the number of shortest paths between nodes j and k.

Parts value: The engineering change of different parts needs different change cost. And different nodes have different values. Therefore, the node’s own attribute: the value of node is used as the evaluation index of node importance. It can be expressed: C_{v
_i} = c_{(v_m)} + c_{(v_l)} + c_{(v_e)} + c_{(v_o)}. Where, c_i is the total value of node i. C_{(v_m)} is the material cost if node i: It refers to the cost of product standard consumption, supporting raw materials, product accessories and various materials used for production or providing services. It mainly includes the purchase price, related taxes, freight, loading and unloading fees, insurance premiums and other costs that can be directly attributable to the acquisition of materials. C_{(v_i)} is the labor cost: It refers to the remuneration and other expenses paid to employees which in order to obtain the services provided by employees. It mainly includes the salary, bonus, allowance, welfare, education fund and so on. C_{(v_m)} is the manufacturing cost: It refers to energy consumption, manufacturing accessories, labor insurance, office and fixed expenses. C_{(v_o)} is the other cost: Some consumption including fuel cost, power cost, office cost and depreciation consumed by each production unit. The Node importance evaluation indexes are as shown in Fig. 3.

Fig. 3

Evaluation indexes of node importance.

5 Research methodology

5.1 Three-parameter interval grey number

From the definition of three-parameter interval grey number, it can be known that it refers to the interval grey number where the center of gravity point with the greatest possible value is known. It can be marked as $A (\otimes) = [\underline{a}, \tilde{a}, \bar{a}]$ , where, $\underline{a} ⩽ \tilde{a} ⩽ \bar{a}$ , $\underline{a}$ , $\bar{a}$ are the the upper and lower limits of the interval respectively. $\tilde{a}$ is called the “center of gravity” point ([27]).

When two of the three parameters $\underline{a}$ , $\tilde{a}$ , $\bar{a}$ are the same, the three-parameter interval grey number degenerates to the interval grey number. When $\underline{a} = \tilde{a} = \bar{a}$ , the three parameter interval grey number degenerates to the real number. In fact, the interval grey number and the real number.are special cases of the three-parameter interval grey number.

Its algorithm is similar to interval grey number. Let three-parameter interval grey number $A (\otimes) = [\underline{a}, \tilde{a}, \bar{a}]$ , $B (\otimes) = [\underline{b}, \tilde{b}, \bar{b}]$ , then $A (\otimes) + B (\otimes) = [\underline{a} + \underline{b}, \tilde{a} + \tilde{b}, \bar{a} + \bar{b}]$ $A (\otimes) / B (\otimes) \in [min {\underline{a} / \underline{b}, \underline{a} / \bar{b}, \bar{a} / \underline{b}, \bar{a} / \bar{b}},$ $\tilde{a} / \tilde{b}, max {\underline{a} / \underline{b}, \underline{a} / \bar{b}, \bar{a} / \underline{b}, \bar{a} / \bar{b}}]$ $λ A (\otimes) = [\underline{λ a}, λ \tilde{a}, λ \bar{a}], λ > 0$

5.2 Three-parameter interval grey number grey relational model

Suppose that there are n alternative engineering change schemes. They constituted by evaluation schemes set A ={ a₁, a₂, ⋯ , a_n }. The index set S ={ s₁, s₂, ⋯ , s_m } is composed of m attributes. The index value of scheme a_i under the evaluation index s_j can be expressed as $u_{ij} (\otimes) = [{\underline{u}}_{ij}, {\tilde{u}}_{ij}, {\bar{u}}_{ij}] ({\underline{u}}_{ij} ⩽ {\tilde{u}}_{ij} ⩽ {\bar{u}}_{ij}, i = 1, 2, \dots, n; j = 1, 2, \dots, m)$ . The effect evaluation vector of each scheme is u_i (⊗) = (u_i1 (⊗) , u_i2 (⊗) , ⋯ , u_im (⊗)) , i = 1, 2, ⋯ , n. The weight of index under each scheme is w_i1, w_i2, ⋯ , w_im, and $\sum_{j = 1}^{m} w_{ij} = 1 (i = 1, 2, . . ., n)$ . There are different attribute indexes with different dimensions and measurement standards. In order to increase the comparability of alternatives, it is necessary to normalize the effect evaluation vector of decision alternatives. In this paper, we use the range transformation method to normalize the decision matrix.

For profitable attribute values: ${\underline{x}}_{ij} = \frac{{\underline{u}}_{ij} - {\underline{u}}_{j}^{\nabla}}{{\bar{u}}_{j}^{*} - {\underline{u}}_{j}^{\nabla}}, {\tilde{x}}_{ij} = \frac{{\tilde{u}}_{ij} - {\underline{u}}_{j}^{\nabla}}{{\bar{u}}_{j}^{*} - {\underline{u}}_{j}^{\nabla}}, {\bar{x}}_{ij} = \frac{{\bar{u}}_{ij} - {\underline{u}}_{j}^{\nabla}}{{\bar{u}}_{j}^{*} - {\underline{u}}_{j}^{\nabla}}$ (1)

For cost attribute values: ${\underline{x}}_{ij} = \frac{{\bar{u}}_{j}^{*} - {\bar{u}}_{ij}}{{\bar{u}}_{j}^{*} - {\underline{u}}_{j}^{\nabla}}, {\tilde{x}}_{ij} = \frac{{\bar{u}}_{j}^{*} - {\tilde{u}}_{ij}}{{\bar{u}}_{j}^{*} - {\underline{u}}_{j}^{\nabla}}, {\bar{x}}_{ij} = \frac{{\bar{u}}_{j}^{*} - {\bar{u}}_{ij}}{{\bar{u}}_{j}^{*} - {\underline{u}}_{j}^{\nabla}}$ (2)

Where, ${\bar{u}}_{j}^{*} = max_{1 ⩽ i ⩽ n} {{\bar{u}}_{ij}}, {\underline{u}}_{j}^{\nabla} = min_{1 ⩽ i ⩽ n} {{\underline{u}}_{ij}}, j = 1, 2, \dots, m$ .

Let the normalized effect evaluation vector be:

$x_{i} (\otimes) = (x_{i 1} (\otimes), x_{i 2} (\otimes), \dots, x_{im} (\otimes)), i = 1, 2, \dots, n$ (3)

Where, $x_{ij} (\otimes) \in [{\underline{x}}_{ij}, {\tilde{x}}_{ij}, {\bar{x}}_{ij}]$ is a three-parameter interval grey number in [0, 1]

Recorded that ${\underline{x}}_{j}^{+} = max_{1 ⩽ i ⩽ n} {{\underline{x}}_{ij}}$ , ${\tilde{x}}_{j}^{+} = max_{1 ⩽ i ⩽ n} {{\tilde{x}}_{ij}}$ , ${\bar{x}}_{j}^{+} = max_{1 ⩽ i ⩽ n} {{\bar{x}}_{ij}}$ , ${\underline{x}}_{j}^{-} = min_{1 ⩽ i ⩽ n} {{\underline{x}}_{ij}}$ , ${\tilde{x}}_{j}^{-} = min_{1 ⩽ i ⩽ n} {{\tilde{x}}_{ij}}$ , ${\bar{x}}_{j}^{-} = min_{1 ⩽ i ⩽ n} {{\bar{x}}_{ij}} (j = 1, 2, \dots, m)$ . Then the m-dimensional three-parameter non negative interval grey number vectors

$\begin{matrix} x^{+} (\otimes) = {x_{1}^{+} (\otimes), x_{2}^{+} (\otimes), \dots, x_{m}^{+} (\otimes)}, x^{-} (\otimes) \\ = {x_{1}^{-} (\otimes), x_{2}^{-} (\otimes), \dots, x_{m}^{-} (\otimes)} \end{matrix}$ (4) are called ideal optimal scheme effect evaluation vectors and critical scheme effect evaluation vectors respectively. Sun et al. (2020) [29].

We assume that the grey interval relational degree of the normalized effect evaluation vector x_i (⊗) of scheme A_i with respect to the ideal optimal scheme effect evaluation vector x ⁺ (⊗) is G (x ⁺ (⊗) , x_i (⊗)). And the grey interval relational degree of critical scheme effect evaluation vector x ^- (⊗) is G (x ^- (⊗) , x_i (⊗)). Assume that the weights of two grey relational degrees are α₁, α₂ (α₁ + α₂ = 1). Then,

$\begin{matrix} G (x_{i} (\otimes)) = α_{1} G (x^{+} (\otimes), x_{i} (\otimes)) + α_{2} \\ [1 - G (x^{-} (\otimes), x_{i} (\otimes))], i = 1, 2, \dots, n \end{matrix}$ (5) is the three-parameter grey interval linear relational degree of the effect evaluation vector x_i (⊗).

$\begin{matrix} G (x_{i} (\otimes)) = {[G (x^{+} (\otimes), x_{i} (\otimes))]}^{α_{1}} \\ + {[1 - G (x^{-} (\otimes), x_{i} (\otimes))]}^{α_{2}}, i = 1, 2, \dots, n \end{matrix}$ (6) is the three-parameter grey interval product relational degree of the effect evaluation vector x_i (⊗).

The distribution probability of barycenter point with the highest probability of taking the value of three-parameter interval grey number $x_{ij} (\otimes) \in [{\underline{x}}_{ij}, {\tilde{x}}_{ij}, {\bar{x}}_{ij}]$ is $f ({\tilde{x}}_{ij}) ⩾ σ$ . Normally, σ⩾ 60 %. If σ⩽ 60 % it indicates that the decision is wrong, and the most likely value needs to be determined again. Based on the center of gravity, we can build a three-parameter interval grey number relational degree evaluation model. Yan et al. ([28]).

Definition 3. for three-parameter interval grey number $x_{ij} (\otimes) \in [{\underline{x}}_{ij}, {\tilde{x}}_{ij}, {\bar{x}}_{ij}]$ , then

$\begin{matrix} γ_{ij}^{+} = \frac{3}{5} \times \frac{{\tilde{m}}^{+} + η {\tilde{M}}^{+}}{{\tilde{Δ}}_{ij}^{+} + η {\tilde{M}}^{+}} + \frac{2}{5} \\ [(1 - β) \frac{{\underline{m}}^{+} + η {\underline{M}}^{+}}{{\underline{Δ}}_{ij}^{+} + η {\underline{M}}^{+}} β \frac{{\bar{m}}^{+} + η {\bar{M}}^{+}}{{\bar{Δ}}_{ij}^{+} + η {\bar{M}}^{+}}] \end{matrix}$ (7) is called the three-parameter grey interval relational coefficient of sub factor. x_ij with respect to ideal factor $x_{j}^{+}$ . η ∈ (0, 1) is the resolution coefficient. β ∈ (0, 1) is the decision preference coefficient. Where,

$\begin{matrix} {\underline{Δ}}_{ij}^{+} = | {\underline{x}}_{j}^{+} - {\underline{x}}_{ij} |, {\tilde{Δ}}_{ij}^{+} = | {\tilde{x}}_{j}^{+} - {\tilde{x}}_{ij} |, {\bar{Δ}}_{ij}^{+} = | {\bar{x}}_{j}^{+} - {\bar{x}}_{ij} |, \\ i = 1, 2, \dots, n; j = 1, 2, \dots, m \\ {\underline{m}}^{+} = min_{1 ⩽ i ⩽ n} min_{1 ⩽ j ⩽ m} {\underline{Δ}}_{ij}^{+}, {\tilde{m}}^{+} = min_{1 ⩽ i ⩽ n} min_{1 ⩽ j ⩽ m} {\tilde{Δ}}_{ij}^{+}, {\bar{m}}^{+} \\ = min_{1 ⩽ i ⩽ n} min_{1 ⩽ j ⩽ m} {\bar{Δ}}_{ij}^{+} \\ {\underline{M}}^{+} = max_{1 ⩽ i ⩽ n} max_{1 ⩽ j ⩽ m} {\underline{Δ}}_{ij}^{+}, {\tilde{M}}^{+} = max_{1 ⩽ i ⩽ n} max_{1 ⩽ j ⩽ m} {\tilde{Δ}}_{ij}^{+}, {\bar{M}}^{+} \\ = max_{1 ⩽ i ⩽ n} max_{1 ⩽ j ⩽ m} {\bar{Δ}}_{ij}^{+} \\ G (x^{+} (\otimes), x_{i} (\otimes)) = \sum_{j = 1}^{m} w_{ij} γ_{ij}^{+}, i = 1, 2, \dots, n \end{matrix}$ (8)

is called the three-parameter grey interval relational degree of the effect evaluation vector x_i (⊗) about the ideal optimal scheme effect evaluation vector x ⁺ (⊗).

Definition 4. For three-parameter interval grey number $x_{ij} (\otimes) \in [{\underline{x}}_{ij}, {\tilde{x}}_{ij}, {\bar{x}}_{ij}]$ ,

$\begin{matrix} γ_{ij}^{-} = \frac{3}{5} \times \frac{{\tilde{m}}^{-} + ɛ {\tilde{M}}^{-}}{{\tilde{Δ}}_{ij}^{-} + ɛ {\tilde{M}}^{-}} + \frac{2}{5} \\ \times [(1 - δ) \frac{{\underline{m}}^{-} + ɛ {\underline{M}}^{-}}{{\underline{Δ}}_{ij}^{-} + ɛ {\underline{M}}^{-}} + δ \frac{{\bar{m}}^{-} + ɛ {\bar{M}}^{-}}{{\bar{Δ}}_{ij}^{-} + ɛ {\bar{M}}^{-}}] \end{matrix}$ (9) is called the three-parameter grey interval relational coefficient of sub factor. x_ij with respect to ideal facto $x_{j}^{-}$ . η ∈ (0, 1) is the resolution coefficient. δ ∈ (0, 1) is the decision preference coefficient. Where,

$\begin{matrix} {\underline{Δ}}_{ij}^{-} = | {\underline{x}}_{ij} - {\underline{x}}_{j}^{-} |, {\tilde{Δ}}_{ij}^{-} = | {\tilde{x}}_{ij} - {\tilde{x}}_{j}^{-} |, {\bar{Δ}}_{ij}^{-} = | {\bar{x}}_{ij} - {\bar{x}}_{j}^{-} |, \\ i = 1, 2, \dots, n; j = 1, 2, \dots, m \\ {\underline{m}}^{-} = min_{1 ⩽ i ⩽ n} min_{1 ⩽ j ⩽ m} {\underline{Δ}}_{ij}^{-}, {\tilde{m}}^{-} = min_{1 ⩽ i ⩽ n} min_{1 ⩽ j ⩽ m} {\tilde{Δ}}_{ij}^{-}, \\ {\bar{m}}^{-} = min_{1 ⩽ i ⩽ n} min_{1 ⩽ j ⩽ m} {\bar{Δ}}_{ij}^{-} \\ {\underline{M}}^{-} = max_{1 ⩽ i ⩽ n} max_{1 ⩽ j ⩽ m} {\underline{Δ}}_{ij}^{-}, {\tilde{M}}^{-} = max_{1 ⩽ i ⩽ n} max_{1 ⩽ j ⩽ m} {\tilde{Δ}}_{ij}^{-}, \\ {\bar{M}}^{-} = max_{1 ⩽ i ⩽ n} max_{1 ⩽ j ⩽ m} {\bar{Δ}}_{ij}^{-} \\ G (x^{-} (\otimes), x_{i} (\otimes)) = \sum_{j = 1}^{m} w_{ij} γ_{ij}^{-}, i = 1, 2, \dots, n \end{matrix}$ (10)

is called the three-parameter grey interval relational degree of the effect evaluation vector x_i (⊗) about the critical scheme effect evaluation vector x ^- (⊗).

5.3 Determination of weight

At present, scholars attach great importance to the development and application of subjective and objective empowerment methods in the research of evaluation. The subjective weight reflects the subjective willingness of the evaluation subject, and highlights the degree of distinction between the evaluation objects through index data information. The combination of them will make the result more objective. In this paper, the simplified BWM subjective weighting method and the Gini coefficient weighting method which can better reflect the data difference information are selected for combination weighting.

5.3.1 Determination of weight based on BWM

BWM (best-worst method) is a new method to determine the subjective weight of index proposed by Rezaei in 2014. The most frequently used method in the multiple indexex evaluation is AHP method. In AHP method, any two indexes are usually compared with each other to get the evaluation matrix of indexes, which needs n (n - 1)/2 times of comparison. The calculation process of it is complicated and will cause certain errors. However, BWM only needs 2n - 3 calculations by selecting the best and the worst indexes and comparing them with other indexes. It simplifies the complicated process of AHP, greatly reduces the amount of data, reduces the mistakes caused by too much data, makes it easier to pass the consistency test, and improves the reliability. The calculation steps are as follows: ([29])

The best index X_B and the worst index X_W are selected according to experts’ opinions in index set X ={ x₁, x₂, . . . x_n }.

Experts use 1–9 point scale to score and determine the importance of other indexes relative to the optimal indexes. We construct the comparison vector C_B = (C_{B
₁}C_{B
₂} . . . , C_{B
_j}). C_{B
_j} represents the importance of the optimal index compared with index j. 1 means C_B and C_{B
_j} are equally important. 9 means C_B is extremely important than C_{B
_j}.

We need to determine the unimportance of other indexes relative to the worst indexes and construct a comparison vector C_w = (C_1wC_{B
_2w} . . . , C_jw ^T). Where, C_jw represents the least importance of the worst index compared with index j. 1 means C_jw and C_w are equally unimportant. 9 means C_jw and C_w are extremely unimportant.

From the goal programming model, a mathematical programming formula is established and solved to obtain the optimal index weight $ω_{j}^{*} = (ω_{1}^{*}, ω_{2}^{*}, . . ., ω_{n}^{*})$ . $\begin{matrix} min max_{j} {| \frac{ω_{j}}{ω_{W}} - {aB}_{j} |} \\ s . t . {\begin{matrix} \sum_{j = 1}^{n} ω_{j} \\ ω_{j} ⩾ 0, j = 1, 2, . . . n \end{matrix} = 1 \end{matrix}$ (11)

Where, ω_B is the weight of C_B, C_j is the criterion vector. ω_j is the weight of C_j. ω_B is the weight of C_W. a_{B
_j} represents the importance of C_B to C_j; a_jW represents the importance of C_j to C_W.

It can be transformed to $\begin{matrix} min k \\ s . t . {\begin{matrix} | \frac{ω B}{ω_{j}} - aBj | ⩽ k \\ | \frac{ω_{j}}{ω W} - ajW | ⩽ k \\ \sum_{j = 1}^{n} ω_{j} = 1 \\ ω_{j} ⩾ 0 \end{matrix}, j = 1, 2, . . ., n \end{matrix}$ (12)

Calculate the consistency ratio. The obtained K can be represented by K ^*, and the consistency ratio Cr (C1 is the given value) can be obtained from $C_{R} = \frac{k^{*}}{C_{1}}$ .

The closer of the value is to 0, the better the consistency. When it is 0, it is consistent. If there are P experts participate in the judgment, the final weight will be calculated by weighted average, and the final weight is ${\bar{ω}}_{j}^{*} = \frac{\sum_{a = 1}^{p} ω_{j}^{a}}{p}$ .

5.3.2 Weight determination method based on Gini coefficient

(1) Principle of Gini coefficient weighting method

Gini coefficient weighting method is an objective weighting method by calculating Gini coefficient of evaluation index and normalizing Gini coefficient of each index. First of all, the different data of n evaluation objects of a specific evaluation index can be regarded as the income of different levels people. Then the Gini coefficient of a certain index can be calculated. The value of Gini coefficient can reflect the data difference between different evaluation objects. Then, In order to ensure that weight of all indexes are in the range of 0 to 1 and the sum is 1, the Gini coefficient value of each index will be normalized to get the Gini coefficient weight of the evaluation index. [30].

(2) Gini coefficient weight’ calculation of evaluation index.

We assume that G_k is the Gini coefficient of the kth index, Y_ki is the ith data of the kth index, and μ K. μK is the expected value of all data of the kth index. Then the Gini coefficient G_k of the kth index is shown as follows: $G_{k} = \sum_{i = 1}^{n} \sum_{j = 1}^{n} | Y_{ki} - Y_{kj} | / 2 n^{2} μ_{k}$ (13) $G_{k} = \sum_{i = 1}^{n} \sum_{j = 1}^{n} | Y_{ki} - Y_{kj} | / n^{2} - n$ (14)

Especially, when the mean value of index data is not 0, the Gini coefficient is calculated by the improved formula (13). When the mean value of the index data is 0, the Gini coefficient of the index is calculated by the original formula (14). Gini coefficient of the index truly reflects the data changes of different evaluation objects of the index.

Gini coefficient weight g_k of the kth index can be obtained by normalizing the Gini coefficient value of each index: $g_{k} = G_{k} / (\sum_{i = 1}^{m} G_{i})$ (15)

Where, g_k is Gini coefficient weight of the kth index, G_k is Gini coefficient value of the kth index, and m is the number of indexes.

The advantages of Gini coefficient weighting method are as follows: first, the weight calculation is not affected by the unit dimension of the index, the definition of Gini coefficient itself eliminates the dimensional influence. Second, Gini coefficient value of the evaluation index reflects the difference between any two evaluation objects. Gini coefficient weight reflects the difference between the data of different evaluation objects of an index. And the weight reflects the data information of the index, which meets the requirements of the objective weighting method.

5.3.3 Combination weighting method based on BWM-Gini coefficient

The BWM method determines the index weight according to the subjective preference of the evaluator, and the method of Gini coefficient determines the objective index weight. In order to fully reflect the advantages of the two methods, from the subjective and objective point of view, this paper combines BWM method and Gini coefficient method to determine the comprehensive weight of the evaluation index by linear weighting:

The weight vector of linear combination is calculated and proved as follows:

Suppose N kinds of subjective and objective methods to get l kinds of weights. The weight vectors are W₁, W₂, . . . , W_L. The k-th weight direction quantity is $W^{k} = (ϖ_{1}^{k}, ϖ_{2}^{k}, . . ., ϖ_{m}^{k})$ ,It satisfies that $\sum_{j = 1}^{m} ϖ_{j}^{k} = 1, ϖ_{j}^{k} ⩾ 0 (j = 1, 2, . . . m; k = 1, 2, . . . L)$ . The vector obtained by linear combination of multiple weight vectors is called linear combination weight vector. Let $W^{*} = (ϖ_{1}^{*}, ϖ_{2}^{*}, . . ., ϖ_{m}^{*})^{T}$ be the linear combination weight vector of L weight vectors. It can be expressed in vector form as $W^{*} = \sum_{k = 1}^{L} x_{k} w^{k}$ . Where x_k is the linear combination coefficient. It satisfies $\sum_{k = 1}^{L} x_{k} = 1, x_{k} ⩾ 0$ . The solution of linear combined weight vector is a multi-objective optimization problem. On the one hand, the sum of weighted generalized distances between all schemes and ideal schemes should be minimized. On the other hand, the uncertainty of combination coefficient should be eliminated as far as possible. According to the Jaynes maximum entropy principle, the comprehensive weight coefficient of the index should be determined so that the Shannon entropy takes the maximum value. Therefore, the following optimization decision model is established:

$min [μ \sum_{i = 1}^{n} \sum_{j = 1}^{m} \sum_{k = 1}^{L} x_{k} ϖ_{j}^{k} (1 - r_{ij}) + (1 - μ) \sum x_{k} ln x_{k}]$ (17)

$s . t . \sum_{k = 1}^{L} x_{k} = 1, x_{k} ⩾ 0$ (18)

Where, 0 ⩽ μ ⩽ 1. It can be used to express the balance coefficient between two targets, which can be given in advance according to practical problems.

According to reference [3], there is a unique solution for the single objective optimization problem (SP), The solution is as follows: $x = [\frac{s_{1}}{\sum_{k = 1}^{l} s_{k}}, \frac{s_{2}}{\sum_{k = 1}^{l} s_{k}}, . . ., \frac{s_{L}}{\sum_{k = 1}^{L} s_{k}}]$ (19)

Where, $s_{k} = exp {- [1 + μ \sum_{i = 1}^{n} \sum_{j = 1}^{m} ϖ_{j}^{k} (1 - r_{ij}) / (1 - μ) \sum_{i = 1}^{n}} (k = 1, 2, . . . L)$ , N is the number of evaluation schemes; M is the number of evaluation indexes.

5.3.4 Steps of complex product node importance evaluation based on three-parameter grey relational degree

To sum up, the three-parameter interval grey relational evaluation algorithm of complex product node importance is as follows:

Constructed the model of complex product presentation based on multi-stage complex network model.

Determine the evaluation index.

Standardize the original three-parameter interval grey number effect evaluation index, and obtain the standardized effect evaluation vector formula of each scheme.

Solve the ideal optimal solution for the decision-making problem and the effect evaluation vector x ⁺ (⊗) and the critical solution x ^- (⊗).

Obtain the three-parameter grey interval relational coefficient vector of each scheme, the ideal scheme and the critical scheme.

Solve the BWM and Gini coefficient models, combine the weights, and obtain the weight of each scheme under different attributes.

Calculate the three-parameter interval grey number relational degree G (x ⁺ (⊗) , x_i (⊗)) and G (x ^- (⊗) , x_i (⊗)) (i = 1, 2, ⋯ , n) between each scheme and ideal scheme and critical scheme, and calculate the three-parameter grey interval linear relational degree or three-parameter grey interval product relational degree G (x_i (⊗)) (i = 1, 2, ⋯ , n) of each scheme.

The schemes are sorted according to the relevance degree G (x_i (⊗)) (i = 1, 2, ⋯ , n). The scheme corresponding to the maximum correlation degree is the optimal one.

The flow of this algorithm is shown in the Fig. 4.

Fig. 4

Three-parameter interval grey number grey relational evaluation model algorithm.

6 Case study

The high-speed permanent magnet synchronous frequency conversion centrifugal high-power chiller of G enterprise is a typical complex product. (It is shown in the Fig. 5a.) The unit has reached the “international leading”level, the annual comprehensive efficiency of the unit has been increased by more than 65%, and the energy saving has been more than 40%. Large units involve many parts and complex products. When engineering change occurs, there are many related impacts occur. This paper analyzes the node importance evaluation of high speed permanent magnet synchronous variable frequency centrifugal compressor of its large unit. The composition and structure of main parts are shown in the Fig. 5b. The main parts of permanent magnet synchronous centrifugal compressor are shown in Table 1.

Fig. 5

Large permanent magnet synchronous centrifugal unit and permanent magnet synchronous centrifugal compressor.

Table 1

Main parts and node name of permanent magnet synchronous centrifugal compressor

Node	Parts	Node	Parts
V₁	Mainshaft	V₁₃	Bend casing
V₂	Impeller rim1	V₁₄	Curved separator
V₃	Roulette1	V₁₅	Refluxer separator
V₄	Blade1	V₁₆	Refluxer flow channels
V₅	Shrink-ring	V₁₇	Volute
V₆	Fixed collar	V₁₈	Impeller rim2
V₇	Balance disc	V₁₉	Roulette2
V₈	Reinforcement on the back of impeller	V₂₀	Blade2
V₉	Thrust disc	V₂₁	Stator winding
V₁₀	Axle sleeve	V₂₂	Stator core
V₁₁	Suction chamber	V₂₃	Foundation
V₁₂	Diffuser	V₂₄	p-m rotor

First of all, we analyze the relationship between process, design and manufacturing network of the direct drive variable frequency centrifugal compressor. The multi-stage knowledge association is shown from Table 2 to Table 4.

Table 2

Design knowledge network connection information

Design knowledge network
Source	Target	Weight	Source	Target	Weight	Source	Target	Weight
1	2	0.43	4	7	0.24	13	14	0.45
1	3	0.55	4	8	0.26	13	15	0.52
1	4	0.64	4	9	0.16	13	16	0.5
1	5	0.64	4	10	0.59	13	17	0.56
1	6	0.55	4	13	0.35	14	15	0.51
2	3	0.53	4	14	0.36	14	16	0.55
2	4	0.38	4	15	0.24	14	17	0.43
2	5	0.49	5	6	0.59	15	16	0.64
2	6	0.33	5	7	0.49	15	17	0.43
2	7	0.37	5	8	0.44	16	17	0.52
2	10	0.55	5	9	0.43	18	19	0.68
2	11	0.31	6	7	0.35	18	20	0.53
2	12	0.22	6	8	0.61	18	21	0.64
3	4	0.51	6	9	0.49	18	22	0.51
3	5	0.34	7	8	0.56	18	23	0.51
3	6	0.43	7	9	0.45	18	24	0.68
3	7	0.45	8	10	0.51	19	20	0.54
3	8	0.59	9	10	0.42	19	21	0.38
3	9	0.53	11	12	0.39	19	22	0.58
3	10	0.52	11	13	0.35	19	23	0.59
3	11	0.54	11	14	0.52	19	24	0.56
3	12	0.35	11	15	0.59	20	21	0.42
3	13	0.34	11	16	0.5	20	23	0.41
3	14	0.35	11	17	0.62	20	24	0.63
3	15	0.33	12	13	0.6	21	22	0.62
3	16	0.42	12	14	0.59	21	24	0.58
4	5	0.16	12	15	0.53	22	23	0.43
4	6	0.33	12	16	0.44	22	24	0.58
4	6	0.33	12	17	0.36	23	24	0.58

Table 3

Process knowledge network connection information

Process knowledge network
Source	Target	Weight	Source	Target	Weight	Source	Target	Weight
1	2	0.51	4	9	0.42	13	15	0.61
1	3	0.63	4	10	0.46	13	16	0.55
1	4	0.61	4	13	0.39	13	17	0.5
1	5	0.58	4	14	0.36	14	16	0.63
1	20	0.31	4	15	0.38	14	17	0.49
1	23	0.54	4	24	0.51	15	16	0.61
1	24	0.32	5	6	0.43	15	17	0.55
2	3	0.67	5	7	0.45	16	17	0.54
2	4	0.55	5	8	0.39	18	19	0.65
2	5	0.23	6	7	0.37	18	20	0.59
2	6	0.46	6	8	0.39	18	21	0.61
2	7	0.33	6	9	0.41	18	22	0.53
2	10	0.52	7	8	0.51	18	23	0.61
2	24	0.33	7	9	0.56	18	24	0.45
3	4	0.69	8	10	0.61	19	20	0.62
3	5	0.58	9	10	0.58	19	21	0.63
3	6	0.55	11	12	0.57	19	22	0.59
3	7	0.51	11	13	0.58	19	23	0.55
3	8	0.61	11	14	0.63	19	24	0.46
3	9	0.63	11	15	0.56	20	21	0.61
3	10	0.55	11	16	0.46	20	22	0.58
3	11	0.41	11	17	0.57	20	23	0.63
3	15	0.32	12	13	0.59	20	24	0.59
3	16	0.31	12	14	0.53	21	22	0.66
4	5	0.29	12	15	0.58	21	23	0.63
4	6	0.33	12	16	0.49	21	24	0.58
4	7	0.38	12	17	0.44	22	23	0.71
4	8	0.37	13	14	0.51	22	24	0.68
						23	24	0.65

Table 4

Manufacturing knowledge network connection information

Manufacturing knowledge network
Source	Target	Weight	Source	Target	Weight	Source	Target	Weight
1	2	0.52	4	9	0.34	13	16	0.56
1	3	0.56	4	10	0.55	13	17	0.51
1	4	0.61	4	13	0.31	14	15	0.48
1	5	0.62	4	14	0.38	14	16	0.55
1	6	0.55	4	24	0.34	14	17	0.59
1	21	0.32	5	6	0.33	15	16	0.65
1	22	0.54	5	7	0.42	15	17	0.68
2	3	0.61	5	8	0.45	16	17	0.58
2	4	0.49	5	9	0.42	18	19	0.61
2	5	0.32	6	7	0.33	18	20	0.43
2	6	0.44	6	8	0.45	18	21	0.62
2	7	0.38	6	9	0.49	18	22	0.49
2	10	0.51	7	8	0.51	18	23	0.58
2	23	0.32	7	9	0.49	18	24	0.63
3	4	0.61	8	10	0.46	19	20	0.58
3	5	0.55	9	10	0.39	19	21	0.49
3	9	0.51	11	12	0.36	19	22	0.63
3	10	0.62	11	14	0.39	19	23	0.58
3	11	0.58	11	15	0.48	19	24	0.63
3	12	0.34	11	16	0.51	20	21	0.51
3	13	0.36	11	17	0.69	20	22	0.61
3	14	0.33	12	13	0.59	20	23	0.46
3	15	0.35	12	14	0.58	20	24	0.65
3	16	0.38	12	15	0.54	21	22	0.65
4	5	0.22	12	16	0.69	21	23	0.59
4	6	0.45	12	17	0.48	21	24	0.56
4	7	0.36	13	14	0.59	22	23	0.46
4	8	0.32	13	15	0.58	22	24	0.61
						23	24	0.59

The multi-stage complex network diagram can be referred to Fig. 3.

Through the calculation of index system, we can get the three parameter interval grey number of the evaluation index as follows:

$\begin{matrix} X (\otimes)_{O} \\ = [\begin{matrix} [4.1781, 4.5991, 5.4411] & [156, 157.20, 158] & [1.3137, 1.6297, 1.8404] & [2.5416, 2.5509, 2.5571] \\ [3.5181, 3.4272, 3.2455] & [48, 49.8, 51] & [3.2943, 3.4309, 3.5220] & [0.6285, 0.6882, 0.7281] \\ [2.3663, 2.5245, 2, 8409] & [95, 97.4, 99] & [3.2742, 3.4899, 3.6337] & [0.8974, 0.9317, 0.9545] \\ [4.4447, 4.5927, 4.8886] & [136, 140.20, 143] & [3.3289, 3.7150, 3.9724] & [1.9269, 1.9524, 1.9694] \\ [2.2597, 2.4163, 2.7293] & [32, 35.60, 38] & [3.6374, 3.7329, 3.7966] & [0.3591, 0.3739, 0.3838] \\ [2.2752, 2.3264, 2.4290] & [34, 38.20, 41] & [3.1403, 3.3597, 3.5060] & [0.3235, 0.3434, 0.3566] \\ [3.2383, 3.3600, 3.6033] & [31, 33.4, 35] & [2.5416, 2.7445, 2.8797] & [0.9380, 0.9692, 0.9901] \\ [3.4278, 3.5670, 3.8453] & [23, 25.40, 27] & [2.1679, 2.5098, 2.7377] & [0.6393, 0.6690, 0.6888] \\ [2.7432, 2.5140, 2.0557] & [28, 32.2, 35] & [3.2829, 3.4382, 3.5417] & [0.4668, 0.4713, 0.4743] \\ [1.6317, 1.7398, 1.9561] & [20, 24.2, 27] & [4.1485, 4.3141, 4.4245] & [0.5028, 0.5522, 0.5850] \\ [2.0845, 2.3590, 2.9080] & [35, 37.4, 39] & [3.4995, 3.5945, 3.6578] & [0.9661, 0.9703, 0.9731] \\ [2.4083, 2.5894, 2.9515 & [40, 44.8, 48] & [3.2912, 3.4439, 3.5457] & [0.8069, 0.8170, 0.8238] \\ [2.0329, 2.1851, 2.4895] & [33, 36, 38] & [3.4864, 3.5814, 3.6448] & [0.9503, 0.9728, 0.9878] \\ [2.3614, 2.4547, 2.6412] & [25, 26.8, 28] & [3.3701, 3.5552, 3.6786] & [0.7959, 0.8087, 0.8173] \\ [1.2285, 1.3942, 1.7257] & [36, 39.6, 42] & [4.1601, 4.2609, 4.3281] & [0.7216, 0.7562, 0.7793] \\ [1.3403, 1.4243, 1.5624] & [30, 33, 35] & [4.0210, 4.1503, 4.2365] & [0.5566, 0.5653, 0.5712] \\ [1.4725, 1.5819, 1.8007] & [24, 26.4, 28] & [4.2381, 4.3856, 4.4839] & [0.6795, 0.6864, 0.6909] \\ [4.0817, 4.3127, 4.7747] & [87, 89.4, 91] & [1.5818, 1.7080, 1.7921] & [1.2446, 1.3339, 1.3934] \\ [4.1011, 4.3729, 4.9166] & [83, 90.20, 95] & [1.2761, 1.6654, 1.9249] & [1.2464, 1.3010, 1.3373] \\ [2.0611, 2.2563, 2.6466] & [29, 33.8, 37] & [3.3842, 3.6593, 3.8426] & [0.5592, 0.5700, 0.5773] \\ [2.2505, 2.3673, 2.6010] & [36, 40.2, 43] & [3.3725, 3.5150, 3.6100] & [0.4429, 0.4619, 0.4745] \\ [3.2548, 3.3191, 3.4478] & [38, 42.2, 45] & [2.2182, 2.3415, 2.4238] & [0.9868, 0.9926, 0.9966] \\ [2.7880, 2.7961, 2.8124] & [37, 39.4, 41] & [3.0972, 3.2275, 3.3144] & [0.3241, 0.3376, 0.3465] \\ [4.5861, 4.6429, 4.7565] & [56, 59.6, 62] & [1.5417, 1.8103, 1.9893] & [0.9790, 0.9862, 0.9911] \end{matrix}] \end{matrix}$

The normalized three-parameter interval grey number evaluation matrix is:

$X (\otimes) = [\begin{matrix} [0.7002, 0.8001, 1.0000] & [0.9855, 0.9942, 1.0000] & [0.9883, 0.8898, 0.8241] & [0.9931, 0.9972, 1.0000] \\ [0.5435, 0.5219, 0.4788] & [0.2029, 0.2159, 0.2246] & [0.3708, 0.3283, 0.2999] & [0.1365, 0.1633, 0.1811] \\ [0.2701, 0.3077, 0.3828] & [0.5435, 0.5609, 0.5725] & [0.3771, 0.3099, 0.2650] & [0.2569, 0.2723, 0.2825] \\ [0.7635, 0.7986, 0.8689] & [0.8406, 0.8710, 0.8913] & [0.3600, 0.2397, 0.1595] & [0.7179, 0.7293, 0.7369] \\ [0.2448, 0.2820, 0.3563] & [0.0870, 0.1130, 0.1304] & [0.2639, 0.2341, 0.2143] & [0.0159, 0.0226, 0.0270] \\ [0.2485, 0.2606, 0.2850] & [0.1014, 0.1319, 0.1522] & [0.4188, 0.3504, 0.3048] & [0.0000, 0.0089, 0.0148] \\ [0.4771, 0.5060, 0.5637] & [0.0797, 0.0971, 0.1087] & [0.6055, 0.5423, 0.5001] & [0.2751, 0.2891, 0.2984] \\ [0.5221, 0.5551, 0.6212] & [0.0217, 0.0391, 0.0507] & [0.7220, 0.6154, 0.5444] & [0.1414, 0.1547, 0.1635] \\ [0.3596, 0.3052, 0.1964] & [0.0580, 0.0884, 0.1087] & [0.3744, 0.3260, 0.2937] & [0.0641, 0.0662, 0.0675] \\ [0.0957, 0.1214, 0.1727] & [0.0000, 0.0304, 0.0507] & [0.1046, 0.0529, 0.0185] & [0.0803, 0.1024, 0.1171] \\ [0.2032, 0.2684, 0.3987] & [0.1087, 0.1261, 0.1377] & [0.3069, 0.2773, 0.2575] & [0.2877, 0.2896, 0.2908] \\ [0.2801, 0.3231, 0.4090] & [0.1449, 0.1797, 0.2029] & [0.3718, 0.3242, 0.2925] & [0.2164, 0.2209, 0.2240] \\ [0.1910, 0.2271, 0.2993] & [0.0942, 0.1159, 0.1304] & [0.3110, 0.2813, 0.2616] & [0.2806, 0.2907, 0.2874] \\ [0.2689, 0.2911, 0.3354] & [0.0362, 0.0493, 0.0580] & [0.3472, 0.2813, 0.2616] & [0.2115, 0.2172, 0.2210] \\ [0.0000, 0.0393, 0.1180] & [0.1159, 0.1420, 0.1594] & [0.3472, 0.2895, 0.2510] & [0.1782, 0.1937, 0.2040] \\ [0.0265, 0, 0441, 0, 0793] & [0.0725, 0.0942, 0.1087] & [0.1010, 0.0695, 0.0486] & [0.1043, 0.1083, 0.1109] \\ [0.0579, 0.0839, 0.1358] & [0.0290, 0.0464, 0.0580] & [0.1443, 0.1040, 0.0771] & [0.1594, 0.1624, 0.1645] \\ [0.6773, 0.7321, 0.8418] & [0.4855, 0.5029, 0.5145] & [0.0766, 0.0306, 0.0000] & [0.4124, 0.4523, 0.4790] \\ [6819, 0.7464, 0.8755] & [0.4565, 0.5087, 0.5435] & [0.9047, 0.8654, 0.8391] & [0.4132, 0.4376, 0.4539] \\ [0.1976, 0.2440, 0.3366] & [0.0652, 0.1000, 0.1232] & [1.0000, 0.8786, 0.7977] & [0.1055, 0.1104, 0.1136] \\ [0.2426, 0.2703, 0.3258] & [0.1159, 0.1464, 0.1667] & [0.3428, 0.2571, 0.1999] & [0.0535, 0.0619, 0.0676] \\ [0.4810, 0.4963, 0.5268] & [0.1304, 0.1609, 0.1812] & [0.3465, 0.3021, 0.2724] & [0.2969, 0.2996, 0.3013] \\ [0.3702, 0.3721, 0.3760] & [0.1232, 0.1406, 0.1522] & [0.7063, 0.6679, 0.6422] & [0.0002, 0.0063, 0.0103] \\ [0.7970, 0.8105, 1.0000] & [0.2609, 0.2870, 0.3043] & [0.4323, 0.3917, 0.3646] & [0.2935, 0.2967, 0.2989] \end{matrix}]$

According to formula (4), the effect evaluation vectors of ideal optimal scheme and critical scheme are obtained: $\begin{matrix} x^{+} (\otimes) = ([0.7970, 0.8105, 1.0000], [0.9855, \\ 0.9942, 1.0000], [0.0766, 0.0306, 0.0000], [0.0000, \\ 0.0063, 0.0103]) \end{matrix}$ $\begin{matrix} x^{-} (\otimes) = ([0.0000, 0.0393, 0.0793], [0.0000, \\ 0.0304, 0.0507], [0.0766, 0.0306, 0.0000], [0.0000, \\ 0.0063, 0.0103]) \end{matrix}$

The weight matrix obtained by expert BWM method is as follows: W = (w₁, w₂, w₃, w₄) = (0 . 21, 0 . 36, 0 . 20, 0 . 23).

The weight obtained from Gini coefficient is as follows: W = (w₁, w₂, w₃, w₄) = (0 . 28, 0 . 31, 0 . 19, 0 . 22).

Then we can calculate the comprehensive weight. W = (w₁, w₂, w₃, w₄) = (0 . 245, 0 . 335, 0 . 195, 0 . 225).

According to Equation (8) and Equation (10), the grey interval relational degree of each scheme with ideal optimal scheme and critical scheme can be obtained. And we also calculate the node importance ranking through the method of reference 9. The above results are shown in the Table 5 and Fig. 6.

Table 5

Calculation results and comparison

Node	G (x⁺ (⊗) , x_i (⊗))	G (x^- (⊗) , x_i (⊗))	Linear relational degree	Ranking of this article	Relational degree calculated in ref. 15	Ranking of paper [15]
1	0.8923	0.6226	0.6349	2	0.7426	2
2	0.7867	0.5950	0.5959	12	0.7360	8
3	0.8272	0.6244	0.6014	8	0.7344	9
4	0.8940	0.6228	0.6356	1	0.7259	1
5	0.8193	0.6536	0.5829	14	0.7223	10
6	0.8503	0.6887	0.5808	15	0.7216	12
7	0.8686	0.6146	0.6270	3	0.7164	3
8	0.8219	0.5846	0.6187	6	0.7152	5
9	0.8394	0.5873	0.6261	4	0.7134	4
10	0.8184	0.7173	0.5506	19	0.7070	20
11	0.8406	0.7116	0.5645	16	0.7057	15
12	0.6741	0.6962	0.4890	24	0.7008	24
13	0.8394	0.7190	0.5602	18	0.6978	14
14	0.8576	0.6823	0.5877	13	0.6843	13
15	0.7064	0.6223	0.5421	20	0.6781	16
16	0.6651	0.6711	0.4970	22	0.6752	23
17	0.7181	0.7309	0.4936	23	0.6723	22
18	0.8920	0.6997	0.5962	11	0.6690	18
19	0.8788	0.6864	0.5962	10	0.6648	19
20	0.6887	0.6931	0.4978	21	0.6618	21
21	0.8421	0.7156	0.5633	17	0.6268	17
22	0.8604	0.6590	0.6007	9	0.6233	11
23	0.8349	0.6117	0.6116	7	0.6134	7
24	0.8426	0.5960	0.6233	5	0.6063	6

Fig. 6

Comparison of relational degree results.

From the Table 5 and Fig. 6, it can be seen that the top 8 node importance degrees calculated by this method are: blade1 > volute>thrust disc > p-m rotor > reinforcement on the back of impeller > foundation>roulette1 > stator core. The top 8 node importance degrees calculated by literature [15] are:blade1 > mainshaft>thrust disc > reinforcement on the back of impeller > p-m rotor > foundation>impeller rim1. From the ranking of node importance obtained by the research method in this paper, we can see that the ranking in this paper is in line with common sense and reasonable.And the results of this paper are similar to those of traditional important node identification, which shows the effectiveness of this method. However, the method proposed in this paper is more simple and fast in calculation, and conforms to the data characteristics of complex products.

7 Conclusion

In order to evaluate the complex product nodes more quickly and accurately, this paper improves the three parameter interval grey number association model and applies it to the ranking of important nodes. It simplifies the problem of identifying important nodes of complex products, and improves the speed of calculating important nodes. Firstly, this paper constructs a multi-stage complex network model based on multi-stage production process. Then, the evaluation index system of node importance is constructed. The improved three parameter grey relational model is used to evaluate the node importance of complex products, and the effectiveness of the proposed method is verified by case analysis and result comparison.

This paper improves the accuracy of the node importance evaluation of complex products, and makes up for the problem that the settlement is complex and not suitable for the actual situation. And compared with interval grey number, it highlights the “center of gravity” point with the greatest possibility of grey number value, and makes up for the lack of “information” of grey number. Based on the analysis of the classic three-parameter interval grey relational evaluation model, this paper studies the problem of the weight value. Considering from both subjective and objective aspects, the scientific combination of BWM weighting method and Gini coefficient weighting method is feasible and more fair and objective.

The deficiency of this paper is that the multi-stage knowledge network is not perfect and meticulous. In the next research, we can combine the multi-stage network with product engineering, behavior, structure and other elements to further refine the multi-stage network. Through the multi-stage knowledge network, the efficiency of node importance evaluation can be further improved.

Declarations

Ethical Approval: This article does not involve the content of violating ethics and morality.

Consent to Participate and Consent to Publish: All authors have read and agreed to participate and publish the manuscript.

Funding: The work was supported by National Natural Science Foundation of China (No. 72072072), National Natural Science Foundation of China (No. 71672074), Natural Science Foundation of Guangdong Province of China (No. 2019A1515010045) and 2018 Guangzhou Leading Innovation Team Program (China) (No. 201909010006).

Competing Interests: The authors declare that they have no conflict of interest.

Availability of data and materials: All data generated and materials or analyzed during this study are included in this published article.

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