# Research on Importance Evaluation of Complex Product Parts Based on Multilayer Complex Network

Authors: Weiming Yang; Congdong Li; Yinyun Yu; Mingsheng Zhong
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

The identification and evaluation of important parts play an important role in effective production arrangement and shortening the product development in the product design stage. In this paper, a complex product node importance evaluation method based on multilayer network is proposed to identify and evaluate importance parts faster. First, a complex product design expression network based on “function behavior structure (FBS)” multilayer complex network is established. Second, the evaluation index system of important design parts for complex products based on multilayer network is constructed. Third, a three-parameter grey relational model based on the fuzzy analytic hierarchy process and the Gini coefficient method is proposed. Finally, this method is available and feasible through taking the large permanent magnet synchronous centrifugal unit as an example.


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## Body

## 1. Introduction

Complex product design process has the characteristics of diversity, nonlinearity, uncertainty, and evolution, which needs the efficient support and cooperation of various resources. Its architecture is highly complex and high dimensional. In this complex design process, on the one hand, companies have to invest a lot of manpower, material resources, and financial resources and require precise budgets and production arrangements. On the other hand, it is often interdisciplinary and accompanied by grey information. The experience and knowledge of experts in various fields are needed. Complex network can well represent the complex product design process. Many scholars use complex networks to realize the design control of complex products in recent years. Among them, the identification of important parts of complex products is of great value to improve the efficiency and sensitivity of the design [1, 2].The identification of important nodes has attracted the attention of researchers and practitioners. Many diffusion models have been proposed to identify important nodes in complex networks [3–5]. The method commonly used to identify the most influential node set is the central node. Many methods can be used to measure the centrality of complex network structure, such as network degree, intermediate number, and tightness. However, measuring the centrality of the network structure to identify the nodes with maximum influence is not the best way to solve this problem. There are few studies on the evaluation of important nodes related to complex product engineering design. However, the research on the important nodes of complex products is of great significance for the design knowledge of control nodes. Therefore, this paper proposes a new important node identification method based on the multilayer network. Firstly, a multilevel design expression network considering the function, structure, and behavior of a complex product design is constructed. Then, a comprehensive new evaluation index system about network characteristics and component value is established. In addition, an improved three-parameter interval grey number model is used for evaluation considering the multidomain heterogeneity and difficult acquisition of knowledge. In the improvement, in order to better adapt to the research of this paper, we integrate the fuzzy analytic hierarchy process and the Gini coefficient method to improve the method.The remainder of this paper is organized as follows. After reviewing some relevant literature in Section2, we describe the research problem in Section 3 and give determination of evaluation index system in Section 4. In Section 5, we give the research methodology in this paper. In Section 6, we provide a case study about large permanent magnet synchronous centrifugal unit. Section 7 concludes this work.


## 2. Literature Review

Many scholars have proposed many methods for important node identification in recent years. Lu et al. [6] designed a cascade discount algorithm to solve the influence maximization problem. Gong et al. [7] proposed a discrete particle swarm optimization algorithm to optimize the local influence criterion. The representations and update rules for the particles are redefined in the proposed algorithm. Bao et al. [8] proposed a heuristic clustering (HC) algorithm based on the similarity index to classify nodes into different clusters, and finally, the center nodes in clusters are chosen as the multiple spreaders. HC algorithm not only ensures that the multiple spreaders are dispersively distributed in networks but also avoids the selected nodes to be very “negligible.” Cui et al. [9] proposed a degree-descending search strategy (DDS) and developed an evolutionary algorithm that is capable of improving the efficiency significantly by eliminating the time-consuming simulations of the greedy algorithms. Li et al. [10] proposed an improved label propagation algorithm named LPA-MNI by combining the modularity function and node importance with the original LPA. Berberler et al. [11] conducted node importance analysis in wheel-related networks by a method of evaluating node importance by node contraction based on network agglomeration in communication networks. Sun et al. [12] proposed an entropy-based self-adaptive node importance evaluation method to evaluate node importance objectively.Some of the above studies are aimed at pure networks, and most of them only study some characteristics of networks. It is difficult to reflect the importance of nodes in an all-round way and cannot effectively identify the important nodes in practical application. We start with multilayer network and establish a new index. This index system includes the value of parts in the design, which is more comprehensive.There are many research studies on the description network of complex products, mainly focusing on the research of single-layer and weighted complex networks. Bencherif and Mouss [13] proposed a product development process model in an innovation context and strategy framework of design process and project management. The process modelling is based on complex network theory, to improve characterization analysis for product development process modelling. Li et al. [14] focused on the impacts of organization-component executing patterns on the risk propagation in CPD interdependent networks considering the limited risk resisting capacity of organizations. Li et al. [15] developed a general model based on complex network to depict the dynamic design change propagation. Numerical simulations are conducted to explore the general law of the propagation and investigate the influences of design change tolerance capacity distribution (alpha, beta), attack strategies, and recovery capacity (gamma). Li et al. [16], for the first time, applied the SoV into the research on quality risk propagation of complex product collaborative manufacturing supply chain network. Yin et al. [17] proposed a method for identifying the influential parts of a CMP based on complex network theory from the perspective of reliability. It is used to identify the influential parts in each community of products. Yu et al. [18] built a directed weighted complex product network model to represent the product structure under given requirement by defining interconnections among parts. Zhang et al. [19] proposed a novel HEO modelling method based on complex networks’ theory. Zhang et al. [20] proposed a new four-phase routing approach based on weighted and directed complex networks for multisource design change propagation.It can be seen from the existing design networks that single-layer networks are often used to study their properties. However, some design factors will be ignored. So, this paper uses a comprehensive design process description model, FBS model, to describe the design process. The comprehensive evaluation index system is established. Then, the three-parameter interval grey number grey relational model is used to evaluate the important nodes.There are many studies on three-parameter interval grey number relational model. Considering the value information of the three-parameter interval grey number and the risk attitude of decision maker, Zhang et al. [21] advanced a multiattribute group grey target decision-making method based on a three-parameter interval grey number. From the perspective of three-way decision space, Li and Zhang [22] combined the theories of interval concept lattice and three-way decision and then put forward interval three-way decision space theory. To solve decision-making problems having the characteristics of grey systems, Li et al. [23] introduced the three-parameter interval grey number to measure the evaluation index. Chen and Chen [24] proposed a dynamic multiattribute decision-making method based on the prospect theory for dealing with the dynamic multiattribute decision-making problem with the three-parameter interval grey number. Aiming at the multiple attribute decision-making problem with three-parameter interval grey numbers, Li and Zhang [25] proposed a grey-incidence clustering decision-making method based on regret theory. In consideration of the fuzziness and the uncertainty in decision information, Li et al. [26] proposed a three-parameter interval grey linguistic variable decision-making method based on projection model and prospect theory. It can be seen from the previous literature that scholars have carried out a lot of research studies on three-parameter interval grey number relational model. However, there are still some areas to be studied about the determination of its weight. In order to make it more suitable for this paper, this study comprehensively considers its weight setting.To sum up, we can see that the single-layer network is mostly used to analyze the network center from the expression of the design process. The research on the identification method of important network nodes mainly adopts the search algorithm. In the process of subcomplex product design, the identification of important nodes is very important, which determines the speed and quality of the following links. Firstly, this paper establishes a multilayer network model considering the functional behavior structure knowledge in the process of the product design. Then, a new evaluation index is determined for the network model, including the network characteristics and the value of parts. We improve the three-parameter interval grey number relational model to adapt to the research. Finally, we use the three-parameter grey relational model based on TFAHP and Gini coefficient method to evaluate and rank the important nodes in engineering change. The research framework is shown in Figure1.Figure 1
Framework of the proposed complex product node importance evaluation method.


## 3. Problem Description

Previous studies have shown that the FBS model is a more comprehensive description model of the design process, in which the function represents what the design object is used to do, behavior describes what the design object does, and structure specifies what the design object is. In the process of complex product design, the FBS model can be used to formalize the parameters of the parts according to the new function, behavior, and structure requirements. In addition, Hamraz et al. also emphasized that the use of the FBS model to express the relationships in complex products can effectively improve the product design. Therefore, this paper considers the FBS relationship to formalize the complex product design process.The realization of function is based on the exchange of material, information, and energy between the product and the external environment. When two parts share the functions in the form of material flow, information flow, and energy flow, we define that there is a functional relationship between the parts. Product function is an abstract concept, which can make the function relationship have strong concealment and often cannot be perceptual cognition by designers.Behavior is a bridge between the function and the structure and an objective description of function realization. It contains three independent implicit behavior attributes: mechanical (strength, inertia, elasticity, etc.), electrical characteristics (conduction, resistance, charging, etc.), and thermal effects (conduction, temperature change, absorption, etc.). Due to the behavior requirements of products, the combination of behavior attributes can be formed between parts with the same behavior attributes.Structure is the physical carrier of function, which emphasizes the specific composition of the physical entity to realize the function. In particular, structural attributes are explicit attributes used to describe physical entities, which mainly include geometry (size, shape, etc.), material (type, volume, density, etc.), surface (surface roughness, texture, etc.), and controller (microchip, relay, etc.). Similarly, due to the structural requirements of products, the combination of structural attributes exists between components with the same structural attributes (Figure2). The multilayer network construction process is shown in Figure 2.Figure 2
Multilayer complex network analysis of complex product design knowledge.Then components of a complex product and their associated relationships are represented as a single-layer subnetwork Gα=V,Eα,Wα, where α = 1, 2, 3 are the network levels. It represents the network level and represents the single-layer subnetwork type (functional network, behavioral network, and structural network). V=vi|i=1,2,…,N is the set of network nodes. It represents the components set. Eα = {ei α, j | i = 1, 2, …, N; j = 1, 2, …, N; i ≠ j} is the set of connected edges of the network. It represents the association set. Wα=wiα,j|i=1,2,…,N;j=1,2,…,N;i≠j is the weight set of connected edges. It represents the association strength set. For node vi and node vj in the network, if there is an association relationship between the corresponding components, then there is a connected edge ei α, j = 1.


## 4. Determination of Evaluation Index System

Compared with the traditional complex network, the multinetwork model can better reflect the diversity of association relations in complex products and the rich topology characteristics in complex products. In order to evaluate the influence of each node, we need to comprehensively consider the local attributes, global attributes, and their specific significance in complex products from multiple perspectives. At the same time, we consider the indicators that can reflect the properties of parts of complex products to form a comprehensive evaluation index. The detailed indicators are as follows:(1)
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 nodei 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: Ci=Ei/kk−1/2∗k, where k is the degree of node i, which is equivalent to counting the number of friends of node i and Ei 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: Di=1/N−1∑i,j≠idij, where dij is the length of the shortest path from node i to node j.(2)
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.(3)
Node betweenness: the number of paths that all shortest paths in the network pass through this node. The betweennessBi of vertex i is defined as Bi=∑j,k∈Vnjki/njk, where njki is the number of shortest paths between nodes j and k.(4)
Parts value: the engineering change of different parts needs different change costs. And, different nodes have different values. Therefore, the node’s own attribute is the value of node which is used as the evaluation index of node importance. It can be expressed asCvi=cvm+cvl+cve+cvo, where ci is the total value of node i and Cvm 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. Cvi is the labor cost. It refers to the remuneration and other expenses paid to employees in order to obtain the services provided by employees. It mainly includes the salary, bonus, allowance, welfare, and education fund. Cvm is the manufacturing cost. It refers to energy consumption, manufacturing accessories, labor insurance, and office and fixed expenses. Cvo is the other costs, some consumption including fuel cost, power cost, office cost, and depreciation consumed by each production unit.


## 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 asA⊗=a¯,a˜,a¯, where a¯≤a˜≤a¯, and a¯ and a¯ are the upper and lower limits of the interval, respectively. a˜ is called the “center of gravity” point ([27, 28]).When two of the three parametersa¯, a˜, and a¯ are the same, the three-parameter interval grey number degenerates to the interval grey number. When a¯=a˜=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 the interval grey number. Let three-parameter interval grey number beA⊗=a¯,a˜,a¯ and B⊗=b¯,b˜,b¯; then,(i)
A⊗+B⊗=⌊a¯+b¯,a˜+b˜,a¯+b¯⌋(ii)
A⊗/B⊗∈mina¯/b¯,a¯/b¯,a¯/b¯,a¯/b¯,a˜/b˜,maxa¯/b¯,a¯/b¯,a¯/b¯,a¯/b¯(iii)
λA⊗=λa¯,λa˜,λa¯,λ>0


### 5.2. Three-Parameter Interval Grey Number Grey Relational Model

Suppose that there aren alternative engineering change schemes. They constituted by evaluation schemes’ set A=a1,a2,…,an. The index set S=s1,s2,…,sm is composed of m attributes. The index value of scheme ai under the evaluation index sj can be expressed as uij⊗=⌊u¯ij,u˜ij,u¯ij⌋u¯ij≤u˜ij≤u¯ij,i=1,2,…,n;j=1,2,…,m. The effect evaluation vector of each scheme is ui⊗=ui1⊗,ui2⊗,…,uim⊗,i=1,2,…,n. The weight of index under each scheme is wi1,wi2,…,wim, and ∑j=1mwij=1i=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,(1)x¯ij=u¯ij−u¯j∇u¯j∗−u¯j∇,x˜ij=u˜ij−u¯j∇u¯j∗−u¯j∇,x¯ij=u¯ij−u¯j∇u¯j∗−u¯j∇.For cost attribute values,(2)x¯ij=u¯j∗−u¯iju¯j∗−u¯j∇,x˜ij=u¯j∗−u˜iju¯j∗−u¯j∇,x¯ij=u¯j∗−u¯iju¯j∗−u¯j∇,where u¯j∗=max1≤i≤nu¯ij,u¯j∇min1≤i≤nu¯ij,j=1,2,…,m.Let the normalized effect evaluation vector be(3)xi⊗=xi1⊗,xi2⊗,…,xim⊗,i=1,2,…,n,where xij⊗∈x¯ij,x˜ij,x¯ij is a three-parameter interval grey number in 0,1.It is recorded thatx¯j+=max1≤i≤nx¯ij, x˜j+=max1≤i≤nx˜ij, x¯j+=max1≤i≤nx¯ij, x¯j−=min1≤i≤nx¯ij, x˜j−=min1≤i≤nx˜ij, and x¯j−=min1≤i≤nx¯ijj=1,2,…,m. Then, the m-dimensional three-parameter nonnegative interval grey number vectors,(4)x+⊗=x1+⊗,x2+⊗,…,xm+⊗,x−⊗=x1−⊗,x2−⊗,…,xm−⊗,are called ideal optimal scheme effect evaluation vectors and critical scheme effect evaluation vectors, respectively [29].We assume that the grey interval relational degree of the normalized effect evaluation vectorxi⊗ of scheme Ai with respect to the ideal optimal scheme effect evaluation vector x+⊗ is Gx+⊗,xi⊗. And, the grey interval relational degree of the critical scheme effect evaluation vector x−⊗ is Gx−⊗,xi⊗. Assume that the weights of two grey relational degrees are α1, α2α1+α2=1. Then,(5)Gxi⊗=α1Gx+⊗,xi⊗+α21−Gx−⊗,xi⊗,i=1,2,…,n,is the three-parameter grey interval linear relational degree of the effect evaluation vector xi⊗.(6)Gxi⊗=Gx+⊗,xi⊗α1+1−Gx−⊗,xi⊗α2,i=1,2,…,n,is the three-parameter grey interval product relational degree of the effect evaluation vector xi⊗ (according to [28] and experts’ opinion, this paper is set to λ=0.5, ξ=0.5, and β1=β2=0.5).The distribution probability of barycenter point with the highest probability of taking the value of three-parameter interval grey numberxij⊗∈x¯ij,x˜ij,x¯ij is fx˜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.Definition 3.
For the three-parameter interval grey numberxij⊗∈x¯ij,x˜ij,x¯ij,(7)γij+=35×m˜++ηM˜+Δ˜ij++ηM˜++251−βm¯++ηM¯+Δ¯ij++ηM¯+βm¯++ηM¯+Δ¯ij++ηM¯+,is called the three-parameter grey interval relational coefficient of subfactor xij with respect to ideal factorxj+. η∈0,1 is the resolution coefficient. β∈0,1 is the decision preference coefficient (according to [28] and experts’ opinion, normally, η=β=0.5), where(8)Δ¯ij+=x¯j+−x¯ij,Δ˜ij+=x˜j+−x˜ij,Δ¯ij+=x¯j+−x¯ij,i=1,2,…,n;j=1,2,…,m,m¯+=min1≤i≤nmin1≤j≤mΔ¯ij+,m˜+=min1≤i≤nmin1≤j≤mΔ˜ij+,m¯+=min1≤i≤nmin1≤j≤mΔ¯ij+,M¯+=max1≤i≤nmax1≤j≤mΔ¯ij+,M˜+=max1≤i≤nmax1≤j≤mΔ˜ij+,M¯+=max1≤i≤nmax1≤j≤mΔ¯ij+,Gx+⊗,xi⊗=∑j=1mwijγij+,i=1,2,…,n,is called the three-parameter grey interval relational degree of the effect evaluation vector xi⊗ about the ideal optimal scheme effect evaluation vector x+⊗.Definition 4.
For three-parameter interval grey numberxij⊗∈x¯ij,x˜ij,x¯ij,(9)γij−=35×m˜−+εM˜−Δ˜ij−+εM˜−+25×1−δm¯−+εM¯−Δ¯ij−+εM¯−+δm¯−+εM¯−Δ¯ij−+εM¯−,is called the three-parameter grey interval relational coefficient of sub factor xij with respect to ideal facto xj−. η∈0,1 is the resolution coefficient. δ∈0,1 is the decision preference coefficient (according to [28] and experts’ opinion, normally, ε=β=0.5), where(10)Δ¯ij−=x¯ij−x¯j−,Δ˜ij−=x˜ij−x˜j−,Δ¯ij−=x¯ij−x¯j−,i=1,2,…,n;j=1,2,…,m,m¯−=min1≤i≤nmin1≤j≤mΔ¯ij−,m˜−=min1≤i≤nmin1≤j≤mΔ˜ij−,m¯−=min1≤i≤nmin1≤j≤mΔ¯ij−,M¯−=max1≤i≤nmax1≤j≤mΔ¯ij−,M˜−=max1≤i≤nmax1≤j≤mΔ˜ij−,M¯−=max1≤i≤nmax1≤j≤mΔ¯ij−,Gx−⊗,xi⊗=∑j=1mwijγij−,i=1,2,…,n,is called the three-parameter grey interval relational degree of the effect evaluation vector xi⊗ about the critical scheme effect evaluation vector x−⊗.


### 5.3. Determination of Weight

In order to better reflect the balance and scientificity of expert evaluation and objective evaluation in the evaluation process, this paper adopts a combined weight method. Firstly, the triangular fuzzy analytical hierarchy process method is used to determine the subjective weight. Then the Gini coefficient method is used to determine the objective weight. Finally, the linear weighting method is used to combine the two methods, so as to increase its scientificity.


#### 5.3.1. Determination of Weight Based on Triangular Fuzzy Analytic Hierarchy Process (TFAHP)

Analytic hierarchy process (AHP) is a multicriteria decision analysis method which transforms subjective judgment into objective one. It makes people's subjective judgment process hierarchical and mathematical and provides a simple decision-making method for multiattribute decision-making problems.Triangular fuzzy number has obvious advantages in solving the problems of complexity, fuzziness, and uncertainty. Fuzzy numbers have been studied extensively. [30–36] TFAHP (triangular fuzzy analytic hierarchy process) is a combination of fuzzy theory and AHP. It fully considers the subjective judgment of the evaluator, the fuzziness of decision, and the preference, which makes the decision result more objective and reasonable. The calculation steps are as follows:(1)
According to the above evaluation index system construction, the hierarchical model is established, as shown in Table1. The top layer is target layer A. The middle layer is criterion layer B. The lowest layer is index layer C.(2)
Each expert can get the triangular fuzzy number judgment matrix of the target layer to the criterion layer and the criterion layer to the index layer by comparing each element. Letx˜ijt be the triangular fuzzy number judgment matrix of the expert t.Then, the triangular fuzzy number is x˜ijt=l˜ijt,m˜ijt,u˜ijt, where l˜ijt and u˜ijt are the upper and lower bounds of fuzzy numbers and m˜ijt is the fuzzy median. The smaller the value of u˜ijt−l˜ijt, the more fuzzy the judgment. When u˜ijt−l˜ijt is 0, the judgment is not fuzzy.Each element value of the fuzzy judgment matrix represents the importance of one factor relative to another. Because it is difficult to construct the consistency matrix by 1–9 and its reciprocal scaling method, the ranking result is far from the actual thinking 0.1–0.9 scale method which can better reflect the actual estimation of experts. Therefore, the 0.1–0.9 scale method is used in this study, as shown in Table2.(3)
The judgment matrix of each expert is synthesized, and the triangular fuzzy number in the judgment matrix is changed into nonfuzzy number. Suppose the total number of experts isT and the weight of each expert is qt; then, the synthetic judgment matrix is X˜=∑t=1TqtX˜t.According to the method of reference 35, the most likely estimated value of each factor in fuzzy judgment matrixA by pairwise comparison are extracted. The fuzzy complementary judgment matrix M: M=∑t=1Tmijtqt is obtained. It can be transformed into fuzzy consistency matrix M′ through formula mi′=∑k=1nmik and mi′=mi′−mj′/2n−1+0.5. If M′ satisfies the requirement of consistency, then X˜ also satisfies the requirement of consistency. For the fuzzy consistent matrix M′ of 0.1–0.9 scale, the consistency is checked by calculating δ and σ to judge whether the fuzzy complementary judgment matrix is consistent with the actual situation:(11)δ=maxmij′−mij,σ=∑i=1n∑j=1nmij′−mij2n.δ<0.2 and σ<0.1, and the fuzzy matrix passes the consistency test, which is in line with the actual situation. Otherwise, the judgment matrix should be adjusted.The fuzzy judgment matrixX˜ is transformed into nonfuzzy matrix F″:(12)f″=lij2lij+2+2mij3+uij6.(4)
In order to determine the index weight of each factor, the comprehensive importance of each element is calculated according to each fuzzy judgment matrix. It can be solved according to the weight formula:(13)ωi=∑j=1nfij″−0.5∑∑j=1nfij″−0.5.Table 1
Index system hierarchical model of important nodes in the complex product design.
ANode importance evaluationBLink degree (B1)Average path length (B2)Shortest route number (B3)Value attributes of complex product (B4)CClustering coefficient (C1)Erdos (C2)Node betweenness (C3)Value of parts (C4)Table 2
0.1–0.9 quantitative scale.
ScaleMeaning0.1When one factor is compared with another, the latter is extremely important0.2When one factor is compared with another, the latter is strongly important0.3When one factor is compared with another, the latter is obviously important0.4When one factor is compared with another, the latter is slightly important0.5When one factor is compared with another, the two factors are of equal importance0.6When one factor is compared with another, the former is slightly more important0.7When one factor is compared with another, the former is obviously important0.8When one factor is compared with another, the former is strongly important0.9When one factor is compared with another, the former is extremely important


#### 5.3.2. Weight Determination Method Based on Gini Coefficient

(1) Principle of Gini Coefficient Weighting Method. The 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 level 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 [37].(2) Gini Coefficient Weight Calculation of Evaluation Index. We assume that Gk is the Gini coefficient of the kth index, Ykiis the ith data of the kth index, and μK is the expected value of all data of the kth index. Then, the Gini coefficient Gk of the kth index is shown as follows:(14)Gk=∑i=1n∑j=1nYki−Ykj2n2μk,(15)Gk=∑i=1n∑j=1nYki−Ykjn2−n.Especially, when the mean value of index data is not 0, the Gini coefficient is calculated by the improved formula (14). When the mean value of the index data is 0, the Gini coefficient of the index is calculated by the original formula (15). The Gini coefficient of the index truly reflects the data changes of different evaluation objects of the index.Gini coefficient weightgk of the kth index can be obtained by normalizing the Gini coefficient value of each index:(16)gk=Gk∑i=1mGi,where gk is Gini coefficient weight of the kth index, Gk is Gini coefficient value of the kth index, and m is the number of indexes.The advantages of the Gini coefficient weighting method are as follows. First, the weight calculation is not affected by the unit dimension of the index, and the definition of Gini coefficient itself eliminates the dimensional influence. Second, the 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 TFAHP-Gini Coefficient

In order to be more scientific, we adopt the linear weighting method to combine the two methods after the weight determining based on subjective and objective.The weight vector of linear combination is calculated and proved as follows.SupposeN kinds of subjective and objective methods to get l kinds of weights. The weight vectors are W1,W2,…,WL. The kth weight direction quantity is Wk=ϖ1k,ϖ2k,…,ϖmk. It satisfies that ∑j=1mϖjk=1,ϖjk≥0j=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∗=ϖ1∗,ϖ2∗,…,ϖm∗T be the linear combination weight vector of L weight vectors. It can be expressed in the vector form as W∗=∑k=1Lxkwk, where xk is the linear combination coefficient. It satisfies ∑k=1Lxk=1,xk≥0. The solution of linear combined weight vector is a multiobjective 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:(17)minμ∑i=1n∑j=1m∑k=1Lxkϖjk1−rij+1−μ∑xklnxk,s.t.∑k=1Lxk=1,xk≥0,where rij is the relational coefficient, 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 [3], there is a unique solution for the single objective optimization problem (SP). The solution is as follows:(18)x=s1∑k=1lsk,s2∑k=1lsk,…,sL∑k=1Lsk,where sk=exp−1+μ∑i=1n∑j=1mϖjk1−rij/1−μk=1,2,…,L, N is the number of evaluation schemes, and 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:Step 1: construct the model of complex product presentation based on the multilayer complex network model.Step 2: determine the evaluation index.Step 3: standardize the original three-parameter interval grey number effect evaluation index, and obtain the standardized effect evaluation vector formula of each scheme.Step 4: solve the ideal optimal solution for the decision-making problem and the effect evaluation vectorx+⊗ and the critical solution x−⊗.Step 5: obtain the three-parameter grey interval relational coefficient vector of each scheme, the ideal scheme and the critical scheme.Step 6: solve the TFAHP and Gini coefficient models, combine the weights, and obtain the weight of each scheme under different attributes.Step 7: calculate the three-parameter interval grey number relational degreeGx+⊗,xi⊗ and Gx−⊗,xi⊗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 Gxi⊗i=1,2,…,n of each scheme.Step 8: the schemes are sorted according to the relevance degreeGxi⊗i=1,2,…,n. The scheme corresponding to the maximum relational degree is the optimal one.The flow of this algorithm is shown in Figure3.Figure 3
Three-parameter interval grey number grey relational evaluation model algorithm.


## 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 asA⊗=a¯,a˜,a¯, where a¯≤a˜≤a¯, and a¯ and a¯ are the upper and lower limits of the interval, respectively. a˜ is called the “center of gravity” point ([27, 28]).When two of the three parametersa¯, a˜, and a¯ are the same, the three-parameter interval grey number degenerates to the interval grey number. When a¯=a˜=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 the interval grey number. Let three-parameter interval grey number beA⊗=a¯,a˜,a¯ and B⊗=b¯,b˜,b¯; then,(i)
A⊗+B⊗=⌊a¯+b¯,a˜+b˜,a¯+b¯⌋(ii)
A⊗/B⊗∈mina¯/b¯,a¯/b¯,a¯/b¯,a¯/b¯,a˜/b˜,maxa¯/b¯,a¯/b¯,a¯/b¯,a¯/b¯(iii)
λA⊗=λa¯,λa˜,λa¯,λ>0


## 5.2. Three-Parameter Interval Grey Number Grey Relational Model

Suppose that there aren alternative engineering change schemes. They constituted by evaluation schemes’ set A=a1,a2,…,an. The index set S=s1,s2,…,sm is composed of m attributes. The index value of scheme ai under the evaluation index sj can be expressed as uij⊗=⌊u¯ij,u˜ij,u¯ij⌋u¯ij≤u˜ij≤u¯ij,i=1,2,…,n;j=1,2,…,m. The effect evaluation vector of each scheme is ui⊗=ui1⊗,ui2⊗,…,uim⊗,i=1,2,…,n. The weight of index under each scheme is wi1,wi2,…,wim, and ∑j=1mwij=1i=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,(1)x¯ij=u¯ij−u¯j∇u¯j∗−u¯j∇,x˜ij=u˜ij−u¯j∇u¯j∗−u¯j∇,x¯ij=u¯ij−u¯j∇u¯j∗−u¯j∇.For cost attribute values,(2)x¯ij=u¯j∗−u¯iju¯j∗−u¯j∇,x˜ij=u¯j∗−u˜iju¯j∗−u¯j∇,x¯ij=u¯j∗−u¯iju¯j∗−u¯j∇,where u¯j∗=max1≤i≤nu¯ij,u¯j∇min1≤i≤nu¯ij,j=1,2,…,m.Let the normalized effect evaluation vector be(3)xi⊗=xi1⊗,xi2⊗,…,xim⊗,i=1,2,…,n,where xij⊗∈x¯ij,x˜ij,x¯ij is a three-parameter interval grey number in 0,1.It is recorded thatx¯j+=max1≤i≤nx¯ij, x˜j+=max1≤i≤nx˜ij, x¯j+=max1≤i≤nx¯ij, x¯j−=min1≤i≤nx¯ij, x˜j−=min1≤i≤nx˜ij, and x¯j−=min1≤i≤nx¯ijj=1,2,…,m. Then, the m-dimensional three-parameter nonnegative interval grey number vectors,(4)x+⊗=x1+⊗,x2+⊗,…,xm+⊗,x−⊗=x1−⊗,x2−⊗,…,xm−⊗,are called ideal optimal scheme effect evaluation vectors and critical scheme effect evaluation vectors, respectively [29].We assume that the grey interval relational degree of the normalized effect evaluation vectorxi⊗ of scheme Ai with respect to the ideal optimal scheme effect evaluation vector x+⊗ is Gx+⊗,xi⊗. And, the grey interval relational degree of the critical scheme effect evaluation vector x−⊗ is Gx−⊗,xi⊗. Assume that the weights of two grey relational degrees are α1, α2α1+α2=1. Then,(5)Gxi⊗=α1Gx+⊗,xi⊗+α21−Gx−⊗,xi⊗,i=1,2,…,n,is the three-parameter grey interval linear relational degree of the effect evaluation vector xi⊗.(6)Gxi⊗=Gx+⊗,xi⊗α1+1−Gx−⊗,xi⊗α2,i=1,2,…,n,is the three-parameter grey interval product relational degree of the effect evaluation vector xi⊗ (according to [28] and experts’ opinion, this paper is set to λ=0.5, ξ=0.5, and β1=β2=0.5).The distribution probability of barycenter point with the highest probability of taking the value of three-parameter interval grey numberxij⊗∈x¯ij,x˜ij,x¯ij is fx˜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.Definition 3.
For the three-parameter interval grey numberxij⊗∈x¯ij,x˜ij,x¯ij,(7)γij+=35×m˜++ηM˜+Δ˜ij++ηM˜++251−βm¯++ηM¯+Δ¯ij++ηM¯+βm¯++ηM¯+Δ¯ij++ηM¯+,is called the three-parameter grey interval relational coefficient of subfactor xij with respect to ideal factorxj+. η∈0,1 is the resolution coefficient. β∈0,1 is the decision preference coefficient (according to [28] and experts’ opinion, normally, η=β=0.5), where(8)Δ¯ij+=x¯j+−x¯ij,Δ˜ij+=x˜j+−x˜ij,Δ¯ij+=x¯j+−x¯ij,i=1,2,…,n;j=1,2,…,m,m¯+=min1≤i≤nmin1≤j≤mΔ¯ij+,m˜+=min1≤i≤nmin1≤j≤mΔ˜ij+,m¯+=min1≤i≤nmin1≤j≤mΔ¯ij+,M¯+=max1≤i≤nmax1≤j≤mΔ¯ij+,M˜+=max1≤i≤nmax1≤j≤mΔ˜ij+,M¯+=max1≤i≤nmax1≤j≤mΔ¯ij+,Gx+⊗,xi⊗=∑j=1mwijγij+,i=1,2,…,n,is called the three-parameter grey interval relational degree of the effect evaluation vector xi⊗ about the ideal optimal scheme effect evaluation vector x+⊗.Definition 4.
For three-parameter interval grey numberxij⊗∈x¯ij,x˜ij,x¯ij,(9)γij−=35×m˜−+εM˜−Δ˜ij−+εM˜−+25×1−δm¯−+εM¯−Δ¯ij−+εM¯−+δm¯−+εM¯−Δ¯ij−+εM¯−,is called the three-parameter grey interval relational coefficient of sub factor xij with respect to ideal facto xj−. η∈0,1 is the resolution coefficient. δ∈0,1 is the decision preference coefficient (according to [28] and experts’ opinion, normally, ε=β=0.5), where(10)Δ¯ij−=x¯ij−x¯j−,Δ˜ij−=x˜ij−x˜j−,Δ¯ij−=x¯ij−x¯j−,i=1,2,…,n;j=1,2,…,m,m¯−=min1≤i≤nmin1≤j≤mΔ¯ij−,m˜−=min1≤i≤nmin1≤j≤mΔ˜ij−,m¯−=min1≤i≤nmin1≤j≤mΔ¯ij−,M¯−=max1≤i≤nmax1≤j≤mΔ¯ij−,M˜−=max1≤i≤nmax1≤j≤mΔ˜ij−,M¯−=max1≤i≤nmax1≤j≤mΔ¯ij−,Gx−⊗,xi⊗=∑j=1mwijγij−,i=1,2,…,n,is called the three-parameter grey interval relational degree of the effect evaluation vector xi⊗ about the critical scheme effect evaluation vector x−⊗.


## 5.3. Determination of Weight

In order to better reflect the balance and scientificity of expert evaluation and objective evaluation in the evaluation process, this paper adopts a combined weight method. Firstly, the triangular fuzzy analytical hierarchy process method is used to determine the subjective weight. Then the Gini coefficient method is used to determine the objective weight. Finally, the linear weighting method is used to combine the two methods, so as to increase its scientificity.


### 5.3.1. Determination of Weight Based on Triangular Fuzzy Analytic Hierarchy Process (TFAHP)

Analytic hierarchy process (AHP) is a multicriteria decision analysis method which transforms subjective judgment into objective one. It makes people's subjective judgment process hierarchical and mathematical and provides a simple decision-making method for multiattribute decision-making problems.Triangular fuzzy number has obvious advantages in solving the problems of complexity, fuzziness, and uncertainty. Fuzzy numbers have been studied extensively. [30–36] TFAHP (triangular fuzzy analytic hierarchy process) is a combination of fuzzy theory and AHP. It fully considers the subjective judgment of the evaluator, the fuzziness of decision, and the preference, which makes the decision result more objective and reasonable. The calculation steps are as follows:(1)
According to the above evaluation index system construction, the hierarchical model is established, as shown in Table1. The top layer is target layer A. The middle layer is criterion layer B. The lowest layer is index layer C.(2)
Each expert can get the triangular fuzzy number judgment matrix of the target layer to the criterion layer and the criterion layer to the index layer by comparing each element. Letx˜ijt be the triangular fuzzy number judgment matrix of the expert t.Then, the triangular fuzzy number is x˜ijt=l˜ijt,m˜ijt,u˜ijt, where l˜ijt and u˜ijt are the upper and lower bounds of fuzzy numbers and m˜ijt is the fuzzy median. The smaller the value of u˜ijt−l˜ijt, the more fuzzy the judgment. When u˜ijt−l˜ijt is 0, the judgment is not fuzzy.Each element value of the fuzzy judgment matrix represents the importance of one factor relative to another. Because it is difficult to construct the consistency matrix by 1–9 and its reciprocal scaling method, the ranking result is far from the actual thinking 0.1–0.9 scale method which can better reflect the actual estimation of experts. Therefore, the 0.1–0.9 scale method is used in this study, as shown in Table2.(3)
The judgment matrix of each expert is synthesized, and the triangular fuzzy number in the judgment matrix is changed into nonfuzzy number. Suppose the total number of experts isT and the weight of each expert is qt; then, the synthetic judgment matrix is X˜=∑t=1TqtX˜t.According to the method of reference 35, the most likely estimated value of each factor in fuzzy judgment matrixA by pairwise comparison are extracted. The fuzzy complementary judgment matrix M: M=∑t=1Tmijtqt is obtained. It can be transformed into fuzzy consistency matrix M′ through formula mi′=∑k=1nmik and mi′=mi′−mj′/2n−1+0.5. If M′ satisfies the requirement of consistency, then X˜ also satisfies the requirement of consistency. For the fuzzy consistent matrix M′ of 0.1–0.9 scale, the consistency is checked by calculating δ and σ to judge whether the fuzzy complementary judgment matrix is consistent with the actual situation:(11)δ=maxmij′−mij,σ=∑i=1n∑j=1nmij′−mij2n.δ<0.2 and σ<0.1, and the fuzzy matrix passes the consistency test, which is in line with the actual situation. Otherwise, the judgment matrix should be adjusted.The fuzzy judgment matrixX˜ is transformed into nonfuzzy matrix F″:(12)f″=lij2lij+2+2mij3+uij6.(4)
In order to determine the index weight of each factor, the comprehensive importance of each element is calculated according to each fuzzy judgment matrix. It can be solved according to the weight formula:(13)ωi=∑j=1nfij″−0.5∑∑j=1nfij″−0.5.Table 1
Index system hierarchical model of important nodes in the complex product design.
ANode importance evaluationBLink degree (B1)Average path length (B2)Shortest route number (B3)Value attributes of complex product (B4)CClustering coefficient (C1)Erdos (C2)Node betweenness (C3)Value of parts (C4)Table 2
0.1–0.9 quantitative scale.
ScaleMeaning0.1When one factor is compared with another, the latter is extremely important0.2When one factor is compared with another, the latter is strongly important0.3When one factor is compared with another, the latter is obviously important0.4When one factor is compared with another, the latter is slightly important0.5When one factor is compared with another, the two factors are of equal importance0.6When one factor is compared with another, the former is slightly more important0.7When one factor is compared with another, the former is obviously important0.8When one factor is compared with another, the former is strongly important0.9When one factor is compared with another, the former is extremely important


### 5.3.2. Weight Determination Method Based on Gini Coefficient

(1) Principle of Gini Coefficient Weighting Method. The 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 level 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 [37].(2) Gini Coefficient Weight Calculation of Evaluation Index. We assume that Gk is the Gini coefficient of the kth index, Ykiis the ith data of the kth index, and μK is the expected value of all data of the kth index. Then, the Gini coefficient Gk of the kth index is shown as follows:(14)Gk=∑i=1n∑j=1nYki−Ykj2n2μk,(15)Gk=∑i=1n∑j=1nYki−Ykjn2−n.Especially, when the mean value of index data is not 0, the Gini coefficient is calculated by the improved formula (14). When the mean value of the index data is 0, the Gini coefficient of the index is calculated by the original formula (15). The Gini coefficient of the index truly reflects the data changes of different evaluation objects of the index.Gini coefficient weightgk of the kth index can be obtained by normalizing the Gini coefficient value of each index:(16)gk=Gk∑i=1mGi,where gk is Gini coefficient weight of the kth index, Gk is Gini coefficient value of the kth index, and m is the number of indexes.The advantages of the Gini coefficient weighting method are as follows. First, the weight calculation is not affected by the unit dimension of the index, and the definition of Gini coefficient itself eliminates the dimensional influence. Second, the 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 TFAHP-Gini Coefficient

In order to be more scientific, we adopt the linear weighting method to combine the two methods after the weight determining based on subjective and objective.The weight vector of linear combination is calculated and proved as follows.SupposeN kinds of subjective and objective methods to get l kinds of weights. The weight vectors are W1,W2,…,WL. The kth weight direction quantity is Wk=ϖ1k,ϖ2k,…,ϖmk. It satisfies that ∑j=1mϖjk=1,ϖjk≥0j=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∗=ϖ1∗,ϖ2∗,…,ϖm∗T be the linear combination weight vector of L weight vectors. It can be expressed in the vector form as W∗=∑k=1Lxkwk, where xk is the linear combination coefficient. It satisfies ∑k=1Lxk=1,xk≥0. The solution of linear combined weight vector is a multiobjective 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:(17)minμ∑i=1n∑j=1m∑k=1Lxkϖjk1−rij+1−μ∑xklnxk,s.t.∑k=1Lxk=1,xk≥0,where rij is the relational coefficient, 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 [3], there is a unique solution for the single objective optimization problem (SP). The solution is as follows:(18)x=s1∑k=1lsk,s2∑k=1lsk,…,sL∑k=1Lsk,where sk=exp−1+μ∑i=1n∑j=1mϖjk1−rij/1−μk=1,2,…,L, N is the number of evaluation schemes, and 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:Step 1: construct the model of complex product presentation based on the multilayer complex network model.Step 2: determine the evaluation index.Step 3: standardize the original three-parameter interval grey number effect evaluation index, and obtain the standardized effect evaluation vector formula of each scheme.Step 4: solve the ideal optimal solution for the decision-making problem and the effect evaluation vectorx+⊗ and the critical solution x−⊗.Step 5: obtain the three-parameter grey interval relational coefficient vector of each scheme, the ideal scheme and the critical scheme.Step 6: solve the TFAHP and Gini coefficient models, combine the weights, and obtain the weight of each scheme under different attributes.Step 7: calculate the three-parameter interval grey number relational degreeGx+⊗,xi⊗ and Gx−⊗,xi⊗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 Gxi⊗i=1,2,…,n of each scheme.Step 8: the schemes are sorted according to the relevance degreeGxi⊗i=1,2,…,n. The scheme corresponding to the maximum relational degree is the optimal one.The flow of this algorithm is shown in Figure3.Figure 3
Three-parameter interval grey number grey relational evaluation model algorithm.


## 5.3.1. Determination of Weight Based on Triangular Fuzzy Analytic Hierarchy Process (TFAHP)

Analytic hierarchy process (AHP) is a multicriteria decision analysis method which transforms subjective judgment into objective one. It makes people's subjective judgment process hierarchical and mathematical and provides a simple decision-making method for multiattribute decision-making problems.Triangular fuzzy number has obvious advantages in solving the problems of complexity, fuzziness, and uncertainty. Fuzzy numbers have been studied extensively. [30–36] TFAHP (triangular fuzzy analytic hierarchy process) is a combination of fuzzy theory and AHP. It fully considers the subjective judgment of the evaluator, the fuzziness of decision, and the preference, which makes the decision result more objective and reasonable. The calculation steps are as follows:(1)
According to the above evaluation index system construction, the hierarchical model is established, as shown in Table1. The top layer is target layer A. The middle layer is criterion layer B. The lowest layer is index layer C.(2)
Each expert can get the triangular fuzzy number judgment matrix of the target layer to the criterion layer and the criterion layer to the index layer by comparing each element. Letx˜ijt be the triangular fuzzy number judgment matrix of the expert t.Then, the triangular fuzzy number is x˜ijt=l˜ijt,m˜ijt,u˜ijt, where l˜ijt and u˜ijt are the upper and lower bounds of fuzzy numbers and m˜ijt is the fuzzy median. The smaller the value of u˜ijt−l˜ijt, the more fuzzy the judgment. When u˜ijt−l˜ijt is 0, the judgment is not fuzzy.Each element value of the fuzzy judgment matrix represents the importance of one factor relative to another. Because it is difficult to construct the consistency matrix by 1–9 and its reciprocal scaling method, the ranking result is far from the actual thinking 0.1–0.9 scale method which can better reflect the actual estimation of experts. Therefore, the 0.1–0.9 scale method is used in this study, as shown in Table2.(3)
The judgment matrix of each expert is synthesized, and the triangular fuzzy number in the judgment matrix is changed into nonfuzzy number. Suppose the total number of experts isT and the weight of each expert is qt; then, the synthetic judgment matrix is X˜=∑t=1TqtX˜t.According to the method of reference 35, the most likely estimated value of each factor in fuzzy judgment matrixA by pairwise comparison are extracted. The fuzzy complementary judgment matrix M: M=∑t=1Tmijtqt is obtained. It can be transformed into fuzzy consistency matrix M′ through formula mi′=∑k=1nmik and mi′=mi′−mj′/2n−1+0.5. If M′ satisfies the requirement of consistency, then X˜ also satisfies the requirement of consistency. For the fuzzy consistent matrix M′ of 0.1–0.9 scale, the consistency is checked by calculating δ and σ to judge whether the fuzzy complementary judgment matrix is consistent with the actual situation:(11)δ=maxmij′−mij,σ=∑i=1n∑j=1nmij′−mij2n.δ<0.2 and σ<0.1, and the fuzzy matrix passes the consistency test, which is in line with the actual situation. Otherwise, the judgment matrix should be adjusted.The fuzzy judgment matrixX˜ is transformed into nonfuzzy matrix F″:(12)f″=lij2lij+2+2mij3+uij6.(4)
In order to determine the index weight of each factor, the comprehensive importance of each element is calculated according to each fuzzy judgment matrix. It can be solved according to the weight formula:(13)ωi=∑j=1nfij″−0.5∑∑j=1nfij″−0.5.Table 1
Index system hierarchical model of important nodes in the complex product design.
ANode importance evaluationBLink degree (B1)Average path length (B2)Shortest route number (B3)Value attributes of complex product (B4)CClustering coefficient (C1)Erdos (C2)Node betweenness (C3)Value of parts (C4)Table 2
0.1–0.9 quantitative scale.
ScaleMeaning0.1When one factor is compared with another, the latter is extremely important0.2When one factor is compared with another, the latter is strongly important0.3When one factor is compared with another, the latter is obviously important0.4When one factor is compared with another, the latter is slightly important0.5When one factor is compared with another, the two factors are of equal importance0.6When one factor is compared with another, the former is slightly more important0.7When one factor is compared with another, the former is obviously important0.8When one factor is compared with another, the former is strongly important0.9When one factor is compared with another, the former is extremely important


## 5.3.2. Weight Determination Method Based on Gini Coefficient

(1) Principle of Gini Coefficient Weighting Method. The 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 level 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 [37].(2) Gini Coefficient Weight Calculation of Evaluation Index. We assume that Gk is the Gini coefficient of the kth index, Ykiis the ith data of the kth index, and μK is the expected value of all data of the kth index. Then, the Gini coefficient Gk of the kth index is shown as follows:(14)Gk=∑i=1n∑j=1nYki−Ykj2n2μk,(15)Gk=∑i=1n∑j=1nYki−Ykjn2−n.Especially, when the mean value of index data is not 0, the Gini coefficient is calculated by the improved formula (14). When the mean value of the index data is 0, the Gini coefficient of the index is calculated by the original formula (15). The Gini coefficient of the index truly reflects the data changes of different evaluation objects of the index.Gini coefficient weightgk of the kth index can be obtained by normalizing the Gini coefficient value of each index:(16)gk=Gk∑i=1mGi,where gk is Gini coefficient weight of the kth index, Gk is Gini coefficient value of the kth index, and m is the number of indexes.The advantages of the Gini coefficient weighting method are as follows. First, the weight calculation is not affected by the unit dimension of the index, and the definition of Gini coefficient itself eliminates the dimensional influence. Second, the 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 TFAHP-Gini Coefficient

In order to be more scientific, we adopt the linear weighting method to combine the two methods after the weight determining based on subjective and objective.The weight vector of linear combination is calculated and proved as follows.SupposeN kinds of subjective and objective methods to get l kinds of weights. The weight vectors are W1,W2,…,WL. The kth weight direction quantity is Wk=ϖ1k,ϖ2k,…,ϖmk. It satisfies that ∑j=1mϖjk=1,ϖjk≥0j=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∗=ϖ1∗,ϖ2∗,…,ϖm∗T be the linear combination weight vector of L weight vectors. It can be expressed in the vector form as W∗=∑k=1Lxkwk, where xk is the linear combination coefficient. It satisfies ∑k=1Lxk=1,xk≥0. The solution of linear combined weight vector is a multiobjective 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:(17)minμ∑i=1n∑j=1m∑k=1Lxkϖjk1−rij+1−μ∑xklnxk,s.t.∑k=1Lxk=1,xk≥0,where rij is the relational coefficient, 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 [3], there is a unique solution for the single objective optimization problem (SP). The solution is as follows:(18)x=s1∑k=1lsk,s2∑k=1lsk,…,sL∑k=1Lsk,where sk=exp−1+μ∑i=1n∑j=1mϖjk1−rij/1−μk=1,2,…,L, N is the number of evaluation schemes, and 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:Step 1: construct the model of complex product presentation based on the multilayer complex network model.Step 2: determine the evaluation index.Step 3: standardize the original three-parameter interval grey number effect evaluation index, and obtain the standardized effect evaluation vector formula of each scheme.Step 4: solve the ideal optimal solution for the decision-making problem and the effect evaluation vectorx+⊗ and the critical solution x−⊗.Step 5: obtain the three-parameter grey interval relational coefficient vector of each scheme, the ideal scheme and the critical scheme.Step 6: solve the TFAHP and Gini coefficient models, combine the weights, and obtain the weight of each scheme under different attributes.Step 7: calculate the three-parameter interval grey number relational degreeGx+⊗,xi⊗ and Gx−⊗,xi⊗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 Gxi⊗i=1,2,…,n of each scheme.Step 8: the schemes are sorted according to the relevance degreeGxi⊗i=1,2,…,n. The scheme corresponding to the maximum relational degree is the optimal one.The flow of this algorithm is shown in Figure3.Figure 3
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 ofG enterprise has been unanimously recognized as the world's first high-speed and high-power word synchronous frequency conversion centrifugal chiller after being certified by many experts. Its technology has reached the international leading level. Its design process will involve many parts, among which the parts have complex connections in many aspects. In this paper, multilayer network is used to sort out its relationship in detail, and the identification of key design parts is studied. The product drawing and component composition are shown in Figure 4 and Table 3.Figure 4
Large permanent magnet synchronous centrifugal unit.Table 3
Main parts and node name of permanent magnet synchronous centrifugal compressor.
NodePartsV1Principal axisV2Inspiratory chamberV3Import regulation structureV4ImpellerV5DiffuserV6VoluteV7Retaining ringV8Tight ringV9Thrust diskV10SeparatorV11Balance plateV12RefluxerV13SealingV14BendV15BearingV16Electrical machineV17Lubricated gearsV18Bearing with lubricationV19Curved casingV20Pressure-regulating valveV21Temperature sensorV22Extraction coolerV23Pressure sensorV24EvaporatorV25Return evaporatorV26Lubricating oilV27CryogenV28Oil coolerV29Electronic expansion valveV30Drier filterV31CondenserFirst of all, combined with enterprise product knowledge base, design base, case base, and interviews with designers, we analyze the relationship between functional behavior and structure network of the large permanent magnet synchronous centrifugal unit. The multilayer network knowledge association is shown from Tables4 to 6.Table 4
Function knowledge network connection information.
Function knowledge networkSourceTargetWeightSourceTargetWeightSourceTargetWeight140.529210.2718230.39170.599220.5018240.65180.5610110.9318250.70190.2810120.5918290.521110.5610130.5219200.121150.4610140.4819210.481170.5610150.5819220.561180.6911130.5519230.23230.4511150.5019250.67250.4411160.6219280.56260.4911170.4020230.472120.4411200.5820290.39340.2511210.5520310.58350.2512130.6421220.43390.4212140.5621230.583120.6912150.3521240.603140.5412160.6021270.22450.6613140.5121280.68460.5913150.5221290.364110.2713160.6121300.544120.4413210.4121310.414140.3613230.4422240.454210.1914150.4922250.204230.5714160.3222270.35560.3114190.2922280.52570.4814250.6822290.43580.5015160.5522300.385100.3915200.3722310.385230.5815230.3123270.42670.6716190.4623280.23680.6916200.3724250.216100.6016210.2524270.306120.6316230.3724280.606140.5816290.5524290.40780.5917180.5924300.51790.5417190.2924310.577100.6817200.6125270.357110.6317210.2025280.497130.7217230.6925300.517140.5117240.6825310.408110.2217250.4026290.258120.5217280.1927280.178130.8117310.5627300.409110.2718190.4727310.639120.4118200.3928310.429140.5118210.6129300.3530310.68Table 5
Behavior knowledge network connection information.
Behavior knowledge networkSourceTargetWeightSourceTargetWeightSourceTargetWeight160.459110.3417210.63170.349120.2717230.50180.439130.3717240.53190.569210.4417250.791110.289230.6018190.291150.3310110.6418200.60230.5310130.5618210.26240.5010140.2218250.45260.1610150.3219200.792130.4311130.4919210.46340.3711150.4619220.23350.5311160.3219250.66390.6211170.3120230.493120.5011200.3620290.283140.5211210.6621220.32450.2212130.5021230.59460.5212140.4221240.504110.4712150.5121300.454120.6712160.2921310.264130.5913140.2622240.464200.6113150.6922250.274210.6413160.5322290.23560.6313210.3622300.68570.5913230.5322310.24580.5614150.2123270.33590.2614160.3023280.415210.2114190.5324250.34670.6114250.5824270.42690.5115160.6324290.766100.6715200.6124300.416120.6915230.7024310.316130.4616190.6725270.51780.3316200.2625280.32790.4416210.2325310.367110.2316220.6126290.567120.4016280.2127300.427130.3917180.6627310.507140.6116280.2128310.578100.3417180.6629300.468120.3517190.4930310.378130.4117200.31Table 6
Structure knowledge network connection information.
Structure knowledge networkSourceTargetWeightSourceTargetWeightSourceTargetWeight140.2310120.4918290.21170.3110130.5019200.30180.2910150.3319210.79190.5011130.3219220.331110.4611150.5219250.661180.5211160.3019280.42230.4411170.4720230.38250.5711200.6520310.20340.4612130.4021230.44350.6312140.5421240.59390.4412150.6821270.143140.2113140.6321280.11450.5913150.6621300.23460.4513160.2821310.534110.4113230.2922240.444120.4014150.7922250.464230.4814160.4822270.41560.3114190.5822280.65570.3514250.3422300.71590.3015160.3122310.495230.2815200.4623270.42670.3815230.2423280.45680.6516190.7024250.756120.2816200.6724280.876140.2216210.2324290.49780.4716230.5424300.35790.6116290.4324310.497100.6117180.4125270.477110.3817190.4425300.337140.4317200.6725310.558110.4717210.5426290.568120.3817250.2827280.198130.4617280.6227300.459110.3617310.2627310.879120.4618190.6028310.689140.4418200.5529300.569220.5218230.3130310.3910110.5518240.45Through the calculation of index system, we can get the three-parameter interval grey number of the evaluation index as follows. This paper combines the enterprise product knowledge base, design library, and case library, as well as interviews with designers, to analyze the relationship and the FBS complex network between the parts of the large permanent magnet synchronous centrifugal unit. Then, the network characteristic analysis is carried out. According to the internal expert experience of the enterprise, the FBS multilayer network is assigned (1/3, 1/3, 1/3), (1/2, 1/4, 1/4), and (1/2, 1/4), 1/4) for obtaining three-parameter index data. The three-parameter value of the value information of the node is obtained through enterprise research. Then, we can get the three-parameter interval grey number of the evaluation index as follows:(19)X⊗O=5.27725.37555.4411139142.61451.57011.70901.80162.28122.46582.58893.69594.02574.24554347.2502.93993.21293.39480.72820.75120.76653.83584.43894.84098790923.52533.78953.96560.81950.84550.86284.20194.37264.4863128131.61343.49823.79583.99422.03472.13632.20402.06562.58272.92743133.4353.33493.68143.91240.37460.38370.38983.51863.46483.42903236.2393.18013.35713.47510.34700.36830.38253.25303.66873.94583034.2372.74203.09893.33680.90960.94050.96113.61483.75313.84532125.2282.02112.46752.76510.14580.45380.65913.55153.75443.88962629.6323.68573.83823.93980.99850.80670.67881.28761.86672.25282327.8313.83364.10624.28790.53790.55670.56922.36932.73212.97403639413.39773.82904.11660.83600.85810.87282.84363.24673.51543841432.70093.13483.42400.72790.77020.79852.17372.63982.95063235373.46803.64693.76620.92410.96120.98592.45602.78032.99642629313.68453.84223.94730.70170.77020.81581.98522.14862.25743540.4444.67994.81294.90150.81260.84920.87372.67773.20853.56242932343.15673.55323.81750.37070.39470.41071.45772.17852.65902627.8293.07783.48013.74830.61010.66280.69803.92264.25124.47037683.8892.91473.23873.45461.00311.17341.28694.67844.97075.16557886.4923.61774.03684.31611.02111.18841.29992.06722.53482.84663033.6363.93654.21574.40190.57930.58980.59682.51802.79852.98563538.6414.08274.53554.83730.42040.43330.44202.97063.43963.75223740.6433.41843.61103.73950.49230.50400.51182.65692.82982.94513639.6423.85624.07284.21730.47350.50340.52343.90194.46994.84865558.6612.99483.40433.67730.81010.83830.85723.12643.58693.89393438.2413.38303.73073.96250.70720.78740.84093.60974.19824.59063034.2373.34963.53093.65170.49950.52180.53673.81474.06914.23862932343.39144.12874.62020.54350.56530.57992.84573.27983.56922830.4323.39913.52783.61360.43070.48190.51613.13863.82434.28143739.4413.99174.15194.25870.50290.54330.57034.12644.73215.13582932.6353.60024.34664.84420.41900.42620.43103.60973.96174.19642628.4303.72394.16494.45900.53030.54740.5587.The normalized three-parameter interval grey number evaluation matrix is(20)X⊗=0.96060.98421.00000.95160.98061.00001.00000.96340.93540.55160.64130.70110.57980.65920.71220.17740.21130.23390.59190.50960.45460.28300.29410.30160.61350.75870.85550.53230.55650.57260.41530.33560.28240.32730.33990.34840.70170.74270.77010.86290.89190.91130.42350.33370.27380.91780.96711.00000.18730.31180.39480.08060.10000.11290.47270.36820.29850.11120.11560.11850.53710.52420.51560.08870.12260.14520.51950.46600.43040.09780.10810.11500.47320.57330.64000.07260.10650.12900.65160.54400.47220.37110.38610.39610.56030.59360.61580.00000.03390.05650.86920.73450.64470.00000.14960.24940.54510.59390.62650.04030.06940.08870.36690.32090.29020.41430.32110.25900.00000.13940.23240.01610.05480.08060.32230.24000.18520.19050.19960.20570.26040.34780.40600.12100.14520.16130.45380.32360.23690.33540.34610.35320.37460.47170.53640.13710.16130.17740.66400.53310.44580.28280.30340.31710.21330.32560.40040.08870.11290.12900.43260.37860.34260.37810.39620.40820.28130.35940.41140.04030.06450.08060.36730.31970.28790.27010.30340.32550.16800.20730.23350.11290.15650.18550.06690.02680.00000.32400.34180.35360.33470.46250.54770.06450.08870.10480.52650.40690.32710.10930.12090.12870.04090.21450.33020.04030.05480.06450.55030.42890.34800.22560.25120.26830.63440.71350.76630.44350.50650.54840.59950.50180.43660.41650.49930.55440.81640.88680.93370.45970.52740.57260.38740.26090.17670.42530.50660.56070.18770.30030.37530.07260.10160.12100.29120.20690.15080.21060.21570.21910.29620.36380.40880.11290.14190.16130.24710.11050.01940.13340.13970.14390.40520.51810.59340.12900.15810.17740.44750.38940.35070.16830.17400.17780.32970.37130.39910.12100.15000.16940.31540.25010.20650.15920.17380.18340.62940.76620.85740.27420.30320.32260.57540.45180.36940.32270.33650.34560.44270.55360.62750.10480.13870.16130.45820.35330.28330.27280.31170.33770.55910.70080.79520.07260.10650.12900.46830.41360.37710.17190.18270.18990.60840.66970.71050.06450.08870.10480.45570.23320.08490.19320.20380.21090.37510.47960.54930.05650.07580.08870.45340.41450.38860.13840.16330.17990.44560.61070.72080.12900.14840.16130.27450.22620.19400.17350.19310.20620.68350.82930.92650.06450.09350.11290.39270.16740.01730.13270.13620.13850.55910.64380.70030.04030.05970.07260.35540.22230.13350.18680.19510.20060.96060.98421.00000.95160.98061.00001.00000.96340.93540.91780.96711.00000.00000.13940.23240.00000.03390.05650.06690.02680.00000.00000.10810.1150.According to formula (4), the effect evaluation vectors of ideal optimal scheme and critical scheme are obtained:(21)X+⊗=0.96060.98421.0000,0.95160.98061.0000,1.00000.96340.9354,0.91780.96711.0000,X−⊗=0.00000.13940.2324,0.00000.03390.0565,0.06690.02680.0000,0.00000.10810.1150.Then, we use TFAHP to calculate the weight:TheA-B comprehensive fuzzy judgment matrix is obtained, as shown in Table 7.Table 7
A-B comprehensive fuzzy judgment matrix.
AB1B2B3B4B10.50.50.50.2920.3420.3920.5170.5670.6170.4630.5130.563B20.6110.6610.7110.50.50.50.5630.6130.6630.5440.5940.644B30.3920.4420.4920.3410.3910.4410.50.50.50.4330.4830.533B40.4420.4920.5420.3650.4150.4650.4710.5210.5710.50.50.5The fuzzy complementary matrix is shown in Table8.Table 8
Fuzzy complementary matrix.
B1B2B3B4B10.5000.3420.5670.513B20.6610.5000.6130.594B30.4420.3910.5000.483B40.4920.4150.5210.500The relative weights of several indexes obtained by TFAHP are as follows:(22)W=w1,w2,w3,w4=0.19,0.32,0.21,0.28.The weight obtained from the Gini coefficient is as follows:W=w1,w2,w3,w4=0.24,0.29,0.17,0.30.Then, we can calculate the comprehensive weightW=w1,w2,w3,w4=0.235,0.30,0.18,0.285.According to equations (8) and (10), the grey interval relational degree of each scheme with the ideal optimal scheme and the critical scheme can be obtained. Then, the linear relational degree is calculated by formula (5) (according to [28] and experts’ opinion, λ=0.5, ξ=0.5, and β1=β2=0.5).And, we also calculate the node importance ranking through the method of [17]. The above results are shown in Tables 9 and 10 and Figure 5.Table 9
Calculation results and comparison.
NodeGx+⊗,xi⊗Gx−⊗,xi⊗Linear correlation degreeRanking of this article10.72270.50260.6101220.61960.59500.51231630.65420.62440.51491540.83670.42280.7070150.69100.48360.6037360.59020.68870.45083070.70860.61460.54701180.58710.58460.50132090.59330.58730.503019100.62200.71730.452428110.74880.71160.518613120.67410.53620.56905130.62330.71900.452229140.81760.70230.55778150.80650.69800.55439160.73190.57110.58044170.71810.73090.493625180.69020.69970.495323190.67560.68640.494624200.79510.69310.551010210.66260.71560.473526220.67710.65900.509117230.62600.61170.507218240.61720.48060.56836250.66110.65910.501021260.67770.64290.517414270.66950.72290.473327280.67930.78520.447031290.62310.62450.499322300.65830.61320.522512310.67550.54650.56457Table 10
Ranking of node importance calculated in [17].
NodeCorrelation degree calculated in [17]Ranking of [17]10.6842220.59331630.59841440.6952150.6807360.51633070.61251080.57052090.585918100.532827110.602013120.65075130.524629140.62149150.62677160.66904170.537626180.550523190.568021200.607811210.526328220.582319230.593117240.62158250.557022260.594915270.543024280.502331290.542025300.602312310.63776Figure 5
Comparison of relational degree results.According to the method of [17], the ranking of node importance is shown in Table 10.Then, we compare the results of the two methods, as shown in Figure5.From the calculation results of this method and the comparison with other methods, we can see that the calculation results of the two methods are in the top several, respectively, impeller≻ principal axis ≻ diffuser ≻ electrical machine ≻ evaporator ≻ evaporator ≻ condenser ≻ bend and impeller ≻ principal axis ≻ diffuser ≻ electrical machine ≻ refluxer ≻ condenser ≻ bearing ≻ evaporator. We can see that the ranking results of the two are roughly the same, except for some subtle differences. The ranking results are realistic and roughly the same. However, we can see from the graphic difference in Figure 5 that the ranking difference obtained by the method proposed in this paper is stronger. This method makes the ranking more obvious and easy to identify. The evaluation method proposed in this paper increases the operability and identifiability of the importance ranking of design elements in the design process.


## 7. Conclusion

In order to comprehensively describe the complex product design process, this paper uses the complex product FBS model to describe the complex detailed design relationship. Then, the identification index system of important parts in the complex product design stage is constructed according to the characteristics of network. The three-parameter interval grey number relational model is improved based on the fuzzy analytic hierarchy process. The comparison of experimental results shows that the results calculated by the proposed method are more representative. It also shows that the method is practical and effective.The multilevel description model of the design process based on FBS proposed in this paper can describe the design process in more detail, and the analysis of the detailed design process is more specific. The evaluation of important parts based on this is more scientific. In addition, the weight improvement of the three-parameter interval grey number based on the fuzzy analytic hierarchy process and the Gini coefficient method is in line with the law of actual data and is more effective. The research comparison of this paper also shows the effectiveness and feasibility of the proposed method.The next research prospect of this paper is to further improve the objectivity of data acquisition in the design stage and reduce the proportion of experience data of designers. Subsequently, we will try to establish a more scientific design database, standardize the management of its design process, and make its data acquisition more accurate and scientific. It will provide service structure for each department to obtain information, which makes the research more practical.



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*Source: 7185830-2021-09-02.xml*