# Supply-Demand Dynamic Matching in Cloud Manufacturing Based on Hypernetwork Model
## 3 Modeling of the SDM Problem Based on the Hypernetwork Model
### Initial Matching Model
#### 3.3.1 Network Resilience Indicator Modeling  
(1) Network connectivity consideration  
From the perspective of network connectivity, nodes with stronger connections to other nodes in the network are selected to ensure more efficient information transfer during task execution. Centralised service providers are ideal as they facilitate resource sharing, reduce redundancy, and lower costs.  
The Clustering Coefficient [43] is used to measure the connectivity between nodes in a network, representing the proportion of actual connections among neighbouring nodes. The local clustering coefficient of a node indicates the proportion of actual connections between its neighbours, and it is calculated by Equation (12).  
\[C_{i}=\frac{2D_{act}^{i}}{D^{i}(D^{i}-1)}, \tag{12}\]  
where \(D^{i}\) represents the degree of node \(i\), indicating the number of edges directly connected to the node \(i\), i.e., the number of its neighbors. This can be computed by Equation (13).  
\[D^{i}=\sum_{i^{\prime}=1}^{NoS}e_{ii^{\prime}}^{S}\ \ \ (i=1,2,...,NoS). \tag{13}\]  
The actual number of connections among neighbors of node \(i\) is denoted as \(D_{act}^{i}\), and \(I_{i}\) represents the set of all neighbor nodes of node \(i\). The calculation is given by Equation (14).  
\[D_{act}^{i}=\frac{1}{2}\sum_{i^{\prime}\in I_{i}}\sum_{i^{\prime}\in I_{i}}e_{ i^{\prime}i^{\prime\prime}}^{S}\ \ \ (i=1,2,...,NoS). \tag{14}\]  
The Average Clustering Coefficient reflects the proportion of edges that exist among the selected services relative to all edges. It is defined as the average of the clustering coefficients across all nodes, calculated by Equation (15).  
\[U_{\rm C}=\frac{\sum_{j=1}^{NoT}\sum_{i=1}^{NoS}(x_{ij}*C_{i})}{NoT}. \tag{15}\](2) Network load balance consideration  
A reasonable load distribution can enhance the manufacturing service network's capacity and improve resource utilisation, which is also one of the important concepts of CMfg [44].  
The load on each service is defined as the total task demands it receives. \(B_{T}^{j}\) represents the quantity of task demands received by service \(i\), the service load \(L_{i}\) for service \(i\) is expressed as Equation (16).  
\[L_{i}=\sum_{j=1}^{NoT}(x_{ij}*B_{T}^{j})\ \ \ (i=1,2,...,NoS). \tag{16}\]  
The average service load (\(\overline{L}\)) for all services is defined as Equation (17).  
\[\overline{L}=\frac{\sum_{i=1}^{NoS}L_{i}}{NoS}. \tag{17}\]  
Load balance is defined as the variance of service loads, reflecting the degree of balance in load distribution. The load balance (\(U_{L}\)) can be expressed as Equation (18).  
\[U_{L}=\frac{\sum_{i=1}^{NoS}(L_{i}-\overline{L})^{2}}{NoS}. \tag{18}\]  
To evaluate the overall performance of the network resilience considering both connectivity and load balance, weighting coefficients \(\omega_{C}\) and \(\omega_{L}\) are introduced. The Average Clustering Coefficient is a positive correlation indicator, representing the connectivity between selected service nodes in the network. Conversely, load balance is a negative correlation indicator, representing the degree of load balance among service nodes in the network. Typically, a lower value of \(U_{L}\) indicates a more evenly balanced load distribution. Therefore, the negative value of \(U_{L}\) is weighted to facilitate comprehensive maximisation. The composite evaluation function is denoted as \(U_{1}\), which is given by Equation (19).  
\[U_{1}=\omega_{C}*U_{C}-\omega_{L}*U_{L}. \tag{19}\]