## 5 Analysis of Impact of Resource Failures on CPS
The number of arcs constructed in the spatial-temporal network \(STN_{j}(V_{j},A_{j})\) for type \(j\) task, where \(j\in\mathbf{J}\), is bounded by \(2N_{j}(\Pi-1)+2+\Pi\). Therefore, the complexity to construct \(STN_{j}(V_{j},A_{j})\) and the number of nodes in \(STN_{j}(V_{j},A_{j})\) is bounded by \(O(N_{j}\Pi)\). As the structure of \(RSTN_{j}(V_{j},A_{j})\) is the same as \(STN_{j}(V_{j},A_{j})\), the complexity to construct \(RSTN_{j}(V_{j},A_{j})\) and the number of nodes in \(RSTN_{j}(V_{j},A_{j})\) is bounded by \(O(N_{j}\Pi)\). The problem to find the solution \(f_{j2}^{r}\) for the FPRSTN (feasibility problem of residual spatial-temporal network) defined by \(RSTN_{j}(V_{j},A_{j})\) with residual capacity constraint \(\widetilde{C}_{rt}\) is a minimal cost flow problem with additional constraints. Therefore, a lower bound of the computational complexity to find the solution \(f_{j2}^{r}\) for the FPRSTN defined by \(RSTN_{j}(V_{j},A_{j})\) with residual capacity constraint \(\widetilde{C}_{rt}\) is \(O(N_{j}^{2}\Pi^{2}Q_{j}^{\ell})\). Therefore, a lower bound on the computational complexity of the algorithm to construct \(RSTN_{j}(V_{j},A_{j})\) and find a solution is \(O(J\!R^{2}\Pi^{2}+N_{j}^{2}\Pi^{2}Q_{j}^{\ell})\). As \(|J(\Delta(m))|=I\) is no greater than \(J\), a lower bound on the computational complexity of the Algorithm 3 to Check Robustness is \(O(J^{2}R^{2}\Pi^{2}+JN_{j}^{2}\Pi^{2}Q_{j}^{\ell})\), which is of polynomial complexity. Although the above analysis only provides a lower bound on the computational complexity of the algorithm, we will show that the numerical results indicate that computational time grows polynomially, which is consistent with above lower bound on the computational complexity analyzed above. ## 6 Results

In this section, we illustrate the proposed method by examples and present the results to illustrate computational feasibility of the proposed method. We first illustrate the results obtained by applying the proposed method to two examples that represent two application scenarios. In Example 1, a scenario in which resource failures influence only one type of product is considered. The patterns of resource failures in the scenario of Example 2 influence two types of products. We illustrate how the proposed method works to check the robustness of CPS with respect to resource failures. Following Example 1 and Example 2, we will present the results to study computational feasibility of the proposed method. **Example 1**. : _Consider a CPS that can produce two types of products. The production processes of the two types of products and the resources are modeled by task subnets, \(G\!I_{1}\) and \(G\!I_{2}\) in Figure 1 and the resource models in Figure 2, respectively. The CPS model is shown in Figure 3. The models can be download from the following link: [[https://drive.google.com/drive/folders/1](https://drive.google.com/drive/folders/1) riBYKMoMDJOU1 KtylDGlz_nm1 ffXOwdY?usp=sharing]([https://drive.google.com/drive/folders/1](https://drive.google.com/drive/folders/1) riBYKMoMDJOU1 KtylDGlz_nm1 ffXOwdY?usp=sharing) (accessed on 15 May 2021)._

Suppose an order which requires three type-1 products and deadline AM 9:20 has arrived. A time horizon starting from AM 8:00 is considered to handle the order. Thetime horizon is divided into time periods and duration of each period is ten minutes. For this example, the order deadline AM 9:20 is presented by \(\theta=8\) and we set \(\Pi=10\) in our problem formulation. The data of the order described based on the above parameters is summarized in Table 2. As the order only requires type-1 products, only the operations with associated transitions in the type-1 task subnet and the associated processing time of the resources are shown in Table 3.

By applying Algorithm 2 to check feasibility of nominal CPS, we obtain a nominal solution as follows: \(f_{1,12}=1\), \(f_{4,15}=1\), \(f_{5,16}=1\), \(f_{12,24}=1\), \(f_{15,27}=1\) and \(f_{16,28}=1\).