## 5 Analysis of Impact of Resource Failures on CPS
Note that the flows \(f_{j1}\), \(j\in\mathbf{J}\), are the parts of the nominal solution corresponding to the flows in \(STN_{j}(V_{j},A_{j})\) not influenced by the resource failures \(\Delta(m)\). As the nominal solution meets the deadline, \(f_{j1}\), \(j\in\mathbf{J}\), meet the deadline. We show that \(f_{j_{k}2}^{\prime}\), \(j_{k}\in\mathbf{J}(\Delta(m))\), meet the deadline. Note that \(f_{j_{k}2}^{\prime}\) is obtained by solving the FPRSTN of \(RSTN_{j_{k}}(V_{j_{k}},A_{j_{k}})\) with residual capacity constraint \(\widetilde{C}_{rt}\) for each \(j\in\mathbf{J}(\Delta(m))\). As \(Robustness_Indicator_{j}=\) \"true\" for each \(j_{k}\in\mathbf{J}(\Delta(m))\), where \(k\in\{1,2,3,\ldots,I\}\), it implies that the solution \(f_{j_{k}2}^{\prime}\) meets the deadline for each \(j\in\mathbf{J}(\Delta(m))\). Therefore, \(f_{j_{k}2}^{\prime}\), \(j_{k}\in\mathbf{J}(\Delta(m))\) and \(f_{j1}\) for each \(j\in\mathbf{J}\) can meet the deadline. Based on the discussions above, if the \(Robustness_Indicator\) is \"true\" by applying the Algorithm 3 to check robustness of \(G=GJ\|GR\) with respect to \(\Delta(m)\), \(f_{j_{k}2}^{\prime}\), \(j\in\mathbf{J}(\Delta(m))\), and \(f_{j1}\), \(j\in\mathbf{J}\), form a solution that satisfies the capacity constraints, flow balance constraints as well as the deadline. ## Appendix C

Example 2: 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 Figures 1 and 2, respectively. The CPS model of the type-1 product is shown in \[\sum_{w\in V}f_{j}(u,w)=0\ \forall u\
otin\{s,e\} \tag{11}\]2 Suppose an order has arrived. The order requires three type-1 products, one type-2 product and deadline AM 9:40. A time horizon starting from AM 8:00 is considered to handle the order. The time horizon is divided into time periods and the duration of each period is ten minutes. For this example, the order deadline AM 9:40 is presented by \(\theta=10\) and we set \(\Pi=10\) in our problem formulation. The data of the order described based on the above parameters is summarized in \[\sum_{w\in V}f_{j}(u,w)=0\ \forall u\
otin\{s,e\} \tag{11}\]3 The operations with associated transitions in the type-1 task subnet and the associated processing time of the resources are shown in \[\sum_{w\in V}f_{j}(u,w)=0\ \forall u\
otin\{s,e\} \tag{11}\]4 The operations with associated transitions in the type-2 task subnet and the associated processing time of the resources are shown in \[\sum_{w\in V}f_{j}(u,w)=0\ \forall u\
otin\{s,e\} \tag{11}\]5

By applying Algorithm 2 to check feasibility of nominal CPS, we obtain a nominal solution for type-1 products as follows: \(f_{1,12}=1\), \(f_{4,15}=1\), \(f_{5,16}=1\), \(f_{12,24}=1\), \(f_{15,27}=1\) and \(f_{16,28}=1\). To present the solution clearly, the above solution is shown in a spatial-temporal network \(STN_{1}(V_{1},A_{1})\) in Figure 10(a). We also obtain a nominal solution for type 2 products as follows: \(f_{2,14}=1\), \(f_{14,25}=1\), \(f_{25,36}=1\). To present the solution clearly, the above solution is shown in a spatial-temporal network \(STN_{2}(V_{2},A_{2})\) in Figure 10(b). Obviously, the above solution can meet the deadline of the order. Suppose a resource failed during the period from AM 8:00. Suppose the failure is expected to be recovered by AM 8:20. The resource failure information is shown in \[\sum_{w\in V}f_{j}(u,w)=0\ \forall u\
otin\{s,e\} \tag{11}\]6 As there are 16 places in the CPS model, we represent the resource failure by the perturbation vector \(\delta=\) [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] and discrete time interval \(\omega_{21}=[\alpha_{21}\ \beta_{21}]=\) [1 2]. To assess the influence of the resource failure, a residual spatial-temporal network is constructed based on \(\Delta(m)=(m,\delta,\omega)\). A solution is found by solving the FPRSTN defined by \(RSTN_{1}(V_{1},A_{1})\) with \(\Delta(m)=(m,\delta,\omega)\).