## 5 Analysis of Impact of Resource Failures on CPS
We prove by showing that the solution \(f^{\prime}_{j2}\) obtained by solving the FPRSTN of \(RSTN_{j}(V_{j},A_{j})\) with residual capacity constraint \(\widetilde{C}^{j}_{rt}\) for each \(j\in J(\Delta(m))\) and the solution \(f_{j1}\) (corresponding to the flows in \(STN_{j}(V_{j},A_{j})\) not influenced by the resource failures \(\Delta(m)\)) for each \(j\in J\) form a solution that satisfies the constraints and meet the deadline for the original problem. We first show that \(f^{\prime}_{j2}\) for each \(j\in J(\Delta(m))\) and \(f_{j1}\) for each \(j\in J\) satisfy the capacity constraints. Suppose \(J(\Delta(m))=\{j_{k}|k\in\{1,2,\ldots,I\}\,\}\). As the _Robustness_Indicator_ is \"true\" by applying the Algorithm 3 to check robustness of \(G=G\|GR\) with respect to \(\Delta(m)\), _Robustness_Indicator_\(j\) = \"true\" for \(j_{k}\), where \(k\in\{1,2,3,\ldots,I\}\). As _Robustness_Indicator_ = \"true\", where \(j=j_{k}\) with \(k\in\{1,2,3,\ldots,I\}\), it implies that the solution \(f^{\prime}_{jk2}\) for the FPRSTN defined by \(RSTN_{j_{k}}(V_{j_{k}},A_{j_{k}})\) with residual capacity constraint \(\widetilde{C}^{j_{k}}_{rt}\) satisfies the condition of Property 1. That is, the flows associated with the solution \(f^{\prime}_{jk2}\) in the arcs from node \(N_{j_{k}}\Pi+t\) to node \(e\) are all zero for all \(t\in\{\theta+1,\theta+2,\ldots,\Pi\}\) and \(f^{\prime}_{j2}\) can meet the due date \(\theta\). Note that the solution \(f^{\prime}_{jk2}\) is found by solving the FPRSTN defined by \(RSTN_{j_{k}}(V_{j_{k}},A_{j_{k}})\) with residual capacity constraint \(\widetilde{C}_{rt}\). Without loss of generality, suppose \(j=j_{1}\) is selected from \(J(\Delta(m))\) for the first iteration. The following inequalities hold:

Figure 6:0 Figure 6:1After processing \(j=j_{1}\), the residual capacity is updated by

Figure 6:2

Figure 6:3

The set \(\textbf{J}(\Delta(m))\) is updated as follows:

Figure 6:4

The algorithm proceeds to the second iteration. Suppose \(j=j_{2}\) is selected from \(\textbf{J}(\Delta(m))\) for the second iteration. Suppose the solution \(f_{j2}^{\prime}\) is found by solving the FPRSTN defined by \(RSTN_{j_{k}}(V_{j_{k}},A_{j_{k}})\) with residual capacity constraint

Figure 6:5

The solution \(f_{j2}^{\prime}\) must satisfy the capacity constraint:

Figure 6:6

After processing \(j=j_{2}\), the residual capacity is updated by

Figure 6:7 Figure 6:8

By following similar procedure, after finishing the iteration \(I-1\), the solution \(f_{j_{(I-1)}^{\prime}}^{\prime}\) must satisfy the capacity constraint:

Figure 6:9

Therefore, the following constraint holds:

Figure 7:0

Figure 7:1

Figure 7:2

Thus the capacity constraint is satisfied. Therefore, \(\sum\limits_{j\in I}f_{j1}+\sum\limits_{j\in I(\Delta(m))}f_{j2}^{\prime}\) is a solution that satisfies the capacity constraints after resource failures of \(\delta\in\Delta(m)\). Next, we show that \(f_{j12}^{\prime},j_{k}\in\textbf{J}(\Delta(m))\), and \(f_{j1},j\in\textbf{J}\), satisfy the flow balance constraints. Let \(Q_{j}^{\delta}\) denote the quantity of the products influenced by the resource failure \(\delta\). Note \(f_{j1}\) must satisfy the following constraints:

Figure 7:3 (A17)Figure 7:4

Figure 7:5

Note \(f_{j_{k}2}^{\prime}\) must satisfy the following constraints:

Figure 7:6

Figure 7:7

Figure 7:8

The following equations hold:

Figure 7:9

\[\sum_{w\in V}f_{j}(u,w)=0\ \forall u\
otin\{s,e\} \tag{11}\]0

\[\sum_{w\in V}f_{j}(u,w)=0\ \forall u\
otin\{s,e\} \tag{11}\]1

Hence \(f_{j_{k}2}^{\prime}\) and \(f_{j1}\), \(j\in\mathbf{J}\), satisfy the flow balance constraints. By following reasoning, \(f_{j_{k}2}^{\prime}\) for each \(j_{k}\in\mathbf{J}(\Delta(m))\) and \(f_{j1}\) for each \(j\in\mathbf{J}\) satisfy the flow balance constraints. Finally, we show that \(f_{j_{k}2}^{\prime}\), \(j\in\mathbf{J}(\Delta(m))\), and \(f_{j1}\), \(j\in\mathbf{J}\), meet the deadline.