# Evaluation of the Robustness for Integrated Production Scheduling and Maintenance Planning Problem
## 3 Completion Time of the Operation
### Impact of PM on Operational Completion Time  
We assume that machines (i.e., \(M1,M2,M3\)) have PM activities executed after operations \(O_{2,2},O_{2,1}\)_and_\(O_{1,3}\). The feasible schedule with preventive maintenance is shown in Figure 2. A feasible schedule that has executed PM activities. The PM activities are indicated with dark blocks. From the perspective of operation \(O_{3,3}\), the other operations can be divided into two sets. The set consisting of \(\{O_{1,1},O_{1,2},O_{2,1},O_{2,2},O_{3,1},O_{3,2}\}\) is called the associated set of \(O_{3,3}\). The set consisting of \(\{O_{1,3},O_{2,3}\}\) is called the non-associated set of \(O_{3,3}\). For the operations in the associated set, if PM activity occurs before or after one operation, it may cause the delay of \(O_{3,3}\). However, the completion time of the operation \(O_{3,3}\) can only have its delay prevented when a PM activity is executed after the operation \(O_{3,2}\). As shown in Figure 2, there are four precedence constrain chains to \(O_{3,3}\). In addition, the influence can be delivered to \(O_{3,3}\) through any chain. In chain 1, i.e., \(O_{1,1}\to O_{1,2}\to O_{2,2}\to O_{3,3}\), the idle time between operations can absorb the impact of PM. The idle time between \(O_{2,2}\) and \(O_{3,3}\) can be obtained by \(Idle_{2,2}^{3,3}=S_{3,3}-C_{2,2}\). The completion time of operation \(O_{3,3}\), affected by chain 1, is denoted as \(C_{3,3}^{pm,1}=C_{1,1}+\max\left\{x_{1,1}^{1,2}tp,Idle_{1,1}^{1,2}\right\}+p_ {1,2}+\max\left\{x_{1,2}^{2,2}tp,Idle_{1,2}^{2,2}\right\}+p_{2,2}+\max\left\{x _{2,2}^{3,3}tp,Idle_{2,2}^{3,3}\right\}+p_{3,3}\). In chain 2, i.e., \(O_{1,1}\to O_{2,1}\to O_{2,2}\to O_{3,3}\), the completion time of the operation \(O_{3,3}\), affected by chain 2, is denoted as \(C_{3,3}^{pm,2}=C_{1,1}+\max\left\{x_{1,1}^{2,1}tp,Idle_{1,1}^{2,1}\right\}+p_ {2,1}+\max\left\{x_{2,1}^{2,2}tp,Idle_{2,1}^{2,2}\right\}+p_{2,2}+\max\left\{x _{2,2}^{3,3}tp,Idle_{2,2}^{3,3}\right\}+p_{3,3}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Jobs** & \multicolumn{3}{c}{**Processing Time of the Operations on Machine**} \\ \hline J1 & \(O_{1,1}(M2,2)\) & \(O_{1,2}(M1,3)\) & \(O_{1,3}(M3,1)\) \\ J2 & \(O_{2,1}(M2,3)\) & \(O_{2,2}(M1,2)\) & \(O_{2,3}(M3,4)\) \\ J3 & \(O_{3,1}(M3,4)\) & \(O_{3,2}(M2,3)\) & \(O_{3,3}(M1,3)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Process table for a JSP problem with size \(3\times 3\). Figure 1: A feasible \(3\times 3\) JSP schedule in Table 1.  
Figure 2: A feasible schedule that has executed PM activities. In chain 3, i.e., \(O_{1,1}\to O_{2,1}\to O_{3,2}\to O_{3,3}\), the completion time of the operation \(O_{3,3}\), affected by chain 3, is denoted as \(C_{3,3}^{pm,3}=C_{1,1}+\max\left\{x_{1,1}^{2,1}t\,p,Idle_{1,1}^{2,1}\right\}+p_ {2,1}+\max\left\{x_{2,1}^{3,2}t\,p,Idle_{2,1}^{3,2}\right\}+p_{3,2}+\max\left\{x _{3,2}^{3,3}t\,p,Idle_{3,2}^{3,3}\right\}+p_{3,3}\). In chain 4, i.e., \(O_{3,1}\to O_{3,2}\to O_{3,3}\), the completion time of the operation \(O_{3,3}\), affected by chain 4, is denoted as \(C_{3,3}^{pm,A}=C_{3,1}+\max\left\{x_{3,1}^{3,2}t\,p,Idle_{3,1}^{3,2}\right\}+p_ {3,2}+\max\left\{x_{3,2}^{3,3}t\,p,Idle_{3,2}^{3,3}\right\}+p_{3,3}\). where \(x_{i,j}^{k,s}=\left\{ \begin{array}{c}1\;if\;there\;is\;a\;PM\;activity\; inserted\;between\;O_{i,j}\;and\;O_{k,s}\;,\;x_{2,1}^{3,2},\;x_{2,2}^{3,3}\; \;are\;all\;\;0\;otherwise\end{array} \right.\)  
equal to 1, and the rest of the binary variables are 0. The completion time of the operation \(O_{3,3}\) caused by PM activities is denoted as \(C_{3,3}^{pm}=\max\left\{C_{3,3}^{pm,1},C_{3,3}^{pm,2},C_{3,3}^{pm,3},C_{3,3}^{ pm,A}\right\}\). The sequence of PM is determined when the preventive maintenance plan is developed. In addition, the time of PM is often considered a constant. The calculation of PM's influence on operation completion time is a deterministic problem.