# Evaluation of the Robustness for Integrated Production Scheduling and Maintenance Planning Problem
## 3 Completion Time of the Operation
### Influence of Machine Failure on Operational Completion Time  
The machine breakdowns will lead to the interruption of the processing of the jobs. Since this paper assumes that the jobs continue to be processed after the machines are repaired, it can be considered that the machine breakdowns substantially increase the time the job stays on the machine; at the operational level, it can be regarded as the increase of operation processing time. Assume that these operations (i.e., \(O_{2,1},O_{2,2},O_{1,3}\)) will suffer from breakdowns. The actual schedule is shown in Figure 3, and the breakdowns are indicated with red blocks. Through the same above principle, there are four precedence constrain chains to \(O_{3,3}\). The effect can be delivered to \(O_{3,3}\) through any chain.  
The completion time of \(O_{3,3}\), affected by chain 1, is denoted as \(C_{3,3}^{cm,1}=C_{1,1}+\max\left\{\left(y_{1,1}\cdot tc-Idle_{1,1}^{1,2} \right),0\right\}+p_{1,2}+\max\left\{\left(y_{2,2}\cdot tc-Idle_{1,2}^{3,3} \right),0\right\}+p_{1,2}+\max\left\{\left(y_{2,2}\cdot tc-Idle_{2,2}^{3,3} \right),0\right\}+p_{3,3}+\max\left\{y_{3,3}\cdot tc,0\right\}\).  
The completion time of \(O_{3,3}\), affected by chain 2, is denoted as \(C_{3,3}^{cm,2}=C_{1,1}+\max\left\{\left(y_{1,1}\cdot tc-Idle_{1,1}^{2,1} \right),0\right\}+p_{2,1}+\max\left\{\left(y_{2,1}\cdot tc-Idle_{2,1}^{2,2} \right),0\right\}+p_{2,2}+\max\left\{\left(y_{2,2}\cdot tc-Idle_{2,2}^{3,3} \right),0\right\}+p_{3,3}+\max\left\{y_{3,3}\cdot tc,0\right\}\).  
The completion time of \(O_{3,3}\), affected by chain 3, is denoted as \(C_{3,3}^{cm,3}=C_{1,1}+\max\left\{\left(y_{1,1}\cdot tc-Idle_{1,1}^{2,1} \right),0\right\}+p_{2,1}+\max\left\{\left(y_{2,1}\cdot tc-Idle_{2,1}^{3,2} \right),0\right\}+p_{3,2}+\max\left\{\left(y_{3,2}\cdot tc-Idle_{3,2}^{3,3} \right),0\right\}+p_{3,3}+\max\left\{y_{3,3}\cdot tc,0\right\}\).  
Figure 3: A feasible schedule that has suffered from machine breakdowns.  
The completion time of \(O_{3,3}\), affected by chain 4, is denoted as \(C_{3,3}^{m,A}=C_{3,1}+\max\left\{\left(y_{3,1}\cdot tc-Idle_{3,1}^{3,2}\right),0 \right\}+P_{3,2}+\max\left\{\left(y_{3,2}\cdot tc-Idle_{3,2}^{3,3}\right),0 \right\}+p_{3,3}+\max\left\{y_{3,3}\cdot tc,0\right\}\) where \(y_{i,j}=\left\{1\;if\;a\;failure\;takes\;place\;on\;operation\;O_{i,j}\,,\) and \(y_{2,1},y_{2,2},y_{1,3}\) are all equal to one and the rest of the binary variables are zero.  
The completion time of operation \(O_{3,3}\) caused by machine breakdowns is denoted as \(C_{3,3}^{m}=\max\left(C_{3,3}^{cm,1},C_{3,3}^{cm,2},C_{3,3}^{cm,3},C_{3,3}^{cm,4}\right)\). Compared to the PM activities, the CM activities are not only affected by subsequent operations of machine dimension, but are also affected by subsequent operations of job dimension, which will lead to more serious disorder.  
Due to stochastic failures, the number and time of machine failures in actual scheduling cannot be obtained before the completion of production. Even though the realized number of breakdowns may be infinite, the complexity of computing increases exponentially with the number of associated operations. For example, the number of operations in the associated operation set of \(O_{3,3}\) is six. Meanwhile, machine breakdown may occur during the operation \(O_{3,3}\). Thus, the number of total scenarios is \(2^{7}=128\).