## 4 Genetic Regulatory Network-Based Rescheduling Method
### Regulation Equation and Expression Procedure of Gene \(\boldsymbol{\pi}_{rm}\)  
In HFS scheduling, each machine prefers to operate on the same job type in order to reduce the setup time. For instance, the jobs of type \(m_{1}\) are operated on the 1 st machine (i.e., the jobs on this machine are thus \"\(m_{1}\)-\(m_{1}\)-\(m_{1}\)-\(m_{1}\)-\(\ldots\) \"), and those of \(m_{2}\) and \(m_{3}\) are assigned to the 2 nd machine and the 3 rd machine, respectively. In this case, the job sequence entering this production stage will be \"\(m_{1}\)-\(m_{2}\)-\(m_{3}\)-\(m_{1}\)-\(m_{2}\)-\(m_{3}\)- \(\ldots\) \" because the parallel machines perform their tasks simultaneously. Thereupon, each job type would appear in the job sequence cyclically, and the related cycle time should be accorded with the number of parallel machines at each stage. However, the greedy nature of rule-based sequencing methods will probably keep the HFS scheduling choosing the \"easy\" jobs, which means jobs of a \"hard\" type will be left to the last positions of the job sequence. For instance, four jobs of type \(m_{4}\) might be left for the last four positions of a job sequence, which is not preferred because there is no scheduling flexibility when assigning these jobs to parallel machines. On the contrary, it is much better to leave four jobs \(m_{1}\), \(m_{2}\), \(m_{3}\) and \(m_{4}\) for remaining positions because the job assigning procedure can choose machines with shorter processing time, less changeovers or earlier availability for each job to minimize the makespan further. For this reason, it is necessary to keep the stable production rate of each job type in the job sequence in order to avoid the near sightedness of rule-based sequencing methods. In terms of these facts, following rules should be obeyed:  
1. A job type cannot be selected if its cycle of entering the HFS is not accord with the number of parallel machines at each stage;
2. A job type cannot be selected if its production ratio differs from its demand ratio. No model sequence could satisfy all these rules completely, and each unsatisfied case might increase the makespan. The regulation equation of gene \(\boldsymbol{\pi}_{rm}\) is thus developed as follows:  
\[u_{rm}=\epsilon_{1}\sum_{i=1}^{I}|\frac{\sum_{v=1}^{r-1}y_{vm}+1}{r}-\ \frac{1}{K_{i}}|+\ \epsilon_{2}|\frac{\sum_{v=1}^{r-1}y_{vm}+1}{r}-\ \frac{d_{lm}}{D_{l}}|+H(\sum_{v=W_{l}}^{r-1}y_{vm}-d_{lm})+H( \sum_{m=1}^{M}y_{rm}), \tag{4}\]  
where \(u_{rm}\) represents the inhibition coefficient to gene \(\boldsymbol{\pi}_{rm}\), \(d_{lm}\) represents the number of job type \(m\) entering the HFS within the \(l\)th window, \(D_{l}=\sum_{m=1}^{M}d_{lm}\), \(W_{l}\) represents the number of jobs having entered the HFS before the \(l\)th window, \(\epsilon_{1}\) and \(\epsilon_{2}\) are regulatory parameters, and \(H(x)\) is a step function that satisfies \(H(x)\ =\ 0\,(x\leq 0)\) and \(H(x)\ =\ +\infty\,(x>0)\). The first two terms of the right side of Equation (4) represent the inhibition strength to gene \(\boldsymbol{\pi}_{rm}\) owing to Rules 1 and 2, respectively. The last two terms ensure the job sequence to satisfy the predetermined quantity of each job type. In the expression procedure of gene \(\boldsymbol{\pi}_{rm}\), all the operations are arranged in an ascending sequence in terms of their starting time in the predictive plan. \(p\) operations with their starting time later than  
Figure 5: Outline of the GRN-based method. the rescheduling instant are selected consecutively in the operation sequence. Assuming that \(d_{lm}\) represents the number of operations processed on the first production stage for job type \(m\), there are \(D_{l}=\sum_{m=1}^{M}d_{lm}\) jobs to be arranged for the job sequence within the \(l\)th window (from the \(W_{l}\)th position to the \((W_{l}+D_{l})\)th position).