# FMS Scheduling under Availability Constraint with Supervisor Based on Timed Petri Nets
## 3 Terminology  
In this section are presented the notations used in TPN modeling for FMS scheduling and supervisor synthesis. * \(i,i^{*}\) : indexes for jobs (\(i,i^{*}=1\ldots n\));
* \(j,j^{*}\) : indexes for machines (\(j=0\ldots m+1\));
* \(t_{i,j}\)(if fired) : transition denoting that the job \(i\) leaves the machine \(j\);
* \(p_{i,j}\)(if marked) : place indicating that the job \(i\) is already being performed by machine \(j\);
* \({}^{\circ}p_{i,j}\) (\(p_{i,j}\) ) : designs the set of input transitions (respectively output) of a place \(p_{i,j}\);
* \(h\Big{(}t_{i,j}\Big{)}\in h(T)\) : is the local-clock linked to \(t_{i,j}\); in the proposed modeling method, \(h\Big{(}t_{i,j}\Big{)}\) defines the processing duration of \(i\) through \(j\);
* \(H\Big{(}t_{i,j}\Big{)}\) : is the global-clock of transition firing \(t_{i,j}\); physically, it defines the end time of job \(i\) through machine \(j\);
* \(A\Big{(}t_{i,j},t_{i^{*},j-1}\Big{)}\) : is a decision variable defining whether the arc from \(t_{i,j}\) to \(t_{i^{*},j-1}\) exists;
* \(Z_{i,j+1}\) : is a decision variable defining whether job \(i\) is performed after the maintenance task on machine \(j+1\);
* \(\varphi_{t_{i,j}}\Big{(}Z_{i,j+1}\Big{)}\) : is the extra time linked to \(t_{i,j}\) depending on the \(Z(i,j+1)\) value;
* \(\pi_{(t_{i,j},t_{i^{*},j-1})}\) : unmarked timed elementary path between two transitions;
* \(\theta_{(i,i^{*})}\) : duration associated with \(\pi_{(t_{i,j},t_{i^{*},j-1})}\);
* \([B_{j},E_{j}]\) : preventive maintenance period where \(B_{j}\) (respectively \(E_{j}\)) is the beginning time (respectively ending time). In our proposed model for FMS using TPN tools (Section 5), each TMG\({}^{i}\) designs the free-dynamics of job \(i\) through the manufacturing system. Relations between two different timed marked graphs TMG\({}^{i}\) and TMG\({}^{i^{*}}\) (\(i,i^{*}=1\ldots n\) and \(i\
eq i^{*}\)) are modeled by timed direct arcs \(\big{(}t_{i,j},t_{i^{*},j-1}\big{)}\) from transition \(t_{i,j}\) to transition \(t_{i^{*},j-1}\). Physically, an established arc \(\big{(}t_{i,j},t_{i^{*},j-1}\big{)}\) is interpreted to mean that the job \(i\) had to be treated on machine \(j\) before that the job \(i^{*}\) finishes its treatment on machine \(j-1\). \(A\big{(}t_{i,j},t_{i^{*},j-1}\big{)}\) is analytically expressed as follows:  
\[A\big{(}t_{i,j},t_{i^{*},j-1}\big{)}=\left\{ \begin{array}{c}1\text{ if the timed direct arc }\big{(}t_{i,j},t_{i^{*},j-1}\big{)}\text{ has taken place}\\ 0\text{ else}\end{array} \right. \tag{1}\]  
**Remark 1**. : _It is obvious that a timed direct arc between two transitions contains \((2\times s)\), \(s\in\mathbb{N}\), places and \((2\times s-1)\) transitions. In the following, we consider \(s=1\)._  
Another important parameter associated with TMG\({}^{i}\) is the extra time \(\varphi_{t_{i,j}}\big{(}Z_{i,j+1}\big{)}\in\mathbb{N}\) added to local-clock \(h\big{(}t_{i,j}\big{)}\). \(Z_{i,j+1}\) corresponds to the job \(i\) position, if it is just after the maintenance action on machine \(j+1\). The analytical expression of \(Z_{i,j+1}\) is:  
\[Z_{i,j+1}=\left\{ \begin{array}{c}1\text{ if the transition }t_{i,j}\text{ should not be fired}\\ 0\text{ else}\end{array} \right. \tag{2}\]  
Indeed, \(\varphi_{t_{i,j}}\big{(}Z_{i,j+1}\big{)}\) is the job \(i\) time delay on machine \(j\) until the availability of machine \(j+1\). Figure 2 illustrates all of the explained parameters. The main contribution of this paper is to carry out an efficient manufacturing supervisor based on an optimal sequence of transitions determined by a mathematical model. In this perspective, the derived supervisor is assumed to be commissioned by two types of controllers defined as follows. TPN controller: A TPN controller is an unmarked timed elementary path \(\pi_{(t_{ij},t_{i^{*},j-1})}:\left(p_{1,(i^{*})},t_{(i^{*})}\left[\theta_{(i,i^{ *})}\right],p_{2,(i^{*})}\right)\) between two transitions \(t_{i,j}\) and \(t_{i^{*},j-1}\) (i.e., see Remark 1).