## 3 Improvement of the Effectiveness of the SOCP Method--Use Cases
First of all, let us consider an example with bidirectional guide paths presented in Figure 3. It should be noted that in the work [2], stations S1-S18 were treated as deadlock-risk which resulted from the fact that only the zone presented in Figure 1a was considered to be a deadlock-free zone. By adding the zone shown in Figure 1b to the set of deadlock-free zones, these stations can be considered to be deadlock-free zones. \begin{table}
\begin{tabular}{c c l} \hline \hline
**Work-Piece** & **Batch Size** & **Routing** \\ \hline Part1 & 20 & I1-S7-**S14**-(B8-M4-B7)-**S13-S5**-(B5-M3-B6)-**S6-S12**-O3 \\ Part2 & 20 & I2-**S10-S3**-(B3-M2-B4)-**S4-S16**-(B10-M5-B9)-**S15-S9**-O2 \\ Part3 & 20 & I3-**S11-S1**-(B1-M1-B2)-**S2-S18**-(B12-M6-B11)-**S17-S8**-O1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The batch size and routing. Let us consider the problem of process reservation for the transport tasks carried out in this example. All tasks shown in Table 1 can be divided into three stages. Stage one concerns the AGV's drive from the parking area to the pick-up location. Stage two is the movement from the pick-up location to the delivery location. Stage three is the return of the AGV to the parking area. There are two deadlock-free segments along the route of each transport task, each consisting of one zone. This means that before the start of each task, it is necessary, according to the SOCP method, to reserve both zones. Taking into account the issue discussed above regarding the possibility of staging process reservations, let us analyze whether this way of execution does not lead to a restricted deadlock. Thus, it is necessary to check whether a waiting cycle for process reservations can be created. In the transportation system under consideration, we have a set of deadlock-free zones from which we can distinguish the subset  
\[PU=\{S2,S4,S6,S7,S10,S11,S13,S15,S17\} \tag{1}\]  
containing the places (zones) from which the parts are taken and the subset of the places  
\[DE=\{S1,S3,S5,S8,S9,S12,S14,S16,S18\} \tag{2}\]  
(zones) containing to which they are transported. In addition, an important role is played by the set \(ZP=\{c_{e}:1\leq e\leq 10\}\) containing the dedicated car parking areas from which vehicles \(v_{e}\) leave after the start of the transport task and return to after its completion. These vehicles form the set \(V=\{v_{e}:1\leq e\leq 10\}\). All parking spaces in the considered system are deadlock-free zones. Assuming that \(c_{e}\in ZP\), \(pu_{j}\in PU\), and \(de_{k}\in DE\) then the transport tasks can be divided into three consecutive stages \(c_{e}\to pu_{j}\), \(pu_{j}\to de_{k}\), and \(de_{k}\to c_{e}\). However, it should be noted that, from the point of view of the SOCP method, the process reservation of the parking space is not necessary. This is due to the assumption made in the considered examples that each parking place \(c_{e}\in ZP\) is dedicated to a specific vehicle \(v_{e}\in V\). This leaves the issue of analyzing the possibility of the emergence of a cycle  
Figure 3: The layout with bidirectional flow paths. of process reservations in the case of staged execution exclusively for elements \(pu_{j}\in PU\) and \(de_{k}\in DE\). Note that for the considered example, the following relation is satisfied. \[PU\cap DE=\emptyset \tag{3}\]  
This means that the \(PU\) and \(DE\) sets are disjoint and this is a sufficient condition for the impossibility of creating a restricted deadlock. This results from the fact that a cycle can only be formed by the zones from which parts are taken and to which parts are returned, i.e., the transport operation is started and the transport operation is finished. This means that the process reservations necessary for each transport task can be carried out in stages without creating a restricted deadlock.