## 3 Improvement of the Effectiveness of the SOCP Method--Use Cases
Accepting this conclusion, it should, however, be noted that there are certain groups of transport tasks, which we can call use cases, and for which there is a possibility to relax the restrictiveness of the discussed method. The staged execution of process reservations, due to a more efficient use of resources will make it possible to reduce the execution time of transport tasks. To work out how to implement this possibility, we must answer the question of whether there are deadlock-free segments on the AGV path on which the reservation process can be completed without causing a restricted deadlock when further deadlock-free segments have to be reserved. To answer this question, it is necessary to look at this issue in the same way as the issue of deadlock but considering only deadlock-free segments. Restricteddeadlock occurs only when there is a cycle of expectations for process reservations, which is an analogy of the cycle of resource dependencies which is a necessary condition for the occurrence of deadlock. To simplify the analysis, let us assume that each deadlock-free segment consists of one zone. The cycle of expectations for process reservations will occur when the transportation processes have made a process reservation in a given deadlock-free zone, and to leave this zone, they need to make another reservation which they cannot make because this zone is already reserved by another process. This sequence of process reservations may lead to a cycle consisting of two processes, as shown in Figure 2, but can also arise in the case of more transportation processes. As we know that it is worth implementing process reservations in a staged manner and we know the risks associated with such an action, we should analyze certain cases in terms of the possibility of using the concept discussed above. Such an action will be demonstrated using three examples taken from [2]. The examples presented there have features similar to industrial systems and taking into account the considered concept of solving the problem we can qualify them as use cases. These examples differ only in the type of roads used (bidirectional--Figure 3, unidirectional--Figure 5 and mixed--Figure 7). The remaining issues concerning the transportation system and the tasks performed are relevant for all examples, so they will be presented together. The principle adopted for all transportation system layouts consists of deadlock-free zones marked in green and deadlock-risk zones in yellow. The length of the side of the squares is set to 1[m]. There are 18 stations. Close to the stations are buffers marked in blue. Buffers: \(B1,B3,B5,B8,B10\), and \(B12\) are input points to machines. Buffers: \(B2,B4,B6,B7,B9\), and \(B11\) are output points from machines. There are 6 machines: \(M1\), \(M2\), \(M3\), \(M4\), \(M5\), and \(M6\), marked in black. Operations are performed on these machines. There are 3 sources of work-pieces: \(I1,I2\), and \(I3\), marked in light-blue. Each source generates 20 parts (a part per second) at the beginning of the simulation. There are 3 sinks for work-pieces (disposal points): \(O1,O2\), and \(O3\), also marked in light-blue. In this manufacturing system, 3 different workpieces: \(Part1,Part2\), and \(Part3\) are produced. On each part, two operations are performed. The process cycle time on each machine for each operation is set to 20[s]. The batch size and routing assigned to each work-piece are presented in Table 1. Finally, there are 10 AGVs, signed \(C1-C10\), parked in their parking places, marked in orange. The AGV transport task consists of three steps. In the first, the AGV travels from its parking (origin point) to the pick-up point, next from the pick-up point to the delivery point, and finally from the delivery point to the origin point.