## 1 Introduction  
The control of automated systems has long been a matter of great interest to the scientific world. The tools for studying discrete event systems (DES) are numerous, but Petri nets (PNs) [1] are nowadays most used because of their various and interesting properties for complex systems, such as airplanes, nuclear power plants or oil refineries, etc. This kind of system with a high level of safety and precision requires formal control methods to ensure operation in conformance with the specifications. Given a manufacturing system, we are interested in imposing on it a function respecting a desired behavior. For less complex systems, the easiest way for the control is the direct design of a supervised model of the system. However, when the complexity of the system increases, this method becomes very laborious and cannot guarantee compliance with control specifications. It is therefore necessary to opt for a formal method allowing the synthesis of the controller. The works of [2, 3] forms the basis of the general theory of control by supervision. Although the formalism used by the authors allows the modeling of a large class of DES, it is not adapted to real systems because of its high sensitivity regarding the combinatorial explosion of states. For this reason, many attempts have been made to solve this problem [4; 5; 6; 7; 8]. In fact, to model complex systems, the use of PNs seems more rigorous and beneficial, since this formalism is characterized by richer modeling structures. Nevertheless, the advantage of representation conciseness is counterbalanced by the low result optimality. One of the most efficient methods that generate maximum permissive PN controllers is the theory of regions (TR) developed by [PERSON] et al. [9; 10; 11; 12]. This theory gives an optimal control solution regardless of the process controllability level. The synthesis of the PN supervisor is based on the reachability graph (RG) and relies on the [2-3] approach to compute the controller. Its applicability is general for bounded PN models. Unfortunately, the TR suffers from a great technical obstacle; In fact, the steps of generating the RG and its analysis in order to find the existing cycles, the legal/forbidden states, weigh down heavily on the process of calculating PN controllers. Moreover, a possible combinatorial explosion of states in RG for complex systems increases significantly the computational burden. Since the calculation time of the control costs money according to the imposed constraints, the main contribution of this paper is to develop a control method which enables us to use the advantages of the PN properties and the TR while avoiding its technical obstacles, i.e., the generation and the analysis of the RG. As a result, several studies have focused on RG analysis to synthesize near-optimal/optimal PN controllers [13; 14; 15; 16]. Indeed, a reachability graph-based approach was developed to design an optimal PN controller with the fewest control places for flexible manufacturing systems. The graph was divided into a live zone and deadlock zone, with the states in the deadlock zone already deadlocked or leading to a deadlock and those in the live zone as legal ones. In addition, [PERSON] et al. [17] proposed a vector-covering approach to decrease the number of states to consider in the synthesis process of the PN supervisor. These methods can reduce the number of PN monitors. However, they have a disadvantage; the computation load is extremely hard, principally for large models. In [PERSON] et al. [18], a new approach for control synthesis called divide-and-conquer strategy was developed to improve the computational efficiency of the PN supervisor developed by the traditional methods of [PERSON] and [PERSON] [19; 20]. Unfortunately, too many subsystems must be disposed. Thus, if the considered model is big, the improvement of these methods will be impeded. In addition, many studies also focused on reducing the computational complexity of TR.