## 1 Introduction
To reduce computation, first-met bad markings (FBMs) are adopted in these works. FBMs are the set of markings in the DZ and the first entrance from the LZ to the DZ. However, as indicated by [38], this process requires the repeated calculation of reachability graphs. [PERSON] et al. [39] consider selective siphons to reduce redundancy problems and provide small-size controllers. The control policy solves the deadlock problems of FMSs successfully. It also makes FMSs optimal. However, as indicated by [40], the process of eliminating all critical markings is time-consuming, since all legal markings must be checked in each iterative step. According to the above descriptions, it is clear that determining how to obtain optimal (maximally permissive) controllers for solving the FMS deadlock problem seems an extremely time-consuming and difficult issue. Therefore, in [33] [PERSON] et al. also propose a policy called the enhancement of selective siphon control method to improve the efficiency of deadlock prevention in FMSs. For improving the above disadvantage, [PERSON] et al. [1] brought forward a new concept named MFFP (i.e., the Maximal number of Forbidding First Bad Marking (FBM) Problems) to obtain maximally permissive reachable markings with a much smaller number of controllers based on the reachability graph analysis method and Place Invariant (PI) [41]. It successfully solves the problem of the excessive consumption of time. However, when the complexity of a flexible manufacturing system increases, the computational burden rises rapidly. Besides, in our survey the MFFP could only be applied in certain special nets, and outside S\({}^{3}\)PR [6], the method would fail. Therefore, in our previous paper [42], we proposed an iterative method called IMFFP-1 to solve the disadvantage of [1] so that it can be used in all general cases whether the controlled net is optimal or suboptimal. Subsequently, for reducing computational time, in [43] we proposed two novel concepts called Pre Idle Transitions (PIT) and Pre Idle Places (PIP). However, the proposed definitions seemed rough and immature. Therefore, in this paper, we aim to redefine and give the formal definitions of PIT and PIP. Please note that we can bypass all PIP once they can be identified from a deadlock flexible manufacturing system under the process of solving MFFP. From the resulting data based on three classical examples, the further improved MFFP (called IMFFP-2) method seems to achieve a better computational efficiency with the same controllers. Please note that in this paper we focus on enhancement of computational efficiency for the conventional MFFP. Therefore, in this paper we consider the same kind of Petri Net (PN) class as that in [1,2]. On the other hand, some deadlock preventions that can recover all reachable markings using transition-based controllers [44; 45; 46; 47; 48; 49] and inhibitor arcs [50] are also proposed. In the future, we will follow the work in [51] to discuss a divide-and-conquer-method and try to enhance their computational efficiency. Section 2 gives a description of the basic definitions and properties of PNs. Section 3 presents the proposed policy. Section 4 shows examples and related experimental data. Section 5 makes comparisons with the existing literature. Finally, conclusions are drawn in Section 6.