A livelock control policy for a flexible manufacturing system modeling with a subclass of generalized Petri nets

: Livelocks, like deadlocks, can result in the serious problems in running process of flexible manufacturing systems (FMSs) as well. Current deadlock control policies based on the approaches of siphon detection and control, cannot cope with livelocks in a system of sequential systems with shared resources (S 4 R), a typical subclass of Petri nets that can model FMSs. On the basis of the mixed integer programming method, this study proposes a livelock control policy (LCP) that can not only solve the new smart siphons (NSSs) associated with livelocks or deadlocks in an S 4 R system directly, but also make the solved NSSs max ′ -controlled by adding the corresponding control places (CPs). As a result, an original S 4 R system with livelocks or deadlocks can be turned into the live controlled one in which no NSSs can be found. The related theoretical analysis and several examples are given to demonstrate the proposed LCP. Compared with the existing methods in the literature, the proposed one is more general and powerful.

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ABOUT THE AUTHORS
The research group of system control and automation of School of Civil Engineering, Lanzhou University of Technology was founded in 2009. There are about eight members with the degree of Master or PhD. Their research interests include control engineering and application of supervisory control of discrete event systems. About 10 papers on the corresponding research results have been published in the Journals indexed by SCI and EI until now. The results in this paper focus on the problem of deadlock as well as livelock that is, the hot topic of flexible manufacturing systems by Petri net theory.

PUBLIC INTEREST STATEMENT
The paper is appropriate for researchers, graduate students, and engineers who are interested in the control problems arising from flexible manufacturing systems (FMSs) modeling with Petri nets (PNs). Livelocks, like deadlocks, can result in the serious problems in running process of flexible manufacturing systems as well. The research of deadlock and livelock control has become a hot topic in the field of supervisory control theory (SCT). On the basis of the previous results and the mixed integer programming (MIP) method, a livelock control policy (LCP) is proposed for an S4R system belonged to a subclass of PNs in this paper. By solving the new smart siphons (NSSs) associated with livelocks or deadlocks and making them max′-controlled, an original S4R system with livelocks or deadlocks can be turned into the live controlled one. The proposed LCP exploit a new path to the deadlock and livelock control problems.

Introduction
A flexible manufacturing system (FMS) is a novel production mode in which there is some amount of flexibility that allows the system to response to market requirement changes in as timely a manner as possible, no matter predicted or unpredicted. An FMS is also a computer controlled configuration in which different operations are executed concurrently, and therefore, have to compete a limited number of resources such as robots, machines, storage devices, fixtures, and conveyors (Li & Zhou, 2009;Wu & Zhou, 2009). This competition can cause deadlocks that are highly undesirable phenomena, in which a set of processes keeps waiting indefinitely for other processes in the set to release resources (Chao, 2006;Ezpeleta, Colom, & Martinez, 1995;Huang, Jeng, Xie, & Chung, 2001;Li, Li, & Hu, 2011;Li & Wang, 2011;Li & Zhou, 2004). In a deadlock state, the global or local FMS is blocked or crippled, causing unnecessary cost or catastrophic results. When an FMS is at a livelock state, a special case of deadlocks, some processes continuously change their state, while the other processes are deadlocked (Chao, 2007;Liu & Li, 2010). That is to say, a livelocked system still does some (although not useful) work in which some execution that never makes effective progress cycle infinitely. As a result, deadlocks and livelocks must be considered for designing an FMS.
In order to effectively handle the above problems in an FMS; digraphs, automata, and Petri nets are among the most popular mathematical tools available. Among them, Petri nets have been widely used to model an FMS, because of their own features (Li & Zhou, 2009;Wu & Zhou, 2009). They can be well used to describe behavioral properties such as reversibility, boundedness, and liveness. A siphon, as a structural object of a Petri net, is a set of places (where tokens can leak out or inject in) of a Petri net model. Once a siphon has lost all its tokens, i.e. it is emptiable at a reachable marking, output transitions of places in the siphon can never be executed and hence the net is not live. Previous research indicates that siphons are closely related to the deadlocks in Petri net models of FMSs. It is well known that the number of strict minimal siphons (SMSs) in a Petri net grows quickly or may grow exponentially with respect to its size (Li & Zhou, 2009) such that it is difficult and time consuming to find out these SMSs. Hence, how to design or select the proper method to solve the siphons is very important for developing new deadlock control policies (DCPs). Many scholars focus on solving and controlling siphons and have developed a large number of siphon-based controlled (SC) policies (Chao, 2006;Huang et al., 2001;Huang, Jeng, Xie, & Chung, 2006;Li & Li, 2012a, 2012bLi, Li, & Al-Ahmari, 2012;Li, Liang, Lun, & Wang, 2004;Li, Hu, & Wang, 2007;Li, Wang, & Wei, 2006;Li & Zhao, 2008;Li & Zhou, 2004, 2006a, 2006bLi, Zhou, & Jeng, 2008;Piroddi, Cordone, & Fumagalli, 2009;Uzam & Zhou, 2006;Zhong & Li, 2010). However, the above DCPs still cannot detect the siphons that cause livelocks and then cannot deal with the livelock problems in OPNs or S 4 R. When a system is at a livelock state, the overall states of the system continue to change, while parts of the system are deadlocked. In other words, once a system is at livelock, other processes cannot be processed smoothly. Actually, livelock is a special case of resource starvation although the system may be deadlock-free. As far as the authors know, Chao (2007) first points out that an S 4 R without deadly marked siphons (DMSs) that contribute to deadlocks (Park & Reveliotis, 2001) is deadlock-free but may be in a livelock state where both live and dead transitions exist, implying that the representative mixed integer programming (MIP) methods in Chu and Xie (1997) and Park and Reveliotis (2001) are no longer valid to identify DMSs, since the net may have livelocks even though it is deadlock-free. Accordingly, the concept of max′controlled siphons is proposed in Chao (2007), but the corresponding approach to detect siphons associated with livelocks is not given. Subsequently, based on the max″-controllability condition, Liu and Li (2010) present a more general MIP to detect deadlocks or livelocks caused by siphons that are called extended DMSs, and open a new path to analysis deadlocks and livelocks in view of siphons. The limits of the proposed MIP using max″-controlled siphons in Liu and Li (2010) are non-linear and more complicated, which increases the computational load. Similarly, no corresponding SC policy is presented in Liu and Li (2010).
Considering the shortcomings of the above SC policies, this paper proposes a novel MIP method to directly solve the smart siphons associated with livelocks or deadlocks in Petri net models. By means of the structural analysis of Petri nets and the concept of smart siphons (Li & Li, 2012b), new constrains are generated and then added to the MIP method in Li and Li (2012b), called revised MIP (RMIP). This RMIP can detect smart siphons that cause deadlocks or livelocks in Petri net models directly. Accordingly, a proper control place (CP) Li & Li, 2012b) is added for each solved smart siphon to make itself max′-controlled (Chao, 2007;Zhong & Li, 2010) during an iterative process. This research claims that if there is no feasible solution (NSF) of an RMIP problem, then the corresponding Petri net system is live. That is to say, livelocks or deadlocks are eliminated in the controlled Petri net system until no new smart siphons (NSSs) can be found by the proposed method in this paper.
The rest of this work is organized as follows. Section 2 reviews preliminaries of Petri nets that are used throughout this work. In Section 3, by adding new constraints to an MIP method in Li and Li (2012b), we present an RMIP method to directly solve NSSs associated with livelocks or deadlocks. By the proposed RMIP method and max′-controlled NSSs, an iterative livelock control policy (LCP) is developed in the same section. In Section 4, three examples are introduced to demonstrate the proposed LCP. We have a discussion in Section 5. Finally, Section 6 concludes this paper and discusses future work.

Petri nets
A Petri net is a four-tuple N = (P, T, F, W), where P and T are finite and non-empty sets. P is a set of places and T is a set of transitions with P ∪ T ≠ � and P ∩ T = �. F ⊆ (P × T)∪(T × P) is called the flow relation or the set of directed arcs. W : (P × T)∪(T × P) → ℕ is a mapping that assigns a weight to an arc: W(x, y) > 0 if (x, y) ∈ F, and W(x, y) = 0 otherwise, where x, y ∈ P ∪ T and ℕ denotes the set of nonnegative integers. N = (P, T, F, W) is called an ordinary net, denoted as N = (P, T, F), if ∀f ∈ F, W(f) = 1. N = (P, T, F, W) is called a generalized net, if ∃f ∈ F, W(f) > 1. Given a node x ∈ P ∪ T, • x = {y ∈ P ∪ T | (y, x) ∈ F} is called the preset of x, while x • = {y ∈ P ∪ T | (x, y) ∈ F} is called the postset of x. A marking is a mapping M : P → ℕ. (N, M 0 ) is called a marked Petri net or a net system, where N is a net and M 0 is an initial marking. The set of markings reachable from M in N is denoted as R(N, M). A string x 1 , …, x n is path from x 1 to x n is a path whose nodes are all different (except, perhaps, x 1 and x n ). A path x 1 , … , x n is called a circuit if it is an elementary path with x 1 = x n . An ordinary Petri net N is called a state machine if ∀t ∈ T,| • t| = |t • | = 1. A state machine component N′ = (P′, T′, F′, W′) of a Petri net N = (P, T, F, W) is a state machine and is a subnet of N consisting of places in P′, their presets and postsets, and related arcs. A Petri net is said to be state machine decomposable if it is covered by state machine components (Li & Zhou, 2009).
A P(T)-vector is a column vector I(J) : P(T) → Z indexed by P(T), where Z is the set of integers. I is a P-invariant (P-inv for short) if I ≠ 0 and I T [N] = 0 T . P-inv I is said to be a P-semiflow if every element of (Li & Zhou, 2009)

Basic definitions and theorems
Definition 2.1 ) Let {r 1 , r 2 , … , r m } ⊆ P R (m ≥2) be a set of resources in N. An elementary circuit C(r 1 , t 1 , r 2 , t 2 , … , r m , t m ), denoted as C R , is called a resource circuit if the following conditions are satisfied: (4) ∀j ∈ ℕ n , the subnet N j derived from P A j ∪{p 0 j }∪T j is a strongly connected state machine such that every circuit contains idle place p 0 j . Furthermore, (6) N is pure and strongly connected.

Definition 2.5 (Li & Zhou, 2009) Let r be a resource place and S be an SMS in an S 4 R net. The operation places that use r is called the set of holders of r, denoted as H(r). The holders of resource r is defined
as the difference of two multisets I r and r : H(r) = I r − r. As a multiset, cates that siphon S loses h s (p) tokens if the number of tokens in p increases by one.
Definition 2.7 (Zhong & Li, 2010) Let S be a siphon of a well-marked S 4 R (N, M 0 ). S is said to be max'controlled iff S is max′-marked at any reachable marking, that is, If all SMSs of the net are max'controlled, the net is live.
Note that an S 4 R is more general than ES 3 PR, S 3 PR, and L-S 3 PR (Li & Zhou, 2009), i.e. L-S 3 PR ⊆ S 3 PR ⊆ ES 3 PR ⊆ S 4 R. Theorem 2.8 is hence suitable for verifying liveness of ES 3 PR, S 3 PR, and L-S 3 PR.
For example, an FMS layout is shown in Figure 1(a) (Chao, 2007). There are three loading buffers I1-I3 and three unloading buffers O1-O3 to load and unload the FMS corresponding to three raw product types, P1-P3, to be processed by machine M1 and moved by robot R1.

RMIP method
By analyzing the relationship between the states of deadlocks or livelocks and the presence of smart siphons, and then adding new constraints to an MIP method in Li and Li (2012b), the RMIP method is described below.
s. t. (1) In a live (resp. dead) subnet, all transitions are enabled (resp. disabled) whose z t = 1 (resp. z t = 0). Note that if υ p = 1 for each input place of a transition t in an OPN, then z t = 1. When υ p = 1 but p is unmarked, z t keeps 1, t is said to be pseudo-enabled. For a GPN, arc (p, t) from each input place of a transition t may not be enabled even though p is marked (υ p = 1). That is to say, it is need for p to be sufficiently marked and then υ p = 1 can lead to z t = 1. In order to distinguish the solution of the MIP method in Li and Li (2012b), the solution of the above RMIP method is called NSS.
The RMIP method can be explained as follows: Equation 8 expresses the fact if the number of tokens at a place p is no less than the weight of arc(p, t), then arc(p, t) is enabled and υ p = 1. Physically, when a transition t is enabled and then fired, tokens can flow along arc (t, p) and enter into each output place p. This is consistent with f pt ≥ υ p (Equation 3) where f pt = 1 if z t = 1. In addition, if t is an output, rather than input, transition of a place p, f pt = 1 does not mean z t = 1; f pt = 0 means z t = 0 since t is disabled if any input arc is disabled. Equation 9 expresses the fact if all output arcs of resource place r are enabled, then r must be max′-marked and υ r = 1 due to Definition 2.6. Otherwise r is not max′-marked and υ r = 0. The objective function and the other constraints Equations 2-7 and Equation 10 are the same as those in the MIP method in Li and Li (2012b). In summary, this RMIP method can find a set of non-max′-marked places that can form an NSS under a marking M ∈ R(N, M 0 ). Comparing with the MIP method in Li and Li (2012b), main difference is that this RMIP method can detect a set of non-max′-marked but non-max-marked places that forms a siphon under a marking M there are both live and dead transitions. In other words, the former can only solve a smart siphon causing deadlocks, and the latter can solve an NSS associated with deadlocks and livelocks. In addition, NSF of the proposed RMIP method is obtained, which means no any NSS exists in an S 4 R.
Furthermore, an important variable, denoted as U z t , with respect to the sum of the values of z t i (i ∈ ℕ + = {1, 2, … , }) is introduced to identify the characteristic of the solved NSSs. After executing the above RMIP method, both G MIP (M) and the value of each z t can be solved. Hence, is obtained accordingly, where |T| denotes the total number of transitions in an S 4 R. From Section 2.1, it is known that if an S 4 R is deadlocked under a marking M in which all transitions are disabled, then z t i = 0, and U z t = 0 is obtained, implying that the characteristic of corresponding NSS is about deadlock. When an S 4 R is livelocked under a marking M in which some transitions are disabled and the rest of transitions are fired, z t i ≥ 0 and 0 < U z t < |T| are obtained, implying that the characteristic of corresponding NSS is about livelock, denoted as NSS ⋆ . Obviously, z t i = 1 and U z t = |T| are true when an S 4 R is live, implying that no any NSS or NSS ⋆ exists in an S 4 R.

An iterative LCP combing the RMIP method with max′-controlled NSSs
After solving an NSS (resp. NSS ⋆ ), its complementary set (denoted as Th(NSS) (resp. Th(NSS) ⋆ ) can be found by Definition 2.5. Depending on Th(NSS) (resp. Th(NSS) ⋆ ), two types of CPs in Li and Li (2012b) and  are properly selected and then added for the solved NSS (resp. NSS ⋆ ). From the above discussion, we add CP for the solved NSS or NSS ⋆ to make it max′ not max-controlled. Thus, the key is to determine the initial marking of added CPs (denoted as M 0 (V)). From Li and Zhou (2009) Similarly, NSS ⋆ is used to denote the control depth variable with respect to an NSS ⋆ , where NSS ⋆ ∈ ℕ m and NSS ⋆ ≥ max t∈(r • ∩ T h(NSS ⋆ ) • ) {W(p, t)}. In general, decreasing NSS (resp. NSS ⋆ )intends to relax the control of NSS (resp. NSS ⋆ ), which may reduce the restriction of the behavioral permissiveness of the controlled system. Moreover, the max′ but not max-controlled condition for NSSs (resp. NSSs*) relaxes the controllability condition and may lead to a live controlled system with more behavioral permissiveness (Liu & Li, 2010;Zhong & Li, 2010).
An NSS or NSS ⋆ in a plant net model (N 0 , M 0 ) can be found by the aforementioned RMIP approach. However, a control policy needs to find all NSSs and NSSs ⋆ and then adds the proper CPs for them in the augmented net (N*, M*) such that the final controlled system is live. Hence, an iterative LCP using the proposed RMIP method, which can find a An NSS or NSS ⋆ and then add a proper CP for it at each iteration until a live controlled system is obtained, is developed in this subsection and described below, where |P| = |P A | + |P 0 | + |P R |, j ∈ ℕ + , Π, and |V| denote the total number of the places in an original net, the iteration step number, the set of solved NSSs and NSSs* in the augmented net At the jth iteration, the proposed LCP also solves NSS ⋆ j or NSS j and then computes G MIP (M j ). Similarly, NSS ⋆ j or NSS j : = � if NFS for G MIP (M j ) holds. Therefore, the corresponding controlled system is live and Policy 1 terminates at this iteration. Otherwise, the proposed LCP solves NSS ⋆ j or NSS j and then adds a proper V j for it. The iteration proceeds until NFS for G MIP (M j ) holds, implying that the corresponding controlled system is live.

Theorem 3.3 Let (N, M 0 ) be a well-marked S 4 R in which deadlocks and livelocks exist. The proposed LCP is applied to it, which leads to a live augmented net (N * , M * ).
Proof Suppose that an NSSj is solved at the jth iteration and the augmented net after the jth iteration is denoted by (N j , M j ), where ∀p ∈ P A ∪ P 0 ∪ P R , M 0, j (p) = M 0 (p), M 0, j (V) = M 0 (NSS j )− NSS j , and NSS j ≥ max t∈(r • ∩ T h(NSS j ) • ) {W(p, t)}. By Definition 2.5, Th(NSS j ) is then computed. Depending on h NSS j (p), a proper V j is added for NSS j . As stated in Li and Li (2012b) and , the addition of both ordinary and general CPs to the original net can make the solved NSSs or NSS ⋆ max′-controlled in the iteration. That is to say, an iterative control process can eliminate all existing NSSs and NSS ⋆ until NFS for G MIP (M j ) holds, which means that no NSSs and NSS ⋆ can be found in the finally augmented net (N*, M*). As a result, the final controlled system (N*, M*) is live owing to Theorem 2.8.
From Chao (2006), Ezpeleta et al. (1995), Huang et al. (2001), Zhou (2004), Li and Zhou (2009), and Zhong and Li (2009), it is known that the number of SMSs in a Petri net grows quickly fast, in the worst case, grows exponentially with respect to its size. Moreover, the computational complexity, structural complexity, and behavior permissiveness are usually used to criticize the different DCPs. Since the proposed LCP can iteratively solve NSSs ⋆ (NSSs) that are belonged to a class of SMSs, this LCP is NP-hard in theory. Due to the fact that any siphon that has more resource places can be composed by those containing less ones (Chao, 2006;Li & Li, 2012b;, the solved NSSs ⋆ (NSSs) contain the minimal number of resource places. Hence, this LCP may reduce the possibility of adding redundant CPs to some degree. In summary, the proposed LCP has high computational efficiency and can generally obtain a live controlled Petri net system with a simple structure and more behavior permissiveness, which can be verified via two examples in Section 4. Take an S 4 R shown in Figure 1(b) as an illustrative example, where |P| = |P 0 | + |P A | + |P R | = 10 and |T| = 8. By Definition 2.2, C R = {p 9 , t 5 , p 10 , t 2 } and T C R = {t 2 , t 5 } can be obtained, and INA (2003) indicates that this S 4 R has 29⋇ reachable states, where ⋇ denotes maximally permissive behavior.
At the second iteration, Policy 1 obtains NFS for G MIP (M 2 ), which means that no NSSs or NSS ⋆ s can be found and Policy 1 terminates at this iteration. As a result, the finally controlled system is live. The results of Policy 1 are briefly shown in Table 1.
From Table 1, the proposed LCP solves one NSS ⋆ and accordingly adds a GCP for it. Policy 1 terminates in the second iteration until NFS for G MIP (M 2 ) is declared. Finally, the controlled system (N*, M*) is live due to Theorems 2.2 and 3.2. In addition, by INA (2003), (N*, M*) has 29 reachable states and R = 100%, where R denotes the ratio of the number of reachable states of the live controlled system (N*, M*) to that of maximally permissive behavior of the original S 4 R (N, M 0 ).

Examples
In this section, two examples are presented to demonstrate the proposed ICP in this work. Moreover, its control performance comparison with the existing approaches presented in Zhong and Li (2009) Zhong and Li (2009), respectively. There are two uploading buffers I1 and I2 and three downloading buffers O1 and O2 to upload and download the FMS corresponding to two raw product types, P1 and P2, to be processed by machine M1 and moved by robot R1 (resp. R2) and R2 (resp. R1).
According to two production cycles, a raw product P1 is taken from I1 by R2 and put in M1. After being processed by M1, it is downloaded by R1 and output to O1. A raw product P2 is taken from I2 by R1 and put in M1 to be processed, and then moved from M1 to O2 by R2. For Figure 2(b), we have Table 1. Results of the proposed LCP for a marked S 4 R shown in Figure 1 P 0 = {p 1 , p 5 }, P A = {p 2 , p 3 , p 4 , p 6 , p 7 , p 8 }, P R = {p 9 , p 10 , p 11 }, |P| = |P 0 | + |P A | + |P R | = 11, and |T| = 8.
By Definition 2.1, C R 1 = {p 9 , t 7 , p 10 , t 2 }, T C R 1 = {t 2 , t 7 ,}, C R 2 = {p 10 , t 6 , p 11 , t 3 }, and T C R 2 = {t 3 , t 6 } can be obtained. The number of its maximally permissive states is 194⋇ by executing INA (2003). Tables 2 and  3 show the results of the proposed LCP and its control performance comparison with the existing approaches, respectively.
Example 2 Considering another FMS layout and its Petri net model S 4 R (N, M 0 ) depicted in Figure 3(a) and (b) (Zhong & Li, 2009), respectively. This FMS consists of four robots R1-R4, each of which can hold one or three products every time, and three machines M1-M3, each of which can process two or three products every time. There are three loading buffers I1-I3 and three unloading buffers O1-O3 to load and unload the FMS. There are three raw product types, namely P1, P2, and P3, to be processed.
From Tables 3 and 5, the number of necessary (redundant) added CPs obtained by this LCP is fewer (equal to zero), leading to the live controlled S 4 R with simple structures. Moreover, this LCP achieves Table 2. Results of the proposed LCP for a non-live marked S 4 R shown in Figure 2  more permissive states compared with the existing methods. That is to say, the proposed LCP can result in a liveness-enforcing Petri net supervisor with a simple structure and more permissive behavior in terms of structural complexity and behavior permissiveness.

Discussion
Different types of multiple resources can be requested by different processes in an FMS. As a typical class of generalized Petri nets, an S 4 R can model more complicated resource allocation systems with multiple concurrent processes. Thus, an S 4 R has better modeling power than ES 3 PR, S 3 PR, and L-S 3 PR (Li & Zhou, 2009). An insufficiently marked siphon in an S 4 R can lead to a dead state. Many DCPs (Huang et al., 2006;Li & Li, 2012b;Li et al., , 2006Piroddi et al., 2009;Zhong & Li, 2009) for S 4 R are developed based on the concept of max-controlled siphons. However, these policies cannot solve livelock problems that may lead to undesirable results for in running process of an FMS.   As far as the authors know, only limited work of (Chao, 2007;Liu & Li, 2010;Zhong & Li, 2010) is reported on the siphon controllability conditions causing livelock states, i.e. max′ or max″-controlled siphons. But no corresponding SC policies are given and the approach in Liu and Li (2010) is nonlinear and more complicate. By means of the study of Chao (2007), Zhong and Li (2010) and the structural analysis of Petri nets, the presence of a non-max-marked siphon is not a sufficient condition for an S 4 R to be not live. Actually, a set of non-max′-marked places under a marking M ∈ R(N, M 0 ) does form a problematic siphon to make an S 4 R be not live. Thus, we add new constrains to the MIP method in Li and Li (2012b) to create a RMIP. The proposed RMIP method in Section 3.1 can solve NSSs in an S 4 R directly and then identify the solved ones causing deadlocks or livelocks depending on the value of U z t . Compared with the method in Liu and Li (2010), this RMIP method is relatively simple and its computation is lower. Moreover, an iterative LCP using the RMIP method and max′-controlled siphon controllability is designed in Section 3.2. Section 4 shows two examples to illustrate that the proposed RMIP method and corresponding LCP are feasible in theory and have potential for development of the theory to cope with both livelocks and deadlocks than the prior theory based on siphon controllability.

Conclusions
On the basis of the structural analysis of Petri nets and the concept of a set of non-max′-marked places, this research presents an RMIP approach to directly solve NSSs associated with deadlocks or livelocks in an S 4 R. Moreover, an LCP adopting the RMIP method and max′-controlled siphon controllability is designed to eliminate deadlocks and livelocks in a non-live S 4 R. The results in Section 4 show that the proposed LCP has a certain advantage over the existing policies in terms of behavioral permissiveness and structural complexity of the final live controlled system. The extension of the proposed RMIP and the corresponding LCP to more general classes of Petri nets is the focus of our future research.