A Set Covering Approach to Design Maximally Permissive Supervisors for Flexible Manufacturing Systems

Abstract

The supervisory control of Petri nets aims to enforce the undesired behavior as unreachable by designing a set of control places. This work presents a set cover approach to design maximally permissive supervisors. For each first-met bad marking, an integer linear programming problem is developed to obtain a control place to prohibit it. An objective function is formulated to make the maximal number of first-met bad markings forbidden. Then, we develop a set covering approach to minimize the number of selected control places. The proposed approach can guarantee the maximal permissiveness of the obtained supervisor and provide a trade-off between structural complexity and computational cost. Several examples are considered to validate the proposed method.

1. Introduction

Discrete event systems (DESs) are a class of systems whose state evolution is driven by the occurrence of events. Flexible manufacturing systems (FMSs) are a typical kind of DESs that can process various kinds of jobs with limited system resources. However, if the allocation of the system resources by different processes is not well handled, it is possible to have a deadlock problem [1,2,3,4,5,6,7,8,9,10]. Deadlocks in an FMS always make a system blocked such that the productivity is degraded. Hence, plenty of work has been carried out to handle deadlock problems in FMSs [11,12,13,14,15,16,17,18,19,20].

Petri nets are a significant tool to model, analyze, and control FMSs [21,22,23,24,25,26]. Based on a reachability graph analysis, it is possible to compute all states of an FMS [27]. Then, a Petri net supervisor is proposed to control the system by preventing it from deadlock states. For different control policies, three criteria are always considered to evaluate their performance: behavioral permissiveness, computational cost, and structural complexity. By a given control specification, the markings of a Petri net that represent the states of a system are classified into two collections: legal and illegal ones, where the legal ones represent the desired behavior and illegal ones indicate the ones that should be forbidden. If a designed supervisor can keep all legal markings of a Petri net model in the final net model, it is considered to be maximally permissive or optimal. This work focuses on the design of behaviorally optimal supervisors, as well as improves the structural complexity and computational costs.

Based on Petri nets, a number of researchers and engineers have performed a host of efforts to design maximally permissive supervisors. By the theory of regions, Uzam [28] provides a maximally permissive supervisor for FMSs. Ghaffari et al. [29] develop an integer linear programming problem (ILPP) to design a maximally permissive control place to disable a transition at its particular marking. Then, the controlled system cannot run out of the desired marking space. However, both methods need to solve ILPPs with too many constraints for sizable systems since there is one constraint for each legal marking. In order to avoid the solving of ILPPs, Uzam and Zhou [30,31] develop a straightforward technique to derive optimal or suboptimal Petri net supervisors. They consider a reachability graph as two partitions: a live zone and a deadlock zone. The live (dead) zone includes all the legal (illegal) markings. Then, they propose the definition of first-met bad markings (FBMs). An FBM M indicates that: (1) M is an illegal marking; and (2) there exists legal markings and a transition such that M is immediately reached from the legal marking by the transition. Then, they design a control place to make an FBM unreachable by restricting the number of tokens in a subset of places. Once the designed supervisor precludes all FBMs, the system finds it impossible to reach any markings in the deadlock zone. Unfortunately, it cannot ensure the maximal permissiveness of the obtained supervisor.

In order to make improvement on the behavioral permissiveness and computational burden for the computation of Petri net supervisors for FMSs, Chen et al. [32] consider deriving a control place to prohibit an FBM and make all legal markings reachable by formulating constraints in an ILPP. Meanwhile, by only considering the operation places, the reachability condition of two legal markings and the forbidding condition of two FBMs are well studied to extraordinarily decrease the number of involved legal markings and FBMs. In the following work [33,34], the objective function of ILPPs is well designed to minimize the structure of the obtained supervisor, that is to say, select the fewest control places. However, the developed ILPPs have too many constraints and variables since all control places are figured out by solving only one ILPP.

In order to trade off the computational cost and structural complexity, Chen et al. [35] work on the computation of maximally permissive supervisors by developing an iterative approach. At each iteration, they formulate an ILPP to compute a control place. All legal markings are ensured to be reachable by the well-constructed constraints in the ILPP. Meanwhile, the number of prohibited FBMs is maximized by an objective function. The main advantage is that the formulated ILPP has fewer constraints and variables but its disadvantage is that the obtained supervisor may not be structurally minimal.

This paper tries to proposes a good trade-off between computational cost and structural complexity. By using a set covering approach, we develop an ILPP to select the fewest control places, aiming to derive a behaviorally optimal and structurally simple supervisor. With the comparison of the work in [33], we do not need to solve ILPPs with too many constraints and variables. By considering the study in [35], our approach can obtain a smaller number of control places and even the minimal number of control places.

The remainder of the paper has the following contents. Section 2 contributes a set covering approach to select a minimal number of maximally permissive control places and an illustrative example to explain the proposed technique. We also consider a widely used example from the literature to make a comparison in Section 3. Finally, we make conclusions in Section 4. Appendix A outlines Petri nets, reachability analysis, a control place design for linear constraints, and a vector covering approach.

2. A Set Covering Approach to Design a Maximally Permissive Supervisor

This section proposes a set covering approach to synthesize a maximally permissive supervisor by minimizing the number of selected control places. As a result, the obtained supervisor is of a simple structure.

2.1. Synthesis of a Maximally Permissive Supervisor

By designing a control place, an FBM of a Petri net model can be prevented by enforcing a linear constraint on the reachable markings. This section outlines the approach in [35] to demonstrate the synthesis of a control place for an FBM.

Let M be the FBM of a net model and $p_s$ be the control place for the enforcement of a linear constraint $(L,\beta )$, as shown in Equation (A7). Marking M can be forbidden if L and $\beta$ satisfy

$$\beta =\sum _{i\in {\mathbb{N}}_A}{l}_i·M\left(p_i\right)-1$$

(1)

Equation (1) indicates that M does not meet the linear constraint $(L,\beta )$. In this case, once $(L,\beta )$ is enforced on the net model, M cannot be reached anymore.

Optimal control should not exclude any legal marking in the controlled net model. Therefore, all legal markings should satisfy the linear constraint $(L,\beta )$, as shown below:

$$\sum _{i\in {\mathbb{N}}_A}{l}_i·M_l\left(p_i\right)\le \beta ,\forall M_l\in {\mathcal{M}}_L$$

(2)

By considering Corollary A2, only the reachability of the markings in the minimal covering set ${\mathcal{M}}_L^{★}$ of legal markings is sufficient for optimal control. Hence, Equation (2) is modified to be

$$\sum _{i\in {\mathbb{N}}_A}{l}_i·M_l\left(p_i\right)\le \beta ,\forall M_l\in {\mathcal{M}}_L^{★}$$

(3)

We often call Equation (3) the reachability condition. By Equation (1), Equation (3) becomes

$$\sum _{i\in {\mathbb{N}}_A}{l}_i·(M_l\left(p_i\right)-M\left(p_i\right))\le -1,\forall M_l\in {\mathcal{M}}_L^{★}$$

(4)

Equation (4) is used to determine the linear constraint $(L,\beta )$. Then, we formulate a control place to enforce $(L,\beta )$. As a result, the designed control place can forbid M but no legal markings, which indicates that the designed control place is maximally permissive. By Corollary A1, if all FBMs in the minimal covered set ${\mathcal{M}}_F^{★}$ of FBMs are forbidden, then no FBM can be reached. However, the supervisor structure may be too complex if we derive a control place for each FBM in ${\mathcal{M}}_F^{★}$. Hence, we show how to design a behaviorally optimal supervisor, but the number of control places in it is much smaller than that of FBMs in ${\mathcal{M}}_F^{★}$.

A control place can generally prohibit more than one FBM. Hence, the work in [35] proposes to formulate a control place to forbid as many FBMs as possible. We briefly outline it as follows.

Let $p_{s_j}$ be a control place to prohibit an FBM $M_j\in {\mathcal{M}}_F^{★}$ by enforcing a linear constraint $(L_j,{\beta }_j)$:

$$\sum _{i\in {\mathbb{N}}_A}{l}_{j,i}·{\mu }_i\le {\beta }_j$$

(5)

with

$${\beta }_j=\sum _{i\in {\mathbb{N}}_A}{l}_{j,i}·M_j\left(p_i\right)-1$$

(6)

For any FBM $M_f\in {\mathcal{M}}_F(f\ne j)$, it is forbidden by $p_{s_j}$ if it satisfies the following constraint:

$$\sum _{{\mathbb{N}}_A}{l}_i·M_f\left(p_i\right)\ge {\beta }_j+1$$

(7)

By substituting Equation (6) into Equation (7), we have

$$\sum _{{\mathbb{N}}_A}{l}_i·M_f\left(p_i\right)\ge \sum _{i\in {\mathbb{N}}_A}{l}_{j,i}·M_j\left(p_i\right)-1+1$$

(8)

By reformulating Equation (8), we have

$$\sum _{{\mathbb{N}}_A}{l}_i·(M_f\left(p_i\right)-M_j\left(p_i\right))\ge 0$$

(9)

For each FBM $M_f\in {\mathcal{M}}_F^{★}$, we use a binary variable $x_{j,f}\in \{0,1\}$ to indicate whether it is forbidden by $p_{s_j}$. Specifically, $x_{j,f}=1$ means that $M_f$ is forbidden by $p_{s_j}$ and $x_{j,f}=0$ indicates that $M_f$ is not forbidden by $p_{s_j}$. Then, we have

$$\sum _{{\mathbb{N}}_A}{l}_i·(M_f\left(p_i\right)-M_j\left(p_i\right))\ge -Q·(1-x_{j,f}),\forall M_f\in {\mathcal{M}}_F^{★}\mathrm{and}f\ne j$$

(10)

where Q is a big enough constant. Then, we formulate an objective function to maximize the number of forbidden FBMs as follows:

$$max{x}_j=\sum _{M_f\in {\mathcal{M}}_F^{★},f\ne j}{x}_{j,f}$$

(11)

By combining Equations (10) and (11), we have an ILPP to determine a control place to forbid as many FBMs as possible, which is denoted as the maximal number of forbidden FBMs problem (MNFFP($M_j$)).

MNFFP($M_j$):

$$\begin{array}{cc}\hfill & max{x}_j=\sum _{M_f\in {\mathcal{M}}_F^{★},f\ne j}{x}_{j,f}\hfill \\ \hfill & \mathrm{subject}\mathrm{to}\hfill \\ \hfill & \sum _{i\in {\mathbb{N}}_A}{l}_{j,i}·(M_l\left(p_i\right)-M_j\left(p_i\right))\le -1,\forall M_l\in {\mathcal{M}}_L^{★}\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & \sum _{{\mathbb{N}}_A}{l}_i·(M_f\left(p_i\right)-M_j\left(p_i\right))\ge -Q·(1-x_{j,f}),\forall M_f\in {\mathcal{M}}_F^{★},f\ne j\hfill \\ \hfill & l_{j,i}\in \{0,1,2,\dots \},\forall i\in {\mathbb{N}}_A\hfill \\ \hfill & x_{j,f}\in \{0,1\},\forall M_f\in {\mathcal{M}}_F^{★}\mathrm{and}f\ne j\hfill \end{array}$$

(13)

Theorem 1.

Let $l_{j,i}^{★}$$(i\in {\mathbb{N}}_A)$ be an optimal solution of MNFFP($M_j$). Then, the linear constraint $(L^{★},{\beta }^{★})$ with $L^{★}={[l_{j,1}^{★},l_{j,2}^{★},\dots ,l_{j,|P_A|}^{★}]}^T$ and ${\beta }^{★}={\sum }_{i\in {\mathbb{N}}_A}{l}_{j,i}^{★}·M_j\left(p_i\right)-1$ can forbid $M_j$ but no legal markings.

Proof. 

By ${\beta }^{★}={\sum }_{i\in {\mathbb{N}}_A}{l}_{j,i}^{★}·M_j\left(p_i\right)-1$, $M_j$ is forbidden by the linear constraint $(L^{★},{\beta }^{★})$ since it does not satisfy the constraint. By Equation (12), no legal marking in ${\mathcal{M}}_L^{★}$ is unreachable since all legal markings in ${\mathcal{M}}_L^{★}$ satisfy the linear constraint $(L^{★},{\beta }^{★})$. By Corollary A2, all legal markings are reachable. Finally, the conclusion holds.    □

MNFFP($M_j$) can be used to compute a control place to prohibit the maximal number of FBMs. Then, an iterative algorithm is applied to design an optimal supervisor with a simple structure [35]. However, the number of control places is not minimized. Hence, we develop a set covering problem to reduce the number of control places.

2.2. A Set Covering Approach to Simplify a Supervisor Structure

By MNFFP($M_j$), we can design a control place to maximize the number of prohibited FBMs. However, it is not necessary to add all the control places obtained by MNFFP($M_j$). In the following, we provide a set covering approach to select the minimum number of control places to suppress all the FBMs. Let $l_{j,i}^{★}$’s ($i\in {\mathbb{N}}_A$) and ${\beta }_j^{★}$ be the optimal solution of MNFFP($M_j$). Then, a control place $p_{s_j}$ is designed by solution $l_{j,i}^{★}$’s ($i\in {\mathbb{N}}_A$) and ${\beta }_j^{★}$. Let $n_F^{★}$ be the number of elements in ${\mathcal{M}}_F^{★}$, i.e., $n_F^{★}=|{\mathcal{M}}_F^{★}|$. Then, the relationship between an FBM $M_k\in {\mathcal{M}}_F^{★}$ and $p_{s_j}$ can be represented by an $n_F^{★}\times n_F^{★}$ matrix A, where

$$A_{jk}=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\sum _{i\in {\mathbb{N}}_A}{l}_{j,i}^{★}·M_k\left(p_i\right)\ge {\beta }_j^{★}+1\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(14)

In Equation (14), $A_{jk}$ indicates the element in the jth row and kth column of A. Specifically, $A_{jk}=1$ means that an FBM $M_k$ is suppressed by $p_{s_j}$, and $A_{jk}=0$ indicates that $M_j$ is not suppressed by $p_{s_j}$. For minimizing the number of selected control places, we present a binary variable ${\zeta }_j$ for each control place $p_{s_j}$. The variable ${\zeta }_j$ is used to represent whether a control place $p_{s_j}$ is selected or not. Specially, ${\zeta }_j=1$ indicates that $p_{s_j}$ is selected and ${\zeta }_j=0$ shows that $p_{s_j}$ should be removed.

For each FBM $M_k\in {\mathcal{M}}_F^{★}$, it must be prohibited by at least one selected control place. Hence, we have

$$\sum _{j=1}^{n_F^{★}}{A}_{j,k}·{\zeta }_j\ge 1$$

(15)

According to Equation (15), it means that there exists $j\in \{1,2,\dots ,n_F^{★}\}$ such that $A_{j,k}·{\zeta }_j=1$. The fact $A_{j,k}·{\zeta }_j=1$ means that $A_{j,k}=1$ and ${\zeta }_j=1$, that is to say, $M_k$ is forbidden by a selected control place $p_{s_j}$. In summary, Equation (15) can ensure that at least one control place is selected to suppress an FBM $M_k$. Then, an objective function is given to select the fewest control places, as shown below:

$$min\zeta =\sum _{j=1}^{n_F^{★}}{\zeta }_j$$

(16)

By combining Equations (15) and (16), an ILPP is created to minimize the number of control places to prohibit all FBMs, which is denoted as the set covering problem of control places (SEPCP).

SEPCP:

$$\begin{array}{cc}\hfill & min\zeta =\sum _{j=1}^{n_F^{★}}{\zeta }_j\hfill \\ \hfill & \mathrm{subject}\mathrm{to}\hfill \\ \hfill & \sum _{j=1}^{n_F^{★}}{A}_{j,k}·{\zeta }_j\ge 1,k\in \{1,2,\dots ,n_F^{★}\}\hfill \\ \hfill & {\zeta }_j\in \{0,1\},j\in \{1,2,\dots ,n_F^{★}\}\hfill \end{array}$$

(17)

Theorem 2.

SEPCP can select the minimal number of control places to forbid all FBMs in ${\mathcal{M}}_F^{★}$.

Proof. 

For an FBM $M_k$, Equation (17) can ensure that there exists $j\in \{1,2,\dots ,n_F^{★}\}$ such that $A_{j,k}·{\zeta }_j=1$, which means that $A_{j,k}=1$ and ${\zeta }_j=1$. By $A_{j,k}=1$, $M_k$ is prohibited by a control place $p_{s_j}$. According to ${\zeta }_j=1$, $p_{s_j}$ is selected. In summary, at least one selected control place forbids $M_k$. By considering all FBMs in ${\mathcal{M}}_F^{★}$, Equation (17) ensures that each FBM is forbidden by at least one selected control place. Finally, the objective function can minimize the number of control places to be selected.    □

2.3. Synthesis of a Maximally Permissive Supervisor

This section proposes an algorithm to obtain a maximally permissive supervisor by applying MNFFP($M_j$) and SEPCP.

Algorithm 1 first generates a reachability graph of a Petri net model. Then, the legal markings and FBMs to be considered are reduced by the vector covering approach. An ILPP, namely, MNFFP($M_j$), is formulated to design a control place for each FBM in the minimal covered set. Then, a set covering approach is applied to select the fewest control places by solving an ILPP, i.e., SEPCP. Finally, a maximally permissive supervisor with a simple structure is obtained.

Theorem 3.

Algorithm 1 obtains a maximally permissive supervisor for a Petri net model of an FMS if MNFFP($M_j$) has an optimal solution for all $M_j\in {\mathcal{M}}_F^{★}$.

Algorithm 1: Design of an optimal supervisor.

Input: Petri net model $(N,M_0)$ of an FMS with $N=(P^0\cup P_A\cup P_R$, T, F, $W)$.     Output: A maximally permissive controlled Petri net system $(N_1,M_1)$.     1. ${\mathcal{M}}_L=\left\{M\right|M\in R(N,M_0)\wedge M_0\in R(N,M)\}$ and ${\mathcal{M}}_F=\left\{M\right|M\in {\mathcal{M}}_{\overline{L}}\wedge (\exists M^{\prime }\in {\mathcal{M}}_L)(\exists t\in T)M^{\prime }[t\rangle M\}$.     2. Compute ${\mathcal{M}}_{\mathrm{FBM}}^{★}$ and ${\mathcal{M}}_L^{★}$ for $(N,M_0)$.     3. $V_M:=\varnothing$. /* $V_M$ denotes the set of control places to be computed.*/     4. foreach $M_j\in {\mathcal{M}}_{\mathrm{FBM}}^{★}$ do             Formulate MNFFP($M_j$).             if solve(MNFFP($M_j$))=false do                  Exit. //* The approach fails to find a maximally permissive supervisor. *//             else                  ($l_{j,i}^{★}(i\in {\mathbb{N}}_A)$, ${\beta }_j^{★}$)=solve(MNFFP($M_j$))                  Compute $p_{s_j}$ for ($l_{j,i}^{★}(i\in {\mathbb{N}}_A)$, ${\beta }_j^{★}$) by Equations (A5) and (A6).             end if         end foreach     5. Let $n_F^{★}=|{\mathcal{M}}_F^{★}|$ and compute A by Equation (14).     6. ${\zeta }_j$ $(j\in \{1,2,\dots ,n_F^{★}\})=solve\left(SEPCP\right)$.     7. foreach ${\zeta }_j=1$ $(j\in \{1,2,\dots ,n_F^{★}\})$ do             $V_M:=V_M\cup \{p_{s_j}\}$.         end foreach     8. $(N_1,M_1)$=$V_M\oplus (N,M_0)$ //*Add all control places in $V_M$ to $(N,M_0)$. *//     9. Output $(N_1,M_1)$.     10. End.

Proof. 

If MNFFP($M_j$) has an optimal solution for all $M_j\in {\mathcal{M}}_F^{★}$, then each obtained control place $p_{s_j}$ is maximally permissive according to Theorem 1. With Theorem 2, SEPCP can minimize the number of control places to prohibit all FBMs in ${\mathcal{M}}_F^{★}$. By Corollary A1, all FBMs are prohibited. As a result, the controlled system is live since it can never reach any illegal markings. In summary, the conclusion holds. □

Next, we discuss the computational complexity of the proposed approach. The proposed method requires the generation of the reachability graph of a Petri net model and solve ILPPs. ILPPs are NP-hard, which means that the computational complexity of the proposed method is NP-hard in theory. In this sense, the related studies in [33,34,35] suffer from the same computational complexity since they all need to generate the reachability graph of a Petri net model and solve ILPPs. However, from the viewpoint of computational cost, we can perform the comparison with the numbers of constraints and variables in the formulated ILPPs. Compared with the work in [33,34], the proposed method is much more efficient since the proposed ILPP has many fewer constraints and variables. Compared with the work in [35], the proposed approach has a very similar computational cost since the ILPPs have a similar numbers of constraints and variables. On the other hand, from the viewpoint of the structural complexity, the proposed work can find a structurally simpler supervisor to the study in [35]. Furthermore, the experimental results show that the proposed method can always obtain the structurally minimal supervisor, as obtained by the work in [33,34].

2.4. An Illustrative Example

In this section, a widely used example in the literature is provided to illustrate the proposed method. An FMS is shown in Figure 1a. It consists of two robots R1-2, each of which can hold one part at a time; four machines M1-4, each of which can process one part at a time; two loading buffers I1-2; and two unloading buffers O1-2. Two part types are considered in the system: P1 and P2. The production sequences are shown in Figure 1b.

The Petri net model of Figure 1 is shown in Figure 2. It has been well-studied in [28,32,33,35,36,37,38]. The net model has 282 reachable markings, 205 and 54 of which are legal markings and FBMs, respectively. By using the vector covering approach, the numbers of legal markings and FBMs to be considered are reduced to be 26 and 8, respectively. Then, we can formulate an ILPP MNFFP($M_j$) for each FBM $M_j\in {\mathcal{M}}_F^{★}$. The obtained constraints $(L_j,{\beta }_j)$’s by MNFFP($M_j$) are presented in

Table 1. The linear constraints obtained by MNFFP($M_j$).

j	${\mathit{M}}_{\mathit{j}}$	$({\mathit{L}}_{\mathit{j}},{\mathit{\beta }}_{\mathit{j}})$
1	$p_2+p_3+p_4$	${\mu }_2+2{\mu }_3+{\mu }_4+2{\mu }_{11}+2{\mu }_{12}\le 3$
2	$p_3+p_5+p_9+p_{10}$	$4{\mu }_2+8{\mu }_3+4{\mu }_4+5{\mu }_5+{\mu }_9+{\mu }_{10}+8{\mu }_{11}+7{\mu }_{12}\le 14$
3	$p_3+p_6+p_9+p_{10}$	${\mu }_2+2{\mu }_3+{\mu }_4+2{\mu }_5+2{\mu }_6+3{\mu }_9+3{\mu }_{10}\le 9$
4	$p_5+p_6+p_9+p_{10}$	${\mu }_2+2{\mu }_3+{\mu }_4+2{\mu }_5+2{\mu }_6+3{\mu }_9+3{\mu }_{10}\le 9$
5	$p_2+p_4+p_6+p_9+p_{10}$	${\mu }_2+2{\mu }_3+{\mu }_4+2{\mu }_5+2{\mu }_6+3{\mu }_9+3{\mu }_{10}\le 9$
6	$p_{11}+p_{12}$	$4{\mu }_2+8{\mu }_3+4{\mu }_4+5{\mu }_5+{\mu }_9+{\mu }_{10}+8{\mu }_{11}+7{\mu }_{12}\le 14$
7	$p_2+p_4+p_{12}$	$4{\mu }_2+8{\mu }_3+4{\mu }_4+5{\mu }_5+{\mu }_9+{\mu }_{10}+8{\mu }_{11}+7{\mu }_{12}\le 14$
8	$p_3+p_{11}$	${\mu }_2+2{\mu }_3+{\mu }_4+2{\mu }_{11}+2{\mu }_{12}\le 3$

, where the first column j is the index of an FBM $M_j$, the second column indicates FBM $M_j$, and the last column provides the obtained linear constraint $(L_j,{\beta }_j)$.

With the solution of MNFFP($M_j$), we can design a control place $p_{s_j}$ to enforce $(L_j,{\beta }_j)$$(j\in \{1,2,\dots ,8\left\}\right)$. Then, we can decide the relationship between $p_{s_j}$ and $M_k\in {\mathcal{M}}_F^{★}$, which is represented by matrix A, as shown in

Table 2. Matrix A for the net shown in Figure 2.

${\mathit{p}}_{{\mathit{s}}_{\mathit{j}}}$	${\mathit{M}}_1$	${\mathit{M}}_2$	${\mathit{M}}_3$	${\mathit{M}}_4$	${\mathit{M}}_5$	${\mathit{M}}_6$	${\mathit{M}}_7$	${\mathit{M}}_8$
$p_{s_1}$	1	0	0	0	0	1	1	1
$p_{s_2}$	1	1	0	0	0	1	1	1
$p_{s_3}$	0	1	1	1	1	0	0	0
$p_{s_4}$	0	1	1	1	1	0	0	0
$p_{s_5}$	0	1	1	1	1	0	0	0
$p_{s_6}$	1	1	0	0	0	1	1	1
$p_{s_7}$	1	1	0	0	0	1	1	1
$p_{s_8}$	1	0	0	0	0	1	1	1

.

Next, we can formulate an ILPP, SEPCP, to select the fewest control places from the obtained eight. SEPCP is presented below:

SEPCP:

$$\begin{array}{c}main\zeta ={\zeta }_1+{\zeta }_2+{\zeta }_3+{\zeta }_4+{\zeta }_5+{\zeta }_6+{\zeta }_7+{\zeta }_8\hfill \\ \mathrm{subject}\mathrm{to}\hfill \\ {\zeta }_1+{\zeta }_2+{\zeta }_6+{\zeta }_7+{\zeta }_8\ge 1\hfill \\ {\zeta }_2+{\zeta }_3+{\zeta }_4+{\zeta }_5+{\zeta }_6+{\zeta }_7\ge 1\hfill \\ {\zeta }_3+{\zeta }_4+{\zeta }_5\ge \hfill \\ {\zeta }_1+{\zeta }_2+{\zeta }_6+{\zeta }_7+{\zeta }_8\ge 1\hfill \\ {\zeta }_j\in \{0,1\},\forall j\in \{1,2,\dots ,8\}\hfill \end{array}$$

Note that we have already removed the redundant constraints from SEPCP since the constraints (represented as Equation (17)) for the three markings $M_3, M_4, M_5$ are the same, and the constraints for $M_6, M_7, M_8$ are the same. By solving SEPCP, we derive an optimal solution with ${\zeta }_2=1$, ${\zeta }_5=1$, and ${\zeta }_j=0(j\in \{1,3,4,6,7,8\})$. Hence, we only need to select two control places $p_{s_2}$ and $p_{s_5}$ to forbid all FBMs. Specifically, from Table 1, it can be seen that $p_{s_2}$ and $p_{s_5}$ are used to enforce the linear constraints $4{\mu }_2+8{\mu }_3+4{\mu }_4+5{\mu }_5+{\mu }_9+{\mu }_{10}+8{\mu }_{11}+7{\mu }_{12}\le 14$ and ${\mu }_2+2{\mu }_3+{\mu }_4+2{\mu }_5+2{\mu }_6+3{\mu }_9+3{\mu }_{10}\le 9$, respectively. As a result, for $p_{s_2}$, we have ${}^{•}p_{s_2}=\{3t_5,5t_6,t_{12},7t_{13}\}$, $p_{s_2}^{•}=\{4t_1,4t_2,t_4,t_9,7t_{11}\}$, and $M_0\left(p_{s_2}\right)=14$. For $p_{s_5}$, we have ${}^{•}p_{s_5}=\{t_1,t_2,t_4,3t_9\}$, $p_{s_5}^{•}=\{2t_7,3t_{11}\}$, and $M_0\left(p_{s_5}\right)=9$. By adding $p_{s_2}$ and $p_{s_5}$ to the original net model, the obtained net model is live with all 205 legal markings. That is to say, the original net model is optimally controlled by the two obtained control places.

The net model has been widely studied in the literature. We compare the performance of some classical policies for this example in

Table 3. Performance comparison for the Petri net in Figure 2.

Parameters	[38]	[36]	[32]	[33]	Alg. 1 in [37]	Alg. 2 in [37]	This Work
No. monitors	9	5	8	2	5	2	2
No. states	205	205	205	205	205	205	205

. Note that we only choose some maximally permissive control policies to perform the comparison. The supervisor obtained by the proposed approach has the same number of control places as that obtained in [33], which is the smallest one among all control policies in this table. However, the ILPP proposed in [33] has 328 constraints and 152 variables. However, in this work, MNFFP($M_j$) has only 33 constraints and 18 variables, and SEPCP has 4 constraints and 8 variables. Hence, the proposed approach can remarkably improve the efficiency of the one in [33] but also leads to the minimum number of control places.

3. Experimental Results

In this section, the proposed approach is applied to a more complex example. An FMS is shown in Figure 3a. It consists of three robots R1-3, each of which can hold one part at a time; four machines M1-4, each of which can process two parts at a time; three loading buffers I1-3; and three unloading buffers O1-3. Three part types are considered in the system: P1, P2, and P3. The production sequences are shown in Figure 3b.

The Petri net model of Figure 3 is shown in Figure 4, which has been widely studied [21,30,32,33,35,36,39,40]. The net model has 26 places and 20 transitions. The places has the following set partitions: $P^0=\{p_1,p_5,p_{14}\}$, $P_A=\{p_2-p_4,p_6-p_{13},p_{15}-p_{19}\}$, and $P_R=\{p_{20}-p_{26}\}$. The net model has 26,750 reachable markings; the numbers of legal markings and FBMs are 21,581 and 4211, respectively. By using the marking reduction approach, the numbers of legal markings and FBMs are reduced to be 393 and 34, respectively. Then, we can formulate an ILPP MNFFP($M_j$) for each FBM $M_j\in {\mathcal{M}}_F^{★}$. MNFFP($M_j$) is solved by an ILPP solver Lingo [41] in a computer with the Windows operating system with an Intel CPU Core 2.7 GHz and 8 GB memory. The obtained constraints $(L_j,{\beta }_j)$’s by MNFFP($M_j$) are presented in

Table A1. The linear constraints obtained by MNFFP($M_j$).

j	${\mathit{M}}_{\mathit{j}}$	$\mathit{\tau }$ (S)	$({\mathit{L}}_{\mathit{j}},{\mathit{\beta }}_{\mathit{j}})$
1	$2p_{11}+2p_{16}$	2	$8{\mu }_2+8{\mu }_3+2{\mu }_6+2{\mu }_7+9{\mu }_8+9{\mu }_9+75{\mu }_{11}+25{\mu }_{12}+25{\mu }_{13}+50{\mu }_{15}+75{\mu }_{16}+{\mu }_{17}+{\mu }_{18}\le 299$
2	$p_6+2p_7+p_8+p_9+p_{11}+p_{15}+2p_{16}+p_{18}$	4	$6{\mu }_2+6{\mu }_3+{\mu }_6+{\mu }_7+7{\mu }_8+7{\mu }_9+{\mu }_{11}+19{\mu }_{12}+20{\mu }_{13}+20{\mu }_{15}+20{\mu }_{16}+{\mu }_{18}\le 78$
3	$p_6+2p_7+2p_9+p_{11}+p_{15}+2p_{16}+p_{18}$	4	${\mu }_2+2{\mu }_3+{\mu }_4+2{\mu }_5+2{\mu }_6+3{\mu }_9+3{\mu }_{10}\le 9$
4	$p_{12}+2p_{16}$	2	$15{\mu }_2+15{\mu }_3+3{\mu }_6+3{\mu }_7+17{\mu }_8+17{\mu }_9+3{\mu }_{11}+97{\mu }_{12}+47{\mu }_{13}+50{\mu }_{15}+97{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 290$
5	$p_6+2p_7+p_8+p_9+p_{11}+p_{13}+p_{15}+p_{16}+p_{18}$	1	$12{\mu }_2+12{\mu }_3+2{\mu }_6+2{\mu }_7+14{\mu }_8+14{\mu }_9+2{\mu }_{11}+38{\mu }_{12}+38{\mu }_{13}+39{\mu }_{15}+39{\mu }_{16}+2{\mu }_{18}\le 153$
6	$p_6+2p_7+2p_9+p_{11}+p_{13}+p_{15}+p_{16}+p_{18}$	5	$12{\mu }_2+12{\mu }_3+2{\mu }_6+2{\mu }_7+14{\mu }_8+14{\mu }_9+2{\mu }_{11}+38{\mu }_{12}+38{\mu }_{13}+39{\mu }_{15}+39{\mu }_{16}+2{\mu }_{18}\le 153$
7	$p_3+p_8+p_9+p_{13}+p_{15}+p_{16}$	8	$13{\mu }_2+13{\mu }_3+3{\mu }_6+3{\mu }_7+15{\mu }_8+15{\mu }_9+3{\mu }_{11}+43{\mu }_{12}+46{\mu }_{13}+46{\mu }_{15}+46{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 180$
8	$p_2+2p_9+p_{13}+p_{15}+p_{16}$	5	$13{\mu }_2+13{\mu }_3+3{\mu }_6+3{\mu }_7+15{\mu }_8+15{\mu }_9+3{\mu }_{11}+43{\mu }_{12}+46{\mu }_{13}+46{\mu }_{15}+46{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 180$
9	$p_6+2p_7+p_8+p_9+p_{15}+2p_{16}+2p_{18}$	3	$15{\mu }_2+15{\mu }_3+3{\mu }_6+3{\mu }_7+17{\mu }_8+17{\mu }_9+3{\mu }_{11}+47{\mu }_{12}+50{\mu }_{13}+50{\mu }_{15}+50{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 196$
10	$p_6+2p_7+2p_9+p_{15}+2p_{16}+2p_{18}$	4	$15{\mu }_2+15{\mu }_3+3{\mu }_6+3{\mu }_7+17{\mu }_8+17{\mu }_9+3{\mu }_{11}+47{\mu }_{12}+50{\mu }_{13}+50{\mu }_{15}+50{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 196$
11	$p_2+2p_9+p_{15}+2p_{16}$	5	$13{\mu }_2+13{\mu }_3+3{\mu }_6+3{\mu }_7+15{\mu }_8+15{\mu }_9+3{\mu }_{11}+43{\mu }_{12}+46{\mu }_{13}+46{\mu }_{15}+46{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 180$
12	$p_2+p_3+p_9+p_{15}+2p_{16}$	2	$15{\mu }_2+15{\mu }_3+3{\mu }_6+3{\mu }_7+17{\mu }_8+17{\mu }_9+3{\mu }_{11}+47{\mu }_{12}+50{\mu }_{13}+50{\mu }_{15}+50{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 196$
13	$p_8+2p_9+p_{15}+2p_{16}$	5	$3{\mu }_6+3{\mu }_7+13{\mu }_8+13{\mu }_9+3{\mu }_{11}+39{\mu }_{12}+42{\mu }_{13}+42{\mu }_{15}+42{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 164$
14	$p_3+p_8+p_9+p_{15}+2p_{16}$	8	$13{\mu }_2+13{\mu }_3+3{\mu }_6+3{\mu }_7+15{\mu }_8+15{\mu }_9+3{\mu }_{11}+43{\mu }_{12}+46{\mu }_{13}+46{\mu }_{15}+46{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 180$
15	$p_6+2p_7+2p_9+p_{11}+p_{12}+p_{15}+p_{16}+p_{18}$	2	$27{\mu }_2+27{\mu }_3+5{\mu }_6+5{\mu }_7+31{\mu }_8+31{\mu }_9+5{\mu }_{11}+85{\mu }_{12}+86{\mu }_{13}+88{\mu }_{15}+88{\mu }_{16}+{\mu }_{17}+4{\mu }_{18}\le 346$
16	$p_6+2p_7+2p_9+p_{11}+p_{15}+2p_{16}+p_{17}$	2	$15{\mu }_2+15{\mu }_3+3{\mu }_6+3{\mu }_7+17{\mu }_8+17{\mu }_9+3{\mu }_{11}+47{\mu }_{12}+50{\mu }_{13}+50{\mu }_{15}+50{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 196$
17	$p_2+p_3+2p_9+p_{13}+p_{15}+p_{16}$	3	$15{\mu }_2+15{\mu }_3+3{\mu }_6+3{\mu }_7+17{\mu }_8+17{\mu }_9+3{\mu }_{11}+47{\mu }_{12}+50{\mu }_{13}+50{\mu }_{15}+50{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 196$
18	$p_3+p_8+p_9+2p_{11}+p_{15}+p_{16}$	4	$26{\mu }_2+26{\mu }_3+2{\mu }_6+2{\mu }_7+26{\mu }_8+27{\mu }_9+9{\mu }_{11}+27{\mu }_{12}+18{\mu }_{13}+27{\mu }_{15}+27{\mu }_{16}+{\mu }_{17}+{\mu }_{18}\le 150$
19	$p_8+2p_9+2p_{11}+p_{15}+p_{16}$	7	$27{\mu }_2+27{\mu }_3+2{\mu }_6+2{\mu }_7+26{\mu }_8+28{\mu }_9+9{\mu }_{11}+28{\mu }_{12}+19{\mu }_{13}+28{\mu }_{15}+28{\mu }_{16}+{\mu }_{17}+{\mu }_{18}\le 155$
20	$p_8+2p_9+p_{13}+p_{15}+p_{16}$	4	$28{\mu }_2+28{\mu }_3+2{\mu }_6+2{\mu }_7+27{\mu }_8+29{\mu }_9+10{\mu }_{11}+29{\mu }_{12}+19{\mu }_{13}+29{\mu }_{15}+29{\mu }_{16}+{\mu }_{17}+{\mu }_{18}\le 161$
21	$p_6+2p_7+p_8+p_9+p_{13}+p_{15}+p_{16}+2p_{18}$	3	$15{\mu }_2+15{\mu }_3+3{\mu }_6+3{\mu }_7+17{\mu }_8+17{\mu }_9+3{\mu }_{11}+47{\mu }_{12}+50{\mu }_{13}+50{\mu }_{15}+50{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 196$
22	$p_6+2p_7+p_9+p_{13}+p_{15}+p_{16}+2p_{18}$	7	$15{\mu }_2+15{\mu }_3+3{\mu }_6+3{\mu }_7+17{\mu }_8+17{\mu }_9+3{\mu }_{11}+47{\mu }_{12}+50{\mu }_{13}+50{\mu }_{15}+50{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 196$
23	$p_2+2p_9+2p_{11}+p_{15}+p_{16}$	3	$26{\mu }_2+26{\mu }_3+2{\mu }_6+2{\mu }_7+27{\mu }_8+27{\mu }_9+9{\mu }_{11}+28{\mu }_{12}+19{\mu }_{13}+28{\mu }_{15}+28{\mu }_{16}+{\mu }_{17}+{\mu }_{18}\le 153$
24	$p_{12}+p_{13}+p_{15}+p_{16}$	3	$15{\mu }_2+15{\mu }_3+3{\mu }_6+3{\mu }_7+17{\mu }_8+17{\mu }_9+3{\mu }_{11}+47{\mu }_{12}+50{\mu }_{13}+50{\mu }_{15}+50{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 196$
25	$p_2+2p_3$	1	$11{\mu }_2+11{\mu }_3+11{\mu }_8+3{\mu }_9+{\mu }_{11}+2{\mu }_{13}+3{\mu }_{15}+3{\mu }_{16}\le 32$
26	$p_6+2p_7+p_{11}+p_{17}+p_{18}$	2	$17{\mu }_6+17{\mu }_7+2{\mu }_8+2{\mu }_9+17{\mu }_{11}+3{\mu }_{12}+3{\mu }_{13}+4{\mu }_{15}+4{\mu }_{16}+15{\mu }_{17}+17{\mu }_{18}\le 99$
27	$2p_3+p_8$	1	$11{\mu }_2+11{\mu }_3+11{\mu }_8+3{\mu }_9+{\mu }_{11}+2{\mu }_{13}+3{\mu }_{15}+3{\mu }_{16}\le 32$
28	$p_2+p_3+p_9+2p_{11}+p_{15}+p_{16}$	3	$28{\mu }_2+28{\mu }_3+2{\mu }_6+2{\mu }_7+29{\mu }_8+29{\mu }_9+10{\mu }_{11}+30{\mu }_{12}+20{\mu }_{13}+30{\mu }_{15}+30{\mu }_{16}+{\mu }_{17}+{\mu }_{18}\le 164$
29	$2p_{11}+p_{13}+p_{15}+p_{16}$	2	$8{\mu }_2+8{\mu }_3+2{\mu }_6+2{\mu }_7+9{\mu }_8+9{\mu }_9+75{\mu }_{11}+25{\mu }_{12}+25{\mu }_{13}+50{\mu }_{15}+75{\mu }_{16}+{\mu }_{17}+{\mu }_{18}\le 299$
30	$2p_{11}+p_{12}+p_{15}+p_{16}$	3	$8{\mu }_2+8{\mu }_3+2{\mu }_6+2{\mu }_7+9{\mu }_8+9{\mu }_9+75{\mu }_{11}+25{\mu }_{12}+25{\mu }_{13}+50{\mu }_{15}+75{\mu }_{16}+{\mu }_{17}+{\mu }_{18}\le 299$
31	$p_6+2p_7+p_{17}+2p_{18}$	1	$21{\mu }_6+21{\mu }_7+2{\mu }_8+2{\mu }_9+21{\mu }_{11}+5{\mu }_{13}+5{\mu }_{15}+5{\mu }_{16}+19{\mu }_{17}+20{\mu }_{18}\le 121$
32	$2p_{11}+p_{17}$	2	$5{\mu }_2+4{\mu }_3+{\mu }_6+{\mu }_7+5{\mu }_8+5{\mu }_9+125{\mu }_{11}+14{\mu }_{12}+14{\mu }_{13}+28{\mu }_{15}+42{\mu }_{16}+84{\mu }_{17}\le 333$
33	$2p_{13}+p_{15}$	2	$15{\mu }_2+15{\mu }_3+3{\mu }_6+3{\mu }_7+17{\mu }_8+17{\mu }_9+3{\mu }_{11}+47{\mu }_{12}+97{\mu }_{13}+97{\mu }_{15}+50{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 290$
34	$p_6+2p_7+2p_9+p_{11}+p_{13}+p_{15}+p_{16}+p_{17}$	3	$15{\mu }_2+15{\mu }_3+3{\mu }_6+3{\mu }_7+17{\mu }_8+17{\mu }_9+3{\mu }_{11}+47{\mu }_{12}+50{\mu }_{13}+50{\mu }_{15}+50{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 196$

, where the first column shows the index of FBM $M_j$, the second indicates $M_j$, the third provides the CPU time for MNFFP($M_j$), and the last presents the obtained constraints $(L_j,{\beta }_j)$.

With the solution of MNFFP($M_j$), we can design a control place $p_{s_j}$ to enforce $(L_j,{\beta }_j)$$(j\in \{1,2,\dots ,8\left\}\right)$. Then, we can decide the relationship between $p_{s_j}$ and $M_k\in {\mathcal{M}}_F^{★}$, which is represented by matrix A, as shown in

Table A2. Matrix A for the net shown in Figure 4.

${\mathit{p}}_{{\mathit{s}}_{\mathit{j}}}$	${\mathit{M}}_1$	${\mathit{M}}_2$	${\mathit{M}}_3$	${\mathit{M}}_4$	${\mathit{M}}_5$	${\mathit{M}}_6$	${\mathit{M}}_7$	${\mathit{M}}_8$	${\mathit{M}}_9$	${\mathit{M}}_{10}$	${\mathit{M}}_{11}$	${\mathit{M}}_{12}$	${\mathit{M}}_{13}$	${\mathit{M}}_{14}$	${\mathit{M}}_{15}$	${\mathit{M}}_{16}$	${\mathit{M}}_{17}$
$p_{s_1}$	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0	1	0
$p_{s_2}$	0	1	1	0	1	1	1	1	1	1	1	1	1	1	0	0	1
$p_{s_3}$	0	1	1	0	1	1	1	1	1	1	1	1	1	1	0	0	1
$p_{s_4}$	0	1	1	1	0	0	0	0	1	1	1	1	1	1	1	1	0
$p_{s_5}$	0	1	1	0	1	1	1	1	1	1	1	1	1	1	1	0	1
$p_{s_6}$	0	1	1	0	1	1	1	1	1	1	1	1	1	1	1	0	1
$p_{s_7}$	0	1	1	0	1	1	1	1	1	1	1	0	1	1	0	1	0
$p_{s_8}$	0	1	1	0	1	1	1	1	1	1	1	0	1	1	0	1	0
$p_{s_9}$	0	1	1	0	1	1	1	1	1	1	1	1	1	1	0	1	1
$p_{s_{10}}$	0	1	1	0	1	1	1	1	1	1	1	1	1	1	0	1	1
$p_{s_{11}}$	0	1	1	0	1	1	1	1	1	1	1	0	1	1	0	1	0
$p_{s_{12}}$	0	1	1	0	1	1	1	1	1	1	1	1	1	1	0	1	1
$p_{s_{13}}$	0	1	1	0	1	1	0	0	1	1	0	0	1	0	0	1	0
$p_{s_{14}}$	0	1	1	0	1	1	1	1	1	1	1	0	1	1	0	1	0
$p_{s_{15}}$	0	1	1	0	1	1	1	1	1	1	1	1	1	1	1	1	1
$p_{s_{16}}$	0	1	1	0	1	1	1	1	1	1	1	1	1	1	0	1	1
$p_{s_{17}}$	0	1	1	0	1	1	1	1	1	1	1	1	1	1	0	1	1
$p_{s_{18}}$	0	0	1	0	0	0	1	1	0	0	1	1	1	1	1	1	1
$p_{s_{19}}$	0	0	1	0	0	0	1	1	0	0	1	1	1	1	1	1	1
$p_{s_{20}}$	0	0	1	0	0	0	0	1	0	0	1	1	1	1	1	1	1
$p_{s_{21}}$	0	1	1	0	1	1	1	1	1	1	1	1	1	1	0	1	1
$p_{s_{22}}$	0	1	1	0	1	1	1	1	1	1	1	1	1	1	0	1	1
$p_{s_{23}}$	0	1	1	0	0	0	1	1	0	0	1	1	1	1	1	1	1
$p_{s_{24}}$	0	1	1	0	1	1	1	1	1	1	1	1	1	1	0	1	1
$p_{s_{25}}$	0	0	0	0	0	0	1	0	0	0	0	1	0	1	0	0	1
$p_{s_{26}}$	0	1	1	0	1	1	0	0	1	1	0	0	0	0	1	0	0
$p_{s_{27}}$	0	0	0	0	0	0	1	0	0	0	0	1	0	1	0	0	1
$p_{s_{28}}$	0	1	1	0	0	0	1	1	0	0	1	1	1	1	1	1	1
$p_{s_{29}}$	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0	1	0
$p_{s_{30}}$	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0	1	0
$p_{s_{31}}$	0	1	1	0	1	1	0	0	1	1	0	0	0	0	0	1	0
$p_{s_{32}}$	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0
$p_{s_{33}}$	0	0	0	0	1	1	1	1	0	0	0	0	0	0	0	0	1
$p_{s_{34}}$	0	1	1	0	1	1	1	1	1	1	1	1	1	1	0	1	1

and

Table A3. Matrix A for the net shown in Figure 4.

${\mathit{p}}_{{\mathit{s}}_{\mathit{j}}}$	${\mathit{M}}_{18}$	${\mathit{M}}_{19}$	${\mathit{M}}_{20}$	${\mathit{M}}_{21}$	${\mathit{M}}_{22}$	${\mathit{M}}_{23}$	${\mathit{M}}_{24}$	${\mathit{M}}_{25}$	${\mathit{M}}_{26}$	${\mathit{M}}_{27}$	${\mathit{M}}_{28}$	${\mathit{M}}_{29}$	${\mathit{M}}_{30}$	${\mathit{M}}_{31}$	${\mathit{M}}_{32}$	${\mathit{M}}_{33}$	${\mathit{M}}_{34}$
$p_{s_1}$	1	1	0	0	0	1	0	0	0	0	1	1	1	0	0	0	0
$p_{s_2}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	0
$p_{s_3}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	0
$p_{s_4}$	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0
$p_{s_5}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	0
$p_{s_6}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	0
$p_{s_7}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	1
$p_{s_8}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	1
$p_{s_9}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	1
$p_{s_{10}}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	1
$p_{s_{11}}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	1
$p_{s_{12}}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	1
$p_{s_{13}}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	1
$p_{s_{14}}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	1
$p_{s_{15}}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	0
$p_{s_{16}}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	1
$p_{s_{17}}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	1
$p_{s_{18}}$	1	1	1	0	0	1	0	0	0	0	1	0	0	0	0	0	0
$p_{s_{19}}$	0	1	1	0	0	1	0	0	0	0	1	0	0	0	0	0	0
$p_{s_{20}}$	1	1	1	0	0	1	0	0	0	0	1	0	0	0	0	0	0
$p_{s_{21}}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	1
$p_{s_{22}}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	1
$p_{s_{23}}$	1	1	1	0	0	1	0	0	0	0	0	0	0	0	0	0	0
$p_{s_{24}}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	1
$p_{s_{25}}$	1	0	0	0	0	0	0	1	0	1	1	0	0	0	0	0	0
$p_{s_{26}}$	0	0	0	1	1	0	0	0	1	0	0	0	0	1	0	0	0
$p_{s_{27}}$	1	0	0	0	0	0	0	1	0	1	1	0	0	0	0	0	0
$p_{s_{28}}$	1	1	1	0	0	1	0	0	0	0	1	0	0	0	0	0	0
$p_{s_{29}}$	1	1	0	0	0	1	0	0	0	0	1	1	1	0	0	0	0
$p_{s_{30}}$	1	1	0	0	0	1	0	0	0	0	1	1	1	0	0	0	0
$p_{s_{31}}$	0	0	0	1	1	0	0	0	1	0	0	0	0	1	0	0	1
$p_{s_{32}}$	1	1	0	0	0	1	0	0	0	0	1	1	1	0	1	0	0
$p_{s_{33}}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	1	1
$p_{s_{34}}$	0	0	1	1	1	0	1	0	0	0	0	0	0	0	0	0	1

.

Next, we can formulate an ILPP, SEPCP, to select the fewest control places from the obtained 34. SEPCP is presented below:

SEPCP:

$\begin{array}{c}min{\displaystyle \sum _{j=1}^{34}}{\zeta }_j\hfill \\ \mathit{subject}\mathit{to}\hfill \\ {\zeta }_1+{\zeta }_{29}+{\zeta }_{30}+{\zeta }_{32}\ge 1\hfill \\ {\zeta }_1+{\zeta }_2+{\zeta }_3+{\zeta }_4+{\zeta }_5+{\zeta }_6+{\zeta }_7+{\zeta }_8+{\zeta }_9+{\zeta }_{10}+{\zeta }_{11}+{\zeta }_{12}+{\zeta }_{13}+{\zeta }_{14}+{\zeta }_{15}+{\zeta }_{16}\hfill \\ +{\zeta }_{17}+{\zeta }_{21}+{\zeta }_{22}+{\zeta }_{23}+{\zeta }_{24}+{\zeta }_{26}+{\zeta }_{28}+{\zeta }_{29}+{\zeta }_{30}+{\zeta }_{31}+{\zeta }_{34}\ge 1\hfill \\ {\zeta }_1+{\zeta }_2+{\zeta }_3+{\zeta }_4+{\zeta }_5+{\zeta }_6+{\zeta }_7+{\zeta }_8+{\zeta }_9+{\zeta }_{10}+{\zeta }_{11}+{\zeta }_{12}+{\zeta }_{13}+{\zeta }_{14}+{\zeta }_{15}+{\zeta }_{16}\hfill \\ +{\zeta }_{17}+{\zeta }_{18}+{\zeta }_{19}+{\zeta }_{20}+{\zeta }_{21}+{\zeta }_{22}+{\zeta }_{23}+{\zeta }_{24}+{\zeta }_{26}+{\zeta }_{28}+{\zeta }_{29}+{\zeta }_{30}+{\zeta }_{31}+{\zeta }_{34}\hfill \\ \ge 1\hfill \\ {\zeta }_4\ge 1\hfill \\ {\zeta }_2+{\zeta }_3+{\zeta }_5+{\zeta }_6+{\zeta }_7+{\zeta }_8+{\zeta }_9+{\zeta }_{10}+{\zeta }_{11}+{\zeta }_{12}+{\zeta }_{13}+{\zeta }_{14}+{\zeta }_{15}+{\zeta }_{16}+{\zeta }_{17}\hfill \\ +{\zeta }_{21}+{\zeta }_{22}+{\zeta }_{24}+{\zeta }_{26}+{\zeta }_{31}+{\zeta }_{33}+{\zeta }_{34}\ge 1\hfill \\ {\zeta }_2+{\zeta }_3+{\zeta }_5+{\zeta }_6+{\zeta }_7+{\zeta }_8+{\zeta }_9+{\zeta }_{10}+{\zeta }_{11}+{\zeta }_{12}+{\zeta }_{14}+{\zeta }_{15}+{\zeta }_{16}+{\zeta }_{17}+{\zeta }_{18}\hfill \\ +{\zeta }_{19}+{\zeta }_{21}+{\zeta }_{22}+{\zeta }_{23}+{\zeta }_{24}+{\zeta }_{25}+{\zeta }_{27}+{\zeta }_{28}+{\zeta }_{33}+{\zeta }_{34}\ge 1\hfill \\ {\zeta }_2+{\zeta }_3+{\zeta }_5+{\zeta }_6+{\zeta }_7+{\zeta }_8+{\zeta }_9+{\zeta }_{10}+{\zeta }_{11}+{\zeta }_{12}+{\zeta }_{14}+{\zeta }_{15}+{\zeta }_{16}+{\zeta }_{17}+{\zeta }_{18}\hfill \\ +{\zeta }_{19}+{\zeta }_{20}+{\zeta }_{21}+{\zeta }_{22}+{\zeta }_{23}+{\zeta }_{24}+{\zeta }_{28}+{\zeta }_{33}+{\zeta }_{34}\ge 1\hfill \\ {\zeta }_2+{\zeta }_3+{\zeta }_4+{\zeta }_5+{\zeta }_6+{\zeta }_7+{\zeta }_8+{\zeta }_9+{\zeta }_{10}+{\zeta }_{11}+{\zeta }_{12}+{\zeta }_{13}+{\zeta }_{14}+{\zeta }_{15}+{\zeta }_{16}+{\zeta }_{17}\hfill \\ +{\zeta }_{21}+{\zeta }_{22}+{\zeta }_{24}+{\zeta }_{26}+{\zeta }_{31}+{\zeta }_{34}\ge 1\hfill \\ {\zeta }_2+{\zeta }_3+{\zeta }_4+{\zeta }_5+{\zeta }_6+{\zeta }_7+{\zeta }_8+{\zeta }_9+{\zeta }_{10}+{\zeta }_{11}+{\zeta }_{12}+{\zeta }_{14}+{\zeta }_{15}+{\zeta }_{16}+{\zeta }_{17}+{\zeta }_{18}\hfill \\ +{\zeta }_{19}+{\zeta }_{20}+{\zeta }_{21}+{\zeta }_{22}+{\zeta }_{23}+{\zeta }_{24}+{\zeta }_{28}+{\zeta }_{34}\ge 1\hfill \\ {\zeta }_2+{\zeta }_3+{\zeta }_4+{\zeta }_5+{\zeta }_6+{\zeta }_9+{\zeta }_{10}+{\zeta }_{12}+{\zeta }_{15}+{\zeta }_{16}+{\zeta }_{17}+{\zeta }_{18}+{\zeta }_{19}+{\zeta }_{20}+{\zeta }_{21}\hfill \\ +{\zeta }_{22}+{\zeta }_{23}+{\zeta }_{24}+{\zeta }_{25}+{\zeta }_{27}+{\zeta }_{28}+{\zeta }_{34}\ge 1\hfill \\ {\zeta }_2+{\zeta }_3+{\zeta }_4+{\zeta }_5+{\zeta }_6+{\zeta }_7+{\zeta }_8+{\zeta }_9+{\zeta }_{10}+{\zeta }_{11}+{\zeta }_{12}+{\zeta }_{13}+{\zeta }_{14}+{\zeta }_{15}+{\zeta }_{16}+{\zeta }_{17}\hfill \\ +{\zeta }_{18}+{\zeta }_{19}+{\zeta }_{20}+{\zeta }_{21}+{\zeta }_{22}+{\zeta }_{23}+{\zeta }_{24}+{\zeta }_{27}+{\zeta }_{28}+{\zeta }_{34}\ge 1\hfill \\ {\zeta }_2+{\zeta }_3+{\zeta }_4+{\zeta }_5+{\zeta }_6+{\zeta }_7+{\zeta }_8+{\zeta }_9+{\zeta }_{10}+{\zeta }_{11}+{\zeta }_{12}+{\zeta }_{14}+{\zeta }_{15}+{\zeta }_{16}+{\zeta }_{17}+{\zeta }_{18}\hfill \\ +{\zeta }_{19}+{\zeta }_{20}+{\zeta }_{21}+{\zeta }_{22}+{\zeta }_{23}+{\zeta }_{24}+{\zeta }_{25}+{\zeta }_{27}+{\zeta }_{28}+{\zeta }_{34}\ge 1\hfill \\ {\zeta }_4+{\zeta }_5+{\zeta }_6+{\zeta }_{15}+{\zeta }_{18}+{\zeta }_{19}+{\zeta }_{20}+{\zeta }_{23}+{\zeta }_{26}+{\zeta }_{28}\ge 1\hfill \\ {\zeta }_1+{\zeta }_4+{\zeta }_7+{\zeta }_8+{\zeta }_9+{\zeta }_{10}+{\zeta }_{11}+{\zeta }_{12}+{\zeta }_{13}+{\zeta }_{14}+{\zeta }_{15}+{\zeta }_{16}+{\zeta }_{17}+{\zeta }_{18}+{\zeta }_{19}\hfill \\ +{\zeta }_{20}+{\zeta }_{21}+{\zeta }_{22}+{\zeta }_{23}+{\zeta }_{24}+{\zeta }_{28}+{\zeta }_{29}+{\zeta }_{30}+{\zeta }_{31}+{\zeta }_{32}+{\zeta }_{34}\ge 1\hfill \\ {\zeta }_2+{\zeta }_3+{\zeta }_5+{\zeta }_6+{\zeta }_9+{\zeta }_{10}+{\zeta }_{12}+{\zeta }_{15}+{\zeta }_{16}+{\zeta }_{17}+{\zeta }_{18}+{\zeta }_{19}+{\zeta }_{20}+{\zeta }_{21}+{\zeta }_{22}\hfill \\ +{\zeta }_{23}+{\zeta }_{24}+{\zeta }_{25}+{\zeta }_{27}+{\zeta }_{28}+{\zeta }_{33}+{\zeta }_{34}\ge 1\hfill \\ {\zeta }_1+{\zeta }_{18}+{\zeta }_{20}+{\zeta }_{23}+{\zeta }_{25}+{\zeta }_{27}+{\zeta }_{28}+{\zeta }_{29}+{\zeta }_{30}+{\zeta }_{32}\ge 1\hfill \\ {\zeta }_1+{\zeta }_{18}+{\zeta }_{19}+{\zeta }_{20}+{\zeta }_{23}+{\zeta }_{28}+{\zeta }_{29}+{\zeta }_{30}+{\zeta }_{32}\ge 1\hfill \\ {\zeta }_2+{\zeta }_3+{\zeta }_5+{\zeta }_6+{\zeta }_7+{\zeta }_8+{\zeta }_9+{\zeta }_{10}+{\zeta }_{11}+{\zeta }_{12}+{\zeta }_{13}+{\zeta }_{14}+{\zeta }_{15}+{\zeta }_{16}+{\zeta }_{17}\hfill \\ +{\zeta }_{18}+{\zeta }_{19}+{\zeta }_{20}+{\zeta }_{21}+{\zeta }_{22}+{\zeta }_{23}+{\zeta }_{24}+{\zeta }_{28}+{\zeta }_{33}+{\zeta }_{34}\ge 1\hfill \\ {\zeta }_2+{\zeta }_3+{\zeta }_4+{\zeta }_5+{\zeta }_6+{\zeta }_7+{\zeta }_8+{\zeta }_9+{\zeta }_{10}+{\zeta }_{11}+{\zeta }_{12}+{\zeta }_{13}+{\zeta }_{14}+{\zeta }_{15}+{\zeta }_{16}+{\zeta }_{17}\hfill \\ +{\zeta }_{21}+{\zeta }_{22}+{\zeta }_{24}+{\zeta }_{33}+{\zeta }_{34}\ge 1\hfill \\ {\zeta }_{25}+{\zeta }_{27}\ge 1\hfill \\ {\zeta }_{26}+{\zeta }_{31}\ge 1\hfill \\ {\zeta }_1+{\zeta }_{18}+{\zeta }_{19}+{\zeta }_{20}+{\zeta }_{25}+{\zeta }_{27}+{\zeta }_{28}+{\zeta }_{29}+{\zeta }_{30}+{\zeta }_{32}\ge 1\hfill \\ {\zeta }_{32}\ge 1\hfill \\ {\zeta }_{33}\ge 1\hfill \\ {\zeta }_7+{\zeta }_8+{\zeta }_9+{\zeta }_{10}+{\zeta }_{11}+{\zeta }_{12}+{\zeta }_{13}+{\zeta }_{14}+{\zeta }_{16}+{\zeta }_{17}+{\zeta }_{21}+{\zeta }_{22}+{\zeta }_{24}+{\zeta }_{31}+{\zeta }_{33}\hfill \\ +{\zeta }_{34}\ge 1\hfill \\ {\zeta }_j\in \{0,1\},\forall j\in \{1,2,\dots ,34\}\hfill \end{array}$

Note that we have already removes the redundant constraints from SEPCP since the constraints (represented as Equation (17)) for markings in sets $\{M_1, M_{29}, M_{30}\}$, $\{M_5, M_6, M_{21}, M_{22}\}$, $\{M_9, M_{10}\}$, $\{M_{19}, M_{23}\}$, and $\{M_{25}, M_{27}\}$, $\{M_{26}, M_{31}\}$ are, respectively, the same. As a result, SEPCP has only 25 constraints and 34 variables. Then, we obtain an optimal solution with ${\zeta }_4=1$, ${\zeta }_{27}=1$, ${\zeta }_{31}=1$, ${\zeta }_{32}=1$, ${\zeta }_{33}=1$, and ${\zeta }_j=0(j\in \{1-3,5-26,28-29,34\})$. Hence, we only need to select five control places $p_{s_4}$, $p_{s_{27}}$, $p_{s_{31}}$, $p_{s_{32}}$, and $p_{s_{33}}$ to control the net model. Specifically, from Table A1, it can be seen that the five control places are used to implement the five linear constraints: $15{\mu }_2+15{\mu }_3+3{\mu }_6+3{\mu }_7+17{\mu }_8+17{\mu }_9+3{\mu }_{11}+97{\mu }_{12}+47{\mu }_{13}+50{\mu }_{15}+97{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 290$, $11{\mu }_2+11{\mu }_3+11{\mu }_8+3{\mu }_9+{\mu }_{11}+2{\mu }_{13}+3{\mu }_{15}+3{\mu }_{16}\le 32$, $21{\mu }_6+21{\mu }_7+2{\mu }_8+2{\mu }_9+21{\mu }_{11}+5{\mu }_{13}+5{\mu }_{15}+5{\mu }_{16}+19{\mu }_{17}+20{\mu }_{18}\le 121$, $5{\mu }_2+4{\mu }_3+{\mu }_6+{\mu }_7+5{\mu }_8+5{\mu }_9+125{\mu }_{11}+14{\mu }_{12}+14{\mu }_{13}+28{\mu }_{15}+42{\mu }_{16}+84{\mu }_{17}\le 333$, and $15{\mu }_2+15{\mu }_3+3{\mu }_6+3{\mu }_7+17{\mu }_8+17{\mu }_9+3{\mu }_{11}+47{\mu }_{12}+97{\mu }_{13}+97{\mu }_{15}+50{\mu }_{16}+{\mu }_{17}+2{\mu }_{18}\le 290$, respectively. As a result, we present the preset (${}^{•}p_{s_j}$), postset ($p_{s_j}^{•}$), and initial marking ($M_0(p_{s_j})$) for the selected control place $p_{s_j}$ ($j\in \{4,27,31,32,33\}$) in

Table 4. The selected control places for the net in Figure 4.

${\mathit{p}}_{{\mathit{s}}_{\mathit{j}}}$	${}^{•}\mathit{p}_{{\mathit{s}}_{\mathit{j}}}$	${\mathit{p}}_{{\mathit{s}}_{\mathit{j}}}^{•}$	${\mathit{M}}_0\left({\mathit{p}}_{{\mathit{s}}_{\mathit{j}}}\right)$
$p_{s_4}$	$17t_5,50t_9,47t_{10},15t_{13},96t_{17},2t_{19}$	$3t_1,14t_3,94t_8,15t_{11},50t_{15},47t_{16},t_{18}$	290
$p_{s_{27}}$	$8t_4,3t_5,t_8,11t_{13},2t_{10},3t_{17}$	$11t_3,t_7,2t_9,11t_{11},3t_{15}$	32
$p_{s_{31}}$	$19t_3,2t_5,21t_8,5t_{10},20t_{19}$	$21t_1,5t_9,5t_{15},14t_{17},t_{18}$	121
$p_{s_{32}}$	$5t_5,111t_8,14t_{10},t_{12},4t_{13},84t_{18}$	$t_1,4t_3,124t_7,5t_{11},28t_{15},14t_{16},42t_{17}$	333
$p_{s_{33}}$	$17t_5,97t_{10},15t_{13},47t_{16},49t_{17},2t_{19}$	$3t_1,14t_3,44t_8,50t_9,15t_{11},97t_{15},t_{18}$	290

. The five control places can make the net model live with all 21,581 legal markings. That is to say, the original net model is optimally controlled by the five obtained control places.

The net model has been widely studied in the literature.

Table 5. Performance comparison of some deadlock control policies.

Parameters	[21]	[40]	[39]	[30]	[36]	[32]	[33]	[35]	This Work
No. monitors	18	6	16	19	13	17	5	6	5
No. states	6287	6287	12656	21562	21581	21581	21581	21581	21581

compares the proposed method with some classical policies by this example. Note that only the last five control polices are behaviorally maximally permissive. The supervisor obtained by the proposed approach has the same number of control places as that proposed in [33], which is the smallest one among all control policies in this table. However, the ILPP proposed in [33] has 15,640 constraints and 1700 variables, which is solved in approximately 44 h, while, in this work, MNFFP($M_j$) has only 392 constraints and 49 variables, and SEPCP has 26 constraints and 34 variables. All 35 ILPPs can be solved in 116 s in total. Hence, we conclude that the proposed approach can remarkably improve the efficiency of the one in [33] but also find the minimal number of control places.

4. Conclusions

This paper presents a deadlock resolution method in Petri nets by using a set covering approach. First, for each FBM of a given Petri net, an ILPP is proposed to derive a control place to prohibit the maximal number of FBMs. Then, we use a set covering approach to obtain the fewest control places to forbid all FBMs. By applying the proposed approach to some well-studied examples, the results verify that it can lead to the minimal number of control places. Compared to the work in [33], the proposed one can remarkably reduce the computation time for solving ILPPs but keep the structural minimality of the obtained supervisor. The main bottleneck is that it is required to compute the reachability graph, which is not efficient for large-sized systems. In the future, we will try to avoid the complete enumeration of reachable markings. It is possible to find all the FBMs of a Petri net model by structural analysis, such as the resource requirement graphs proposed in [42]. On the other hand, some efficient approaches are also possible to use to analyze large-sized systems, such as the computation of reachability graphs by binary decision diagrams, parallel algorithms, and graphic processing unit (GPU) computing.