One novel control policy named selective siphon control policy is proposed to solve for deadlock problems of flexible manufacturing systems (FMSs). The new policy not only solves the deadlock problem successfully but also obtains maximally permissive controllers. According to our awareness, the policy is the first one to achieve the goal of obtaining maximally permissive controllers for all S^{3}PR (one system of simple sequential processes with resources, S^{3}PR) models in existing literature. However, one main problem is still needed to solve in their algorithm. The problem is that the proposed policy cannot check the exact number of maximally permissive states of a deadlock net in advance. After all iterating steps, the final maximally permissive states can then be known. Additionally, all legal markings are still to be checked again and again until all critical markings vanished. In this paper, one computationally improved methodology is proposed to solve the two problems. According to the experimental results, the computational efficiency can be enhanced based on the proposed methodology in this paper.
Generally, deadlock prevention of FMS includes structural analysis and reachability graphs [
For improving above disadvantage of conventional siphon control, Piroddi et al. [
On the other hand, under their policy, all reachable markings are needed to check again and again until all critical markings are vanished. This is time consuming since all legal markings are needed to check in each iterative step.
In this paper, one computationally improved methodology is proposed to solve above two problems. First of all, we use the developed algorithm [
In this paper, we adopt Petri nets theory to model FMS. Due to limitation of paper space, however, we don’t show detailed information of Petri nets theory here. The readers can refer to [
Piroddi et al. [
In the following, uncontrolled siphons, critical markings, and selected siphons are defined.
Let
The set
The set
For any
The set
A
Table
The relation between critical markings and uncontrolled siphons.


 


v  

v  v  

v  v  

v  v  

v  

v 
In our previous works [
In this paper, one computationally improved methodology is present to enhance the computational efficiency of conventional selective siphon control technology. The methodology shown in Figure
Flow chart of the proposed deadlock prevention.
Two classical flexible manufacturing systems are used to check the proposed deadlock prevention. They are taken from [
An FMS is shown in Figure
A classical FMS PN model [
The reachability graph of Example
First of all, in the “reachability graph (RG) stage,” 20 reachable markings (i.e.,
The property of dead and quasidead markings.
Marking number  Information of marking  Classification 
















Therefore, the total number of
In siphon identification stage, firstly, three sets of strict minimal siphons
The detailed information of selective siphons in Example
Critical marking  Minimal siphon  



 

v  v  

v  v  

v  



v 
In siphon control stage, two control places are then obtained by siphon control method [
The first additional control places in Example
Control places 

●( 
( 


1 



1 


In second iteration, new four sets of strict minimal siphons are identified. They are
The new incidence matrix of Example







 


−1  0  0  1  0  0  0  0 

1  −1  0  0  0  0  0  0 

0  1  −1  0  0  0  0  0 

0  0  1  −1  0  0  0  0 

0  0  0  0  −1  0  0  1 

0  0  0  0  1  −1  0  0 

0  0  0  0  0  1  −1  0 

0  0  0  0  0  0  1  −1 

−1  1  0  0  0  0  −1  1 

0  −1  1  0  0  −1  1  0 

0  0  −1  1  −1  1  0  0 

−1  1  0  0  0  −1  1  0 

0  −1  1  0  −1  1  0  0 
The relation between minimal siphons and critical markings in the second iteration of Example
Critical marking  Minimal siphon  




 

v  v  v  v 
The second additional control place in Example
Control places 

●( 
( 


1 


When the three control places are added into Example
The new reachability graph of Example
Another classical FMS is shown in Figure
Another classical FMS PN model [
Therefore, the maximally permissive number of states is 205 since 2101661 = 205. In the following, five sets of minimal siphons are calculated and obtained in siphon identification stage. They are
The first additional control places in Example
Control places 

●( 
( 


1 



2 



3 


After adding the three controllers, ten new minimal siphons appear in the second stage. The detailed information of them is that
The relation between minimal siphons and critical markings in the second iteration of Example









 


v  v  

v  v  

v  v  v  v  

v  

v  v  v  v 
The second additional control places in Example
Control places 

●( 
( 


6 



3 


Based on Tables
The comparison results in Example
Important parameter comparison  [ 
This work 

Optimal (maximally permissive) control?  Yes  Yes 


Can make sure the exact number of optimal states in first stage?  No  Yes 


How many reachable markings are needed to check in first stage?  20  5 


How many reachable markings are needed to check in second stage?  16  1 
The comparison results in Example
Important parameter comparison  [ 
This work 

Optimal (maximally permissive) control?  Yes  Yes 


Can make sure the exact number of optimal states in first stage?  No  Yes 


How many reachable markings are needed to check in first stage?  282  77 


How many reachable markings are needed to check in second stage?  210  5 
The proposed policy can be implemented for the deadlock problems of FMSs based on the selective siphons method and critical markings. The underlying notion of the prior work is that it cannot check the exact number of maximally permissive states of a deadlock net in advance. After all iterating steps, the final maximally permissive can then be known. Additionally, all legal markings are still to be checked again and again until all critical markings vanished. In this work, one computationally improved methodology is proposed to solve the two problems. Under the proposed methodology, the reachability graph is only run one time in initial stage. All critical markings are identified based on the reachability graph. One just checks if strict minimal siphons are empty in these markings or not. According to the experimental results, the computational efficiency can be enhanced based on the proposed methodology in this papers.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was partially supported by the National Science Council of Taiwan, under Grant NSC 1022221E013001. Additionally, the authors acknowledge the financial support of MOS T1032221E390026.