Extraction of elementary siphons in a class of generalized Petri nets using graph theory

Abstract

Purpose -Siphon-based deadlock control in a flexible manufacturing system (FMS) suffers from the problems of computational and structural complexity since the number of siphons grows exponentially with respect to the size of its Petri net model. In order to reduce structural complexity of a supervisor, a set of elementary siphons derived from all strict minimal siphons (SMS) is explicitly controlled. The purpose of this paper is through fully investigating the structure of a class of generalized Petri nets, WS 3 PR, to compute all SMS and a compact set of elementary siphons. Design/methodology/approach -Based on graph theory, the concepts of initial resource weighted digraphs and restricted subgraphs are proposed. Moreover, the concept of augmented siphons is proposed to extend the application of elementary siphons theory for WS 3 PR. Consequently, the set of elementary siphons obtained by the proposed method is more compact and well suits for WS 3 PR. Findings -In order to demonstrate the proposed method, an FMS example is presented. All SMS and elementary siphons can be derived from initial resource weighted digraphs. Compared with those obtained by the method in Li and Zhou, the presented method is more effective to design a structural simple liveness-enforcing supervisor for WS 3 PR. Originality/value -This work presents an effective method of computing SMS and elementary siphons for WS 3 PR. Monitors are added for the elementary siphons only, and the controllability of every dependent siphon is ensured by properly supervising its elementary ones. A same set of elementary siphons can be admitted by different WS 3 PR with isomorphic structures.

Deadlock analysis and control based on Petri nets: A siphon approach review

“…Consequently, the augmented elementary siphon theory is established by sufficiently considering the requirement of multiple resource types by an operation and its weight information. In the work by Hou et al, 224 the concept of augmented siphons is first proposed for extending the elementary siphon's application to a class of generalized Petri nets, WS 3 PR. An augmented siphon is a multiset that reflects the weights information of resource places contained by its original siphon.…”

“…[222][223][224]226 Augmented elementary siphons in WS 3 PR Augmented elementary siphons. The augmented siphon and augmented complementary set of a siphon are defined in the work by Hou et al 224 for WS 3 PR and then elementary siphons are redefined. The relationship between augmented siphons and their augmented complementary sets is also investigated by fully considering the weights information.…”

“…224 A multisetS is the augmented version of siphon S such that (1) jjSjj = S; (2) 8r 2 S R ,S(r) = 1 and 8p 2 S P ,S(p) = I S (p), where I S (p) = P r2S R I r (p) and I r is the minimal P-semiflow associated with the resource r.…”

“…A dependent siphon can be implicitly controlled by properly arranging the number of tokens that can stay at its elementary siphons. The results mentioned above have widely applied to a variety of elementary siphon-related deadlock control approaches 139,[217][218][219][220][221][222][223][224][225][226][227][228][229][230][231][232][233] in the literature. This article mainly aims to reveal, first, how the elementary siphon theory of Petri nets can be used to troubleshoot deadlock problems, and second, how the refined concept of elementary siphons in a Petri net improves the existing deadlock control policies.…”

“…Based on graph theory, Wang et al 233 propose a polynomial complexity algorithm to find a set of elementary siphons for a linear system of simple sequential processes with resources (LS The related works can be found in the studies by Hou et al 223,224 Li et al 270 propose an iterative algorithm to extract a set of elementary siphons in S 3 PR. By an MIP method, the algorithm finds a maximal unmarked siphon at each iteration, classifies the places in it, extracts an elementary siphon from the classified places, and adds a new constraint in order to extract the next elementary siphon.…”