On the computation of the minimal siphons of S4PR nets from a generating family of siphons

Minimal siphons in the class of S⁴PR nets have become a conceptual and practical central tool to deal with deadlocks caused by the sharing of resources in Flexible Manufacturing Systems. The availability of efficient algorithms to compute these structural objects is very important. In this paper we take advantage from the particular properties of the siphons in S⁴PR to obtain an efficient algorithm. These properties allow to compute the minimal siphons from a generating family of minimal siphons. This family is composed by the minimal siphons containing only one resource. The computation of the minimal siphons is based in the maximal strongly connected components of a graph representing the relations between the siphons of the generating family. The algorithm is very economic in memory in all intermediate steps with respect to the classical algorithms.

1 Introduction

Flexible Manufacturing Systems (FMS) are a kind of production systems conceived to manufacture a set of different types of products. Each type of product follows a predefined sequence of operations named production plan, requesting, in a competitive way, a set of shared resources of the system (robots, machines, etc). This relation of competition between different production plans executing concurrently in a FMS can cause deadlock situations. Roughly speaking, a deadlock is a system state so that some production plans cannot be finished. The modeling of Flexible Manufacturing Systems by means of Petri Nets has given rise to a successful subclass named nets. This subclass allows to represent the Resource Allocation issues of the FMS in order to study the appearing of deadlocks caused by the shared resources. In these studies minimal siphons of the net play a central role in the analysis tasks [10], [11] and in the techniques for enforcing liveness [9], [12]. Therefore, the availability of efficient algorithms for the computation of the mini- mal siphons or other sets of siphons is indispensable in the frameworks working with this kind of Petri Net models.