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

Abstract

Minimal siphons in the class of S 4 P R 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 S 4 P R 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.

Virtual Control Policy for Binary Ordered Resources Petri Net Class

“…The reduced structure of this new class gives advantages that allow analysis of the entire system’s behavior, which is a prohibitive task for large systems due the complexity of a routing algorithm. These siphons were computed using the pruning graph [ 11 ]. With the information given for the siphons we located the sets of implicated resources and implicated places that produce.…”

“…Pruning labelling function. K G : V → 2 P s , where for all r V, K G (r) ⊆ P S is computed by the algorithm [ 11 ].…”

“…The Pruning Graph of Petri net model of Figure 3 is shown in Figure 4 . This graph was obtained using the algorithm presented in [ 11 ]. The obtained pruning relation functions are as follows: L(R, S) = {t 9 , p 7 }; L(R, T) = {t 9 , p 7 }; L(S, T) = {t 8 , p 6 }; L(T, S) = {t 3 , p 2 }; L(T, R) = {t 3 , p 2 }; L(S, R) = {t 2 , p 1 }.…”

“…Employing the algorithm presented in [ 11 ] over the Pruning Graph (G) of Figure 3 were computed three S trongly Connected Subgraphs called G 1 , G 2 and G 3 . The subgraph contains the following vertexes: G 1 = {R, S}, G 2 = {R, T} and G 3 = {S, T}.…”

“…A new approach is presented in [ 11 ] that works with higher-level objects, avoiding wasted memory in the intermediate steps. The method is based on the graph theory through which the manipulations of the maximum are strongly connected to subgraphs in the graph [ 12 ].…”