Improved Controllability Test for Dependent Siphons in S3PR Based on Elementary Siphons

Abstract

When siphons in a flexible manufacturing system (FMS) modeled by an ordinary Petri net (OPN) become unmarked, the net gets deadlocked. To prevent deadlocks, some control places and related arcs are added to strict minimal siphons (SMS) so that no siphon can be emptied. For large systems, it is infeasible to add a monitor to every SMS since the number of SMS or control elements grows exponentially with respect to the size of a Petri net. To tackle this problem, Li and Zhou propose to add control nodes and arcs for only elementary siphons. The rest of siphons, called dependent ones, may be controlled by adjusting control depth variables of elementary siphons associated with a dependent siphon after the failure of two tests. First, they test a Marking Linear Inequality (MLI); if it fails, then they perform a Linear Integer Programming (LIP) test which is an NP-hard problem. This implies that the MLI test is only sufficient, but not necessary. We propose a sufficient and necessary test for adjusting control depth variables in an S3PR to avoid the sufficient-only time-consuming linear integer programming (LIP) test (NP-complete problem) required previously for some cases. We theoretically prove the following: i) no need for LIP test for Type II siphons; and ii) Type I strongly n-dependent (n>2) siphons being always marked. As a result, the total time complexity to check controllability of all strongly dependent siphons is no longer exponential but reduced to linear if all siphons are of Type I. The total time complexity is O(|ΠE||ΠD|) (order of the product of total number of elementary siphons and total number of dependent siphons) if all siphons are of Type II. A well-known S3PR example has been illustrated to show the advantages.