Incremental Integer Linear Programming Models for Petri Nets Reachability Problems

The operational management of complex systems is characterized, in general, by the existence of a huge number of solutions. Decision-making processes must be implemented in order to find the best results. These processes need suitable modeling tools offering true practical resolution perspectives. Among them, Petri nets (PNs) provide a simple graphical model taking into account, in the same formalism, concurrency, parallelism and synchronization. Their graphical and precise nature, their firm mathematical foundation and the aboundance of analysis methods have made them become a classical modeling tool for the study of discrete event systems, ranging from operating systems to logistic ones. However, their interest in the field of problem solving is still badly known. In this paper, we consider some PN reachability problems. Since PNs can model flows in a natural and efficient way, many operations research problems can be defined using reachability between states, e.g. scheduling (Lee and DiCesare, 1994; Van Der AAlst, 1995), planning (Silva et al., 2000), car-sequencing problems (Briand, 1999). Moreover, research on Petri nets addresses the issue of flexibility: many extensions have been proposed to facilitate the modeling of complex systems, by addition of ``color’’, ``time’’ and ``hierarchy’’ (Jensen, 1992; Wang,1998). For example, it is relatively easy to map scheduling problems onto timed PNs. Their graphical nature reinforce obviously this strength, by allowing a kind of interactivity with the system. At last, a large number of difficult PN analysis problems are equivalent to the reachability problem, or to some of its variants or sub-problems (Keller, 1976). Particularly, model-checking (Latvala, 2001) which represents a key point when dealing with systems analysis is directly linked to an exhaustive traversal of the corresponding PN reachability graph. Various methods have been suggested to handle the PNs reachability problem. In this paper, we propose to use the mathematical programming paradigm. Some PN analysis problems have already been handled using such techniques (Melzer and Esparza, 1996;Silva et al., 1998; Khomenko and Koutny, 2000), but none has considered the general PNs reachability problem. The proposed approach is based on an implicit traversal of the Petri net reachability graph, which does not need its construction. This is done by considering a unique sequence of steps growing incrementally to represent exactly the total behavior of the net. We follow here a