Modelling Systems by Hybrid Petri Nets: an Application to Supply Chains

Abstract

Petri Nets (PNs) are a discrete event model firstly proposed by C. A. Petri in his Ph.D. thesis in the early 1960s (Petri, 1962). The main feature of a (discrete) PN is that its state is a vector of non-negative integers. This is a major advantage with respect to other formalisms such as automata, where the state space is a symbolic unstructured set, and has been exploited to develop many analysis techniques that do not require to enumerate the state space (structural analysis) (Silva et al., 1996). Another key feature of PNs is their capacity to graphically represent and visualize primitives such as parallelism, concurrency, synchronization, mutual exclusion, etc. In the related literature various PN extensions have been proposed. In this paper we focus on Continuous and Hybrid PNs. Continuous Petri Nets (CPNs) originate from the “fluidification” of discrete PNs (David & Alla, 1987). In simple words, the content of places is relaxed to be a real non-negative number rather than an integer non-negative number, and appropriate rules for transitions firings are given. This highly reduces the computational complexity of the analysis and optimization of realistic scale problems, and has been successfully applied to manufacturing systems. The main advantages of fluidification can be summarized in the following four items. • The computational complexity of the analysis and control of complex systems may be significantly reduced. • Fluid approximations provide an aggregate formulation to deal with complex systems, thus reducing the dimension of the state space. The resulting simple structures allow explicit computation and performance optimization. • The design parameters in fluid models are continuous; hence, it is possible to use gradient information to speed up optimization and to perform sensitivity analysis. • Finally, in many cases it has also been shown that fluid approximations do not introduce significant errors when carrying out performance analysis via simulation. In general, different fluid approximations are necessary to describe the same system, depending on its discrete state, e.g., in the manufacturing domain, machines working or down, buffers full or empty, and so on. Thus, the resulting model can be better described as a hybrid model, where a different continuous dynamics is associated to each discrete state. Hybrid Petri Nets (HPNs) keep all those good features that make discrete PNs a valuable