Batch Deterministic and Stochastic Petri Nets and Transformation Analysis Methods

Abstract

Industrial systems such as production systems and distribution systems are often characterized as batch processes where materials are processed in batches and many operations are usually performed in batch modes to take advantages of the economies of scale or because of the batch nature of customer orders. The Batch Deterministic and Stochastic Petri Nets (BDSPN) is a class of Petri nets recently introduced for the modelling, analysis and performance evaluation of such systems which are discrete event dynamic systems with batch behaviours. The BDSPN model enhances the modelling and analysis power of the existing discrete Petri nets. It is able to describe essential characteristics of logistics systems (batch behaviours, batch operational policies, synchronization of various flows, randomness) and more generally discrete event dynamic systems with batch behaviours. The model is particularly adapted for the modelling of flow evolution in discrete quantities (variable batches of different sizes) and is capable of describing activities such as customer order processing, stock replenishment, production and delivery in a batch mode. The capability of the model to meet real needs is demonstrated through applications to modelling and performance optimization of inventory systems (Labadi et al., 2007,2005) and a real-life supply chain (Amodeo et al., 2007; Chen et al., 2005,2003). Graph transformation is a fundamental concept for analysis of the systems described by graphs. The state of the art reporting for languages, tools and applications for graph transformation is given in the “Handbook of Graph Grammars and Computing by Graph Transformation” (Ehrig et al., 1999). In contrast to most applications of the graph transformation approach, where the states of a system are denoted by a graph, and transformation rules describe the state changes and the dynamic behaviour of systems, in the area of Petri nets (Murata, 1989) we apply transformation rules to modify a net in a stepwise way. This kind of transformation for Petri nets is considered to be a vertically structural technique, known as rule-based net transformation. This approach has been applied to various Petri net models such as basic Petri nets (Lee-Kwang et al., 1985, 1987), timed Petri nets (Juan et al., 2001; Wang et al. 2000) , stochastic Petri nets (Ma & Zhou, 1992; Li-Yao et al. 1995), and coloured Petri nets (Haddad, 1988). There are different types of reduction/transformation techniques proposed in the literature. As we know, the reduction