Incremental Optimization of Cyclic Timed Event Graphs

In this paper we deal with the problem of allocating at least one place not shared among the other circuits. Kanban systems belong to this class. Then, we provide necessary and sufficient conditions 1 a given number of tokens, so as to maximize the firing rate of a cyclic event graph with deterministic transition firing delays. We propose a new incremental algorithm that is inspired by the algorithm formulated by Panayiotou and Cassandras for the case of kanban systems. The algorithm can be applied to a special class of nets in which tokens are allocated to places that belong to only one circuit: this class is powerful enough to model kanban systems. We provide necessary and sufficient conditions for the convergence to the optimum also in the case of multiple solutions. 1 I n t r o d u c t i o n Cyclic timed event-graphs (CTEG) are a special class of timed ordinary Petri nets. They are often used for modeling and analyzing manufacturing systems assuming a cyclic manufacturing of the parts, since it has been shown that choice-free job-shop, kanban systems, and assembly systems, can be modeled using event graphs. Marked graphs may be either stochastic or deterministic. The most important results are in the deterministic cyclic production case. In fact, in this case it is possible to evaluate the steady state performance of the net in terms of its critical time. This means that in steady state conditions, at any point of the net, a number of tokens (equal to the inverse of the critical time), pass through that point during each unit of period of time (on the average). Note that , such a performance evaluation can be done assuming that all the elementary circuits are known. Some simple algorithms have been described by Laftit et al. in [4] to obtain all the elementary circuits of a cyclic event graph. In this paper we deal with the problem of allocating a given number of tokens to a CTEG, so as to maximize the firing rate of the net. We propose a new algorithm that is inspired by the previous ones recently formulated by Panayiotou and Cassandras in [6] and denoted as Incremental Optimization (IO) algorithm and Generalized Incremental Optimization (GIO) algorithm, in the case of unique and nmltiple solutions, respectively. In [6] the authors focused their attention on kanban systems and formulated efficient algorithms to allocate a fixed number of kanban to a number of stages so as to maximize the system throughput. Moreover, in the case of uniqueness of the solution, they also provided a necessary and sufficient condition under which their optimization scheme yields an optimal allocation. In this paper we first extend their algorithms to cyclic event graphs whose structure satisfy the following restriction: in each elementary circuit there exists 0 7 8 0 3 5 8 8 6 4 / 0 0 1 5 1 0 . 0 0