Chaos

Chaos in Manufacturing Systems : Study of Different Cases
Patrick Charpentier*, Miguel Alfaro1
^Research Centre for Automatic Control of Nancy CNRS ESA 7039faculty of Science, 54506 Vandoeuvre, France ^Department of Industrial Engineering, University of Santiago, Av. Ecuador 3769, Estacion central, Santiago, Chile

Abstract. Since a few years, an abundant literature has been published inorder to proof the existence of chaotic behaviours both in the field of science and in the field of technique. Until now very few articles studied the conduct of manufacturing production systems. Apparently some production systems let us think that their behaviour might be chaotic. Nevertheless, in our opinion, the proof of existing chaos in the production systems has not been totally confirmed. Theworks presented in this article are aimed to make obvious and to prove the existence of chaotic behaviours in manufacturing production systems. After the presentation of the interest of this study in a manufacturing production environment, we present our analysis method of the dynamic of non-planned production systems. We then justify the choices which have been made regarding in particular : thesub-system in which our study is made, the variable of interest (temporal average of the number of parts in a waiting line), the determinism of the system parameters, and the imposed balance conditions (in the sense that the number of parts is finished regardless of the considered instant). In the second part are presented the results obtained with two manufacturing systems, both very simple andvery similar, although they give very different results. We then compare the results with the rules of assignment and management of different waiting lines. In the last part, we show that an actual system, under certain management conditions, can also present a chaotic behaviour. This study has been realised from the modelling of a flexible assembly cell. Keywords. Chaos, Flexible ManufacturingSystems, Simulation.

INTRODUCTION
Since a few years, an abundant literature has been published in order to show the existence of a chaotic behaviour in the scientific field as well as in the technical field. However very few works studied the conduct of manufacturing production systems. One of the major difficulties attached to these systems is their discrete event character. If an analyticalmodel is possible for continued systems (on continuous time or on discrete time), this is not the case in general for manufacturing systems. Nevertheless, the study of these phenomena is very important for these systems. Even though it can be practically observed, the unpredictable character of the behaviour of some systems is rarely quantified. Right now, one can predict the evolution of stochasticsystems, but not the evolution of chaotic systems. The sensitivity of the later ones to the initial conditions is such that it is impossible to predict the system behaviour on the long term. Thus, this problem is crucial for any manager who wants to be able to plan its production and to know the conditions to get it. These concerns perfectly meet those mentioned by the theory on incursivity andhyperincursivity [1], [2]. Some studies have been performed on these chaos problems for manufacturing systems. Desmukh's works [3] show how a particular system can different behaviours depending on the waiting lines order. However the system working conditions do not allow to conclude in a definitive manner about the chaotic or non-chaotic character of the system. Massotte [4] showed that under someconditions, recurrent relations could represent the part number in the system, equations that can become chaotic for some values of the parameters. Other important works exist in this field, like the ones of Tonshoff [5], Horns [6] which let us think that some chaotic behaviour exist in some manufacturing systems. These different works enable us to make different choices in our study. The first...