Weight-Balanced Timed Event Graphs to Model Periodic Phenomena in Manufacturing Systems

Abstract

Timed event graphs (TEGs) are suitable to model manufacturing systems in which synchronization and delay phenomena appear. Since 1980s, TEGs are studied as a class of linear discrete event systems in idempotent semirings such as the (min,+) algebra. In this paper, we consider the class of weighted TEGs (WTEGs) that corresponds to TEGs where the edges have integer weights. By introducing nonunitary weights, WTEGs widen the class of manufacturing systems that can be modeled, especially systems with batches and duplications. Moreover, a subclass of WTEGs called weight-balanced TEGs (WB-TEGs) can be studied with the algebraic tools that stem from the theory of (min,+) linear systems. In this paper, the focus lies on some modeling issues for manufacturing systems. Besides cutting and palletization operations, it is shown that WB-TEGs are also well adapted to describe periodic routing policies and, in a symmetrical way, how to merge flows similar to a multiplexer. In order to simplify the modeling step, a class of cycloweighted TEGs (CW-TEGs) is introduced. It is an extension of WTEGs where the weights of the edges can change according to a periodic sequence. Finally, we propose some elements of modeling that can be described by CW-TEGs or equivalently with an input–output transfer relation in an appropriate idempotent semiring of operators. Note to Practitioners — The (min,+) linear system theory used in this paper aims at obtaining linear models for a subclass of man-designed systems, such as automated manufacturing systems or traffic networks. This theory has many analogies with the conventional linear system theory (for continuous systems) and it provides the basis to develop a specific control theory for man-made systems. More precisely, the theory of (min,+) linear systems is well suited to systems where the prevailing phenomena are synchronizations, delays, duplications, and batches. These phenomena arise, for example, in operations such as assembly/matching, cutting/lot splitting, and palletization/lot making. Among the possible representations, we can describe these systems by transfer functions obtained by the combination of a finite number of basic operators. This is analogous to block diagram in the conventional system theory, i.e., a transfer function describes the complete input-output behavior of a system. In the context of manufacturing systems, a transfer function describes the way a system maps an input flow of materials (raw part inputs) into an output flow (finished parts), without the necessity of simulation tools to predict this. Moreover, the transfer function thus obtained can be used to compute controllers in order to regulate the internal flows of a system, for instance to decrease internal stocks. In the case of an automated system, the obtained controllers can be implemented on a programmable logic controller as supplementary code. This paper focuses on the use of these algebraic tools in the model process of manufacturing systems and, in particular, on their ability to describe splitting and merging flows of materials.