Optimization of the Time Durations by Exploiting Time Margins in Time Interval Models

Abstract

Considering time interval models, which can describe a large class of models including timed event graphs and P-time event graphs, a general aim is to control the system such that it follows a one-periodic behavior starting from an initial state with a minimal or maximal cycle time. The aim of this article is the optimization of the time durations in order to achieve a given rate of production when the quantity of resources (number of pallets, machines, and so on), usually represented by the initial marking, is assumed to be a fixed datum. A prior step is the determination of the critical subsystems, whose variations can influence these optimal values and affect the obtainment of the relevant trajectories and the noncritical subsystems leading to time margins. Two approaches are proposed in that aim. A first technique is based on an adaptation of the classical Martínez and Silva’s algorithm, where each solution gives a critical subsystem, while a second approach checks each inequality of the system by an optimality verification. Using this partition of the subsystems and particularly exploiting the time margins corresponding to idle times of machines, the approach allows an optimization of the noncritical time durations when the extremum cycle times and the resources are the constant parameters of the problem. In this article, the definition of P-time event graphs is generalized by introducing the model of time supervisor place, which restricts the time behavior of a set of places. The approach is applied to a plant bakery composed of two production lines.