Perturbation analysis of discrete event dynamic systems via minimax algebra

Abstract

In the paper, a discrete event system has been considered, consisting of a set of tasks, organized in precedence-relation graphs, to be executed by a set of machines, in the fulfilment of a certain set of constraints. It has been shown that, using minimax algebra formalism, one is able to: i) express the task completion times as analytic functions of the binary decision variables relevant to the alternative choices and of the continuous variables representing the task activation delays with respect to the earliest activation times; ii) represent analytically all the constraints affecting the decision variables and resulting from the structure of the model considered. Thus, we are able to determine the structure, which means, the cost functional plus the constraints imposed, of a mathematical programming problem having the objective of optimizing a cost functional related to task completion times. The resulting optimization problems are of the mixed-integer type. To write the optimization problem, i.e., the cost functional and the relevant constraints, in terms of the decision variables it is not necessary to list explicitly all the possible perturbed semi-schedules.