Abstract
The optimization problems, such as scheduling or project management, in which the objective function depends on the operations maximum and plus, can be naturally formulated and solved in max-plus algebra. A system of discrete events, e.g., activations of processors in parallel computing, or activations of some other cooperating machines, is described by a systems of max-plus linear equations. In particular, if the system is in a steady state, such as a synchronized computer network in data processing, then the state vector is an eigenvector of the system. In reality, the entries of matrices and vectors are considered as intervals. The properties and recognition algorithms for several types of interval eigenvectors are studied in this paper. For a given interval matrix and interval vector, a set of generators is defined. Then, the strong and the strongly universal eigenvectors are studied and described as max-plus linear combinations of generators. Moreover, a polynomial recognition algorithm is suggested and its correctness is proved. Similar results are presented for the weak eigenvectors. The results are illustrated by numerical examples. The results have a general character and can be applied in every max-plus algebra and every instance of the interval eigenproblem.
Highlights
In many practical problems, the standard algebraic operations “plus” and “product” are inadequate, e.g., in scheduling problems, synchronization problems, or in project management, and binary operations “maximum” and “plus” seem to be more appropriate
The standard algebraic operations “plus” and “product” are inadequate, e.g., in scheduling problems, synchronization problems, or in project management, and binary operations “maximum” and “plus” seem to be more appropriate. This observation leads to the definition and use of so-called max-plus algebra, which has been used by many authors, see e.g., [1,2,3,4]
Max-plus algebras represent a suitable mathematical tool for exploration of systems working in discrete steps called discrete event systems (DES, for short)
Summary
The standard algebraic operations “plus” and “product” are inadequate, e.g., in scheduling problems, synchronization problems, or in project management, and binary operations “maximum” and “plus” seem to be more appropriate This observation leads to the definition and use of so-called max-plus algebra, which has been used by many authors, see e.g., [1,2,3,4]. It is assumed that an eigenvector is different from the ‘zero’ vector with all entries equal to ε In this notation, the intervals between the beginnings of consecutive cycles on every component of DES are equal to a scheduled value λ ∈ B. The strong, strongly universal and weak interval eigenvectors in max-plus algebra are investigated, and polynomial algorithms for the recognition versions of these problems are presented. The results can be applied in every max-plus algebra and every instance of the interval eigenproblem
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