A production–inventory model of HMMS on time scales

Abstract

The aim of this work is to investigate the optimal production and inventory paths of HMMS type models (proposed by Holt, Modigliani, Muth and Simon) on complex time domains. Time scale calculus which is a rapidly growing theory is a main tool for solving and for analyzing the model. This work will enrich management and economics by providing a flexible and capable modelling technique.

The variational calculus on time scales

“…This is the subject of our next section. The nabla calculus seems to be particularly useful as a modeling technique in the calculus of variations with applications to economics [4,7,32].…”

“…Motivated by applications in economics [4,7], a different formulation for the problems of the calculus of variations on time scales has been considered, which involve a functional with a nabla derivative and a nabla integral [3,6,43]:…”

Delta-nabla optimal control problems

“…In order to deal with non-traditional applications in areas such as medicine, economics, or engineering, where the system dynamics are described on a time scale partly continuous and partly discrete, or to accommodate non-uniform sampled systems, one needs to work with systems defined on a so called time scalesee, e.g., [Atici et al (2006)], [Atici and Uysal (2008)], [Malinowska and Torres (2010b)]. The optimal control theory on time scales was introduced in the beginning of the XXI century in the simpler framework of the calculus of variations, and is now a fertile area of research in control and engineering [Seiffertt et al (2008)], [Malinowska and Torres (2010c)].…”

“…The optimal control theory on time scales was introduced in the beginning of the XXI century in the simpler framework of the calculus of variations, and is now a fertile area of research in control and engineering [Seiffertt et al (2008)], [Malinowska and Torres (2010c)]. In the literature there are two different approaches to the problems of optimal control on time scales: some authors use the delta calculus [Bohner (2004)], [Bohner et al (2010)], [Bartosiewicz and Torres (2008)], [Ferreira and Torres (2008)], ], [Malinowska and Torres (2009)], while others prefer the nabla methodology [Almeida and Torres (2009)], [Atici et al (2006)], [Atici and Uysal (2008)], [Martins and Torres (2009)]. In this paper we propose a simple and effective unification of the delta and nabla approaches of optimal control on time scales.…”

A unified approach to the calculus of variations on time scales

“…It has found applications in several different fields that require simultaneous modeling of discrete and continuous data, in particular in the calculus of variations. There are two approaches that are followed in the literature of the calculus of variations on time scales: one is concerned with the minimization of delta integrals with a Lagrangian depending on delta derivatives [1,6,7,11,14]; the other with minimization of nabla integrals with integrands that involve nabla derivatives [2,4]. Both formulations of the problems of the calculus of variations give results that are similar among them and similar to the classical results of the calculus of variations (see, e.g., [16]) but are obtained independently.…”