Observations of Chaotic Behaviour in Nonlinear Inventory Models

Observations of Chaotic Behaviour in Nonlinear Inventory Models

Anthony S. White (Middlesex University, London, UK) and Michael Censlive (School of Science and Technology, Middlesex University, London, UK)
Copyright: © 2019 |Pages: 28
DOI: 10.4018/IJAIE.2019010101

Abstract

This article describes the use of simulation to investigate incipient chaotic behaviour in inventory models. Model structures investigated were either capacity limited or of variable delay time, implemented in discrete and continuous transform algebras. Results indicate the absence of chaos for a continuous time model but gave limited evidence for chaos in both unrestricted discrete models and those with a positive orders only limit. The responses where interaction with the capacity limit occurred did not confirm chaotic behaviour at odds with published results. Using the Liapunov exponent as a measure of chaotic behaviour, the results indicated, where the delay varies in proportion to order rate, a larger fixed delay reduced the Liapunov exponent as did increasing the dependence of delay on order rate. The effect of the model structures showed that the IOBPCS model, produced the largest Liapunov exponent. Reducing the discrete model update time reduced the Liapunov exponent.
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Introduction

In modern engineering, science and management great use is made of models to enable predictions to be made. Studies of the dynamics of physical and human systems are usually based on experiments and on the decision processes used to control them. Such studies of the variation with time depend on the initial conditions and on the parameters to be determined, usually by experiment but also by theoretical analysis. Some deterministic dynamic systems have been shown to be subject to chaotic behavior (Drazin, 1992). Fawcett and Waller (2011) have shown clearly why rigorous theoretical analysis of business processes is as important to progress as more practical discussions. They argue for dual approach but with rigorous evaluation of all research. This paper attempts to answer the question whether evidence for chaos in supply chains is real.

Chaos has been defined by Wilding (1998) as “…aperiodic, bounded dynamics in a deterministic system with sensitivity dependence on initial conditions, and has structure in phase space…” Wilding further outlines these terms as:

  • Aperiodic: The same state is never repeated twice;

  • Bounded: On successive iterations, the state stays within a finite range and does not approach ± infinity;

  • Deterministic: There is a definite rule with no random terms governing the dynamics;

  • Sensitivity to initial conditions: Two points that are initially close will drift apart as time proceeds;

  • Structure in phase space: Nonlinear systems are described by multi-dimensional vectors. The space in which these vectors lie is called a phase space or state space. The dimensions of this phase space are an integer (Abarbanel 1996). Chaotic systems display discernible patterns when viewed.

Chaos is defined as a deterministic behavior of a system governed by fixed rules involving no random elements. The sensitivity to initial conditions is such that a minor change in any variable may result in a totally different response. Thompson and Stewart (2002) state that chaos is unpredictable over long time scales because any two phase space trajectories starting close to a chaotic attractor will separate as they progress in time. The separation rate will depend on the largest Liapunov exponent (Kapitaniak, 1998) that is related to the system eigenvalues. The phenomenon is referred to as Deterministic Chaos.

Chaotic behavior has been found in cardiac systems (Garfinkel et al, 1992), stock market performance (Weis, 1992) and management of telephone exchanges (Erramilli and Forys, 1991).

When a model is created of the chosen system experimental measurements may be used. The ultimate accuracy of experiments is necessarily limited both by time and money and by the character of the parameters themselves. It is therefore imperative to determine what effect small changes in value of parameters have on the overall system performance. This not only has a bearing on whether the system as described behaves as needed but to see if the model is a true representation of the real system i.e. to see how sensitive the system is to parameter changes. A review of the history and progress in understanding chaos is given by Thompson and Stewart (2002).

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