Mathematical and Stochastic Models for Reliability in Repairable Industrial Physical Assets

Mathematical and Stochastic Models for Reliability in Repairable Industrial Physical Assets

Pablo A. Viveros Gunckel (Universidad Técnica Federico Santa María, Chile), Adolfo Crespo Márquez (Universidad de Sevilla, Spain), Fredy A. Kristjanpoller (Universidad Técnica Federico Santa María, Chile), Rene W. Tapia (RelPro SpA, Chile) and Vicente González-Prida (Universidad de Sevilla, Spain)
DOI: 10.4018/978-1-4666-8222-1.ch012
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Abstract

Generally, assets present a varied behaviour in their life cycle, which is related directly to the use given and consequently related to the technical assistance traditionally known as maintenance or maintenance policies. It can be of a diverse nature: perfect, minimum, imperfect, over-perfect, and destructive as appropriate. This feature requires the application of advanced techniques in order to model the behaviour of assets life, adapting ideally to each reality of use and wear out. In this chapter, the stochastic models PRP, NHPP, and GRP are explained with their conceptual, mathematic, and stochastic development. For each model, the conceptualization, parameterizing, and stochastic simulation are analysed. Additionally, complementing the analysis and resolution pattern, these models are concluded with a numeric application that allows one to show step by step the mathematic and stochastic development as appropriate.
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1. Introduction

The model and analysis of repairable equipments are of great importance, mainly in order to increase the performance oriented to reliability and maintenance as part of the cost reduction in this last item. A reparable system is defined as:

A system that, after failing to perform one or more of its functions satisfactorily, can be restored to fully satisfactory performance by any method other than replacement of the entire system. (Ascher & Feingold, 1984)

A system that, after failing in order to develop an activity properly, is possible to restore satisfactorily its functioning by some method. (Ascher & Feingold, 1984)

Depending on the type of maintenance given to an equipment, is possible to find 5 cases (Veber et al., 2008):

  • 1.

    Perfect Maintenance or Reparation: Maintenance operation that restores the equipment to the condition “as good as new”.

  • 2.

    Minimum Maintenance or Reparation: Maintenance operation that restores the equipment to the condition “as bad as old”.

  • 3.

    Imperfect Maintenance or Reparation: Maintenance operation that restores the equipment to the condition “worse than new but better than old”.

  • 4.

    Over-Perfect Maintenance or Reparation: Maintenance operation that restores the equipment to the condition “better than new”

  • 5.

    Destructive Maintenance or Reparation: Maintenance operation that restores the equipment to the condition “worse that old”.

For a perfect maintenance, the most common developed model corresponds to the Perfect Renewal Process (PRP). In it, we assume that repairing action restores the equipment to a condition as good as new and assumes that times between failures in the equipment are distributed by an identical and independent way. The most used and common model PRP is the Homogeneous Processes of Poison (HPP), which considers that the system not ages neither spoils, independently of the previous pattern of failures. That is to say, it is a process without memory. Regarding case b), “as bad as old” is the opposite case to what happens in case a) “as good as new”, since it is assumed that the equipment will stay after the maintenance intervention in the same state than before each failure. This consideration is based that the equipment is complex, composed by hundreds of components, with many failure modes and the fact that replacing or repairing a determined component will not affect significantly the global state and age of the equipment. In other words, the system is subject to minimum repairs, which does not cause any change or considerable improvement. The most common model to represent this case is through Non –Homogeneous Processes of Poison (NHPP), in this case the most used model to represent NHPP is called “Power Law”. In this model, it is assumed a Weibull distribution for the first failure, that later it is modified over time.

Although the models HPP and NHPP are the most used, they have a practical restriction regarding its application, since a more realistic condition after a repairing action is what we find between both: “worse than new but better than old”. In order to find a generalization to this situation and not distinguish between HPP and NHPP it was necessary to create the Generalized Renewal Process (GRP) (Kijima & Sumita, 1986), which establishes an improvement ratio. Unfortunately, the incorporation of this variable can complicate the analytic calculation of parameters and adjustments of probability. Therefore, its applicability in mathematic terms is complex. For this reason, it has been considered solutions through the Monte Carlo simulation (MC) being one of the most validated methods according the proposal developed by author Krivstov (2000) where time series of good functioning are generated through the use of the inverse function of the probability distribution (pdf) that has as a base a random variable.

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