PurposeThis paper models the deterioration process of a multi-component system. Each deterioration process is modelled by the Wiener process. The purposes of this paper are to address these issues and consider the cost process based on the multi-component system.Design/methodology/approachCondition-based Maintenance is a method for reducing the probability of system failures as well as the operating cost. Nowadays, a system is composed of multiple components. If the deteriorating process of each component can be monitored and then modelled by a stochastic process, the deteriorating process of the system is a stochastic process. The cost of repairing failures of the components in the system forms a stochastic process as well and is known as a cost process.FindingsWhen a linear combination of the processes, which can be the deterioration processes and the cost processes, exceeds a pre-specified threshold, a replacement policy will be carried out to preventively maintain the system.Originality/valueUnder this setting, this paper investigates maintenance policies based on the deterioration process and the cost process. Numerical examples are given to illustrate the optimisation process.

The geometric process (GP) is a stochastic process that was an extension of the renewal process. It was introduced by Lam (1988) in 1988 with an intention to model the failure process of a repairable system whose the times between failures become shorter and shorter after repairs and repair times become longer and longer. The GP has been widely studied in the literature of reliability and maintenance and applied in optimisation of maintenance policies. Some authors have proposed various versions of its extensions (or the GP-like models), including the α-series process, the threshold GP, the extended Poisson process, the doubly GP, and the doubly-ratio GP. Some papers also compare the performance of the GP with that of other models, but not with the performance of the extensions of the GP. This paper therefore reviews the GP-like models, compares the performance of the GP and its extensions in terms of the Akaike information criterion (AIC), the corrected AIC (AICc) and the maximum likelihood (ML) based on 25 real-world datasets. Besides, the least square methods for estimating the parameters in some models are discussed, which is used for model performance of GP and GP-like models. The finding is useful for practitioners in their selection of the GP-like models.

Modelling the failure process of a system is one of the most important problems in the reliability and maintenance research community. The geometric process (GP) is widely used for modelling the failure process because it can describe the phenomenon that the working times after repairs become shorter and shorter. This article reviews the geometric process and its extensions based on existing research. It also reviews relevant methods for estimating parameters, model performances, and widely used distributions for times to first failures. Future challenges for the GP-like processes will be discussed.