On the delayed worse-than-minimal repair model and its application to preventive replacement

Cha, Ji Hwan and Finkelstein, Maxim and Levitin, Gregory (2022) On the delayed worse-than-minimal repair model and its application to preventive replacement. IMA Journal of Management Mathematics, 34 (1). pp. 101-122. dpab036. ISSN 1471-678X (https://doi.org/10.1093/imaman/dpab036)

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Minimal repair and other imperfect repair models have been intensively studied in the literature. Much less attention has been payed to the 'worse than minimal' repair problem, although it often occurs in practice due to the adverse effects of previous repairs, environmental and internal shocks, etc. To model this type of repair, we define a new point process that behaves as the non-homogeneous Poisson process up to a certain event or time (minimal repairs) and then it becomes the generalized Polya process of repairs (worse than minimal repairs). The corresponding replacement policy is defined and the optimal solutions that minimize the long run expected cost rate are analyzed. The replacement can be executed univariately either after the given time T or the given number of repairs (on the k-th failure). Moreover, the system can be also replaced by implementing the bivariate strategy, that is, after the time T or on the k-th failure, whichever comes first. The detailed numerical examples illustrate our findings. It is shown that the k-strategy outperforms the T -strategy (lower cost rates), whereas the bivariate strategy is not worse than the best univariate strategy.