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Abstract:
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We consider a class of periodic preventive maintenance (PM ) optimization problems , for a single piece of equipment that deteriorates with time or use , and can be repaired upon failure , through corrective maintenance (CM ) . We develop analytical and simulation -based optimization models that seek an optimal periodic PM policy , which minimizes the sum of the expected total cost of PMs and the risk -averse cost of CMs , over a finite planning horizon . In the simulation -based models , we assume that both types of maintenance actions are imperfect , whereas our analytical models consider imperfect PMs with minimal CMs . The effectiveness of maintenance actions is modeled using age reduction factors . For a repairable unit of equipment , its virtual age , and not its calendar age , determines the associated failure rate . Therefore , two sets of parameters , one describing the effectiveness of maintenance actions , and the other that defines the underlying failure rate of a piece of equipment , are critical to our models . Under a given maintenance policy , the two sets of parameters and a virtual -age -based age -reduction model , completely define the failure process of a piece of equipment . In practice , the true failure rate , and exact quality of the maintenance actions , cannot be determined , and are often estimated from the equipment failure history .
We use a Bayesian approach to parameter estimation , under which a random -walk -based Gibbs sampler provides posterior estimates for the parameters of interest . Our posterior estimates for a few datasets from the literature , are consistent with published results . Furthermore , our computational results successfully demonstrate that our Gibbs sampler is arguably the obvious choice over a general rejection sampling -based parameter estimation method , for this class of problems . We present a general simulation -based periodic PM optimization model , which uses the posterior estimates to simulate the number of operational equipment failures , under a given periodic PM policy . Optimal periodic PM policies , under the classical maximum likelihood (ML ) and Bayesian estimates are obtained for a few datasets . Limitations of the ML approach are revealed for a dataset from the literature , in which the use of ML estimates of the parameters , in the maintenance optimization model , fails to capture a trivial optimal PM policy .
Finally , we introduce a single -stage and a two -stage formulation of the risk -averse periodic PM optimization model , with imperfect PMs and minimal CMs . Such models apply to a class of complex equipment with many parts , operational failures of which are addressed by replacing or repairing a few parts , thereby not affecting the failure rate of the equipment under consideration . For general values of PM age reduction factors , we provide sufficient conditions to establish the convexity of the first and second moments of the number of failures , and the risk -averse expected total maintenance cost , over a finite planning horizon . For increasing Weibull rates and a general class of increasing and convex failure rates , we show that these convexity results are independent of the PM age reduction factors . In general , the optimal periodic PM policy under the single -stage model is no better than the optimal two -stage policy . But if PMs are assumed perfect , then we establish that the single -stage and the two -stage optimization models are equivalent . |