This paper proposes an integrated approach for reliability modelling and maintenance scheduling of repairable parallel systems subject to hidden failures. The system consists of heterogeneous redundant subsystems whose failures are revealed only by inspections. Inspections at periodic times reveal the components state and repair actions are decided by the excursion of a basic state process describing the total number of failed components in each subsystem. Using the standard renewal arguments, the paper aims at minimizing the average cost rate by the joint determination of the optimal inspection interval, the partial repair threshold and the preventive replacement threshold. We illustrate the procedure for the case as the components' lifetimes conform to the Weibull distribution. Numerical examples are used to illustrate the proposed model and the response of the optimal solutions to the model's parameters.

In this paper, a rapid and high accurate numerical method for pricing discrete single and double barrier knock-out call options is presented. With regard to the well-known Black-Scholes model, the price of an option in each monitoring date could be calculated by computing a recursive integral formula that is based on the heat equation solution. We have approximated these recursive solutions with the aid of Lagrange interpolation on Jacobi polynomial nodes. After that, an operational matrix, that makes our computation significantly fast, has been derived. In some theorems, the convergence of the presented method has been shown and the rate of convergence has been derived. The most important benefit of this method is that its complexity is very low and does not depend on the number of monitoring dates. The numerical results confirm the accuracy and efficiency of the presented numerical algorithm.