Optimization issues in k-out-of-n systems

Abstract

• We propose two replacement policies including T -policy and N -policy for a k -out-of- n system. • Optimization of the long term expected cost per unit time for each policy is presented theoretically. • Some numerical examples are shown to fit our research. The k -out-of- n system is a system consisting of n independent components such that the system works if and only if at least k of these n components are successfully running. Each component of the system is subject to shocks which arrive according to a nonhomogeneous Poisson process . When a shock takes place, the component is either minimally repaired (type 1 failure) or lying idle (type 2 failure). Assume that the probability of type 1 failure or type 2 failure depends on age. First, we investigate a general age replacement policy for a k -out-of- n system that incorporates minimal repair, shortage and excess costs. Under such a policy, the system is replaced at age T or at the occurrence of the ( n - k + 1)th idle component, whichever occurs first. Moreover, we consider another model; we assume that the system operates some successive projects without interruptions. The replacement could not be performed at age T . In this case, the system is replaced at the completion of the N th project or at the occurrence of the ( n - k + 1)th idle component, whichever occurs first. For each model, we develop the long term expected cost per unit time and theoretically present the corresponding optimum replacement schedule. Finally, we give a numerical example illustrating the models we proposed. The proposed models include more realistic factors and extend many existing models.