Optimal allocation of two redundancies in a n-component series system with proportional hazard rates lifetimes

In this paper, we deal with the problem of optimally allocating two active [standby] redundancies in a series system consisting of n-components ((Formula presented.)) in the sense of the stochastic ordering. For the case of active redundancy, we establish two results associated with the likelihood ratio and reserved hazard rate orders in the setting of proportional hazard rates (PHR) model. For the case of standby redundancy, we also present some new ordering results under the exponential or PHR framework. The results established here generalize and strengthen corresponding ones in the recent literature. Numerical examples are also provided to explicate the theoretic results.