Bivariate Optimal Repair Replacement Policy for a Deteriorating System

k-out-of-n system with repair: the max(N,T) policy

“…Differentiating C(N, α) in (14) twice with respect to N and with respect to α, we see that ∂ 2 C(N, α)/∂N 2 > 0 for fixed α, and ∂ 2 C(N, α)/∂α 2 > 0 for fixed N.…”

“…ᮀ Note. We observe that when k = 1, C(N, α) in (14) corresponds to the total expected cost, for the parallel system. The optimal (N, α) value corresponding to such a system is then immediately obtained.…”

“…Theorem 2. If n, k and n − k are sufficiently large C(N, α) given by (14) is convex in N(α) for fixed α(N) and hence the global minimum N * (α * ) of N(α) exists for fixed α(N). N * and α * can be obtained by solving the inequalities C(N * , α) ≤ C(N * ± 1, α) for fixed α and C(N, α * ) ≤ C(N, α ± ), for fixed N, for > 0.…”

“…This can be regarded as the optimal N-policy for replacement. Rangan and Sarada [13,14] discuss the optimal strategies for replacements of deteriorating systems with changing failure distributions. Yeh [7,8] analyses a repairable-replacement model and obtains the optimal number of repairs before replacement.…”