This paper investigates the effects of a renewing free-replacement warranty (RFRW) on the age replacement policy for a repairable product with a general failure model. In the general model, there are two types of failure when the product fails. One is type I failure (minor failure), which can be removed by a minimal repair; and the other is type II failure (catastrophic failure), which can be removed only by a replacement. After a minimal repair, the product is operational but the failure rate of the product remains unchanged. For both warranted and nonwarranted products, cost models are developed, and the corresponding optimal replacement ages are derived such that the long-run expected cost rate is minimized. The impacts of the RFRW on the optimal replacement age are investigated analytically. Finally, numerical examples are given for illustration.
AcronymsNHPP Non-homogeneous poisson process IFR Increasing failure rate RFRW Renewing free-replacement warranty, a type of warranty policies PM Preventive maintenance pdf Probability density function Cdf Cumulative distribution function Sf Survival function r.v. Random variable Notations t age to replace the product * implies: the optimal value w warranty period X time to failure of a new product f( Á ), F( Á ), FðÁÞ pdf, Cdf and Sf of the r.v. X r( Á ) failure rate (hazard) function of the r.v. X {N(y); y ! 0} NHPP with intensity r(y) {L(y); y ! 0} NHPP with intensity p Á r(y) {M(y); y ! 0} NHPP with intensity (1 À p) Á r(y) Ã(t) R t 0 rðuÞdu, cumulative hazard function Y waiting time until the first type II failure of a new product g(Á), G(Á), GðÁÞ pdf, Cdf and Sf of the r.v. Ỹ EðYÞ ¼ R 1 0 GðuÞdu, mean time to first type II failure of a new product p Pr{type II failure when the product fails}, 0 < p 1 C d downtime cost for each type II failure of a product C p purchasing cost for a replacement C m minimal repair cost for each type I failure of a product C 0 (t), T 0 (t) cycle cost, cycle time when the age for replacement is t, and w ¼ 0 C 1 (t), T 1 (t) cycle cost, cycle time when the age for replacement is t ! w > 0 C 2 (t), T 2 (t) cycle cost, cycle time when the age for replacement is 0 < t < w CR i (t) expected cost rate, which is E½C i ðtÞ=E½T i ðtÞ for i ¼ 0, 1, 2 (t) intermediate function, which is prðtÞ R t 0 GðuÞdu À GðtÞ t à i optimal age for replacement under i ¼ 0, 1, 2 t à w optimal age for replacement when the warranty period is w>0