Renewal-geometric process is used to describe such a non-homogeneous deteriorating process that a system will deteriorate after several consecutive repairs, not after each repair described by the geometric process. In the maintenance domain, the effect of corrective maintenance after failure is generally not repairable as new (e.g. geometrically deteriorating). Preventive maintenance is critical before a system failure, due to economic losses and security threats caused by a sudden shutdown. Therefore, this article assumes that a system is geometrically deteriorating after corrective maintenance, wherein preventive maintenances sequence in the same repair period form a renewal process since it can restore the system to the initial state of the period. Furthermore, a binary policy $(N,T)$ is utilized to minimize the long-run average cost rate, where $N$ represents the number corrective maintenances and $T$ denotes the time interval between two consecutive preventive maintenances. In particular, pseudo-age replacement model represents a special case of $N=1$, which is considered as a generalization of the traditional age-based replacement model. Subsequently, the optimal policy $N^{*}$ can be verified in theory and an asymptotic optimal policy $(N^{*},T^{*})$ can be obtained based on a heuristic grid search. Finally, numerical examples verify the effectiveness of this proposed model and show that implementation of preventive maintenance for some repairable systems is superior to no preventive maintenance in both economic and reliability aspects.

更新几何过程用于描述这样的非均匀恶化过程，即系统将在几次连续修复后退化，而不是在几何过程描述的每次修复后退化。在维护领域，故障后进行纠正性维护的效果通常无法像新的那样得到修复（例如，几何形状恶化）。由于经济损失和突然关机造成的安全威胁，预防性维护对于系统故障至关重要。因此，本文假设系统在纠正性维护之后在几何上恶化，其中在相同维修期内的预防性维护序列形成了更新过程，因为它可以将系统恢复到该时段的初始状态。此外，二元政策$（ñ，Ť）$ 用于最大程度地降低长期平均成本率，其中 $ñ$ 代表数字校正维护，并且 $Ť$表示两次连续预防性维护之间的时间间隔。特别是伪年龄替代模型代表了一种特殊情况$ñ=\mathrm{1个}$，这是对传统基于年龄的替换模型的概括。随后，最优政策${ñ}^{*}$ 可以在理论上证明和渐近最优策略 $（{ñ}^{*}，{Ť}^{*}）$可以基于启发式网格搜索获得。最后，数值算例验证了该模型的有效性，表明在经济和可靠性方面，对某些可修复系统实施预防性维护优于无预防性维护。