Dynamic Analysis of a Reward Process Defined on a Cyclic Renewal Process with Applications to Preventive Maintenance Problems
Dynamic Analysis of a Reward Process Defined on a Cyclic Renewal Process with Applications to Preventive Maintenance Problems
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DOI：
10.15807/jorsj.52.283
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发表时间：
2009-09
影响因子：
--
通讯作者：
U. Sumita;Kazuki Takahashi
中科院分区：
文献类型：
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作者：
U. Sumita;Kazuki Takahashi
循环更新过程被认为是交替更新过程的一种推广，其中每个基本的独立同分布（i.i.d.）非负随机增量由多个阶段组成。这样的过程可能适合于分析生产管理的最佳预防性维护政策，其中一对两个阶段代表直到出现轻微故障的正常运行时间和随后的最短修复时间，直到决定进行彻底检修。为了解决这类应用中的经济问题，我们还引入了一个定义在循环更新过程上的跳跃奖励过程。当系统运行在阶段j时，利润以ρ（j）的速率线性增长。在一个小故障时，在阶段（j + 1）中的随后的最小修复以ρ（j + 1）的比率产生线性成本。此外，无论是最低限度的维修还是彻底的大修，都可能产生固定成本，从而导致奖励过程的跳跃。然后，问题是确定何时进行彻底检修，以便在时间间隔（0，T ]内最大化总回报。一个多变量的马尔可夫过程产生的循环更新过程和奖励过程进行了广泛的研究，产生各种新的转换结果明确，并推导出其渐近展开。这些结果被用来数值探索生产管理的最佳预防性维护政策。
A cyclic renewal process is considered as an extension of an alternating renewal process, where each of the underlying independently and identically distributed (i.i.d.) nonnegative random increments is composed of multiple stages. Such a process may be appropriate for analyzing optimal preventive maintenance policies for production management, where a pair of two stages representing an uptime until a minor failure and the subsequent minimal repair time would be repeated until it is decided to conduct a complete overhaul. In order to address economic problems in such applications, we also introduce a reward process with jumps defined on the cyclic renewal process. When the system is running in stage j, the profit grows linearly at the rate of ρ(j). Upon a minor failure, the subsequent minimal repair in stage (j + 1) incurs the linear cost at the rate of ρ(j + 1). In addition, the fixed cost may be imposed whenever either a minimal repair or a complete overhaul takes place, resulting in jumps of the reward process. The problem is then to determine when to conduct a complete overhaul so as to maximize the total reward in the time interval (0, T ]. A multivariate Markov process generated from both the cyclic renewal process and the reward process is studied extensively, yielding various new transform results explicitly and deriving their asymptotic expansions. These results are used to numerically explore optimal preventive maintenance policies for production management.