욕조곡선 형태의 고장발생률을 가지는 수리가능시스템에 대한 신뢰도 분석 및 보전 최적화

Abstract

수리가능 시스템 (repairable system)은 고장이 발생하였을 때 고장이 난 부분의 수리 또는 교체를 통하여 운용 가능한 시스템을 의미한다. 수리가능 시스템은 때때로 번인 프로세스 (burn-in process)를 통하여 완전히 제거되지 않는 결함으로 인하여 초기고장 (early failure)이 발생하며, 시간이 지남에 따라 시스템이 악화되어 마모고장 (wear-out failure)이 발생한다. 이러한 수리가능 시스템은 고장발생률이 감소하다가 증가하는 욕조곡선 형태의 고장발생률 (bathtub shaped intensity)을 가지게 되며, 욕조곡선 형태의 고장발생률은 많은 고장모드를 가지는 크고 복잡한 시스템에서 전형적으로 발생한다. 욕조곡선 형태의 고장발생률을 모형화하기 위하여 몇 개의 독립적인 NHPP (nonhomogeneous Poisson process)의 조합으로 표현되는 SPP (superposed Poisson process)가 제안되어 왔다. 가장 많이 사용되는 SPP는 S-PLP (superposed power law process) 및 BBIP (bathtub-bound intensity process) 등이 있지만, 국내의 욕조곡선 형태의 고장발생률을 가지는 수리가능 시스템에 적용하였을 때 적절하게 모형화하지 못하였다. 본 논문에서는 욕조곡선 형태의 고장발생률을 가지는 수리가능 시스템에 대한 신뢰도 분석 및 보전 최적화를 제안하였으며, 자세한 사항은 다음과 같다. 첫 번째, 본 논문에서는 욕조곡선 형태의 고장 패턴을 탐지하기 위하여 정보적 변화점 접근법 (informational change-point approach)을 제안하였으며, 변화점의 수를 결정할 수 있는 절차를 제안하였다. 두 번째, 욕조곡선 형태의 고장발생률을 가지는 수리가능 시스템의 고장을 모형화하기 위하여 S-LLP (superposed log-linear process)를 제안하였으며, S-LLP 기반의 최적 보전정책 (optimal maintenance policy)을 제안하였다. 마지막으로, 다중 수리가능 시스템 (multiple repairable systems)에 대하여 신뢰도를 분석하기 위하여 mixed-effects NHPP model을 제안하였으며, 욕조곡선 형태의 고장발생률을 가지는 다중 수리가능 시스템에 대한 최적 정비정책을 제안하였다.| Modern systems consist of numerous parts working together, and system reliability depends on numerous potential failure modes. Due to the prohibitive cost of testing eld system during the manufacturing phase, equipment is subject to a high risk of early failures. Unless the system is badly damaged, failures are followed by repairs which sometimes help eliminate future failures from the same failure mode. Repair processes of this type can emulate a minimal repair model in which the repair or the substitution of a failed part tends to have a negligible effect on overall system reliability, restoring the system performance to the exact same condition as it was just before the failure. Because the system is restored to its current state (immediately preceding the most recent failure), the assumption of minimal repair reveals a failure pattern governed by a nonhomogeneous Poisson process (NHPP). The NHPP has been widely used in modeling failure frequency for repairable systems. The most popular NHPPs include the power law process (PLP) and the log-linear process (LLP). Both the PLP and the LLP have been introduced to model the failure patterns of a repairable system having monotonic intensity. Sometimes, a repairable system is subject both to early (or infant mortality) failures due to the presence of defective parts or assembly defects that are not screened out completely through the burn-in process, as well as wear-out failures caused by deteriorating phenomena. This causes a non-monotonic trend in the failure data in which the intensity function initially decreases (as defective parts are weeded out of the system), followed by a long period of constant or near constant intensity until wear-out finally occurs, at which time the intensity function begins to increase. This is called the bathtub shaped failure intensity, which is typical for large and complex equipment with a number of different failure modes. The PLP and LLP are too simplistic to accommodate this bathtub characteristic of the failure process. As an alternative, unions of several independent NHPPs called superposed Poisson processes (SPPs) have been developed to model this kind of non-monotonic failure intensity. When any subsystem failure can independently cause the system to break down, the superposed model is a natural model for system failure. The most commonly used SPPs are the superposed power law process (S-PLP) and the bathtub-bound intensity process (BBIP). However, we find that existing models, including the S-PLP and BBIP, do not adequately capture the non-monotonic trend in the failure process. In this thesis, we propose a reliability analysis and maintenance optimization for repairable systems with bathtub shaped intensity. The contributions of this thesis are summarized as follows: First, we propose an informational change-point approach to analyze the bathtub shaped patterns of recurrent failures in repairable systems. The change-point approach enables us to detect the positions of change-points as well as the existence of change-points between failure-times. We also suggest sequential procedures for determining the number of change-point as the dimension of the change-point model. Second, we derive a superposed log-linear process (S-LLP) for modeling repairable systems with bathtub shaped intensity. The method of maximum likelihood is used to estimate model parameters and construct confidence intervals for the expected number of failures. We also propose an optimal maintenance policy based on the S-LLP. The optimality is defined as the minimization of the expected cost per unit of time for each PM policy. Finally, we provide flexible applications of the mixed-effects NHPP model for the reliability analysis of multiple repairable systems. We also propose an optimal maintenance policy for multiple repairable systems with bathtub shaped intensity. The Monte Carlo simulation based on the mixed-effects S-LLP model is used to obtain the preventive maintenance (PM) check point.; Modern systems consist of numerous parts working together, and system reliability depends on numerous potential failure modes. Due to the prohibitive cost of testing eld system during the manufacturing phase, equipment is subject to a high risk of early failures. Unless the system is badly damaged, failures are followed by repairs which sometimes help eliminate future failures from the same failure mode. Repair processes of this type can emulate a minimal repair model in which the repair or the substitution of a failed part tends to have a negligible effect on overall system reliability, restoring the system performance to the exact same condition as it was just before the failure. Because the system is restored to its current state (immediately preceding the most recent failure), the assumption of minimal repair reveals a failure pattern governed by a nonhomogeneous Poisson process (NHPP). The NHPP has been widely used in modeling failure frequency for repairable systems. The most popular NHPPs include the power law process (PLP) and the log-linear process (LLP). Both the PLP and the LLP have been introduced to model the failure patterns of a repairable system having monotonic intensity. Sometimes, a repairable system is subject both to early (or infant mortality) failures due to the presence of defective parts or assembly defects that are not screened out completely through the burn-in process, as well as wear-out failures caused by deteriorating phenomena. This causes a non-monotonic trend in the failure data in which the intensity function initially decreases (as defective parts are weeded out of the system), followed by a long period of constant or near constant intensity until wear-out finally occurs, at which time the intensity function begins to increase. This is called the bathtub shaped failure intensity, which is typical for large and complex equipment with a number of different failure modes. The PLP and LLP are too simplistic to accommodate this bathtub characteristic of the failure process. As an alternative, unions of several independent NHPPs called superposed Poisson processes (SPPs) have been developed to model this kind of non-monotonic failure intensity. When any subsystem failure can independently cause the system to break down, the superposed model is a natural model for system failure. The most commonly used SPPs are the superposed power law process (S-PLP) and the bathtub-bound intensity process (BBIP). However, we find that existing models, including the S-PLP and BBIP, do not adequately capture the non-monotonic trend in the failure process. In this thesis, we propose a reliability analysis and maintenance optimization for repairable systems with bathtub shaped intensity. The contributions of this thesis are summarized as follows: First, we propose an informational change-point approach to analyze the bathtub shaped patterns of recurrent failures in repairable systems. The change-point approach enables us to detect the positions of change-points as well as the existence of change-points between failure-times. We also suggest sequential procedures for determining the number of change-point as the dimension of the change-point model. Second, we derive a superposed log-linear process (S-LLP) for modeling repairable systems with bathtub shaped intensity. The method of maximum likelihood is used to estimate model parameters and construct confidence intervals for the expected number of failures. We also propose an optimal maintenance policy based on the S-LLP. The optimality is defined as the minimization of the expected cost per unit of time for each PM policy. Finally, we provide flexible applications of the mixed-effects NHPP model for the reliability analysis of multiple repairable systems. We also propose an optimal maintenance policy for multiple repairable systems with bathtub shaped intensity. The Monte Carlo simulation based on the mixed-effects S-LLP model is used to obtain the preventive maintenance (PM) check point.