The availability model and parameters estimation method for the delay time model with imperfect maintenance at inspection

The delay time model (DTM) is widely used to model the two-stage failure process and is helpful for developing cost-effective inspection/maintenance plans. Imperfect maintenance is common in practice, but seldom considered in DTM. An improved DTM with imperfect maintenance at inspection has been developed based on the assumption of imperfect inspection maintenance and perfect failure maintenance. The model of the long-run availability for the improved DTM is established. Parameters estimation method and the test for goodness of fit method are given. Numerical simulations are performed to study the influence of imperfect maintenance on the long-run availability and to validate the credibility of the parameters estimation method. The results show that imperfect maintenance will decrease the long-run availability. The existence of the optimal inspection interval regarding the maximum long-run availability is tightly related to the improvement factor, which denotes the maintenance effect. The parameters estimation method proves credible. The maximum likelihood estimations of the reliability parameters can be easily achieved by the Genetic Algorithms (GAs) searching tool.     

Expected present value of total dividends in a delayed claims risk model under stochastic interest rates

In this paper, a compound binomial risk model with a constant dividend barrier under stochastic interest rates is considered. Two types of individual claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim and may be delayed for one time period with a certain probability. In the evaluation of the expected present value of dividends, the interest rates are assumed to follow a Markov chain with finite state space. A system of difference equations with certain boundary conditions for the expected present value of total dividend payments prior to ruin is derived and solved. Explicit results are obtained when the claim sizes are K_n distributed or the claim size distributions have finite support. Numerical results are also provided to illustrate the impact of the delay of by-claims on the expected present value of dividends.     

Bifurcation analysis of a multidelayed HIV model in presence of immune response and understanding of in‐host viral dynamics

In this paper, we proposed a multidelayed in‐host HIV model to represent the interaction between human immunodeficiency virus and immune response. One delay was considered to incorporate the time required by the virus for various intracellular events to make a host cell productively infective, and the second delay was introduced to take into account the time required for adaptive immune system to respond against infection. We extensively analyzed this multidelayed model analytically and numerically. We show that delay may have both destabilizing and stabilizing effects even when the system contains a single immune response delay. It happens when there exists two sequences of critical values of this delay. If the system starts with stable state in absence of delay, then the smallest value of these critical delays causes instability and the next higher value causes stability. The system may also show stability switching for different values of the virus replication factor. These results demonstrate the possible reasons of intrapatients and interpatients variability of CD4⁺ T cells and virus counts in HIV‐infected patients.     

Optimal inspection policy for three-state systems monitored by control charts

This paper presents the formulations of the expected long-run cost per time unit for a system monitored by a static control chart and by an adaptive control chart respectively. The static chart has a fixed sampling interval and a fixed sample size. The adaptive chart has a fixed sample size but variable sampling intervals. The system is supposed to have three states, normal working state, failure delay time state, and failed state. Two levels of repair are used to maintain the system. A minor repair is used to restore the system if a detectable defect is confirmed by an inspection. A major repair will be performed if the system fails. The expected cost per time unit for maintaining such a system is obtained. The objective of such analysis is to find an optimal sampling policy for the inspection process. An artificially generated data example and a real data example are used to compare the expected cost per time unit for both the static and adaptive control charts.     

利用故障数据和估计的检查数据建立维修优化模型

利用时间延迟概念,根据故障记录数据和估计的检查数据建立了预防维修模型.通过对故障记录数据统计分析,提出了模型的假定条件.采用最大似然估计法,估计参数,包括缺陷发生率、不完全检查概率和时间延迟分布.建立了有关预防维修间隔期和总停机时间之间关系的检查模型,并根据估计参数和检查模型,计算最佳维修间隔期.     

Optimal policies for a delay time model with postponed replacement

We develop a delay time model (DTM) to determine the optimal maintenance policy under a novel assumption: postponed replacement. Delay time is defined as the time lapse from the occurrence of a defect up until failure. Inspections can be performed to monitor the system state at non-negligible cost. Most works in the literature assume that instantaneous replacement is enforced as soon as a defect is detected at an inspection. In contrast, we relax this assumption and allow replacement to be postponed for an additional time period. The key motivation is to achieve better utilization of the system’s useful life, and reduce replacement costs by providing a sufficient time window to prepare maintenance resources. We model the preventive replacement cost as a non-increasing function of the postponement interval. We then derive the optimal policy under the modified assumption for a system with exponentially distributed defect arrival time, both for a deterministic delay time and for a more general random delay time. For the settings with a deterministic delay time, we also establish an upper bound on the cost savings that can be attained. A numerical case study is presented to benchmark the benefits of our modified assumption against conventional instantaneous replacement discussed in the literature.     

Optimal production run time for a deteriorating production system under an extended inspection policy

A deteriorating production system is subjected to random deterioration from an in-control state to an out-of-control state with a general shift distribution. In order to reduce the defective items, part inspection policy, under which production inspections are performed only at the end of the production run, and full inspection policy are both considered in the literature. Moreover, the former dominates the latter. Since the product produced towards the end of a production cycle are more likely to be defective, it can further economize the inspection costs that they are directly reworked without inspection. In this paper, we propose an extended product inspection policy for a deteriorating production system. Product inspections are performed in the middle of a production cycle, and after the inspection, all products produced until the end of the production run are fully reworked. Based on the model, we show that there exists a production run time and a corresponding unique inspection policy such that the expected total cost per item per cycle is minimized. Finally, numerical examples are provided to illustrate our extended inspection policy, and indicate that such product inspection model will reduce the quality-related cost than part inspection does.     

On a Simplified Delay Time Model of Reliability of Equipment Subject to Inspection Monitoring

The delay time model developed by Christer to determine the reliability consequences of various inspection periods is shown to be considerably simplified if information is obtained only at the inspection times. It is shown that the average reliability over an inspection period may be approximated by a linear function giving a relative error of the order of 10% and by a cubic resulting in a relative error of less than 1%.     

Stability of a stochastic SEIS model with saturation incidence and latent period

In this paper, a stochastic SEIS epidemic model with a saturation incidence rate and a time delay describing the latent period of the disease is investigated. The model inherits the endemic steady state from its corresponding deterministic counterpart. We first show the existence and uniqueness of the global positive solution of the model. Then, by constructing Lyapunov functionals, we derive sufficient conditions ensuring the stochastic stability of the endemic steady state. Numerical simulations are carried out to confirm our analytical results. Furthermore, our simulation results shows that the existence of noise and delay may cause the endemic steady state to be unstable.     

A least squares identification algorithm for a state space model with multi-state delays

A parameter estimator is presented for a state space model with time delay based on the given input–output data. The basic idea is to expand the state equations and to eliminate some state variables, and to substitute the state equation into the output equation to obtain the identification model which contains the information vector and parameter vector. A least squares algorithm is developed to estimate the system parameter vectors. Finally, an illustrative example is provided to verify the effectiveness of the proposed algorithm.     

Poisson冲击下的$k/n(G)$系统的可靠性分析

本文研究了一类Poisson冲击下的$k/n(G)$系统(即$k$-out-of-$n$: $G$系统). 假定冲击的到达数形成一个参数为$\lambda$的Poisson过程, 且冲击的量服从某一分布. 当每次冲击到达时, 对系统中工作的部件独立地产生影响. 进而假定每一部件以一定的概率故障, 概率值是冲击量的函数. 且各次冲击独立地对系统造成损失, 直到工作部件数少于$k$系统故障为止. 在这些假定下, 我们获得了系统的可靠度函数和系统的平均工作时间. 进一步, 假定系统是可修的, 系统中有一个维修工, 并根据``先坏先修’’的维修规则对故障部件进行维修. 在维修时间服从指数分布的假设下, 系统状态转移服从Markov过程. 对该系统我们建立了状态转移方程, 并求得了系统可用度、稳态下的平均工作时间、平均停工时间和系统失效频率等可靠性指标. 最后, 我们还给出了一个简单例子来演示讨论的模型.     

顾客部分延期付款下两级商业信用易腐品订货策略

针对易腐品的经济订货批量问题展开研究。在供应商给零售商提供延迟还款的同时,零售商也给顾客提供部分延期还款条件。分五种情况分别讨论零售商的成本构成,并由此建立数学模型以求解最优订货周期使得零售商单位时间总成本最小化。通过数学证明得到了目标函数的解析性质,结果显示每种情况下在可行域范围内至多存在一个极小值点。以此为基础给出了相应的命题以有效地确定零售商的最优决策。最后通过数值算例说明了模型的有效性。     

Influence of time delay and nonlinear diffusion on herbivore outbreak

Herbivore outbreaks, a major form of natural disturbance in many ecosystems, often have devastating impacts on their food plants. Understanding those factors permitting herbivore outbreaks to occur is a long-standing issue in conventional studies of plant-herbivore interactions. These studies are largely concerned with the relative importance of intrinsic biological factors and extrinsic environmental variations in determining the degree of herbivore outbreaks. In this paper, we illustrated that how the time delay associated with plant defense responses to herbivore attacks and the spatial diffusion of herbivore jointly promote outbreaks of herbivore population. Using a reaction-diffusion model, we showed that there exists a threshold of time delay in plant-herbivore interactions; when time delay is below the threshold value, there is no herbivore outbreak. However, when time delay is above the threshold value, periodic outbreak of herbivore emerges. Furthermore, the results confirm that during the outbreak period, plants display much lower density than its normal level but higher in the inter-outbreak periods. Our results are supported by empirical findings.     

A two-stage prognosis model in condition based maintenance

We often observe in practice that the life of a piece of production equipment can be divided into two stages. The first stage is referred to as the normal working stage where no significant deviation from the normal operating state is observed. The second stage is called the failure delay period, since a defect may be initiated, and progressively develop into an actual failure, i.e., the equipment is in a defective stage but still working during this stage. With the help of condition monitoring, hidden defects already present in the equipment may be detected, but for maintenance planning purposes, the prediction of the initiation point of the second stage, and more importantly, the residual life thereafter is important. This paper reports on the development of a probability model to predict the initiation point of the second stage and the remaining life based on available condition monitoring information. The method for model parameters estimation is discussed and applied to real data.     

A two-phase inspection policy with imperfect testing

This paper presents an inspection policy to detect failures of a single component system that remain hidden otherwise. Inspection reveals whether the unit is in good or failed state. The possibility of non perfect testing is assumed, thus, successive inspections may fail detecting a failure or result in a false alarm. The occurrence of false alarms is reported in optical fire detectors and inspection of printing circuit boards which are on the basis of electronic systems. A two-phase inspection schedule takes into account the changes in component’s aging. The system may undergo different inspection frequencies to detect both early failures or those due to the natural deterioration in the system as time goes by. The examples reveal the advantages of a two-phase inspection when comparing with the unique interval inspection.     

A simulation-based multivariate Bayesian control chart for real time condition-based maintenance of complex systems

When complex systems are monitored, multi-observations from several sensors or sources may be available. These observations can be fused through Bayesian theory to give a posterior probabilistic estimate of the underlying state which is often not directly observable. This forms the basis of a Bayesian control chart where the estimated posterior probability of the state can be compared with a preset threshold level to assess whether a full inspection is needed or not. Maintenance can then be carried out if indicated as necessary by the inspection. This paper considers the design of such multivariate Bayesian control chart where both the transition between states and the relationship between observed information and the state are not Markovian. Since analytical or numerical solutions are difficult for the case considered in this paper, Monte Carlo simulation is used to obtain the optimal control chart parameters, which are the monitoring interval and the upper control limit. A two-stage failure process characterised by the delay time concept is used to describe the underlying state transition process and Bayesian theory is used to compute the posterior probability of the underlying state, which is embedded in the simulation algorithm. Extensive examples are shown to demonstrate the modelling idea.     

一类具有时滞的捕食者食饵模型的稳定性和Hopf分支

针对种群的发展趋势不仅依赖于当前的状态,而且还依赖于过去某一时间段的状态.研究了一类带时滞的捕食者-食饵模型.首先应用线性化方法分析该模型正平衡点的局部渐近稳定性,然后以时滞为分支参数讨论Hopf分支的存在性,最后对模型进行数值模拟证实了以上相关结果.     

Hopf bifurcations in a reaction-diffusion population model with delay effect

A reaction-diffusion population model with a general time-delayed growth rate per capita is considered. The growth rate per capita can be logistic or weak Allee effect type. From a careful analysis of the characteristic equation, the stability of the positive steady state solution and the existence of forward Hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem, where the time delay is used as the bifurcation parameter. The general results are applied to a “food-limited” population model with diffusion and delay effects as well as a weak Allee effect population model.     

考虑不完美检测的两阶段点检与备件订购策略联合优化

针对设备维修与备件管理相互影响与制约的问题, 在基于延迟时间理论的基础上, 提出了两阶段点检与备件订购策略联合优化。点检是不完美的, 当点检识别设备的缺陷状态时, 进行预防更新; 设备故障时, 进行故障更新。结合设备更新时备件的状态, 采用更新报酬理论建立了以第一阶段点检时间、第二阶段点检周期和备件订购时间为决策变量, 以最小化单位时间期望成本为目标的模型。最后, 通过人工蜂群算法对模型求解, 并在数值分析中将两阶段点检策略与定期点检策略进行比较, 结果表明:两阶段点检策略始终优于定期点检策略, 验证了所建模型的有效性。     

基于产品质量控制的设备维修优化模型

根据产品质量和生产该产品的设备退化状态之间的相关性,设计了周期性设备检测与产品质量控制相结合的设备维修策略。该策略是在对设备进行周期性检测的基础上,利用控制图进行产品质量异常波动的检测,结合对设备退化状态的检测选择设备应采取的维修活动。根据这一设备维修策略,利用更新过程理论和统计过程控制方法,构建了基于产品质量控制的设备维修优化模型,并用遗传算法对其进行求解。通过实例仿真验证了该模型的可行性与有效性。