An optimal replacement policy for complex multi-component systems

Given a reward structure, this paper addresses an optimal replacement problem for complex multi-component systems. To maintain revenue stream resulting from system, the system is inspected according to a homogeneous Poisson process and certain actions are carried out in response to the system state. Decisions are based on a performance measure described by a Squared Bessel process. Given some assumption, we explore the inherent relation between the Squared Bessel process and an extended Gamma (EG) process. Since there are some flow of income and increasing costs due to inspections, the problem is to optimally stop processing the system and carrying out a renewal to maximize the reward functional. To this end, using the local characteristics of the EG process as a stopping criterion and the expected total discounted reward as a measure of policy, this paper aims at determining an optimal operating (stopping) time which truly balances both income and cost and so maximizes the expected discounted reward over a cycle. In support of the model a numerical example is provided to show feasibility of this programme in real application. Attention is restricted to perfect repair and inspection, but the paper provides the structure so that different scenarios can be explored.     

Maintenance scheduling of a manufacturing system subject to deterioration

This paper presents an integrated model for the joint determination of both optimal inspection strategy and optimal repair policy for a manufacturing system whose resulting output is subject to system state. An appropriate maintenance strategy is essential to optimize revenue from a manufacturing system which is in continuous operation and subject to deterioration. The optimum policy balances the amount of maintenance required to increase availability against the loss of revenue arising from the down time: insufficient maintenance leads to an increase in the number of defective items, low profit and low maintenance cost; excessive maintenance results in a reduction in the proportion of defective items, high profit and high maintenance cost. In this paper, an intensity control model adapted to partial information provides an optimal inspection intensity and repair degree of the system as an optimal control process to yield maximum revenue. The solution is obtained through formulating an equivalent deterministic Hamilton-Jacobi equation. A numerical example is provided to illustrate the behavior of the optimal control process. The optimal control process determines a solution to both optimum inspection frequency and optimal replacement policy which results in an optimal production run length of the system.     

A condition-based maintenance policy with non-periodic inspections for a two-unit series system

This paper considers a condition-based maintenance policy for a two-unit deteriorating system. Each unit is subject to gradual deterioration and is monitored by sequential non-periodic inspections. It can be maintained by good as new preventive or corrective replacements. Every inspection or replacement entails a set-up cost and a component-specific unit cost but if actions on the two components are combined, the set-up cost is charged only once. A parametric maintenance decision framework is proposed to coordinate inspection/replacement of the two components and minimize the long-run maintenance cost of the system. A stochastic model is developed on the basis of the semi-regenerative properties of the maintained system state and the associated cost model is used to assess and optimize the performance of the maintenance model. Numerical experiments emphasize the interest of a control of the operation groupings.     

Optimal maintenance decisions under imperfect inspection

The process industry is increasingly making use of Risk Based Inspection (RBI) techniques to develop cost and/or safety optimal inspection plans. This paper proposes an adaptive Bayesian decision model to determine these optimal inspection plans under uncertain deterioration. It uses the gamma stochastic process to model the corrosion damage mechanism and Bayes’ theorem to update prior knowledge over the corrosion rate with imperfect wall thickness measurements. This is very important in the process industry as current non-destructive inspection techniques are not capable of measuring the exact material thickness, nor can these inspections cover the total surface area of the component. The decision model finds a periodic inspection and replacement policy, which minimizes the expected average costs per year. The failure condition is assumed to be random and depends on uncertain operation conditions and material properties. The combined deterioration and decision model is illustrated by an example using actual plant data of a pressurized steel vessel.     

A stochastic dynamic programming model for stream water quality management

This paper deals with development of a seasonal fraction-removal policy model for waste load allocation in streams addressing uncertainties due to randomness and fuzziness. A stochastic dynamic programming (SDP) model is developed to arrive at the steady-state seasonal fraction-removal policy. A fuzzy decision model (FDM) developed by us in an earlier study is used to compute the system performance measure required in the SDP model. The state of the system in a season is defined by streamflows at the headwaters during the season and the initial DO deficit at some pre-specified checkpoints. The random variation of streamflows is included in the SDP model through seasonal transitional probabilities. The decision vector consists of seasonal fraction-removal levels for the effluent dischargers. Uncertainty due to imprecision (fuzziness) associated with water quality goals is addressed using the concept of fuzzy decision. Responses of pollution control agencies to the resulting end-of-season DO deficit vector and that of dischargers to the fraction-removal levels are treated as fuzzy, and modelled with appropriate membership functions. Application of the model is illustrated with a case study of the Tungabhadra river in India.     

Impact of inspection errors on the performance measures of a general repeat inspection plan

Multi-characteristic critical components exist in many systems. Such components may be a part of an aircraft, space shuttle or a gas ignition system. An inspection plan for such components has been proposed in quality control that deals with several types of classification errors made by the inspector. In this paper, performance measures for this plan are defined and the statistical and economic impact of the several types of inspection errors on these measures is investigated. The impact of the errors is studied by conducting sensitivity analysis on the errors utilizing computer software which implements an algorithm that determines the optimal parameters of the model of the plan. The behaviour of the performance measures upon variation in the levels of errors is investigated. The results indicate that these errors have a considerable effect on the performance measures of the inspection plan.     

Economic oriented stochastic optimization in process control using Taguchi’s method

The optimal operating region of complex production systems is situated close to process constraints related to quality or safety requirements. Higher profit can be realized only by assuring a relatively low frequency of violation of these constraints. We defined a Taguchi-type loss function to aggregate these constraints, target values, and desired ranges of product quality. We evaluate this loss function by Monte-Carlo simulation to handle the stochastic nature of the process and apply the gradient-free Mesh Adaptive Direct Search algorithm to optimize the resulted robust cost function. This optimization scheme is applied to determine the optimal set-point values of control loops with respect to pre-determined risk levels, uncertainties and costs of violation of process constraints. The concept is illustrated by a well-known benchmark problem related to the control of a linear dynamical system and the model predictive control of a more complex nonlinear polymerization process. The application examples illustrate that the loss function of Taguchi is an ideal tool to represent performance requirements of control loops and the proposed Monte-Carlo simulation based optimization scheme is effective to find the optimal operating regions of controlled processes.     

带漂移因子及停时的最优脉冲随机控制问题

通过把漂移参数引入到受控于Poisson过程的状态结构中,本文建立了一非对称型最优脉冲随机控制模型。在此模型的目标函数中,首次引进了停时因素。利用随机积分及脉冲控制理论,我们不但给出了最优回报函数应满足的充分性条件,而且在一定条件下得出了其显解及相应的最优控制策略。     

Reliability evaluation and opportunistic maintenance policy based on a novel delay time model

The delay time concept is widely adopted in literature to model the two‐stage failure process of most industrial systems which can be divided into normal stage (from new to an initial point of a defect) and defective stage (from defect arrival point to failure). Most existing delay time models assume that the normal and defective stages are independent. A generalized delay time model is proposed in this paper by considering the dependence between the normal and defective stages which is reflected in the fact that they share the same external shock process. According to the definition of shot‐noise process, external shocks will incur random hazard rate increments in the two stages. The failure state is self‐announcing, whereas the defective state can only be detected by block‐based inspection or opportunistic inspection offered by unexpected shutdown due to unavoidable external factors. The system is correctively replaced upon the occurrence of a system failure or preventively replaced at the detection of a defective state. Based on the stochastic failure model and maintenance policy, this paper evaluates system reliability performance and average long‐run cost rate via a Markov‐chain based approach. Finally, a case study on a steel convertor plant is given to demonstrate the applicability of the proposed model.     

Optimal periodic replacement for a deteriorating production system with inspection and minimal repair

Here we discuss an inspection policy model for a deteriorating production system with minimal repair. A minimal repair is resorted to as and when the system is found to be in a failed state during an inspection unless it is apre-set overhaul/replacement time in which case the system is overhauled or replaced. Using a dynamic programming formulation, and assuming that the cost of minimal repair is a non-decreasing function of age, we arrive at the optimal inspection time that maximizes the profit per unit time for a given overhaul/replacement time. The procedure is then extended to determine the optimal periodic overhaul/replacement time and the corresponding optimal number of inspections and their schedule.     

Optimal decision of an economic production quantity model for imperfect manufacturing under hybrid maintenance policy with shortages and partial backlogging

Considering the characteristics of the stochastic shift of the machine state and the uncertainty of the product quality of production, in this paper, we develop an optimisation decision of economic production quantity model for an imperfect manufacturing system under hybrid maintenance policy with shortages and partial backlogging. We assume that the production process is imperfect stemming from the machine reliability and the probability of out-of-control, a hybrid maintenance policy combined of emergency maintenance and preventive maintenance is executed during each production run. Three decision models based on the scenarios of machine breakdown and repair time are developed. The optimal production quantity and maintenance inspection number during each production run are solved with minimising the expected average cost of the system. Numerical examples are used to demonstrate the effectiveness and feasibility of the model. Sensitivity analysis is conducted to analyse the impacts of key parameters on the optimal decision. Some implications related to the effective and economical execution of maintenance policy for practitioners are derived.     

A stochastic model of a reconfigurable manufacturing system Part 1: A framework

Products required by customers are classified into several product families, each of which is a set of similar products. A reconfigurable manufacturing system (RMS) manages to satisfy customers, with each family corresponding to one configuration of the RMS. Then the products belonging to the same family will be produced by the RMS under the corresponding configuration. The manufacturing system possesses the reconfigurable function for different families. In an RMS there are three important issues: the optimal configurations in the design, the optimal selection policy in the utilization, and the performance measure in the improvement. This paper proposes a framework for a stochastic model of an RMS, which involves the above issues. Two optimization problems and the performance measure stemmed from the issues are formulated. An example is given for illustration. Some discussions are presented for future research work.     

Chaos and nonlinear stochastic dynamics

This article presents a discussion of certain aspects of the interrelationship between the chaotic response of some deterministic nonlinear dynamic system and the stochastic response of the corresponding system obtained by introducing a stochastic perturbation in the form of additive Gaussian white noise. The state space vector of the latter system can then often be represented as a Markov diffusion process. The joint probability density function of the state space vector of this Markov process is closely related to the corresponding chaotic attractor of the underlying deterministic system, and it will be demonstrated that it can be effectively used for prediction purposes.     

How heterogeneity influences condition-based maintenance for gamma degradation process

In many applications, units from the same population exhibit heterogeneity that they degrade with different rates due to random factors. This article studies how this heterogeneity in degradation influences condition-based maintenance (CBM) policy. Many CBM polices are developed based on gamma process because it is popularly used to characterise monotone degradation processes. In this study, we also model the unit’s degradation by gamma process. To account for the heterogeneity among units’ degradation, we incorporate a random effect parameter in the gamma process. Then the optimal policy for CBM is obtained through Markov decision process. We show that when heterogeneity exists, the transition probability of degradation state depends on both unit’s age and observed degradation level. And consequently, the optimal maintenance policy is a monotone control limit policy. We conduct extensive numerical experiments to validate and demonstrate our findings in depth.     

非线性时变结构随机地震响应最优多项式控制

以随机地震动作用下具有Bouc-Wen滞回特性的非线性结构系统为受控对象,开展了最优多项式控制算法研究:包括系统矩阵中Maclaurin展开取初始零值衍生的具有时不变增益矩阵的控制律,和系统矩阵中Maclaurin一阶展开衍生的具有时变增益矩阵的控制律。研究表明,受控结构层间位移响应的变异性明显降低,结构的安全性显著提高。同时,基于时不变增益矩阵的控制律的控制效果在一定程度上受制于控制力施加的大小与系统稳定性之间的平衡关系,而考虑了每一个时间步位移和速度对增益矩阵影响、基于时变增益矩阵的控制律则能以较小的控制出力获得较好的控制效果。     

A physical approach to structural stochastic optimal controls

The generalized density evolution equation proposed in recent years profoundly reveals the intrinsic connection between deterministic systems and stochastic systems by introducing physical relationships into stochastic systems. On this basis, a physical stochastic optimal control scheme of structures is developed in this paper, which extends the classical stochastic optimal control methods, and can govern the evolution details of system performance, while the classical stochastic optimal control schemes, such as the LQG control, essentially hold the system statistics since there is still a lack of efficient methods to solve the response process of the stochastic systems with strong nonlinearities in the context of classical random mechanics. It is practically useful to general nonlinear systems driven by non-stationary and non-Gaussian stochastic processes. The celebrated Pontryagin’s maximum principles is employed to conduct the physical solutions of the state vector and the control force vector of stochastic optimal controls of closed-loop systems by synthesizing deterministic optimal control solutions of a collection of representative excitation driven systems using the generalized density evolution equation. Further, the selection strategy of weighting matrices of stochastic optimal controls is discussed to construct optimal control policies based on a control criterion of system second-order statistics assessment. The stochastic optimal control of an active tension control system is investigated, subjected to the random ground motion represented by a physical stochastic earthquake model. The investigation reveals that the structural seismic performance is significantly improved when the optimal control strategy is applied. A comparative study, meanwhile, between the advocated method and the LQG control is carried out, indicating that the LQG control using nominal Gaussian white noise as the external excitation cannot be used to design a reasonable control system for civil engineering structures, while the advocated method can reach the desirable objective performance. The optimal control strategy is then further employed in the investigation of the stochastic optimal control of an eight-storey shear frame. Numerical examples elucidate the validity and applicability of the developed physical stochastic optimal control methodology.     

Sequential degradation-based burn-in test with multiple periodic inspections

Burn-in has been proven effective in identifying and removing defective products before they are delivered to customers. Most existing burn-in models adopt a one-shot scheme, which may not be sufficient enough for identification. Borrowing the idea from sequential inspections for remaining useful life prediction and accelerated lifetime test, this study proposes a sequential degradation-based burn-in model with multiple periodic inspections. At each inspection epoch, the posterior probability that a product belongs to a normal one is updated with the inspected degradation level. Based on the degradation level and the updated posterior probability, a product can be disposed, put into field use, or kept in the test till the next inspection epoch. We cast the problem into a partially observed Markov decision process to minimize the expected total burn-in cost of a product, and derive some interesting structures of the optimal policy. Then, algorithms are provided to find the joint optimal inspection period and number of inspections in steps. A numerical study is also provided to illustrate the effectiveness of our proposed model.     

Optimization for condition-based maintenance with semi-Markov decision process

The semi-Markov decision model is a powerful tool in analyzing sequential decision processes with random decision epochs. In this paper, we have built the semi-Markov decision process (SMDP) for the maintenance policy optimization of condition-based preventive maintenance problems, and have presented the approach for joint optimization of inspection rate and maintenance policy. Through numerical examples, the improvement of this method is compared with the scheme, which optimizes only over the inspection rate. We also find that under a special case when the deterioration rate at each failure stage is the same, the optimal policy obtained by SMDP algorithm is a dynamic threshold-type scheme with threshold value depending on the inspection rate.     

A delay-time model with safety constraint

We consider the basic delay-time model in which a system has three states, the perfect functioning state, a defective state and the failure state. The system is deteriorating and to reduce the number of failures, preventive replacements are carried out when the system is in the defective state. The time in the defective state is referred to as the delay time. Inspections are required to check whether the system is in the defective state. System failures are safety critical and to control the risk, management considers two types of safety constraints: (i) the probability of at least one failure in the interval [0,A] should not exceed a fixed probability ω₁ and (ii) the fraction of time the system is in the defective state should not exceed a fixed limit ω₂. The problem is to determine optimal inspection intervals T, minimizing the expected discounted costs under the safety constraints. Conditions are established for when the safety constraints affect the optimal inspection time and causes increased costs.     

Integrated dynamic single item lot-sizing and quality inspection planning

This paper proposes an integrated model for single item dynamic lot-sizing problem and Quality Inspection Planning (QIP). The objective is to provide a model of production planning that takes into account a targeted level of outgoing quality or an Acceptable quality level (AQL) when the manufacturing system inherently generates a proportion of defectives that increases significantly when the system switches from the in-control state to the out-of-control state. The average outgoing quality of each period of time of the planning horizon is bounded as a function of the inspection capacity. The effects of integrating QIP are analysed and discussed through several experiments representing different quality control system’s parameters, i.e. inspection capacity, inspection cost and AQL. The simulation results show that it is very important to take into account the inspection process into production planning decisions. This study will help the decision-makers to negotiate service levels or react properly to given customer quality requirements based on cost and lead time parameters in addition to their process characteristics in terms of capability and stability.