Extended optimal replacement policy for a two-unit system with failure rate interaction and external shocks

This study presents an extended replacement policy for a two-unit system which is subjected to shocks and exhibits failure rate from interaction. The external shocks that affect the system are of two types. A type I shock causes a minor failure of unit-A and the damage that is caused by such a failure affects unit-B, whereas a type II shock causes a total failure of the system (catastrophic failure). All unit-A failures can be recovered by making minimal repairs. The system also exhibits the interaction between the failure rates of units: a failure of any unit-A causes an internal shock that increases the failure rate of unit-B, whereas a failure of a unit-B causes instantaneous failure of unit-A. The goal of this study is to derive the long-run cost per unit time of replacement by introducing relative costs as a factor in determining optimality; then, the optimal replacement period, T*, and the optimal number of unit-A failures, n*, which minimise that cost can be determined. A numerical example illustrates the method.     

Optimal age-replacement time with minimal repair based on cumulative repair cost limit and random lead time

In this article, we consider an age-replacement model with minimal repair based on a cumulative repair cost limit and random lead time for replacement delivery. A cumulative repair cost limit policy uses information about a system's entire repair cost history to decide whether the system is repaired or replaced; a random lead time models delay in delivery of a replacement once it is ordered. A general cost model is developed for the average cost per unit time based on the stochastic behaviour of the assumed system, reflecting the costs of both storing a spare and of system downtime. The minimum-cost policy time is derived, its existence and uniqueness is shown, and structural properties are presented. Various special cases are included. Because the framework and analysis are general, the proposed model extends several existing results. Finally, a numerical example is provided for illustration.     

Optimal number of minimal repairs before replacement based on a cumulative repair-cost limit policy

In this paper, we consider a replacement model with minimal repair based on a cumulative repair-cost limit policy, where the information of all repair costs is used to decide whether the system is repaired or replaced. As a failure occurs, the system experiences one of the two types of failures: a type-I failure (repairable) with probability q, rectified by a minimal repair; or a type-II failure (non-repairable) with probability p (=1 − q) that calls for a replacement. Under such a policy, the system is replaced anticipatively at the nth type-I failure, or at the kth type-I failure (k < n) at which the accumulated repair cost exceeds the pre-determined threshold, or any type-II failure, whichever occurs first. The object of this paper is to find the optimal number of minimal repairs before replacement that minimizes the long-run expected cost per unit time of this polish. Our model is a generalization of several classical models in maintenance literature, and a numerical example is presented for illustration.     

Optimal replacement period of a two-unit system with failure rate interaction and external shocks

In this article, a periodical replacement model for a two-unit system which is both subjected to failure rate interaction and external shocks will be presented. Without external shocks, each unit 1, whenever it fails, will act as an interior shock to affect the failure rate of unit 2 and increase the failure rate of unit 2 to a certain degree, while each unit 2 failure causes unit 1 into instantaneous failure. Besides failure rate interaction between units, the system is also subjected to external shocks which can be divided into two types. Type A shock causes unit 1 into failure and then converts the damage of such a failure to unit 2, while type B shock makes the system total breakdown. All unit 1 failures are corrected by minimal repairs. The aim of this article is to derive the expected cost rate per unit time by introducing relative costs as a criterion of optimality, and then the optimal replacement period which minimizes that cost will be determined. A numerical example is given to illustrate the method.     

Optimal number of repairs before replacement for a two-unit system subject to non-homogeneous pure birth process

We consider a discrete replacement model for a two-unit system subject to failure rate interaction and shocks. Two types of shocks occur according to a non-homogeneous pure birth process and can affect the two-unit system. Type I shock causes unit A to fail and can be rectified by a general repair, while type II shock results in a non-repairable failure and must be fixed by a replacement. Two-unit systems also exhibit failure rate interactions between the units: each failure of unit A causes some damage to unit B, while each failure of unit B causes unit A into an instantaneous failure. The occurrence of a particular type of shock is dependent on the number of shocks occurred since the last replacement. The objective of this paper is to determine the optimal number of minor failures before replacement that minimizes the expected cost rate. A numerical example is presented to illustrate application of the model.     

Optimal replacement policy for a two-dissimilar-component cold standby system with different repair actions

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. When failures occur, the repair of both component 1 and component 2 are not ‘as good as new’. The consecutive operating times of component 1 after repair constitute a decreasing geometric process, while the repair times of component 1 are independent and identically distributed. For component 2, its failure is rectified by minimal repair, and the repair time is negligible. Component 1 has priority in use when both components are good. The replacement policy N is based on the failure number of component 1. Under policy N, we derive the explicit expression of the long-run average cost rate C(N) as well as the average number of repairs of component 2 before the system replaced. The optimal replacement policy N*, which minimises the long-run average cost rate C(N), is obtained theoretically. If the failure rate r(t) of component 2 is increasing, the existence and uniqueness of the optimal policy N* is also proved. Finally, a numerical example is given to validate the developed theoretical model. Some sensitivity analyses are provided to show the influence of some parameters, such as the costs for replacement and repair, and the parameters of the lifetime and repair time distributions of both components, to the optimal replacement policy N* and corresponding average cost rate C(N*).     

A generalised maintenance policy with age-dependent minimal repair cost for a system subject to shocks under periodic overhaul

A system is subject to shocks that arrive according to a non-homogeneous Poisson process. As shocks occur, the system has two types of failures: type 1 failure (minor failure) is removed by a minimal repair, whereas type 2 failure (catastrophic failure) is removed by overhaul or replacement. The cost of minimal repair depends on age. A system is overhauled when the occurrence of a type 2 failure or at age T, whichever occurs first. At the N-th overhaul, the system is replaced rather than overhauled. A maintenance policy for determining optimal number of overhauls and optimal interval between overhauls which incorporate minimal repairs, overhauls and replacement is proposed. Under such a policy, an approach which using the concept of virtual age is adopted. It is shown that there exists a unique optimal policy which minimises the expected cost rate under certain conditions. Various cases are considered.     

Optimal number of minimal repairs before replacement of a deteriorating system with inspections

In this study, we propose a generalised replacement model for a deteriorating system with failures that could only be detected through inspection work. The system is assumed to have two types of failures and is replaced at the Nth type I failure (minor failure) or first type II failure (catastrophic failure), depending on whichever occurs first. The probability of type I and II failures depends on the number of failures since the last replacement. Such systems can be repaired upon type I failure, but are stochastically deteriorating, that is, the lengths of the operating intervals are stochastically decreasing, whereas the durations of the repairs are stochastically increasing. Then, the expected net cost rate is obtained. Some special cases are considered. Finally, a numerical example is provided.     

Optimal single-replacement for repairable products with different failure rate under a finite planning horizon

This paper provides a replacement policy for repairable products with free-repair warranty (FRW) under a finite planning horizon from the consumer's viewpoint. Assume that the product is replaced once within a finite planning horizon, and the failure rate of the second product is lower than the failure rate of the first product. Within FRW, the failed product is corrected by minimal repair without any cost to the consumers. After FRW, the failed product is repaired with a fixed repair cost to the consumers. However, each failure incurs a fixed downtime cost to the consumers over a finite planning horizon. In this paper, we derive the three models of the expected total disbursement cost within a finite planning horizon and some properties of the optimal replacement policy under some reasonable conditions are obtained. Finally, numerical examples are given to illustrate the features of the optimal replacement policy under various maintenance costs.     

A geometric process repair model for a two-component system with shock damage interaction

In this article, the repair-replacement problem for a two-component system with shock damage interaction and one repairman is studied. Assume that component 1 will be replaced as soon as it fails, and each failure of component 1 will induce a random shock to component 2. The shock damages may be accumulative, and whenever the total shock damage equals or exceeds a given threshold Δ, component 2 fails and the system breaks down. Component 2 is repairable, and it follows a geometric process repair. Under these assumptions, we consider a replacement policy N based on the number of failures of component 2. Our problem is to determine an optimal replacement policy N* such that the average cost rate (i.e. the long-run average cost per unit time) is minimised. The explicit expression of the average cost rate is derived by the renewal reward theorem, and the optimal replacement policy can be determined analytically or numerically. The existence and uniqueness of the optimal replacement policy N* is also proved under some mild conditions. Finally, two appropriate numerical examples are provided to show the effectiveness and applicability of the theoretic results in this article.     

Optimal strategies for an inventory system with cost functions of general form

For an inventory system with a continuous compact phase state and control set, the existence of the optimal (S, s)-policy is proved and an algorithm for determination of this policy based on stochastic approximation is presented. The cost functions are monotone, continuous, and of a rather general form. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 106–123, July–August, 1999.     

不确定时滞系统的指定衰减度保代价可靠控制

针对一类含有时变时滞的不确定参数线性系统,研究了在执行器发生故障情况下系统具有指定衰减度的保代价可靠控制器设计问题。系统中的参数不确定性满足广义匹配条件,时变时滞及其变化率有界,执行器失效采用增益故障模型。系统的性能函数是含有指数项和故障输入项的积分二次型函数。经过适当的状态变换,将原系统的指定衰减度保代价可靠控制问题转化为另一个等价系统的保代价可靠控制问题。根据Lyapunov稳定性理论,得到了系统存在指定衰减度保代价可靠控制器应满足的一个矩阵不等式,进一步将这个矩阵不等式转化为线性矩阵不等式（LMI）,并给出了系统的保代价表达式。利用论文方法设计的指定衰减度保代价可靠控制器能够使得时滞系统对于任意允许的不确定量以及执行器故障都保持鲁棒可靠指数稳定,并且使系统具有保代价的性能指标。     

Maintenance interval decision models for a system with failure interaction

Based on the three types of failure interactions, two periodical maintenance cost models were presented for a two-state series system and a three-state series system respectively, which all subjected to failure interactions between units. Consider any unit fails would cause damages to other units. The failure interaction influences included instantaneous damages and continuous damages between units. The result indicated that failure interactions will shorten system preventive maintenance interval, if the preventive maintenance strategy is based on the cost.     

Robust stabilization of binary distillation columns with input constraints

This paper examines the problem of robust stabilization of binary distillation columns with input constraints. We develop a controller that takes into account the physical constraints on the inputs and which ensures the global asymptotic stability of the closed loop system, with robustness to modelling errors, and the capability of performing set point tracking and disturbance rejection. We stress that this controller does not require all the internal states to be measurable.     

Optimal guaranteed cost control for an uncertain discrete T-S fuzzy system with time-delay

This paper considers the guaranteed cost control problem for a class of uncertain discrete T-S fuzzy systems with time delay and a given quadratic cost function. Sufficient conditions for the existence of such controllers are derived based on the linear matrix inequalities （LMI） approach by constructing a specific nonquadratic Lyapunov-Krasovskii functional and a nonlinear PDC-like control law. A convex optimization problem is also formulated to select the optimal guaranteed cost controller that minimizes the upper bound of the closed-loop cost function. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed approaches.     

Robust reliable guaranteed cost control of positive interval systems with multiple time delays and actuator failure

This paper addresses the robust reliable guaranteed cost control problem of positive interval systems with multiple time delays and actuator failure for a given quadratic cost function. Through constructing a Lyapunov–Krasovskii functional, a sufficient condition for the existence of robust reliable guaranteed cost controllers is established such that the closed-loop system is positive and asymptotically stable, and the cost function is guaranteed to be no more than a certain upper bound. Based on the linear matrix inequality method, a criterion for the design of robust reliable guaranteed cost controllers is derived which can tolerate all admissible uncertainties as well as actuator failure. Moreover, a convex optimisation problem with linear matrix inequality constraints is formulated to design the optimal robust reliable guaranteed cost controller which minimises the upper bound of the closed-loop system cost. A numerical example is given to show the effectiveness of the proposed methods.     

一类时延网络控制系统的骱容错保成本控制

针对具有时延和参数不确定性的网络控制系统的研究。考虑系统的时延小于一个采样周期,传感器是时钟驱动,控制器和执行器是事件驱动。当执行器发生故障时,研究网络控制系统的Hoo容错保成本控制。依据所描述情况建立系统模型,基于Lyapunov稳定性理论、容错控制理论和线性矩阵不等式（LMIs）处理方法,推导出网络控制系统是渐近稳定的,并且得出系统的Hoo容错保成本控制的充分条件和系统的保成本上界。实例仿真证明结论的有效性。     

An economic manufacturing quantity model for a two-stage assembly system with imperfect processes and variable production rate

This article considers a two-stage assembly system with imperfect processes. The former is an automatic stage in which the required components are manufactured. The latter is a manual stage which deals with taking the components to assemble the end product. In addition, the component processes are independent of each other, and the assembly rate is variable. Shortage is allowed, and the unsatisfied demand is completely backlogged. Then, we formulate the proposed problem as a cost minimization model where the assembly rate and the production run time of each component process are decision variables. An algorithm for the computations of the optimal solutions under the constraint of assembly rate is also provided. Finally, a numerical example and sensitivity analysis are carried out to illustrate the model.     

Optimal service of aging demands in a single-server system with variable service rate and payment for buffer place rent

Necessary and sufficient conditions are obtained for optimal service of aging demands, which minimize the value of an additive performance functional with specified cost of rent of places. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 145–150, May–June, 2000.