Long-run Expected Cost Rate Research Articles

Optimizing multilevel maintenance in multi-component systems under s-dependent competing risks

This paper presents a novel reliability-centered multilevel maintenance optimization model tailored for multicomponent systems subjected to stochastic dependent (s-dependent) competing risks (CRs). Four types of maintenance actions, including minimal repair (MR), medium preventive maintenance (MPM), overhaul preventive maintenance (OPM), and preventive replacement (PR), are incorporated into the development of the multilevel maintenance model. The Copula function is utilized to quantify the s-dependence among failure risks, and subsequently, all unknown parameters are estimated via the inference function for margins (IFM) method. Three types of decision variables are designed to minimize the long-run expected cost rate of the system. Finally, sequential decision process method is utilized to solve the model, and a numerical example of a metro door system illustrates the effectiveness of our proposed model.

On the point process with finite memory and its application to optimal age replacement

There has been extensive study of various repair models in the literature, mostly under the assumption that these repairs are minimal or imperfect/better than minimal. Although this is often a realistic assumption, it may not be sufficient to model instances where the repair is worse than minimal. The generalized Polya process (GPP) that has been used to describe this type of repair takes into account all previous events/repairs, which is not often the case in practice. Therefore, in this paper, we define a new process with finite memory that starts as the GPP but, after a certain number of events or elapsed time, becomes the non-homogeneous Poisson process of repairs (minimal repairs). The corresponding age replacement policy is defined and the optimal solutions that minimize the long-run expected cost rate are analyzed. The detailed numerical examples illustrate our findings.

Erlang loss systems with shortest idle server first service discipline: Maintenance considerations

We consider a variation of an Erlang loss system in which jobs are routed to servers according to the Shortest Idle Server First service discipline. Specifically, we consider a system in which idle servers are arranged in a stack; servers are returned to the top of the stack upon service completion; and arriving jobs are assigned to the server currently at the top of the stack. When busy, servers accumulate age and incur an age-dependent operating cost. For such systems, we (i) formulate a continuous-time Markov chain model to characterize the system’s transient behavior, and (ii) develop maintenance policies consisting of two possible actions: server group replacement and stack inversion. The stack inversion may be performed at any time prior to group replacement to achieve a more evenly distributed utilization among servers. We develop an optimization model to determine the optimal inversion and replacement times so as to minimize the long-run expected cost rate. Because the model is nonlinear and non-convex, we develop a set of algorithms to solve for the optimal replacement and inversion time. Lastly, we establish a lower bound for the inversion cost threshold below which it is optimal to invert the stack of servers before their replacement.

Joint optimization of inspection-based and age-based preventive maintenance and spare ordering policies for single-unit systems

This paper presents a joint optimization policy of preventive maintenance (PM) and spare ordering for single-unit systems, which deteriorate subject to the delay-time concept with three deterioration stages. PM activities that combine a nonperiodic inspection scheme with age-replacement are implemented. When the system is detected to be in the minor defective stage by an inspection for the first time, place an order and shorten the inspection interval. If the system has deteriorated to a severe defective stage, it is either repaired imperfectly or replaced by a new spare. However, an immediate replacement is required once the system fails, the maximal number of imperfect maintenance (IPM) is satisfied or its age reaches to a pre-specified threshold. In consideration of the spare's availability as needed, there are three types of decisions, i.e., an immediate or a delayed replacement by a regular ordered spare, an immediate replacement by an expedited ordered spare with a relative higher cost. Then, some mutually independent and exclusive renewal events at the end of a renewal cycle are discussed, and the optimization model of such a joint policy is further developed by minimizing the long-run expected cost rate to find the optimal inspection and age-replacement intervals, and the maximum number of IPM. A Monte-Carlo based integration method is also designed to solve the proposed model. Finally, a numerical example is given to illustrate the proposed joint optimization policy and the performance of the Monte-Carlo based integration method.

Optimal preventive maintenance for repairable products with three types of failures sold under a renewable hybrid FRW/PRW policy

This paper proposes the optimal preventive maintenance policy for products under a warranty policy, which combines fully renewable free replacement warranty (FRW) and pro-rata warranty (PRW). Replacement age is divided into three separate periods: within FRW, during PRW and post-warranty. Moreover, it is assumed the product is subject to three types of failures arriving according to a non-homogeneous Poisson process. The first type can be fixed by minimal repair, and the third type is a catastrophic failure that results in product replacement. If the failure is of the second type and the product is at age t, it is either minimally repaired with probability p(t) or replaced by a new one with probability 1−p(t). The aim of this study is to develop the long-run expected cost rate for both warranted and non-warranted repairable products as well as to derive the optimal replacement time such that the incurred costs are minimized. The proposed approach provides valuable information for the product customers, managers, or engineers to make better decisions about the preventive replacement time of such products. The proposed model covers some preventive maintenance policies in the existing literature. Numerical computations are also performed to investigate the effect of the model’s parameters on the optimal solutions. Eventually, a real-life application is discussed to demonstrate the applicability of the proposed policy.

An alert-assisted inspection policy for a production process with imperfect condition signals

We study the inspection scheduling decisions for a production process that goes through a hidden defective state before its failure. The production process is equipped with a predictive model, generating alert and no-alert signals. An alert signal indicates that production process is in the defective state, while a no-alert signal indicates it is in the healthy state. The signals are imperfect, meaning that an alert signal can be generated for a healthy process and a no-alert signal can be generated for a defective process. Only a costly inspection can detect the true condition. We introduce a new inspection policy, which generalizes the age-based inspection policy that performs planned inspections at predetermined intervals, by considering that an inspection can also be triggered by a certain number of alerts from the predictive model. To optimize the proposed inspection policy, a stochastic dynamic programming model is formulated with the objective of minimizing the long-run expected cost rate. The performance improvement achieved by the optimal policy is quantified by comparing it to practically relevant benchmark policies. Numerical experiments with a set of realistic problem instances show that adding alert-triggered inspections to traditional age-based inspection scheduling brings up to 44% reduction in the expected cost rate when the predictive model is sufficiently accurate. Characterizing the performance of the optimal policy at a given level of imperfectness is especially useful in practice as it allows making an assessment on how much can be invested to justify a certain level of improvement in the predictive model.

A generalized age-dependent minimal repair with random working times

Two extended preventive replacement models for systems that execute projects at random times are discussed in this paper. The system is subject to shocks at which the system experiences one of two kinds of failures whose probabilities depend on age. A type I failure will cause a minor failure of the system and is fixed by a minimal repair. A type II failure will lead to a catastrophic failure of the system, and a corrective replacement is required at such failure. First, we investigate a preventive replacement policy where the system is replaced at the m-th type I failure, or at the time instant when the n-th working project is completed, or at age τ, or at the first type II failure, whichever occurs first. In addition, we also investigate another preventive replacement model where the system is replaced preventively at the time instant when the n-th working project is completed, or at the m-th type I failure, or at age τ, whichever occurs last, and is replaced correctively at the first type II failure. We formulate the long-run expected cost rate for each replacement policy, and determine analytically the optimum preventive replacement policy. We also show that several previous replacement models in the literature are special cases of our models. Finally, a procedure for finding the optimum preventive replacement schedule is presented and some numerical examples are given illustrating the present policies.

A δ-shock model for reengineering of a repairable supply chain using quasi renewal process

In this study, a quasi-renewal process model with δ-shock is analyzed for a degenerative repairable supply chain with a single supplier source. The supplier of the supply chain system encounters shocks due to crisis events and, as a consequence, tends to fail if the interarrival time between two successive shocks is less than a random threshold. The non negligible repair time of the supplier and the random threshold time of shocks are modeled on a quasi renewal process while the shock arrivals are governed by a renewal process. The long-run average cost per unit time is derived by employing the reengineering strategy N, which is determined by the number of failures of the supplier. The optimal reengineering policy is obtained analytically by minimizing the long-run expected cost rate. Numerical illustrations and sensitivity analysis evidence the consideration of maintenance policy for a repairable supply chain to sustain its smooth running.

Optimal periodic preventive maintenance policy for a system subject to failures/repairs which follow the non-homogeneous pure birth process

ABSTRACT This paper considers a periodic preventive maintenance policy for a deteriorating and repairable system. The system is subject to random failures and is repaired at each failure. To improve the reliability, at periodic times iT (i = 1, 2, … N–1), the system is preventively maintained (PM). At the time NT the system is replaced. The failure/repair process between two PMs follows the Non-Homogeneous Pure Birth Process (NHPBP) which is a generalized version of the Non-homogeneous Poisson process (NHPP). The reliability improvement made by a PM action is stochastically modeled based on the properties of NHPBP. The cost for each PM depends on the reliability condition of the system at that PM time. The PM policy is characterized by the two parameters (N, T). We develop a two-stage procedure to find the optimal (N*, T*) which minimizes the long-run expected cost rate. Numerical examples are presented to illustrate the properties of the optimal policies. Abbreviations: NHPBP: Non-homogeneous pure birth process; NHPP: Non-homogeneous Poisson process; PM: Preventive maintenance; CM: Corrective maintenance; RM: Replacement maintenance

A Stochastic Queueing - Inventory System with Renewal Demands and Positive Lead Time

This article analyses a stochastic inventory system with a service facility. This is an extended work of Yadavalli et al. (2008) by including the positive lead time. The customer arrives according to a renewal process and demanded item is delivered to the customer after performing an exponentially distributed service time. An (s, S) type ordering policy is adopted with exponentially distributed lead times. The stationary probability distribution for number of customers in the system and inventory level at arrival epoch and at arbitrary time point are derived. Some system performance measures in the steady state are computed and using these system performance measures the long-run expected cost rate is calculated. Since the long run expected cost rate is highly complex, the mixed integer distributed ant colony optimisation is used to obtain the optimal values. A sensitivity analysis to illustrate the effects of parameters and cost on the optimal values is also carried out in this work. [Received: 13 December 2018; Accepted: 10 October 2019]

A stochastic queueing - inventory system with renewal demands and positive lead time

This article analyses a stochastic inventory system with a service facility. This is an extended work of Yadavalli et al. (2008) by including the positive lead time. The customer arrives according to a renewal process and demanded item is delivered to the customer after performing an exponentially distributed service time. An (s, S) type ordering policy is adopted with exponentially distributed lead times. The stationary probability distribution for number of customers in the system and inventory level at arrival epoch and at arbitrary time point are derived. Some system performance measures in the steady state are computed and using these system performance measures the long-run expected cost rate is calculated. Since the long run expected cost rate is highly complex, the mixed integer distributed ant colony optimisation is used to obtain the optimal values. A sensitivity analysis to illustrate the effects of parameters and cost on the optimal values is also carried out in this work. [Received: 13 December 2018; Accepted: 10 October 2019]

Optimal periodic replacement policy for a GPP repairable product under the free-repair warranty

This article provides an extension study on the optimal periodic replacement policy for repairable products under the free-repair warranty. A new and more general type of repair, called the ‘GPP repair’, is applied in this study, which is defined based on the Generalized Polya Process and which is more close to practical situations than the ‘minimal repair’. Under the GPP repair assumption, the future reliability performance of a product becomes worse as the number of system failures that occurred in the past increases. Cost models from the user’s perspective are developed for both a GPP repairable product with free-repair warranty and without warranty, and the corresponding optimal replacement periods are derived such that the long-run expected cost rate is minimized. The difference and relationship between GPP repairable and minimal repairable products on the optimal replacement policies are analytically presented. Numerical examples are demonstrated for illustration and verification. Finally, a practical case is provided to explain the potential application areas of this study, and managerial implications and insights are addressed.

Optimal Control of Service Rate in a Service Facility System Maintenance and Retrial Demands

This paper, deals with a problem of optimally controlling service rates for a retrial service facility system. We consider a finite capacity service facility system with Poisson arrivals and exponentially distributed service times. Here, the customer who sees the server busy, joins the orbit and reattempts the facility with exponential distributed time interval. The system is formulated as Markov Decision Process and we find the optimum service rates to be employed at each instant of time. Linear Programming method is implemented in the context of minimizing the long-run expected cost rate.

Quantitative assessments of performance and robustness of maintenance policies for stochastically deteriorating production systems

Over the last few decades, many efforts have been invested in improving the economic performances of maintenance policies for stochastically deteriorating production systems. However, with the development of complex production systems, maintenance managers are interested not only in cost saving, but also in how to trustworthily plan and allocate the required maintenance budget. In this context, the robustness of maintenance policies which is related to the maintenance cost variability from a renewal cycle to another plays a pivotal role. This research deals with a quantitative approach to jointly assess the economic performance and robustness of some representatives of two most well-known classes of maintenance policies: time-based and condition-based maintenance. To this end, we first propose a new cost criterion which combines the long-run expected cost rate and standard deviation of maintenance cost per renewal cycle. Then, we develop and compare the associated mathematical cost models of the considered maintenance policies on the basis of the Gamma degradation process and the theory of stochastic renewal processes. The comparison results under different situations of maintenance costs and system characteristics show that the optimal configuration of maintenance policies gives the best compromise between the performance and robustness, and is mostly affected by the system downtime. Under this aspect, the condition-based maintenance remains more profitable than the time-based maintenance. Still, maintenance managers could implement condition-based maintenance policies that efficiently control the downtime to maximise the maintenance effectiveness of production systems from both performance and robustness viewpoints.

Condition-Based Maintenance With Scheduling Threshold and Maintenance Threshold

In order to arrange maintenance resources according to system condition, the lead time needs to be considered within the context of condition-based maintenance (CBM). Therefore, a scheduling threshold is introduced to replace the time to schedule, which is used as a decision variable in combination with a maintenance threshold and a failure threshold. The long-run expected cost rate for maintenance considers the maintenance cost, the cost of the waiting time of suppliers and customers. In this way, suppliers can schedule maintenance services in advance when the system condition reaches the scheduling threshold, and perform maintenance when the system condition exceeds the maintenance threshold. Furthermore, the optimal maintenance plan is updated dynamically in the framework of Prognostics and Health Management (PHM). Finally, a numerical example is provided to demonstrate the effectiveness and the dynamic nature of the proposed method.

Optimal Control of Service Parameter for a Perishable Inventory System with Service Facility, Postponed Demands and Finite Waiting Hall

This paper deals with the problem of optimally control service rates for a continuous review (s, S) inventory system with a service facility consisting of finite buffer(waiting hall) and a single server. The customers arrive according to a Poisson process. The customer’s demand is satisfied after an exponential service time. An arriving customer, who finds the buffer is full, enters into the pool of finite size or leaves the system according to a Bernoulli trial. The replenishment time of the order is distributed as exponential. The life time of each item in the inventory is assumed to be exponential distribution. Here we determine the service rates to be employed at each instant of time so that the long-run total expected cost rate is minimized. The problem is modelled as a semi-Markov decision problem. The stationary optimal policy is computed using linear programming algorithm and the results are illustrated numerically.

Condition-based switching and replacement policies for a two-unit warm standby redundant system subject to non-instantaneous switchover

This study investigates condition-based switching and replacement policies for a two-unit warm standby redundant system subject to non-instantaneous switchover. The system consists of two identical units: one is operating unit, while the other is standby unit. Two units are under continuous monitoring and degradation described by Gamma processes. Both switching and replacement policies are based on the degradation level of the operating unit. The switching and replacement thresholds are decision variables decided by minimizing the long-run expected cost rate. We first setup the mathematical optimization model and then analyze the global optimal solution of replacement threshold, as well as the existence and uniqueness condition for the local optimal solution of switching threshold (STH). Finally, we find that the optimal replacement threshold is “the bigger the better”, but the optimal STH depends on some optimal conditions which can be easily computed. Numerical examples are provided to verify the policies, and the effects of non-instantaneous switchover and cost ratio on optimal STHs are numerically discussed.

Optimal order-replacement policy for a phase-type geometric process model with extreme shocks

A system is subject to random shocks that arrive according to a phase-type (PH) renewal process. As soon as an individual shock exceeds some given level the system will break down. The failed system can be repaired immediately. With the increasing number of repairs, the maximum shock level that the system can withstand will be decreasing, while the consecutive repair times after failure will become longer and longer. Undergoing a specified number of repairs, the existing system will be replaced by a new and identical one. The spare system for the replacement is available only by sending a purchase order to a supplier, and the duration of spare system procurement lead time also follows a PH distribution. Based on the number of system failures, a new order-replacement policy (also called (K,N) policy) is proposed in this paper. Using the closure property of the PH distribution, the long-run average cost rate for the system is given by the renewal reward theorem. Finally, through numerical calculation, it is determined an optimal order-replacement policy such that the long-run expected cost rate is minimum.

The comparativestudyon the influenceof warranty period to thepractical age-replacementunder two situations

The paper focus on analyzing the impact of warranty periods on the optimal age-replacement from the consumers'perspectives.First we construct themathematical formulations for age -replacement model.After optimizing we find there exists a unique optimal replacement age based the long-run expected cost rate is minimized.Further,a concise numerical example is demonstrated, and the sensitivity analysis of the warranty period to practicalreplacement age is carried out as well.Afterwards, the influnce of warranty periods on the optimal age-replacement under two situations thatpreventive replacement is within and beyond the warranty periods are comparedanalytically. Keywords: Warranty period;Practicalreplacement age;Preventive replacement age;Long-run expected cost rate.

The Effects of a Renewing Free-Replacement Warranty on the Discrete Age Replacement Policy

This paper presents the effects of a renewing free-replacement warranty (RFRW) on the age-replacement policy in a discrete time process. Consider a product that should be operating over an indefinitely long operation cycle n (n=1,2,...); under the discrete age-replacement policy, a product is replaced at cycle N (N=1,2,...) after its installation, or at failure, whichever occurs first. The cost models from the customer's perspective are developed for both warranted, and non-warranted products. The corresponding optimal replacement age N* (i.e., the optimal number of operation cycles for a preventive replacement) is derived such that the long-run expected cost rate is minimized. Under the assumption of the discrete time increasing failure rate, the existence and uniqueness of the optimal N* are shown, and the impacts of a RFRW on the optimal replacement policies are investigated analytically. The optimal N* for a warranted product is closer to the end of the warranty period than for a non-warranted product. Finally, a numerical example is demonstrated for the optimal policy illustration and verification. The observations from the technical analysis and numerical results provide valuable information for a buyer (user) to adjust the optimal age-replacement policy if a product is operating in discrete time under a RFRW.