Expected Cost Rate Research Articles

Optimal maintenance policies for a k-out-of-n system with replacement bias costs

In this paper, the issue of determining an optimal age replacement is explored by incorporating minimal repair, preventive replacement, and corrective replacement into a k-out-of-n system subject to shocks. The k-out-of-n system in question consists of n identical components, and functions if at least k components function. Each shock causes the failure of a random number of components; if at least n-k + 1 components fail, the system fails and implements corrective replacement, otherwise the system is minimally repaired. The system is scheduled to implement preventive replacement before failure at age T or at the complement of a random working cycle, whichever occurs first or last. In order to balance the bias time between preventive replacement and corrective replacement, a replacement bias cost is also taken into account for planning replacement policies. The present paper develops a two-phase maintenance methodology and determines the optimal number of components and preventive replacement age that minimize the expected cost rate functions. For each model, the expected cost rate function is formulated analytically and optimized theoretically, and a numerical example is given to illustrate the applicability of the proposed methodology.

A Condition-based Maintenance Policy in Chance Space

A condition-based maintenance policy is considered for a deteriorating system including both of preventive and corrective maintenance actions. The gamma process is used to model stochastic degradation in the probability space. Although, the cost of preventive maintenance is considered as an uncertain variable due to incomplete information, and its distribution is estimated based on the opinions of some experts using the Delphi method. The optimal policy is determined by minimizing the expected cost rate function. Since in this function, there are both random variables discussing in a probability space, and an uncertain variable, which is considered in an uncertain space, we have to study the optimal policy in a chance space which is a combination of probability and uncertain spaces. The proposed methodology is explained in an illustrative example. Finally, the results are applied to a real data set.

Optimal replacement policy based on number of failures for a system with multiple attempt minimal repairs

This study proposes a failure number-dependent replacement policy for a minimal repairable system with multi-attempt repairs. The motivation for this study is based on the novel notion of "multiple attempts minimal repair," first introduced by Cha and Finkelstein [1,2] recently. Their repair/replacement model, along with useful properties derived and discussed for optimal preventive maintenance (PM) problems, served as the motivation for this study. This paper analyzes the optimal replacement policy based on the number of failures for a multi-attempt minimal repairable system.Our policy's characteristic is that the corrective replacement (CR) of a system is only possible at a system failure instant, rather than implementing a scheduled preventive replacement (PR). Specifically, under our policy, the system is replaced not only after a fixed number of unsuccessful repair attempts (k), but also upon reaching a predetermined failure number (n). We show that the optimal policy (or optimal n*) that minimizes the average cost per unit of time in the long run can be determined by the unique solution of an equation under certain conditions. The numerical results demonstrate that our policy is more efficient in terms of expected cost rate than other policies with the scheduled PR.

A chance-constrained net revenue model for online dynamic predictive maintenance decision-making

Most models for predictive maintenance (PdM) decision-making focus on the expected value of a system performance metric (e.g., the expected cost rate). However, focusing solely on such expected values may overlook the significant risks of resulting PdM decisions. Indeed, it would be practically valuable to make the optimal PdM decision by considering the least probability of achieving a target system performance. Furthermore, emphasizing only maintenance costs without accounting for operational revenues can render the maintenance strategies unappealing, particularly when the revenues across system lifecycles are significant. This study introduces a method that employs a stochastic net revenue model and chance-constrained programming to address these issues in PdM decision-making, wherein the preventive maintenance cost is proportional to the system's current degradation state. By exemplifying system degradation with the Gamma process, we derive the probability density function of stochastic net revenue and present a Bayesian method to simultaneously update the drift and diffusion parameters of the Gamma process model. The effectiveness and practical applicability of our proposed chance-constrained net revenue model and parameter updating method are demonstrated through a numerical example and a case study.

Reliability modeling and warranty optimization for products with self-healing under a dynamic shock environment

As modern products evolve in intelligence, self-healing mechanisms have been gaining prominence in the field of reliability. In actual operation, the performance of a product is inevitably influenced by external environments and user interactions. However, existing warranty studies often overlook these factors, leading to oversimplifications in product reliability modeling. This paper offers a comprehensive and realistic view of the product failure process by integrating the dynamic shock environment, inherent self-healing property, and post-sale consumer utilization. The constructed reliability evaluation model accounts for fatal and non-fatal shocks with a dynamic time-varying failure threshold. We introduce the concept of self-healing trigger/delay time, considering a generalized self-healing process. The approximate product reliability is derived, and its accuracy is validated using Monte Carlo simulations. Then, derived from real-world practices, we propose a novel two-stage warranty policy which combines lemon law principles with preventive maintenance. By minimizing the expected cost rate throughout the warranty service period, a preventive maintenance schedule optimization model is developed from the manufacturer's perspective. Finally, a numerical case is presented to reveal the influences of different factors on product reliability and demonstrate the existence of the optimal solution.

Optimization of a preventive maintenance strategy for a product with limited repair times

This paper proposes a base warranty and replacement warranty strategy for repairable products. It should be emphasized that the product failure rate is low in the early stage of the base warranty. Therefore, in order to reduce the manufacturer’s maintenance cost, the manufacturer does not carry out preventive maintenance in the early stage of the base warranty, and carries out regular preventive maintenance in the later stage of the base warranty. At the same time, if the number of product failures in the base warranty exceeds the specified limit, the manufacturer must replace products for customers free of charge according to the warranty contract. After the product is replaced, if the number of product failures does not exceed the limit, the manufacturer will continue to provide customers with the remaining base warranty period of the original product, and the warranty period will continue to be extended for a period of time. By minimizing the expected cost rate, it is shown that the optimal maintenance policy for warranty products is unique. Finally, we give a numerical example and study the effects of periodic preventive maintenance and other relevant parameters on the optimal solution.

Determining the optimal design for complex systems using a reliability signature

In reliability analysis, the structure-function is a commonly used mathematical representation of the studied system. A signature vector is used for systems with independently and identically distributed (i.i.d.) component lifetimes. Each element in the signature represents the probability that the failure of the corresponding component will fail the entire system. This paper aims to provide a comprehensive understanding of assessing the performance of two complex systems for optimal communication design. The study compares two systems with the same components using signatures, expected cost rate, survival signature, and sensitivity to determine which system is preferred. It also provides several sufficient conditions for comparing the lifetimes of two systems based on the usual stochastic order. The results are applied to two communication systems that have the same components. The mathematical properties presented in the study have been proven to enable efficient weighting of the optimal design.

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.

First hitting time distribution and cost assessment in a two-unit system with dependent degradation processes subject to imperfect maintenance

This paper proposes a degradation model for a two-unit series system, where the components exhibit dependence and are subject to imperfect maintenance. The interdependence between both components is captured using the trivariate reduction method. The system failure occurs when the degradation level of either component exceeds a predetermined threshold. The main motivation of this work is to develop a preventive maintenance strategy for this system, incorporating periodic imperfect maintenance actions in order to extend its useful lifetime. These actions aim to reduce the accumulated degradation level of each component from its installation in a fixed percentage, following the Arithmetic Reduction of Degradation model of infinite order. Another goal is to derive the distribution of time to the system failure, which provides crucial insights into the system’s reliability and performance, especially in the case of bivariate degradation. Additionally, a cost model for this maintenance strategy is developed and several numerical examples are presented to illustrate the practical implications. The maintenance decision variables are optimized in order to obtain the minimal expected cost rate within a finite time horizon.

An intelligent maintenance policy for a latent degradation system

This paper looks at the challenge of making maintenance decisions for deteriorating systems when the degradation process leading to failure cannot be directly observed or measured. In this scenario, the system’s health is monitored by observing the progression of a degradation-related marker index, which can be obtained through inspections. To model this configuration, a bivariate gamma process is employed. One component represents the marker process, while the other represents the degradation process, which dictates the time of failure. Two condition-based maintenance (CBM) policies are proposed and analyzed. The first policy is based on a conventional decision structure, utilizing a fixed preventive threshold directly applied to the measured process. The second policy relies on monitoring data related to the marker process to estimate the level of latent degradation at inspections. We demonstrate that the second policy is equivalent to a policy employing an adaptive preventive threshold that sequentially evolves. We provide insights into some key properties associated with this approach. The expected cost rate is calculated and employed for policy optimization. Additionally, a numerical study is presented that showcases the practical implementation of the method and highlights the effectiveness of the second approach, even when the correlation between degradation and the marker process is low.

Uncertain Random Block Replacement Policy with No Replacement at Failure

The classic block replacement problem assumes the lifetime of component as a random variable, and the downtime cost caused by failed component is a constant. However, when a new type of unit is used in a block replacement system, its lifetime is indeterminate and its distribution cannot be obtained by probability theory due to the lack of historical operational data. And if the component is not replaced immediately when it fails, the downtime cost per unit time is not fixed but always effected by some stochastic factors such as market. Thus, uncertainty and randomness coexist in a block replacement system of new components. In this paper, the lifetime of the component is regarded as an uncertain variable and the downtime cost per unit time is considered as a random variable, and an uncertain random programming model of block replacement with no replacement at failure is developed for a system composed of new type of components, and the formulated model aims to find an optimal replacement time to minimize the expected cost rate in one replacement cycle. An implicit solution of the optimal replacement time is obtained, and the condition ensuring the existence and uniqueness of the optimal replacement time is given. Finally, a numerical example is presented to prove feasibility of the proposed model.

Optimal post-warranty replacement policy based on number of product failures for the second-hand product

This paper deals with a post-warranty maintenance model for the second-hand product as follows. The second-hand product of age [Formula: see text], which is assumed to be positive, is purchased with a fixed length of non-renewing warranty, during which the product is given a fixed number of preventive maintenances periodically by the dealer. At each preventive maintenance, the failure rate of the product is adjusted to some extent for the purpose of reducing the likelihood of product failure. At the expiration of warranty, the user starts to self-maintain the product until a pre-determined number of failures occur, at which time the product is replaced by another one. For each post-warranty failure, only a minimal repair is taken to restore the failed product to its previous functioning state. This paper aims to determine an optimal number of post-warranty product failures that minimizes the expected cost rate during the second-hand product’s life cycle. To this end, we derive a formula to evaluate the expected cost rate during the second-hand product’s life cycle by assuming a certain cost structure for maintaining the product during the life-span of the product and determine an optimal number of post-warranty product failures from the user’s perspective. And we provide a numerical example to illustrate our proposed optimal maintenance model by assuming a Weibull failure distribution. The main contribution of this work is to use the number of post-warranty failures to propose a maintenance model for the second-hand product, where the number of failures is in general easier to observe than the failure times.

Optimal replacement policy for a two-unit system subject to shocks and cumulative damage

This paper investigates an extended replacement policy for a two-unit system subject to shocks and cumulative damage. As a shock occurs, the system suffers two types of effects: type I shock-effect is rectified by a minimal repair and type II shock-effect is removed by a corrective replacement. The probability of type II shock depends on the number of shocks from the last replacement. Each type I shock causes a minor failure of unit 1 and some damage to unit 2. These damages to unit 2 are additive and the system fails when the total damage exceeds a critical level L. A replacement policy (T, m, l) is considered in which the system is preventively replaced at age T or at the mth type I shock or at the time which the total damage to unit 2 exceeds a pre-specified level l; and is correctively replaced either at the first type II shock or when the total damage to unit 2 exceeds a failure level L, whichever occurs first. The optimal policy that minimizes the expected cost rate is determined analytically. The proposed model extends several existing results. Our replacement policy is more general, flexible, and applicable in the real situations.

A novel delay time modelling method for incorporating reuse actions in three-state single-component systems

This paper presents a new delay time modelling method for reusing single-component systems with two defective states and one failure state. It assumes that a component may be reused for the purposes of resource, economic and environmental sustainability. The possibility of reusing industrial components is not generally considered in maintenance models, which represents a knowledge gap in the literature, especially in the delay time related models. To address this gap, this paper proposes a method based on the delay time modelling method to investigate different scenarios of component reusability and uses real-world systems in the mining industry to illustrate its applicability. The paper then derives the expected cost rate, obtains lower and upper bounds of the expected total cost, considers the improving learning rate of correctly classifying defective components and incorporates the environmental impact of disposed components in optimization of the inspection interval. Results discuss when the reuse action may provide economic benefits even when the reused item may have different reliability than new one.

Petri net analysis of a queueing inventory system with orbital search by the server

Abstract In this article, a queueing inventory system with finite sources of demands, retrial demands, service time, lead time, ( s , S ) \left(s,S) replenishment policy, and demands search from the orbit was studied. When the lead time is exponentially distributed (resp. lead time is generally distributed), generalized stochastic Petri net (GSPN) (resp. Markov regenerative stochastic Petri net [MRSPN]) is proposed for this inventory system. The quantitative analysis of this stochastic Petri net model was obtained by continuous time Markov chain for the GSPN model (resp. the supplementary variable method for the MRSPN model). The probability distributions are obtained, witch allowed us to compute performance measures and the expected cost rate of the studied system.

A data-driven approach for condition-based maintenance optimization

Developments in sensor techniques enable the continuous monitoring of the health of an operating system. The resulting condition data provides an opportunity for better prediction of failures and thereby for improving maintenance decisions. In this study, we consider condition-based maintenance for a single unit with an unknown, non-decreasing deterioration process and unknown failure behavior. Building on, but different from the existing maintenance optimization literature, we present the first fully data-driven approach, where the condition threshold triggering maintenance is based purely on past condition data and failures. Numerical results for a gamma deterioration process show that the maintenance threshold resulting from our data-driven approach converges to the optimal threshold. The threshold is set higher during the initial runs-to-failure, and this helps to explore the deterioration process. An encouraging result is that the convergence is especially fast during the first few runs-to-failure so that the expected cost rate quickly converges to the minimum cost rate.

Optimal Policies on Education Training for Employees Using Maintenance Modeling Aspects

Employee training is essential for corporate activities to improve their production efficiency and product quality. The most representative two types of employee training are known, i.e., on-the-job training and off-the-job training. Off-the-job training can be classified into two types, i.e., compulsory training and non-compulsory training. Safety training and compliance are known as compulsory training, and they are needed to ensure that daily work continues without serious problems. Compulsory education is undertaken in a classroom every year by all employees of a department or section. Daily e-learning is effective for complementing and enhancing mandatory education and it helps employees to remember what they have learned during their annual education. In this paper, we discuss optimal employee safety education models that are complemented and enhanced by e-learning. The expected cost rate of education is expressed using the imperfect maintenance model, and the optimal policies that minimize it are discussed.

Optimal bivariate spare ordering policy for a system with imperfect maintenance

In this paper, we investigate an ordering-replacement modelling framework for a system deteriorating with two failure types: a repairable failure (Type-I failure) and an unrepairable failure (Type-II failure). Repairable failures are followed by minimal repair, whereas unrepairable failures are removed by a corrective replacement. A spare unit ordered regularly is associated with the system's operation time T and the number N of successive working times for preventive replacement, or ordered in emergency at any unrepairable failure for corrective replacement, whichever is first to happen. Three kinds of objectives – system availability, expected cost rate, and cost-effectiveness – are optimised to seek for the optimal spare unit ordering schedule (T*, N*), respectively. The existence and uniqueness of each optimal ordering policy are derived analytically and computed numerically.

Cost-Effective Maintenance Policy for Sliding Surfaces of Bridge Bearings Using a Gamma Stochastic Process for Forecasting

To determine the optimal alert threshold for sliding surface replacement of bridge bearings, a cost-effective maintenance policy is proposed in this paper using a gamma stochastic process. First, the sliding surface-triggered run-to-failure process of bridge bearings is discussed based on existing field inspection and maintenance records. Then, the wear thickness of sliding materials is estimated step-by-step based on the cumulative travel distance and wear rate. The gamma stochastic process is used to model degradation of sliding surfaces by using the indicator of wear thickness to depict uncertainties during the degrading process. Next, the optimal alert threshold for replacement of sliding surfaces is determined based on the cost-effective maintenance policy by minimizing the long-term expected maintenance cost rate. Finally, bearings of a long-span suspension bridge are employed to demonstrate the potential effectiveness of the proposed methodology. As a result, the wear thickness of sliding materials approximately follows a linear degrading law. Based on the gamma degrading model, the objective function subject to the long-term expected cost rate is formed. After optimization, the optimal alert threshold for replacement of sliding surfaces is 2.016 mm to achieve a minimum long-term expected maintenance cost rate of US$9,427 per day. In addition, the estimated service life subject to the alert threshold obeys a Gaussian distribution with a mean of 1278 days based on the one-year monitored displacement data.