Long-run Average Cost Rate Research Articles

Opportunistic maintenance for a novel load-sharing system with dependent degradation rate and volatility

For traditional degrading load-sharing systems, a typical assumption lies in the only effect of unit load on its degradation rate. However, motivated by a laser system example with positive correlation between degradation rate and volatility, we present a novel load-sharing system with general degradation rate-volatility-load correlation. Furthermore, to take account for unit economic dependence caused by sharable maintenance set-up cost, and the window period generated by the maintenance on some unit, opportunistic maintenance (OM) strategy is introduced to offer the opportunity of simultaneously maintaining another working unit under control-limit criteria. Based on the semi-regenerative process, the long-run average cost rate is minimized to obtain the optimal inspection interval, preventive maintenance and OM thresholds. Compared to the traditional condition-based maintenance strategy separately for each unit, the effectiveness of OM strategy is validated through numerical examples and case studies, offering broad managerial insights for engineers to reduce maintenance expenses.

A condition-based maintenance policy for a two-component balanced system with dependent degradation processes

Preventive maintenance of balanced systems has garnered increasing research attention due to its significance in production and servicing processes. This paper investigates the optimization problem of a condition-based maintenance policy for a two-component balanced system with dependent degradation processes. The system’s degradation follows a bivariate gamma process. Once the degradation level of one component exceeds a critical value, the system fails and triggers an immediate corrective replacement. Moreover, if the degradation difference between two components exceeds a threshold, the system is in an unbalanced state, which incurs a penalty cost. Periodic inspections are implemented to reveal the degradation of the system. At each inspection epoch, the decision-maker chooses an action among do-nothing, repair, and preventive replacement. Different from previous research, a repair action does not reduce the degradation levels of either component but increases the degradation level of the lower one to rebalance the system. The objective is to determine the optimal condition-based maintenance policy that minimizes the long-run average cost rate. The optimization problem is analyzed in a semi-Markov decision process framework. A policy-iteration dynamic programming algorithm is developed to derive the optimal policy. A numerical example is presented to illustrate the effectiveness of the proposed approach.

Modeling coupling impacts of self-healing mechanisms and dynamic environments on systems subject to dependent failure processes

Modern smart products are endowed with the capability to autonomously repair damage from random shocks, thereby highlighting the critical necessity to incorporate this self-healing attribute in reliability assessments. This paper investigates periodically inspected self-healing systems that are exposed to dependent competing failure processes within dynamically evolving environments. The impact of environmental changes is manifested in that both the natural degradation process and the self-healing behavior are governed by distinct laws in varying environments. Explicit formulas for the system reliability and availability are analytically derived using the stochastic process theory, and their correctness is verified by proposed simulation algorithms respectively. Meanwhile, the inspection period is optimized by minimizing the long-run average cost rate. Lastly, a practical example of semiconductor lasers is applied to showcase the application of the presented methods. The sensitivity analysis reveals that increasing the self-healing time threshold or the shock arrival rate can improve the system performance while lowering the long-run average cost rate.

An integrated optimization of lot-sizing and condition-based maintenance policy of a two-stage deteriorating system

In this study, a two-stage deteriorating production system is considered. The initiation time of the deterioration follows an exponential distribution. A homogeneous Gamma process is used to model the deterioration propagation process. The equipment produces a single product with a constant production rate and demand rate. To ensure the efficiency of the production, inspection is carried out at the end of each production run. The lot-sizing and the condition-based maintenance threshold are optimized by minimizing the long-run average cost rate, which is derived in the framework of a semi-Markov decision process. The traversal algorithm is used to obtain optimal results. A numerical example is presented to illustrate the applicability of the proposed model. Some sensitivity analysis are carried out in order to highlight the interest and properties of the model. It may provide a theoretical reference for production and maintenance schedules.

Analyze periodic inspection and replacement policy of a shock and wear model with phase-type inter-shock arrival times using roots method

We consider a shock and wear model in which the inter-shock arrival process is a phase-type (PH) renewal process, and the system’s lifetime is generally distributed. The system has two competing failure modes. One failure mode is due to random shocks, which cause failure by overloading the system. The other failure mode is owning to wear-out failures, which usually happen after the system has run for many cycles. System failure is not self-announcing and remains undiscovered unless an inspection is performed. The intervals between successive inspections are identical and equal to T time units. If a system failure is detected, the corrective repair or replacement is conducted immediately. If the system is found working at inspection, preventive maintenance will be carried out to prolong its useful life. Furthermore, to model the occurrence of events with an underlying monotonic trend, the extended geometric process (EGP) is employed to account for the impact of different types of failures on the system’s degradation. Moreover, for establishing the cost rate function in our model, the counting process generated from a PH renewal shock process is studied in detail using the roots method and formula for calculating residues. Based on these results, the survival function and other characteristics of the system are further investigated. Finally, numerical examples that determine the optimal inspection period T[Formula: see text] and the optimal replacement policy N[Formula: see text], which minimizes the long-run average cost rate, are presented.

Replacement and Repair Optimization for Production Systems Under Random Production Waits

This study considers a repairable production system operated under an age-based preventive replacement policy that is subject to independent random production waits and failures. In addition to the age-based replacement policy, we propose a maintenance model that uses production waits to schedule preventive replacement. That is, a decision maker can preventively replace the system either during a production wait or at the age threshold, whereas if a failure occurs during production, the decision maker must decide whether to perform a minimal repair or a corrective replacement to restore the system. Under the above setting, we develop a semi-Markov decision process (SMDP) to obtain the optimal maintenance policy that minimizes the long-run average maintenance cost rate. We establish the existence of the optimal maintenance policy and provide an algorithm to numerically obtain the optimal action for each state. We further generalize the model to incorporate imperfect maintenance and the nonhomogeneous arrival of production waits. In the latter case, it is computationally intractable to optimize the SMDP using a value iteration algorithm due to the curse of dimensionality. To address this challenge, we further develop an approximate dynamic programming framework to generate high-quality solutions. An attractive feature of our model is its generality, such that the model includes many existing maintenance models as special cases. A real-world example from a steel factory is used to demonstrate the proposed model.

An Imperfect Repair Model with Delayed Repair under Replacement and Repair Thresholds

Based on the extended geometric process, a repair replacement model of a degradation system is studied, in which the delayed repair time depends on the working time after the last repair. Replacement and repair thresholds describe when the system will be replaced and when the system can be repaired, respectively. Two kinds of replacement policies are studied. One policy is jointly determined by the moment of the Nth failure and the first hitting time of the working time after the last repair for the replacement threshold, and the system is replaced, whichever occurs first; the other is the special case of the first policy, and the system is replaced when the working time after the last repair first hits the replacement threshold. The exact expressions of the long-run average cost rate are obtained. The optimal policies exist and can be ascertained by numerical methods. Finally, numerical examples are presented to demonstrate the application of the results obtained in the paper.

A Preventive Replacement Policy for a System Subject to Bivariate Generalized Polya Failure Process

Numerous studies on preventive maintenance of minimally repaired systems with statistically independent components have been reported in reliability literature. However, in practice, the repair can be worse-than-minimal and the components of a system can be statistically dependent. The existing literature does not cover this important in-practice setting. Therefore, our paper is the first to deal with these issues by modeling dependence in the bivariate set up when a system consists of two dependent parts. We employ the bivariate generalized Polya process to model the corresponding failure and repair process. Relevant stochastic properties of this process have been obtained in order to propose and further discuss the new optimal bivariate preventive maintenance policy with two decision parameters: age and operational history. Moreover, introducing these two parameters in the considered context is also a new feature of the study. Under the proposed policy, the long-run average cost rate is derived and the optimal replacement policies are investigated. Detailed numerical examples illustrate our findings and show the potential efficiency of the obtained results in practice.

A hybrid repair-replacement policy in the proportional hazards model

Cox's proportional hazards model is widely used to describe the hazard rate of a system deteriorating with age and diagnostic covariates. Existing maintenance in the proportional hazards model was primarily concerned with replacement, resulting in excessive maintenance for many repairable systems. This paper develops a novel hybrid repair-replacement model in the proportional hazards model with a stochastically increasing Markovian covariate process. Preventive repair reduces both age and covariate, and the reduction rate decreases as the number of repairs increases. At an inspection epoch where the system age, covariate state, and repair number are available, the decision-maker considers three possible actions, i.e., no maintenance, preventive repair, and preventive replacement. The objective is to derive the optimal policy that minimizes the long-run average maintenance cost rate. The optimization problem is formulated in the semi-Markov decision process (SMDP) framework. The structural properties of the optimal policy are examined to reduce the policy space. Then a policy-iteration algorithm with a backward policy-improvement step is developed for efficiently finding the optimization results. A practical numerical example with sensitivity analysis is conducted to illustrate the effectiveness of the proposed approach. A comparison with two heuristic policies confirms the superiority of the proposed policy in reducing maintenance costs.

Performance analysis of an M/G/1 queue with bi-level randomized (p, N1, N2)-policy

This paper proposes an M/G/1 queueing model with bi-level randomized (p, N1, N2)-policy. That is, after all of the customers in the system are served, the server is closed down immediately. If N1(≥ 1) customers are accumulated in the queue, the server is activated for service with probability p(0 ≤ p ≤ 1) or still left off with probability 1 − p. When the number of customers in the system becomes N2(≥ N1), the server begins serving the waiting customers until the system becomes empty again. Using the total probability decomposition technique and the Laplace transform, we study the transient queue length distribution and obtain the expressions of the Laplace transform of the transient queue-length distribution with respect to time t. Then, employing L’Hospital’s rule and some algebraic operations, the explicit recursive formulas of the steady-state queue-length distribution, which can be used to accurately evaluate the probabilities of queue length, are presented. Moreover, some other important queuing performance indices, such as the explicit expressions of its probability generating function of the steady-state queue-length distribution, the expected queue size and so on, are derived. Meanwhile, we investigate the system capacity optimization design by the steady-state queue-length distribution. Finally, an operating cost function is constructed, and by numerical calculation, we find the minimum of the long-run average cost rate and the optimal bi-level threshold policy (N*1,N*2) that satisfies the average waiting time constraints.

A condition-based maintenance policy for multi-component systems subject to stochastic and economic dependencies

Condition-based maintenance (CBM) optimization utilizing prognostic information and methods for complex systems has attracted increasing attention. This paper presents a bi-level approach to optimize a CBM policy for multi-component systems subject to stochastic and economic dependencies. The objective of the decision rule at the system level is to address if preventive and/or corrective maintenance actions are necessary regarding the predictive reliability of the system. At the component level, the aim is to identify the optimal group of components to be preventively maintained when maintenance is triggered due to the decision made at the system level. A maintenance efficacy indicator is introduced to prioritize components to be maintained by taking into account both the resulting reliability improvement quantity and the maintenance cost incurred. The optimal inter-inspection interval, the thresholds of system and component conditional reliability for triggering maintenance are jointly determined for minimization of the long-run average cost rate. A numerical study and sensitivity analysis on a 2oo3:G system are carried out to validate the performance of the proposed maintenance policy. Compared to another opportunistic maintenance policy without considering the system health condition, the proposed CBM policy leads to more cost gain and effectiveness in the maintenance decision-making.

A dynamic inspection and replacement policy for a two-unit production system subject to interdependence

This paper proposes a dynamic inspection and replacement policy for a production system composed of two nonidentical units and an inventory system. Unit 1 deteriorates gradually and it is subject to soft failure. Unit 2 experiences hard failures that occur following a general lifetime distribution. The degradation processes of the two units are stochastically interdependent. Condition monitoring is performed to reveal the deterioration state of unit 1 when a certain number of production runs complete or when unit 2 fails, whichever comes first. At each inspection epoch, a decision is made on whether to replace unit 1 and/or unit 2. The objective is to jointly optimize the inspection interval and the replacement policy by minimizing the long-run average cost rate. The optimization problem is formulated in the semi-Markov decision process (SMDP) framework. Economic dependence and structural dependence are also integrated into the maintenance model. A matrix-based approximation method is employed to derive the SMDP quantities. The application of the proposed approach to a fuel oil supply system demonstrates the superiority of the proposed approach in saving operation and maintenance costs. The obtained graphical representation of the obtained policy is supportive during maintenance implementation.

Modeling periodically inspected k/r-out-of-n system

A k/r-out-of-n system undergoing periodic inspection with three states viz. normal, degraded and completely failed is presented in this paper. If at inspection, the system ends up degraded or completely failed then a repair is carried out taking random times. Availability and cost analysis of periodically inspected k/r-out-of-n system is performed specifically propositions on point availability, limiting availability and long-run average cost rate are stated and derived in the study. Conditions for determining the optimal inspection period based on cost rate and availability is also discussed. The derived theorems are illustrated with the help of an example.

Modeling a sequentially inspected system prone to degradation and shocks

PurposeIn certain environments, the system may not fail completely, but undergoes degradation, and the system productivity might decrease. Meanwhile, at the same time, the system may be vulnerable to shocks. A single-unit system prone to degradation and shocks is proposed in this study, and emphasis is placed upon determining its availability and cost rate.Design/methodology/approachThe considered single-unit system is expected to have three states, namely, normal, degraded and failed. As the system enters the degraded state, it is said to be partially failed. The degraded state incurs higher degradation than the normal state and is more prone to shocks. Inspections are used to determine the state and failure type of the system. Inspections are predetermined to be carried out sequentially at time where 0 < a ≤ 1, until the detection of degradation/failure. Perfect repairs are conducted instantly on spotting the partial/complete failure. Two cases have been considered of repair taking constant times and random times.FindingsExplicit results on the reliability, availability (both point and limiting availability) and long-run average cost rate (LRACR) of a sequentially inspected single-unit system prone to degradation and shocks under constant and random repair times are given. Numerical example of an oil pipeline system is taken to clarify the acquired results.Originality/valueA sequentially inspected single-unit system prone to degradation and shock is studied unlike done previously.

Minimal repair models with non-negligible repair time

In existing minimal repair models, it is usually supposed that the repair time of systems is negligible, and thus the occurrences of breakdowns obey non-homogeneous Poisson Process within a replacement interval. However, for the case that the mean working time of a system is not far longer than its mean repair time, ignoring repair time may lead to a large error in calculating the system long-run average cost rate. In this paper, taking random repair time into account, we generalize several calendar time-based minimal repair policies, including periodic replacement policy with minimal repair, reference time policy, and bivariate T−N maintenance policy. By using probability decomposition technique and indicator function method, the system long-run average cost rate under each minimal repair policy is gained explicitly. Various special cases are discussed. An application of the periodic replacement policy with minimal repair is presented in warranty field. Finally, a comprehensive sensitivity analysis is conducted to investigate the relative error on the long-run average cost rate caused by ignoring repair time, and to compare calendar time-based minimal repair policies with age-based minimal repair policies in terms of the long-run average cost rate.

Availability and cost assessment of systems with dormant failure undergoing sequential inspections

PurposeThe system encountering dormant failure subject to sequential inspections is modeled and the emphasis is made on determining the availability and long-run average cost rate for the model. The derived results are then utilized to obtain the optimal inspection period minimizing the cost.Design/methodology/approachExplicitly, a system with a functional and a failed state is taken into account. Inspections are performed to reveal the dormant failures and are assumed to be carried out at time T, T + aT, T + aT+a2 T, … where 0 < a = 1 in each cycle. Perfect repairs taking random times are performed if the system is found in a failed state during any inspection.FindingsSome theorems on the point availability, limiting availability and long-run average cost rate are obtained in the study. An illustration is shown to explain the results obtained in the proposed work. The effect of inspection time on the availability and cost rate is also analyzed graphically.Originality/valueThe availability and cost rate for a system with dormant failure under a sequential inspection policy are figured out unlike previous research.

Joint optimization of lot sizing and condition-based maintenance for a production system using the proportional hazards model

This paper optimizes economic production quantity (EPQ) and condition-based maintenance (CBM) simultaneously for a production system subject to aging and deterioration. Different from previous models jointly determining EPQ and CBM, the proposed model is developed based on the proportional hazards model with a continuous-state covariate process, and on a CBM policy with multiple maintenance actions and dynamic control limits. Taking advantage of the opportunity of the downtime when a production run completes, condition monitoring is performed to reveal the system deterioration and then a suitable action is selected from preventive replacement, preventive repair, and no maintenance. Random failures during a production phase can be fixed by corrective replacement, corrective repair, or minimal repair. Considering that in many practical situations condition monitoring cannot be conducted at failure, corrective maintenance actions are determined based on (1) the system age at failure and the deterioration at the beginning of the production run (Scenario 1); or (2) both the system age and deterioration at the beginning of the production run (Scenario 2). The objective is to jointly optimize the production lot size and the CBM policy by minimizing the long-run average cost rate. The optimization problem is solved based on the policy-iteration algorithm in the semi-Markov decision process (SMDP) framework. A numerical example is provided to illustrate the proposed approach. The results show that Scenario 1 produces a more cost-effective but more complex maintenance strategy than Scenario 2 does.

On optimal joint replacement policy for deteriorating series systems

In this paper, a general repair-replacement model for a k-component series system is proposed. Components of the system may break down randomly and can be repaired immediately. Due to the imperfect repair, the successive operating times decrease stochastically while the consecutive repair times increase stochastically. A joint replacement policy is adopted based on the failure number of each component. The explicit expression of the long-run average cost rate is derived. Furthermore, we investigate the optimal policies of series system and single-component system respectively, the explicit relationship between them is determined in the form of a theorem for the first time. It shows that the optimal replacement policy of a component in a series system is not less than that when it forms a single-component system, and the former is non-decreasing with the number of components. Some other analytical results are also obtained. Based on these useful results, an efficient algorithm is devised to search the optimal joint policy which leads to an elegant, mathematical appealing analysis that is easy to use in practice. Finally, illustrative examples are provided to demonstrate the proposed model and the method.

Optimization of a repairable deteriorating system subject to random threshold failure using preventive repair and stochastic lead time

This paper presents a preventive repair / replace strategy for a repairable deteriorating system with stochastic lead time using phase type quasi-renewal processes. The aim of this study is to indicate improvement in the system performance when adopting preventive repair. Employing the standard results of the renewal reward process, an explicit expression for the long-run average cost rate and the operational availability of the system are derived to obtain an optimal replacement policy N* analytically, for the proposed model. Numerical illustrations are provided to show the effectiveness in the long-run average cost rate and operational availability, which depict the advantage of adopting a preventive repair with stochastic lead time.

Optimal Condition-Based Maintenance Strategy for Multi-Component Systems under Degradation Failures

This paper proposes a condition-based maintenance strategy for multi-component systems under degradation failures. The maintenance decision is based on the minimum long-run average cost rate (LACR) and the maximum residual useful lifetime (RUL), respectively. The aim of this paper is to determine the optimal monitoring interval and critical level for multi-component systems under different optimization objectives. A preventive maintenance (PM) is triggered when the degradation of component exceeds the corresponding critical level. Afterwards, the paper discusses the relationship between the critical level and the monitoring interval with regards to the LACR and RUL. Methods are also proposed to determine the optimal monitoring interval and the critical level under two decision models. Finally, the impact of maintenance decision variables on the LACR and RUL is discussed through a case study. A comparison with conventional maintenance policy shows an outstanding performance of the new model.