Expected Cost Rate Research Articles

An inventory system with replacement and refurbishment of failed items

We consider a continuous review inventory system with two types of inventory, namely type-I and type-II. Type-I and type-II inventory, respectively, contain fresh and refurbished items. Two streams of customers, called type-A and type-B customers arrive to the system in which type-A customer demands for an item either from type-I or type-II inventory according to a Bernoulli trial and type-B customer occurs to replace the failed item with new one from type-I inventory. The arrival time points of type-A and type-B customers follow independent Poisson Processes. Failed items are stored in a place, called pool, of infinite size for refurbishment. After refurbishment, these items are kept in type-II inventory. The refurbishment of pooled item is stopped once type-II inventory reaches its maximum level S 2 and is again commenced only when type-II inventory level falls to s 2. The refurbishment time follows exponential distribution. Type-I inventory is replenished according to a (s, S) policy with exponential lead time. The joint pdf of number of items in the pool, type-II inventory level, status of refurbishment and type-I inventory level is obtained in the steady state. Various system performance measures are derived and total expected cost rate is calculated. Results are illustrated numerically.

Determination of optimal post-warranty variable maintenance period for repairable second-hand product based on mean residual life

Due to the increased transactions of second-hand products in the market, the optimization of maintenance strategy for the second-hand product has become very important issue to attract a great attention from many researchers of late. This paper proposes a new post-warranty strategy with a variable self-maintenance period for the second-hand product, assuming that the product is replaced by another one on the first failure following a fixed length of post-warranty self-maintenance period. During the non-renewing warranty period, the product is subject to preventive maintenance periodically at a prorated cost while only minimal repair is implemented at each failure by the dealer. The main goal of this study is to determine an optimal length of post-warranty self-maintenance period which minimizes the expected cost rate per unit time during the product’s life cycle from the user’s perspective. This approach considers not only the periodic preventive maintenance during the warranty period, but also the remaining life distribution of the product after the warranty expires, which is the significant difference of this work from many existing maintenance policies. For this purpose, we formulate the expected length of life cycle and evaluate the expected total cost incurred during the life cycle of the second-had product which is purchased at the age of [Formula: see text] The existence of the optimal self-maintenance period is proved analytically under mild conditions and the proposed maintenance model is compared with an existing model with regard to the expected cost rate. Finally, assuming that the life distribution of the product follows a Weibull distribution, the effect of relevant parameters on the optimal self-maintenance period is analyzed numerically.

Design of biased random walks on a graph with application to collaborative recommendation

This work investigates a paths-based statistical physics formalism, inspired from the bag-of-paths framework, for the design of random walks on a graph in which the transition probabilities (the policy) are biased in favor of some node features. More precisely, given a weighted directed graph G and a non-negative cost assigned to each edge, the biased random walk is defined as the policy minimizing the expected cost rate along the walks while maintaining a constant relative entropy rate. As for the standard bag-of-paths and the randomized shortest paths frameworks, the model assigns a Gibbs–Boltzmann distribution to the set of infinite walks and allows to recover known results from the literature, derived here from a different perspective. Examples of quantities of interest are the partition function of the system, the cost rate, the optimal transition probabilities, etc. In addition, the same formalism allows the introduction of capacity constraints on the expected node visit rates and an algorithm for computing the optimal policy subject to such capacity constraints is developed. Simulation results indicate that the proposed procedure can be effectively used in order to define a Markov chain driving the walk towards nodes having some specific properties, like seniority, education level or low node degree (hub-avoiding walk). An application relying on this last property is proposed as a tool for improving serendipity in collaborative recommendation, and is tested on the MovieLens data.

Optimal renewing free-maintenance warranty period and preventive replacement age for repairable products

This paper proposes a decision model for determining the optimal renewing free-maintenance warranty (RFMW) period and preventive replacement age of a general repairable product with a random workload. To model a maintenance action, if a failure occurs, either a corrective replacement or a minimal repair is performed, in case of catastrophic failure or minor failure, respectively. To avoid catastrophic failures occurring for a degraded product, a preventive replacement is predetermined for implementation at age T or at the completion of a working time, whichever comes last. During RFMW, any maintenance action (either repair or replacement) is free of charge to the customer. However, each catastrophic failure still results in a downtime cost to the customer, while a planned maintenance action increases the initial purchasing cost. The aim of this paper is to determine the optimal warranty period and preventive replacement age, from the collection viewpoint of the manufacturer as well as the customers, while minimising the corresponding expected cost rate functions. The effects of RFMW on the optimal preventive replacement model are further investigated by some comparative studies and demonstrated via a numerical example. It is shown that the proposed model provides a generalised application in maintenance theory and warranty strategy.

Optimizations of Replacement Policies for Random k-out-of-n Systems

In this paper, a random repairable [Formula: see text]-out-of-[Formula: see text] system is studied when random replacement policies are performed, where [Formula: see text] is a stochastic parameter. It is replaced before failure at a planned time [Formula: see text] or at random working cycle [Formula: see text], whichever occurs last or first. Furthermore, optimal replacement policies are obtained for the extended models with working numbers. This paper aims to find the optimal replacement policies when replacement first and last policies are performed. Expected cost rates for the first and last replacement policies are derived, and their optimal replacement policies are derived analytically. Numerical examples are given to illustrate the theoretical results.

On the delayed worse-than-minimal repair model and its application to preventive replacement

Abstract Minimal repair and other imperfect repair models have been intensively studied in the literature. Much less attention has been payed to the ‘worse than minimal’ repair problem, although it often occurs in practice due to the adverse effects of previous repairs, environmental and internal shocks, etc. To model this type of repair, we define a new point process that behaves as the non-homogeneous Poisson process up to a certain event or time (minimal repairs) and then it becomes the generalized Polya process of repairs (worse than minimal repairs). The corresponding replacement policy is defined and the optimal solutions that minimize the long run expected cost rate are analyzed. The replacement can be executed univariately either after the given time T or the given number of repairs (on the k-th failure). Moreover, the system can be also replaced by implementing the bivariate strategy, that is, after the time T or on the k-th failure, whichever comes first. The detailed numerical examples illustrate our findings. It is shown that the k-strategy outperforms the T -strategy (lower cost rates), whereas the bivariate strategy is not worse than the best univariate strategy.

Optimizations of Generational Garbage Collections Based on Continuous Damage Process

In the computer science community, garbage collection is a dynamic storage management technology to ensure the reliability of computer systems. In this paper, we consider two garbage collection policies to meet the goal of time consumption for a generational garbage collector when increase in objects might be unclear at discrete times for the high frequency of computer processes. That is, (a) tenuring collection is triggered at the [Formula: see text]th minor collection preventively or at a threshold amount [Formula: see text] of surviving objects correctively, and (b) major collection is made at discrete times [Formula: see text] for a given [Formula: see text] or at the [Formula: see text]th collection including minor and tenuring collections. Using the damage process and renewal theory, the expected cost rates are obtained, and their optimal policies for tenuring and major collection are discussed analytically and computed numerically.

Optimal Education Plan of Employees Using Maintenance Model

The employee education is indispensable for companies to improve productive efficiency and product quality. In general, the employee education is divided into two types, i.e., On-job trainings and Off-job ones, and Off-job trainings are divided into two types, i.e., compulsory educations and non-compulsory ones. Compulsory educations such as safety program and compliance education, are necessary to maintain daily production without any accidents. Although all employees of a department and a division gather in a classroom annually in conventional compulsory educations, daily e-learning education complements and strengthens conventional compulsory ones today because employees forget what they learn by annual learning. In past studies, the logit model is used for modeling the influence from education receipt to its memory condition and quantitative relationship between the effect of traffic safety education and the accident-related human-errors was clarified. The effectiveness of the safety driving educational program was indicated by Structural Equation Modelling method. In this paper, an annual compulsory which is complemented and strengthened by e-learning, is discussed. The expected cost rate of education is expressed using imperfect maintenance models and optimal policies which minimizes it is considered. Although we use the exponential function which denotes the occurrence probability of accidents which are caused by forgetting lessons, the actual occurrence probability which is approximated from the actual data would be a complicated one. For solving such complex optimization problems, metaheuristic methods can be applied.

Perishable Inventory System with N-Policy, MAP Arrivals, and Impatient Customers

In this study, we consider a perishable inventory system that has an (s, Q) ordering policy, along with a finite waiting hall. The single server, which provides an item to the customer after completing the required service performance for that item, only begins serving after N customers have arrived. Impatient demand is assumed in that the customers waiting to be served lose patience and leave the system if the server’s idle time overextends or if the arriving customers find the system to be full and will not enter the system. This article analyzes the impatient demands caused by the N-policy server to an inventory system. In the steadystate, we obtain the joint probability distribution of the level of inventory and the number of customers in the system. We analyze some measures of system performance and get the total expected cost rate in the steadystate. We present a beneficial cost function and confer the numerical illustration that describes the impact of impatient customers caused by N-policy on the inventory system’s total expected cost rate.

Optimal control-limit maintenance policy for a production system with multiple process states

In this paper, a new control limit maintenance policy is proposed for a production system with multiple process states. The process state can be obtained by quality information or other measurable indicators from the production process, when the process state by itself provides an insufficient indication or measure of the actual machine state. This can make the maintenance activities difficult to implement. Alternatively, machine virtual age can indicate the actual machine state and the maintenance effect on the machine state. Therefore, machine virtual age is used to compensate for the possible weakness of process state as an indication of machine state, and it is also used as part of maintenance decisions for more effective maintenance activities. At each inspection time, maintenance decisions are made for spare parts ordering and maintenance activities. The maintenance activities which are considered in this paper are replacement and two types of imperfect maintenance. The new maintenance planning model considers the different maintenance levels and their effects on system performance. Moreover, different maintenance cost and durations are considered. The system and associated maintenance policy are modeled by a discrete-time Markov decision process, and the long-term expected cost rate is investigated. A modified policy-iterative algorithm is then proposed to accelerate the search speed effectively during cost optimization. Compared to a traditional algorithm, the modified algorithm checks the effects of some possible parameters, rather than all parameters during the policy-iteration process. Simultaneously, it assures that the checked parameters are better than those unchecked. Finally, a numerical example is presented to illustrate the proposed method, and sensitivity analyses of important parameters are also implemented.

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.

Optimizations of discrete garbage collections in computer reliability systems

In computer science community, garbage collection is a dynamic storage management technology to ensure the reliability of computer systems. In this paper, we consider two discrete garbage collection policies to meet the goal of time consumption for a generational garbage collector. That is, garbage collections occur at a nonhomogeneous Poisson process, (a) tenuring collection is triggered at the Nth minor collection preventively or at a threshold amount [Formula: see text] of surviving objects correctively, whichever takes place first, and (b), tenuring collection is triggered at the first collection when the amount of surviving objects has exceeded a threshold level [Formula: see text] and major collection is triggered at discrete times kT for a given T. Using the damage process and renewal theory, the expected cost rates are obtained, and their optimal policies for tenuring and major collections are discussed analytically and computed numerically.

Age and periodic replacement policies with two failure modes in general replacement models

This paper studies age and periodic replacement policies in general replacement models by considering major failures and minor failures, meanwhile, random replacement policies are applied as alternatives for regular policies into replacement first and last models. We obtain the expected cost rates for each planned replacement policies. Optimal regular replacement and random replacement policies are discussed and compared from the point of cost in analytical ways. Furthermore, exponential minor failure, exponential major failure, and Weibull major failure with an increasing hazard rate of minor failure are illustrated in numerical analyses.

Optimal replacement policy under cumulative damage model and strength degradation with applications

In many real-life scenarios, system failure depends on dynamic stress-strength interference, where strength degrades and stress accumulates concurrently over time. In this paper, we consider the problem of finding an optimal replacement strategy that balances the cost of replacement with the cost of failure and results in the minimum expected cost per unit time under cumulative damage model with strength degradation. In the most general setting, we propose to find optimal choices of three thresholds on operation time, number of arriving shocks and amount of cumulative damage such that replacement of the system due to failure or reaching any of the three thresholds, whichever occurs first, results in the minimum expected cost per unit time. The existing recommendations are applicable only under the assumption of Exponential damage distribution including Poisson arrival of shocks and/or with fixed strength. As theoretical evaluation of the expected cost per unit time turns out to be very complicated, a simulation-based algorithm is proposed to evaluate the expected cost rate and find the optimal replacement strategy. The proposed method is easy to implement having wider domain of application including non-Poisson arrival of shocks and non-Exponential damage distributions. For illustration, the proposed method is applied to real case studies on mailbox and cell-phone battery experiments.

A Finite Source Two Commodity Inventory System with Retrial Demands and Multiple Server Vacation

Consider a two commodity continuous review inventory system with a finite number of homogeneous sources generating demands. The maximum storage capacity for the commodity - i is S i (i = 1, 2). We adopt a joint reordering policy for placing orders. The lead time for the delivery of orders is assumed to have an exponential distribution. An arriving demand receives either commodity - 1 or commodity - 2 according to a Bernoulli trial. The commodities are assumed to be substitutable in the sense that during the zero stock of any commodity, the demand for that commodity can be met by the unit of the other commodity. The server goes for a vacation of an exponentially distributed duration whenever the inventory level of both commodities reaches zero. If the server finds empty stock when he returns to the system, he immediately takes another vacation. The demands that occur during stock-out period or during the server vacation period enter into the orbit of finite size. These orbiting demands retry for their demand after a random time, which is assumed to be an exponential distribution. The joint probability distribution of the inventory level and the number of demands in the orbit is obtained in the steady state case. Various system performance measures in the steady state are derived and the long-run total expected cost rate is calculated.

Periodic and sequential inspection policies with mission failure probabilities

AbstractWhen a unit should provide reliability at mission time, the mission success probability or failure probability becomes an important constraint to plan inspection policies. From this viewpoint, this paper considers the optimization problems of periodic and sequential inspection policies with mission failure probabilities. In this paper, a mission time is considered as a renewal point for inspection polices, but the failure cost occurred at mission time is higher than the total cost of inspection and failure detection. We give an upper limit of the failure probability at mission time, that is, mission failure probability, to minimize the total expected costs and the expected cost rates of inspection policies. In addition, the extended models of inspection policies for an interval mission, possible random times, and operating interval are discussed.

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.

Performance-oriented risk evaluation and maintenance for multi-asset systems: A Bayesian perspective

In this article, we present a risk evaluation and maintenance strategy optimization approach for systems with parallel identical assets subject to continuous deterioration. System performance is defined by the number of functional assets, and the penalty cost is measured by the loss of performance. To overcome the practical challenges of information sparsity, we employ a Bayesian framework to dynamically update unknown parameters in a Wiener degradation model. Order statistics are utilized to describe the failure times of assets and the stepwise incurred performance penalty cost. Furthermore, based on the Bayesian parameter inferences, we propose a short-term value-based replacement policy to minimize the expected cost rate in the current planning horizon. The proposed strategy simultaneously considers the variability of parameter estimators and the inherent uncertainty of the stochastic degradation processes. A simulation study and a realistic example from the petrochemical industry are presented to demonstrate the proposed framework.

Optimal maintenance strategies for warranty products with limited repair time and limited repair number

Nowadays, to protect the interests of customers, many warranty policies take into account the limit of repair time and the limit of repair number. Under a rebate warranty policy, the seller has to give a refund to the customer according to the rebate warranty contract if the number of product failures or a single repair time for failure exceeds some certain limits. Minimal repair is conducted when the product fails, and the repair cost is charged to seller during the warranty period. After warranty expires, the product continues operating for a period of time before being replaced. The repair cost during the post-warranty period is undertaken by the customer. From the customer’s perspective, the expected life cycle cost rate of the warranty product is derived. By minimizing the expected cost rate, the existence and uniqueness of the optimal maintenance strategy for the warranty product are proven theoretically. Finally, as an example, the warranty car under the protection of “lemon laws” is studied to illustrate the proposed rebate warranty model.

A Combined Maintenance Strategy Considering Spares, Buffer, and Quality

This paper studies an innovative maintenance model for an upstream machine in a common production system. Multiple system-wise features are integrated in the model, including maintenance, spare parts, buffer inventory, and product quality. The objective is to minimize the system expected cost rate by determining the optimal inspection interval, preventive maintenance threshold, corrective maintenance threshold, reorder level of spare parts, maximum stock level of spare parts, and maximum inventory level of the buffer. In this maintenance model, both the hazard rate and the product qualification rate can trigger maintenance activities. The well-known proportional hazard model (PHM) is adopted to describe the state of the upstream machine. The effect of machine state on product quality is explicitly characterized. Due to the complexity of the mathematical model, simulation and a genetic algorithm are employed to determine the optimal solution. A case study of bearings is presented to demonstrate the performance of the proposed maintenance policy. The results show that this policy is practical and can reduce system cost significantly.