Evaluating the effects of machine breakdowns in stochastic scheduling problems

In most of the stochastic resource-allocation problems discussed in the literature it is supposed that the key resource, herein called the machine, is continuously available until all tasks are completed. Plainly, this will often be an unrealistic assumption. This paper supposes that intermittent availability of the machine is due to a breakdown proces, and describes various approaches to the evaluation of the effect of breakdowns. Firstly, for the case of geometric up times, conditions are given under which breakdowns have no effect on optimal allocation strategies. Secondly, two different procedures are given which yield an upper bound on the loss incurred when a processing strategy is adopted under the assumption of no breakdowns, when in fact breakdowns do occur. The first of these is based on Gittins's indices and is described for the case of geometric up times, and the second uses a bounding argument on the breakdown process.     

Optimization of schedule stability and efficiency under processing time variability and random machine breakdowns in a job shop environment

The ability to cope with uncertainty in dynamic scheduling environments is becoming an increasingly important issue. In such environments, any disruption in the production schedule will translate into a disturbance of the plans for several external activities as well. Hence, from a practical point of view, deviations between the planned and realized schedules are to be avoided as much as possible. The term stability refers to this concern. We propose a proactive approach to generate efficient and stable schedules for a job shop subject to processing time variability and random machine breakdowns. In our approach, efficiency is measured by the makespan, and the stability measure is the sum of the variances of the realized completion times. Because the calculation of the original measure is mathematically intractable, we develop a surrogate stability measure. The version of the problem with the surrogate stability measure is proven to be NP‐hard, even without machine breakdowns; a branch‐and‐bound algorithm is developed for this problem variant. A tabu search algorithm is proposed to handle larger instances of the problem with machine breakdowns. The results of extensive computational experiments indicate that the proposed algorithms are quite promising in performance. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Proactive release procedures for just‐in‐time job shop environments,subject to machine failures

In this paper, we consider just‐in‐time job shop environments (job shop problems with an objective of minimizing the sum of tardiness and inventory costs), subject to uncertainty due to machine failures. We present techniques for proactive uncertainty management that exploit prior knowledge of uncertainty to build competitive release dates, whose execution improves performance. These techniques determine the release dates of different jobs based on measures of shop load, statistical data of machine failures, and repairs with a tradeoff between inventory and tardiness costs. Empirical results show that our methodology is very promising in comparison with simulated annealing and the best of 39 combinations of dispatch rules & release policies, under different frequencies of breakdowns. We observe that the performance of the proactive technique compared to the other two approaches improves in schedule quality (maximizing delivery performance while minimizing costs) with increase in frequency of breakdowns. The proactive technique presented here is also computationally less expensive than the other two approaches. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

An FPTAS for scheduling a two‐machine flowshop with one unavailability interval

We study a deterministic two‐machine flowshop scheduling problem with an assumption that one of the two machines is not available in a specified time period. This period can be due to a breakdown, preventive maintenance, or processing unfinished jobs from a previous planning horizon. The problem is known to be NP‐hard. Pseudopolynomial dynamic programming algorithms and heuristics with worst case error bounds are given in the literature to solve the problem. They are different for the cases when the unavailability interval is for the first or second machine. The existence of a fully polynomial time approximation scheme (FPTAS) was formulated as an open conjecture in the literature. In this paper, we show that the two cases of the problem under study are equivalent to similar partition type problems. Then we derive a generic FPTAS for the latter problems with O(n⁵/ε⁴) time complexity. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

Machine scheduling with an availability constraint and job delivery coordination

In this paper we study the scheduling problem that considers both production and job delivery at the same time with machine availability considerations. Only one vehicle is available to deliver jobs in a fixed transportation time to a distribution center. The vehicle can load at most K jobs as a delivery batch in one shipment due to the vehicle capacity constraint. The objective is to minimize the arrival time of the last delivery batch to the distribution center. Since machines may not always be available over the production period in real life due to preventive maintenance, we incorporate machine availability into the models. Three scenarios of the problem are studied. For the problem in which the jobs are processed on a single machine and the jobs interrupted by the unavailable machine interval are resumable, we provide a polynomial algorithm to solve the problem optimally. For the problem in which the jobs are processed on a single machine and the interrupted jobs are nonresumable, we first show that the problem is NP‐hard. We then propose a heuristic with a worst‐case error bound of 1/2 and show that the bound is tight. For the problem in which the jobs are processed on either one of two parallel machines, where only one machine has an unavailable interval and the interrupted jobs are resumable, we propose a heuristic with a worst‐case error bound of 2/3. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Scheduling jobs that are subject to failure propagation

A single machine is available to process a collection of stochastic tasks. Processing is interrupted when the machine breaks down. We introduce a new model of breakdowns that more realistically incorporates the effects that job processing may have on the machine. This failure propagation model is equivalent to a Bayesian formulation in which learning about breakdown rates occurs as jobs evolve. Optimal scheduling policies are described in terms of Gittins indices and these indices are characterised in two special cases. For example, we obtain conditions which ensure that an optimal policy will only preempt a job's processing either at its completion or at a machine breakdown. We also bound the value lost by simplistic modelling which ignores the learning phenomenon. © 1996 John Wiley & Sons, Inc.     

Monotone optimal preventive maintenance policies for stochastically failing equipment

This paper examines various models for maintenance of a machine operating subject to stochastic deterioration. Three alternative models are presented for the deterioration process. For each model, in addition to the replacement decision, the option exists of performing preventive maintenance. The effect of this maintenance is to “slow” the deterioration process. With an appropriate reward structure imposed on the processes, the models are formulated as continuous time Markov decision processes. the optimality criterion being the maximization of expected discounted reward earned over an infinite time horizon. For each model conditions are presented under which the optimal maintenance policy exhibits the following monotonic structure. First, there exists a control limit rule for replacement. That is, there exists a number i* such that if the state of machine deterioration exceeds i* the optimal policy replaces the machine by a new machine. Secondly, prior to replacement the optimal level of preventive maintenance is a nonincreasing function of the state of machine deterioration. The conditions which guarantee this result have a cost/benefit interpretation.     

The single-machine absolute-deviation early-tardy problem with random completion times

We consider sequencing n jobs on a single machine subject to job completion times arising from either machine breakdowns or other causes. The objective is to minimize an expected weighted combination of due dates, completion times, earliness, and tardiness penalties. The determination of optimal distinct due dates or optimal common due dates for a given schedule is investigated. The scheduling problem for a fixed common due date is considered when random completion times arise from machine breakdowns. The optimality of a V-shaped about (a point) T sequence is established when the number of machine breakdowns follows either a Poisson or a geometric distribution and the duration of a breakdown has an exponential distribution. © 1996 John Wiley & Sons, Inc.     

Scheduling maintenance and semiresumable jobs on a single machine

The majority of scheduling literature assumes that the machines are available at all times. In this paper, we study single machine scheduling problems where the machine maintenance must be performed within certain intervals and hence the machine is not available during the maintenance periods. We also assume that if a job is not processed to completion before the machine is stopped for maintenance, an additional setup is necessary when the processing is resumed. Our purpose is to schedule the maintenance and jobs to minimize some performance measures. The objective functions that we consider are minimizing the total weighted job completion times and minimizing the maximum lateness. In both cases, maintenance must be performed within a fixed period T, and the time for the maintenance is a decision variable. In this paper, we study two scenarios concerning the planning horizon. First, we show that, when the planning horizon is long in relation to T, the problem with either objective function is NP-complete, and we present pseudopolynomial time dynamic programming algorithms for both objective functions. In the second scenario, the planning horizon is short in relation to T. However, part of the period T may have elapsed before we schedule any jobs in this planning horizon, and the remaining time before the maintenance is shorter than the current planning horizon. Hence we must schedule one maintenance in this planning horizon. We show that the problem of minimizing the total weighted completion times in this scenario is NP-complete, while the shortest processing time (SPT) rule and the earliest due date (EDD) rule are optimal for the total completion time problem and the maximum lateness problem respectively. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 845–863, 1999     

Dual criteria preemptive open-shop problems with minimum makespan

In this article we present an algorithm for the minimum makespan preemptive open shop, which is superior to existing algorithms in both space and time requirements. We define the more complex generalized open shop and flexible open shop and address the minimum makespan problem on these shops. We show how we can use the algorithm for the minimum makespan open shop to achieve load balancing in simple and generalized open shops without increasing the complexity of the algorithm. Load balancing dictates that the number of busy machines in each period is as even as possible. We also consider preventive maintenance issues in the open shop, and makespan retains its minimum value. In particular we consider the scenario where a machine can be maintained during any period that it happens to be idle. Also we consider the case that a maintenance schedule is prespecified. We show that this problem can be solved via a linear programming formulation that can also take into account release times for the jobs and ready times for the machines. Faster algorithms are presented for open shops with three machines or less. © 1995 John Wiley & Sons, Inc.     

Scheduling on M parallel machines subject to random breakdowns to minimize expected mean flow time

The problem of scheduling n jobs on m parallel machines is considered when the machines are subject to random breakdowns and job processing times are random variables. An objective function of mean flow time is developed for a general parallel machine system, and an expression of its expected value is derived. The problem is transformed into a deterministic unrelated parallel machine scheduling model with modified processing times when the number of breakdowns is modeled as a generalized Poisson process. © 1994 John Wiley & Sons, Inc.     

Stochastic scheduling subject to machine breakdowns: The preemptive‐repeat model with discounted reward and other criteria

We consider the problem of scheduling a set of jobs on a single machine subject to random breakdowns. We focus on the preemptive‐repeat model, which addresses the situation where, if a machine breaks down during the processing of a job, the work done on the job prior to the breakdown is lost and the job will have to be started from the beginning again when the machine resumes its work. We allow that (i) the uptimes and downtimes of the machine follow general probability distributions, (ii) the breakdown process of the machine depends upon the job being processed, (iii) the processing times of the jobs are random variables following arbitrary distributions, and (iv) after a breakdown, the processing time of a job may either remain a same but unknown amount, or be resampled according to its probability distribution. We first derive the optimal policy for a class of problems under the criterion to maximize the expected discounted reward earned from completing all jobs. The result is then applied to further obtain the optimal policies for other due date‐related criteria. We also discuss a method to compute the moments and probability distributions of job completion times by using their Laplace transforms, which can convert a general stochastic scheduling problem to its deterministic equivalent. The weighted squared flowtime problem and the maintenance checkup and repair problem are analyzed as applications. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Integrated production scheduling and preventive maintenance planning for a single machine under a cumulative damage failure process

This paper finds the optimal integrated production schedule and preventive maintenance plan for a single machine exposed under a cumulative damage process, and investigates how the optimal preventive maintenance plan interacts with the optimal production schedule. The goal is to minimize the total tardiness. The optimal policy possesses the following properties: Under arbitrary maintenance plan when jobs have common processing time, and different due dates, the optimal production schedule is to order the jobs by earliest due date first rule; and when jobs have common due date and different processing times, the optimal production schedule is shortest processing time first. The optimal maintenance plan is of control limit type under any arbitrary production schedule when machine is exposed under a cumulative damage failure process. Numerical studies on the optimal maintenance control limit of the maintenance plan indicate that as the number of jobs to be scheduled increases, the effect of jobs due dates on the optimal maintenance control limit diminishes. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

A single-machine replacement model with learning

The replacement or upgrade of productive resources over time is an important decision for a manufacturing organization. The type of technology used in the productive resources determines how effectively the manufacturing operations can support the product and marketing strategy of the organization. Increasing operating costs (cost of maintenance, labor, and depreciation) over time force manufacturing organizations to periodically consider replacement or upgrade of their existing productive resources. We assume that there is a setup cost associated with the replacement of a machine, and that the setup cost is a nonincreasing function of the number of replacements made so far due to learning in setups. The operating cost of a newer machine is assumed to be lower than the operating cost of an older machine in any given period, except perhaps in the first period of operation of the new machine when the cost could be unusually high due to higher initial depreciation. A forward dynamic programming algorithm is developed which can be used to solve finite-horizon problems. We develop procedures to find decision and forecast horizons such that choices made during the decision horizon based only on information over the forecast horizon are also optimal for any longer horizon problem. Thus, we are able to obtain optimal results for what is effectively an infinite-horizon problem while only requiring data over a finite period of time. We present a numerical example to illustrate the decision/forecast horizon procedure, as well as a study of the effects of considering learning in making a series of machine replacement decisions. © 1993 John Wiley & Sons. Inc.     

Scheduling jobs with random processing times on a single machine subject to stochastic breakdowns to minimize early-tardy penalties

We examine the problem of scheduling n jobs with a common due date on a single machine. The processing time of each job is a random variable, which follows an arbitrary distribution with a known mean and a known variance. The machine is not reliable; it is subject to stochastic breakdowns. The objective is to minimize the expected sum of squared deviations of job completion times from the due date. Two versions of the problem are addressed. In the first one the due date is a given constant, whereas in the second one the due date is a decision variable. In each case, a general form of the deterministic equivalent of the stochastic scheduling problem is obtained when the counting process related to the machine uptime distribution is a generalized Poisson process. A sufficient condition is derived under which optimal sequences are V-shaped with respect to mean processing times. Other characterizations of optimal solutions are also established. Based on the optimality properties, algorithms with pseudopolynomial time complexity are proposed to solve both versions of the problem. © 1996 John Wiley & Sons, Inc.     

装备的可用度与保障资源延误概率

通过对装备可用度进行讨论,尤其对使用可用度Ao给出了深入分析,引入了修复性维修保障资源延误概率及预防性维修保障资源延误概率,使Ao的表达式更加清晰,并指出影响使用可用度的6个因素,最后给出了一个计算实例.     

Maintenance scheduling for modular systems: Modeling and algorithms

We study new models of scheduled maintenance management for modular systems, consisting of multiple components with respective cycle limits. The cycle limit of each component specifies the time interval in which this component must be repaired or replaced. The goal is to compute a feasible maintenance schedule that minimizes the cost associated with component maintenance. Applications of these models arise in Air Force aircraft maintenance as well as in other arenas with required preventive maintenance. The typical cost structures that arise in practical settings are submodular, which make the resulting models computationally challenging. We develop two efficient and operationally tenable approximation algorithms. We prove constant factor worst‐case guarantees for both algorithms, and present computational experiments showing that these algorithms perform within a few percent of optimality on operationally relevant instances. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 472–488, 2014     

基于二维量度的复杂设备预防性维修决策优化

针对使用与维修具有两个测量维度的复杂设备,开展了其预防性维修决策的优化研究。基于二维量度的故障模式,给出了二维故障率的定量描述方法;分析了其预防性更换过程的基本过程,探讨了二维量度下更换周期对维修效果的影响,并从经济性角度建立了二维工龄更换费用模型;最后,采用算例的形式,对某设备维修决策同时考虑日历使用时间和行驶里程的情况,进行了二维更换间隔期的优化求解,从而验证了所建立方法与模型的实用性。     

Nonperiodic preventive maintenance for repairable systems

As a complex system with multiple components usually deteriorates with age, preventive maintenance (PM) is often performed to keep the system functioning in a good state to prolong its effective age. In this study, a nonhomogeneous Poisson process with a power law failure intensity is used to describe the deterioration of a repairable system, and the optimal nonperiodic PM schedule can be determined to minimize the expected total cost per unit time. However, since the determination of such optimal PM policies may involve numerous uncertainties, which typically make the analyses difficult to perform because of the scarcity of data, a Bayesian decision model, which utilizes all available information effectively, is also proposed for determining the optimal PM strategies. A numerical example with a real failure data set is used to illustrate the effectiveness of the proposed approach. The results show that the optimal schedules derived by Bayesian approach are relatively more conservative than that for non‐Bayesian approach because of the uncertainty of the intensity function, and if the intensity function are updated using the collected data set, which indicates more severe deterioration than the prior belief, replacing the entire system instead of frequent PM activities before serious deterioration is suggested. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Assessment of machine tool reliability using a proportional hazards model

In this article we present a proportional hazards model for describing machine tool failures. The model enables us to describe the effect of aging as well as the effect of the machining environment on tool life. We discuss inference for the model and show that computable results can be obtained in a fully Bayesian analysis by using approximation techniques. We illustrate the usefulness of our model by applying it to some actual data on machine tool failures.