A stochastic model for joint spare parts inventory and planned maintenance optimisation

Spare parts demands are usually generated by the need of maintenance either preventively or at failures. These demands are difficult to predict based on historical data of past spare parts usages, and therefore, the optimal inventory control policy may be also difficult to obtain. However, it is well known that maintenance costs are related to the availability of spare parts and the penalty cost of unavailable spare parts consists of usually the cost of, for example, extended downtime for waiting the spare parts and the emergency expedition cost for acquiring the spare parts. On the other hand, proper planned maintenance intervention can reduce the number of failures and associated costs but its performance also depends on the availability of spare parts. This paper presents the joint optimisation for both the inventory control of the spare parts and the Preventive Maintenance (PM) inspection interval. The decision variables are the order interval, PM interval and order quantity. Because of the random nature of plant failures, stochastic cost models for spare parts inventory and maintenance are derived and an enumeration algorithm with stochastic dynamic programming is employed for finding the joint optimal solutions over a finite time horizon. The delay-time concept developed for inspection modelling is used to construct the probabilities of the number of failures and the number of the defective items identified at a PM epoch, which has not been used in this type of problems before. The inventory model follows a periodic review policy but with the demand governed by the need for spare parts due to maintenance. We demonstrate the developed model using a numerical example.     

A multi-objective approach for solving a replacement policy problem for equipment subject to imperfect repairs

This paper proposes a multi-objective approach to model a replacement policy problem applicable to equipment with a predetermined period of use (a planning horizon), which may undergo critical and non-critical failures. Corrective replacements and imperfect repairs are taken to restore the system to operation respectively when critical and non-critical failures occur. Generalized Renewal Process (GRP) is used to model imperfect repairs. The proposed model supports decisions on preventive replacement intervals and the number of spare parts purchased at the beginning of the planning horizon. A Multi-Objective Genetic Algorithm (MOGA) coupled with discrete event simulation (DES) is proposed to provide a set of solutions (Pareto-optimum set) committed to the different objectives of a maintenance manager in the face of a replacement policy problem, that is, maintenance cost, rate of occurrence of failures, unavailability, and investment on spare parts. The proposed MOGA is validated by an application example against the results obtained via the exhaustive approach. Moreover, examples are presented to evaluate the behavior of objective functions on Pareto set (trade-off analysis) and the impact of the repair effectiveness on the decision making.     

考虑不完美检测的两阶段点检与备件订购策略联合优化

针对设备维修与备件管理相互影响与制约的问题, 在基于延迟时间理论的基础上, 提出了两阶段点检与备件订购策略联合优化。点检是不完美的, 当点检识别设备的缺陷状态时, 进行预防更新; 设备故障时, 进行故障更新。结合设备更新时备件的状态, 采用更新报酬理论建立了以第一阶段点检时间、第二阶段点检周期和备件订购时间为决策变量, 以最小化单位时间期望成本为目标的模型。最后, 通过人工蜂群算法对模型求解, 并在数值分析中将两阶段点检策略与定期点检策略进行比较, 结果表明:两阶段点检策略始终优于定期点检策略, 验证了所建模型的有效性。     

Failure replacement and preventive maintenance spare parts ordering policy

This paper addresses inventory policy for spare parts, when demand for the spare parts arises due to regularly scheduled preventive maintenance, as well as random failure of units in service. A stochastic dynamic programming model is used to characterize an ordering policy which addresses both sources of demand in a unified manner. The optimal policy has the form (s(k),S(k)), where k is the number of periods until the next scheduled preventive maintenance operation. The nature of the (s(k),S(k)) policy is characterized through numeric evaluation. The efficiency of the optimal policy is evaluated, relative to a simpler policy which addresses the failure replacement and preventive maintenance demands with separate ordering policies.     

Two-Stage Replacement Strategies

The following replacement problem is considered. N items, which are subject to failure, can be divided into two groups distinguished by the fact that the individual replacement cost in one group is higher than in the other. A strategy is required to minimize replacement costs. In some cases the cheapest policy is to replace each item, when it fails, by a new item. However, the paper shows that this policy can usually be improved upon by what is called a two-stage policy. In a two-stage policy the failures in one group are replaced by new items; those in the other group are replaced by items already operating in the first group. Under some circumstances it is shown to be worth while to create a second group. Formulae are given for calculating the optimum two-stage strategies for any life distribution, but the emphasis is on the formulation of general conditions under which two-stage schemes are preferable to simple replacement. Some extensions and generalizations are briefly indicated.     

Optimum Redundancy with Equipment that Operates and Idles

Despite their widespread exclusion from the literature on reliability and replacement, many items of equipment are subject to failures to idle as well as failures to operate. For such equipment a new optimisation problem arises at the stage of systems design in terms of the optimal use of redundancy to maximise expected systems life. The models hitherto used as the basis for the solution of this optimization problem ignore the dependence that should exist for the solutions upon the operating and idling requirements to which the system is to be exposed. In this paper a simple model for equipment subject to such opposite failure modes is constructed which takes explicit account of the proportion of time the equipment is in use. The implications of this new model for the selection of an optimum redundancy configuration are illustrated for the case where four identical items of equipment are available.     

Analysis of a multi-part spares inventory system subject to ambiguous fault isolation

This paper considers a multi-part spares inventory model for a maintenance system composed of a spares-stocking centre and a repair centre. In the maintenance system, multi-part spares are jointly needed to repair faulty end items determined by ambiguous fault isolation done by the built-in-test-equipment (BITE). If any operating end item breaks down, then all associated parts should be replaced either iteratively (one at a time) or altogether. In the case of iterative replacement, failed parts are detected in the field after fixing the broken end item. However, in the case of group (altogether) replacement, all removed parts are sent to the repair centre where the failed part is detected and fixed. The repaired or non-failed parts are then restocked at the spares-stocking centre. For the system, this paper is to derive the exact expressions for the distribution function and the expected numbers of the backlogged end items under the cannibalization policy. These expressions are then used in the optimization of the associated spares' inventory level. Illustrative numerical examples are also presented.     

Intermediate buffer analysis for a production system

In this paper, a production system consisting of two serial machines and an intermediate buffer is studied. A shortage cost is incurred when the upstream machine is down and the buffer is exhausted. The practical example for this type of system can be an automated work center or an automobile general assembly.Researches on a similar two-machine system have been done in some articles where maintenance and an intermediate buffer are considered, but the spare parts are not involved. Nevertheless, spare parts are essential for maintenance implementation, and there is interaction between the buffer inventory and the spare parts due to maintenance activity. This paper is aimed to investigate three types of cost related to the intermediate buffer inventory, and obtain their expectations as functions of several decision parameters on maintenance, buffer, and spare parts during a renewal cycle, by using mathematical analysis. The proposed method can be an important basis for further study of system cost calculation and decision making optimization.     

New failure and minimal repair processes for repairable systems in a random environment

Most often, minimal repair is defined as a replacement of a failed item by an operable item that has the same distribution of the remaining lifetime as the failed one just prior a failure. This is the so‐called statistical minimal repair extensively explored in the literature. Another well‐known type of minimal repair takes into account the state of a system prior to a failure (the information‐based minimal repair). In this paper, we suggest the new type of minimal repair to be called conditional statistical minimal repair. Our approach goes further and deals with the corresponding minimal repair processes for systems operating in a random environment. Moreover, we also consider heterogeneous populations of items, which makes the model more realistic. Both of these aspects that affect the failure mechanism of items are studied. Environment is modeled by the nonhomogeneous Poisson shock process. Two models for the failure mechanism defined by the extreme shock model and the cumulative shock model, respectively, are considered. Some examples illustrating our findings are presented.     

A model of imperfect preventive maintenance with dependent failure modes

Consider a system subject to two modes of failures: maintainable and non-maintainable. A failure rate function is related to each failure mode. Whenever the system fails, a minimal repair is performed. Preventive maintenances are performed at integer multiples of a fixed period. The system is replaced when a fixed number of preventive maintenances have been completed. The preventive maintenance is imperfect because it reduces the failure rate of the maintainable failures but does not affect the failure rate of the non-maintainable failures. The two failure modes are dependent in the following way: after each preventive maintenance, the failure rate of the maintainable failures depends on the total of non-maintainable failures since the installation of the system. The problem is to determine an optimal length between successive preventive maintenances and the optimal number of preventive maintenances before the system replacement that minimize the expected cost rate. Optimal preventive maintenance schedules are obtained for non-decreasing failure rates and numerical examples for power law models are given.     

A condition-based order-replacement policy for a single-unit system

This paper presents a condition-based order-replacement policy for a single-unit system, aiming to optimize the condition-based maintenance and the spare order management jointly. The concerned system deteriorates stochastically and gradually, and is inspected periodically. Under the proposed policy, both the preventive replacement and the spare order are decided based on the observed deterioration level of the system. Therefore, the decision variables for this order-replacement problem include the inspection interval, the ordering threshold, and the preventive replacement threshold. The analytical modeling of the condition-based order-replacement policy is presented in detail in this paper. The policy performance is evaluated in terms of the long-run average cost per unit time, the mean availability, and the rate of preventive replacement, for which the mathematical models are also derived. Numerical examples illustrate the performance of the condition-based order-replacement policy, especially the influences of the lead time of the spare order over the different performance criteria.     

考虑非周期检测的视情维修与备件订购联合策略优化

针对单部件系统/关键部件提出视情维修与备件订购联合策略,其中系统退化服从两阶段延迟时间过程且采用非周期检测策略,退化初期以检测间隔T₁检查系统状态,而在第一次识别缺陷状态时,缩短检测周期为T₂、订购备件且进行不完美维修;若系统在随后的退化中被识别处于缺陷状态,执行不完美维修直至超过阈值次数N_max并采取预防性更换,但若在检测周期内发生故障则进行更换。根据系统状态和备件状态分析各种可能更新事件及相应的联合决策,利用更新报酬理论构建最小化单位时间内期望成本的目标函数,优化T₁,T₂, N_max。与对比模型策略相比,算例结果表明所提出的联合策略能有效降低单位时间内的期望成本。     

Optimal provisioning strategies for slow moving spare parts with small lead times

When an expensive piece of equipment is bought, spare parts can often be bought at a reduced price. A decision must be made about the initial provisioning of spare parts. Furthermore, if at a certain time the stock drops to zero, because a number of failures have occurred, a decision must be made about the number of parts to be ordered. We focus on one specific expensive slow-moving part that is essential for the functioning of the equipment. The lead time of that part is small compared to its lifetime. We seek an ordering strategy for this part that covers the entire lifetime. Such a strategy should also be applicable to cases where there is uncertainty about the lifetime of the machine, the reliability of the components and the cost of failure. The main result is the development of a simple strategy that performs well in almost all cases. Furthermore, this strategy can easily deal with uncertainty and changes in the parameters.     

An improved stochastic programming model for supply chain planning of MRO spare parts

The maintenance, repair and operation (MRO) spare parts that are vital to machine operations are playing an increasingly important role in manufacturing enterprises. MRO spare parts supply chain management planning must be coordinated to ensure spare part availability while keeping the total cost to a minimum. Due to the specificity of MRO spare parts, randomness and uncertainties in production and storage should be quantified to formulate the problem in a mathematical model. Given these considerations, this paper proposes an improved stochastic programming model for the supply chain planning of MRO spare parts. In our stochastic programming model, the following improvements are made: First, we quantify the uncertain production time capacity as a random variable with a probability distribution. Second, the upper bound of the storage cost is modeled as a multi-choice variable in the constraint. To derive the equivalent deterministic model, the Lagrange interpolating polynomial approach is used. The results of the numerical examples validate the feasibility and efficiency of the proposed model. Finally, the model is tested in the supply chain planning of continuous caster (CC) bearings.     

多货栈及变质情形下两种可替代物品的经济订货批量模型

研究了多货栈及变质情形下两种可替代物品的经济订货批量问题.在计划期内,若某一种易变质物品发生缺货,则可以被另一种易变质物品以一定的替代率代替补充,不同物品有不同的变质率,且要决定租用货栈的数量.以库存系统的总费用最小为目标函数,分别对货栈容量无限与有限的情形建立模型,证明了最优策略存在的唯一性,并分别给出了求解最优订购策略的算法,最后通过一个算例验证了算法的最优性.     

Joint optimization of level of repair analysis and spare parts stocks

In the field of after sales service logistics for capital goods, generally, METRIC type methods are used to decide where to stock spare parts in a multi-echelon repair network such that a target availability of the capital goods is achieved. These methods generate a trade-off curve of spares investment costs versus backorders. Backorders of spare parts lead to unavailability of the capital goods. Inputs in the spare parts stocking problem are decisions on (1) which components to repair upon failure and which to discard, and (2) at which locations in the repair network to perform the repairs and discards. The level of repair analysis (LORA) can be used to make such decisions in conjunction with the decisions (3) at which locations to deploy resources, such as test equipment that are required to repair, discard, or move components. Since these decisions significantly impact the spare parts investment costs, we propose to solve the LORA and spare parts stocking problems jointly. We design an algorithm that finds efficient solutions. In order for the algorithm to be exact and because of its computational complexity, we restrict ourselves to two-echelon, single-indenture problems. In a computational experiment, we show that solving the joint problem is worthwhile, since we achieve a cost reduction of over 43% at maximum (5.1% on average) compared with using a sequential approach of first solving a LORA and then the spare parts stocking problem.     

Consolidating Maintenance Spares

The inventory of spare parts that a firm holds depends on the number of working parts and age of the equipment to be serviced, the expected failure rate associated with each working part, and the acceptable level of service. We model the problem of consolidation of spare parts to reduce overall inventory as an integer program with a nonlinear objective function. A linear reformulation of this model is obtained that helps solve some practical instances. A more compact implicit formulation is developed and solved using a specialized branch-and-price technique. We also demonstrate how this specialized branch-and-price technique is modified to devise a very effective heuristic procedure with a prespecifiable guarantee of quality of solution produced. This provides a practical and efficient methodology for maintenance spare consolidation.     

基于对故障规律统计分析的汽车备件需求预测

汽车备件的需求与汽车故障紧密相关,文章介绍了一种在对汽车故障进行统计分析并确定其分布规律的基础上预测备件需求的方法,预测中需要结合整车保有量的历史数据以及故障与备件的对应表。用统计的方法对某型客车的故障信息进行分析,认为故障的规律可用四种典型的分布进行描述。实例验证了这种方法的准确性高于传统方法,并且在计算机的辅助下可以方便操作。     

提前期不确定条件下的预防性维修备件存储模型研究

在考虑预防性维修周期和提前期不确定的条件下,分别研究备件存储与其相关的维修费用、缺货费用、库存费用以及订购费用等四种费用之间的关系,明确了备件存储量对各项费用的影响.以各项费用总和最小化为目标,构建了提前期不确定条件下的预防性维修备件存储模型.通过备件存储模型的构建,对备件存储过程中的各项成本进行分析,以期对备件库存策略的确定给出一种解决方案.     

An integrated production and preventive maintenance planning model

We are given a set of items that must be produced in lots on a capacitated production system throughout a specified finite planning horizon. We assume that the production system is subject to random failures, and that any maintenance action carried out on the system, in a period, reduces the system’s available production capacity during that period. The objective is to find an integrated lot-sizing and preventive maintenance strategy of the system that satisfies the demand for all items over the entire horizon without backlogging, and which minimizes the expected sum of production and maintenance costs. We show how this problem can be formulated and solved as a multi-item capacitated lot-sizing problem on a system that is periodically renewed and minimally repaired at failure. We also provide an illustrative example that shows the steps to obtain an optimal integrated production and maintenance strategy.