Multi-parameter maintenance optimisation via the marginal cost approach

In this paper we show how the marginal-cost approach can be used to optimise multi-parameter replacement rules. We will illustrate this for an opportunity-based age replacement rule that consists of two parameters. The first parameter is a control limit t, which indicates from what age on a unit is replaced preventively at the first arising opportunity. The second parameter is a planned replacement age T, which indicates at what age the unit is replaced if it has not been replaced yet. The unit can fail and is immediately replaced upon failure. It can be shown that this replacement rule belongs to a class of policies for which the long-run average-cost function is unimodal. The marginal cost approach is based on the following assertion: any point, in which the marginal cost(s) of deferring maintenance equals the average-cost, is an average-cost minimum. Assuming unimodality the minimisation problem can be solved as a root-finding problem, for which there are numerous efficient routines. It appears that the marginal cost approach is very practical for the optimisation of the considered replacement rule, especially because a quick assessment can be made of the optimal parameter values. The marginal cost approach can be used for many other multi-parameter problems, insofar as they can be modelled as a regenerative process.     

Marginal cost analysis of single-item maintenance policies with several decision variables

The marginal cost approach for the analysis of repair/replacementmodels was introduced by Berg in 1980 and has since been appliedto many maintenance policies of various complexity. All modelshitherto analysed in the literature by the marginal cost approachhave one single decision variable only, this being, typically,the age of the current item at the time of ordering or replacement.This paper is concerned with the extension of the marginal costtechnique to maintenance policies with several decision variables.After addressing the general framework appropriate for the multi-parametercase, we exemplify the workings of the technique by analysinga two-variable maintenance model involving replacement and minimalrepair. We demonstrate that the marginal cost approach is anattractive and intuitively appealing technique also for modelswith several decision variables. Just as in the single-parametersituation, the approach is amenable to economic interpretation,a welcome feature for users of maintenance models with a primeinterest in its economic (rather than its mathematical) aspects.As an added bonus of the marginal cost approach, in our example,some otherwise necessary tools from the theory of stochasticprocesses are dispensable.     

Optimal periodic replacement policy for repairable products under free-repair warranty

This paper investigates the effects of a free-repair warranty on the periodic replacement policy for a repairable product. Cost models are developed for both a warranted and a non-warranted product, and the corresponding optimal periodic replacement policies are derived such that the long-run expected cost rate is minimized. For a product with an increasing failure rate function, structural properties of these optimal policies are obtained. By comparing these optimal policies, we show that the optimal replacement period for a warranted product should be adjusted toward the end of the warranty period. Finally, examples are given to numerically illustrate the impact of a product warranty on the optimal periodic replacement policy.     

A general age-replacement model with minimal repair under renewing free-replacement warranty

A general age-replacement model in which incorporates minimal repair, planned and unplanned replacement, is considered in this paper for products under a renewing free-replacement warranty policy. For both warranted and non-warranted products, cost models from the user’s perspective are developed, and the corresponding optimal replacement ages are derived such that the long-run expected cost rate is minimized. The impacts of a product warranty on the optimal replacement model are investigated analytically. Furthermore, we show that the optimal replacement age for a warranted product is closer to the end of the warranty period than for a non-warranted product. Finally, numerical examples are given for illustration.     

Optimization problems of replacement first or last in reliability theory

This paper takes up age and periodic replacement last models with working cycles, where the unit is replaced before failure at a total operating time T or at a random working cycle Y, whichever occurs last, which is called replacement last. Expected cost rates are formulated, and optimal replacement policies which minimize them are discussed analytically. Comparisons between such a replacement last and the conventional replacement first are made in detail. It is determined theoretically and numerically which policy is better than the other according to the ratios of replacement costs and how the mean time of working cycles affects the comparison results. It is also shown that the unit can be operating for a longer time and avoid unnecessary replacements when replacement last is done. For further studies, expected cost rates of modified models and their applications in a standard cumulative damage model with working cycles are obtained and computed numerically. Finally, case studies on replacement last and first in maintaining electronic systems of naval ships under battle and non-battle statuses are given.     

Optimal Replacement Policy for a General Model with Imperfect Repair

A general model is considered which incorporates imperfect repair and repair cost which depends on time and on the number of repairs in the cycle. This model is an extension of models examined previously in the literature. The objective of this paper is to find the optimal replacement policy and compare it with the replacement policies considered earlier for some variants of this model. The form of the optimal replacement policy is found in the general case and the expected average cost per unit time is derived in two special cases. Numerical examples show that the optimal policy is considerably better than the optimal periodic policy. This paper generalizes and unifies previous research in the area.     

Quantifying the risk in age and block replacement policies

Preventive maintenance policies have been studied in the literature without considering the risk due to the cost variability. In this paper, we consider the two most popular preventive replacement policies, namely, age and block replacement policies under long-run average cost and expected unit time cost criteria. To quantify the risk in the preventive maintenance policies, we use the long-run variance of the accumulated cost over a time interval. We numerically derive the Risk-sensitive preventive replacement policies and study the impact of the Risk-sensitive optimality criterion on the managerial decisions. We also examine the performance of the expected unit time cost criterion as an alternative to the traditional long-run average cost criterion.     

Marginal productivity index policies for scheduling a multiclass delay-/loss-sensitive queue

This paper addresses the problem of scheduling a Markovian multiclass queue with a finite dedicated buffer for each class, where class-dependent linear holding and rejection cost rates model differing levels of tolerance to delay and loss. The goal is to design well-grounded and tractable scheduling policies that nearly minimize expected total discounted or long-run average cost. New dynamic index policies are introduced, awarding higher priority to classes with larger index values, where a class’ index measures the marginal productivity of work at its current state. The results are obtained by deploying the work-cost analysis approach to marginal productivity indices (MPIs) for restless bandits developed by the author, which is extended to the bias criterion. The MPI furnishes new insights: for a loss-sensitive class, it is a decreasing function of the number of empty buffer spaces, independent of the buffer size; for a delay-sensitive class, it is a decreasing function of the queue length. Such opposite orderings show that preventive work is more valuable than reactive work for the latter classes, whereas the opposite holds for the former. The results of a computational study on two-class instances are reported, shedding light on how the MPI policy’s relative performance varies with each parameter. Parameter ranges are thus identified where the MPI policy is near optimal, and substantially outperforms conventional benchmark policies. 2000 Mathematics Subject Classification 90B22 · 90B36 · 90B18 · 60K25 · 60K30 · 68M20     

Replacement Policies with Used Items Constraint

Various block replacement policies with used items have appeared in the literature. These policies use new as well as used items (created by the policies) for replacement. In these policies no constraint is placed on the number of used items used in each cycle. This paper considers these policies with a constraint (of practical significance) to ensure that the expected number of used items used per cycle is less than or equal to the expected number of used items created per cycle.     

Optimal Repair Limit Replacement Policies with Imperfect Repair

This paper considers the repair limit replacement policies with imperfect repair. The repair is imperfect in the sense that the mean life of a repaired system is less than the mean life of a new system. Furthermore, we examine the repair limit replacement policy for the case in which there are two types of repair-local and central repair. The local repair is imperfect whilst the central repair is perfect (i.e. the system is as good as new after central repair). The optimal policies are derived to minimize the expected cost per unit of time for an infinite time span. Analytical results are presented along with numerical examples.     

Repair Replacement Modelling over Finite Time Horizons

In this paper an integral equation approach is given for evaluating the expected cost of repair replacement policies over finite time horizons. An asymptotic estimate of this expected cost is also obtained. The policy involving imperfect repair on failure with replacement after N failures is taken as an illustrative example and optimal policies N* are found for both infinite and finite time horizons of use.     

A note on replacement policy for a system subject to non-homogeneous pure birth shocks

A system is subject to shocks that arrive according to a non-homogeneous pure birth process. As shocks occur, the system has two types of failures. Type-I failure (minor failure) is removed by a general repair, whereas type-II failure (catastrophic failure) is removed by an unplanned replacement. The occurrence of the failure type is based on some random mechanism which depends on the number of shocks occurred since the last replacement. Under an age replacement policy, a planned (or scheduled) replacement happens whenever an operating system reaches age T. The aim of this note is to derive the expected cost functions and characterize the structure of the optimal replacement policy for such a general setting. We show that many previous models are special cases of our general model. A numerical example is presented to show the application of the algorithm and several useful insights.     

Economic interpretation of models for the replacement of machines

The purpose of this article is to give an economic interpretation of results derived from mathematical replacement models. It is based on results found in the literature. The conclusion is that the moment of replacement is always determined by the equality of marginal replacement costs to the marginal costs of postponed replacement. This extension gives the models practical applicability by providing an exact decision criterion and a concrete interpretation. In other words: practical decisions are given a theoretical foundation.     

Renewal replacement policies for one unit systems

A simple and unifying presentation of a class of renewal replacement policies is given, to which all well-known policies belong. The introduced expected cost rate function turns out to be a helpful tool when deriving the objective function. Due to it the objective function can be expressed in a basic form from which characteristics common to the different policies can be easily identified.Furthermore an economic interpretation may be given to several results.     

Optimal threshold values of age and two-phase maintenance policy for leased equipments using age reduction method

This paper investigates the optimal threshold values of age to perform preventive maintenance (PM) actions for leased equipment within the lease period. In this paper, we use age reduction method to describe the degree of PM and construct the maintenance cost function. For repairable leased equipment, two maintenance models are proposed: (i) maintenance policy of single-phase and (ii) maintenance policy of two-phase. During the lease period, PM actions are carried out when the age of equipment reaches a certain threshold value. Any failure of the leased equipment is rectified by a minimal repair within the lease period. Under these maintenance schemes, the mathematical models of the expected total cost for maintenance policies of single-phase and two-phase are established, and the optimal maintenance policies are derived such that the expected total cost is minimized. Finally, the features of the optimal maintenance policy are illustrated through numerical examples.     

Simultaneous optimization of capacity and planned lead time in a two-stage production system with different customer due dates

We consider a two-stage make-to-order manufacturing system with random demands, processing times, and distributed customer due dates. The work to each stage is released based on a planned lead time. A general approach to minimize total inventory holding and customer order tardiness cost is presented to find the optimal manufacturing capacities and planned lead times for each manufacturing stage. Expressions are derived for work-in process inventories, finished-goods-inventory and expected backorders under the assumption of a series of M/M/1 queuing systems and exponentially distributed customer required lead times. We prove that the distribution of customer required lead time has no influence on the optimal planned lead times whenever capacity is predefined but it influences the optimal capacity to invest into. For the simultaneous optimization of capacity and planned lead times we present a numerical study that shows that only marginal cost decreases can be gained by setting a planned lead time for the upstream stage and that a considerable cost penalty is incurred if capacity and planned lead time optimization are performed sequentially.     

A unified treatment of single component replacement models

In this paper we discuss a general framework for single component replacement models. This framework is based on the regenerative structure of these models and by using results from renewal theory a unified presentation of the discounted and average finite and infinite horizon cost models is given. Finally, some well-known replacement models are discussed, and making use of the previous results an easy derivation of their cost functions is presented.     

An Economic Replacement Model

The report is based on a study of the replacement of Army vehicles. These vehicles are currently phased out of service by a system of decision rules called repair limits. Existing repair limit systems, although superior to group replacement policies, appear to be sub-optimum.A theory of repair limits is proposed. This leads to the determination of optimum repair limits by a simulation method. A second optimization method is introduced based on the use of certain frequency distributions which are found to represent the repair cost data.The results show that the methods proposed can be expected to lead to financial savings.     

Manufacturing delivery performance for supply chain management

This paper considers single-stage make-to-order production systems. We focus on (1) modeling the appropriate expected costs under a variety of modeling assumptions and (2) characterizing the optimal policies. Our approach to solving the problem is to derive the distribution of actual completion times of the process for individual orders and to compare it to the corresponding quoted due dates in order to obtain the expected total costs. We then show the convexity of the objective cost function for determining the decision variable(s), the planned customer order leadtime.     

自动化车床最优刀具检测模型研究

针对自动化车床工序最优检测和刀具更换问题进行了探讨.将定期检测和将刀具更换作用于同一工序流程,在只考虑刀具故障条件下,通过概率论和更新过程理论建立了以单位时间内期望费用为目标函数的数学模型,以检测间隔和刀具更换间隔为策略,确定最优的策略使得目标函数达到最小,并求出了经长期运行单位时间内期望费用的明显表达式.最后还对结果进行了讨论.