Optimal age-replacement model with age-dependent type of failure and random lead time based on a cumulative repair-cost limit policy

In this paper, we consider an age-replacement model with minimal repair based on a cumulative repair cost limit and random lead time for replacement delivery. A cumulative repair cost limit policy uses information about a system’s entire repair cost history to decide whether the system is repaired or replaced; a random lead time models delay in delivery of a replacement once it is ordered. A general cost model is developed for the average cost per unit time based on the stochastic behavior of the assumed system, reflecting the costs of both storing a spare and of system downtime. The optimal age for preventive replacement minimizing that cost rate is derived, its existence and uniqueness is shown, and structural properties are presented. Various special cases are included, and a numerical example is given for illustration. Because the framework and analysis are general, the proposed model extends several existing results.     

Interrupt and Opportunistic Replacement Strategies for Systems of Deteriorating Components

In some factories production epochs occur that allow system components to be replaced at reduced cost. Over a long production run the unit cost of replacing these stochastically deteriorating components can be controlled by decisions which govern when production is to be interrupted for component replacement and when components are to be replaced at the reduced cost replacement opportunities. This paper develops and analyses models for optimizing "interrupt and opportunistic" replacement strategies in simple systems. Numerical results are given that illustrate the advantages of combining interrupt replacement with opportunistic replacement.     

A marginal cost analysis for prevetive replacement policies

Some well known preventive replacement policies are each characterized by a single parameter which governs the sequence of planned replacements. In this paper we define and compute for any one of these policies the marginal cost of a planned replacement in terms of the policy parameter. This function is used to obtain the equation of optimality, with respect to two common objective functions: the expected cost per unit time and the expected total discounted costs, and to investigate the existence of a unique finite solution. Moreover, the marginal cost notion provides reasoning to the mathematical results of the models and thereby clarifies the effect of the various components of the model on these results. This better insight into models is perhaps the most important benefit of a marginal cost analysis.     

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 maintenance scheduling for a complex manufacturing system subject to deterioration

We address the problem of determining inspection strategy and replacement policy for a deteriorating complex multi-component manufacturing system whose state is partially observable. We develop inspection and replacement scheduling models and other simple maintenance scheduling models via employing an imperfect repair model coupled with a damage process induced by operational conditions. The system state in performance of the imperfectly repaired system is modelled using a proportional intensity model incorporating a damage process and a virtual age process caused by repair. The system is monitored at periodic times and maintenance actions are carried out in response to the observed system state. Decisions to perform imperfect repair and replacement are based on the system state and crossing of a replacement threshold. The model proposed here aims at joint determination of a cost-optimal inspection and replacement policy along with an optimal level of maintenance which result in low maintenance cost and high operational performance and reliability of the system. To demonstrate the use of the model in practical applications a numerical example is provided. Solutions to optimal system parameters are obtained and the response of the model to these parameters is examined. Finally some features of the model are demonstrated. The approach presented provides a framework so that different scenario can be explored.     

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.     

Operational asset replacement strategy: A real options approach

This paper analyses the problem of replacement by investigating the optimal moment of investment replacement in a given tax environment with a given depreciation policy. An operation and maintenance cost minimization model, based on the definition of equivalent annual cost, is applied to a real options paradigm. The developed methodology allows for an innovative evaluation of the flexibility of replacement process analysis. A new two-factor evaluation function is introduced to quantify decisions on asset replacement under a unique cycle environment. This study improves upon previous findings in the literature as it accounts for autonomous salvage value processes. Based on partial differential equations, this model achieves a general analytical solution and particular numerical solution. The results differ significantly from those observed in one-factor models by showing evidence of over-evaluation in optimal levels of replacement, and by confirming suspicions that different types of uncertainties produce non-monotonous effects on the optimal replacement level. The scientific contribution of this study lies in new and stronger approaches to equivalent annual cost literature, supplying an algorithm for operation and maintenance cost minimization that is conditioned by autonomous salvage value. This study also contributes to the real options literature by developing a two-factor model with Brownian processes applied to asset replacement.     

Analysis of the asset replacement level with an uncertain salvage value: a two-factor model

The aim of this article is to analyse the asset replacement problem from the perspective of optimal replacement level, given a specific tax environment and depreciation policy. Using a real options approach, our model minimizes current operation and maintenance costs and allows the definition of a new value of the replacement flexibility within a multi-cycle environment. The innovation on the valuation process comes from adding an autonomous salvage value’s factor. The results from partial differential equations reveal relevant differences from those observed in one-factor models, especially with regard to optimal replacement levels and in the non-monotonous effects of salvage value variation. The numerical case study also confirms that the salvage value is indeed a worthwhile element in the replacement process. It was possible to determine that, in terms of the magnitude of the cost replacement level, the key roles are played by changes in the speed of mean reversion, as well in the salvage factor volatility. This paper provides some improvements to the existing literature in equivalent annual cost by drawing up a cost minimization problem conditioned by a different salvage value dynamics, and contributes to real options literature by introducing a salvage value factor in the pricing model.     

An Equipment Replacement Problem

This paper describes a study undertaken to develop a model for the replacement of a particular type of machine. The dominant operating costs are identified, and existing replacement models reviewed. One of the most important factors is the cost of production stoppages which can sometimes result from the breakdown of these machines. In order to predict the effects of this in terms of the machines' age, a simulation model is developed.The results from the replacement model are investigated in terms of their sensitivity to the variability in the estimates of the parameters required by the model. In particular some interesting results relating the method used for calculating the resale values and the optimal replacement interval are presented.     

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.     

Following the traces:: An introduction to conjoint measurement without transitivity and additivity

This paper presents a self-contained introduction to a general conjoint measurement framework for the analysis of nontransitive and/or incomplete binary relations on product sets. It is based on the use of several kinds of marginal traces on coordinates induced by the binary relation.This framework leads to defining three general families of models depending on the kind of trace that they use. Contrary to most conjoint measurement models, these models do not involve an addition operation. This allows for a simple axiomatic analysis at the cost of very weak uniqueness results.     

First Steps Towards an Institution of Algebra Replacement Systems

Algebra replacement systems are introduced as formal models of state dependent and state transforming systems. The first part of an institution of algebra replacement systems is developed, that is, a model theoretic and logical framework that can be used to describe and reason about such systems. The usual operational understanding of a replacement system as a labeled transition system is then considered as one particular model in the model category. Under appropriate conditions such a constructed replacement system is initial.     

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.     

Optimal repair–replacement policies for a system with two types of failures

In this paper, the optimal replacement problem is investigated for a system with two types of failures. One type of failure is repairable, which is conducted by a repairman when it occurs, and the other is unrepairable, which leads to a replacement of the system at once. The repair of the system is not “as good as new”. The consecutive operating times of the system after repair form a decreasing geometric process, while the repair times after failure are assumed to be independent and identically distributed. Replacement policy N is adopted, where N is the number of repairable failures. The system will be replaced at the Nth repairable failure or at the unrepairable failure, whichever occurs first. Two replacement models are considered, one is based on the limiting availability and the other based on the long-run average cost rate of the system. We give the explicit expressions for the limiting availability and the long-run average cost rate of the system under policy N, respectively. By maximizing the limiting availability A(N) and minimizing the long-run average cost rate C(N), we theoretically obtain the optimal replacement policies N^∗ in both cases. Finally, some numerical simulations are presented to verify the theoretical results.     

Optimal cost and policy for a Markovian replacement problem

We consider the computation of the optimal cost and policy associated with a two-dimensional Markov replacement problem with partial observations, for two special cases of observation quality. Relying on structural results available for the optimal policy associated with these two particular models, we show that, in both cases, the infinitehorizon, optimal discounted cost function is piecewise linear, and provide formulas for computing the cost and the policy. Several examples illustrate the usefulness of the results.This research was supported by the Air Force Office of Scientific Research Grant AFOSR-86-0029, by the National Science Foundation Grant ECS-86-17860, by the Advanced Technology Program of the State of Texas, and by the Air Force Office of Scientific Research (AFSC) Contract F49620-89-C-0044.     

Lower and upper bounds for location-arc routing problems with vehicle capacity constraints

This paper addresses multi-depot location arc routing problems with vehicle capacity constraints. Two mixed integer programming models are presented for single and multi-depot problems. Relaxing these formulations leads to other integer programming models whose solutions provide good lower bounds for the total cost. A powerful insertion heuristic has been developed for solving the underlying capacitated arc routing problem. This heuristic is used together with a novel location–allocation heuristic to solve the problem within a simulated annealing framework. Extensive computational results demonstrate that the proposed algorithm can find high quality solutions. We also show that the potential cost saving resulting from adding location decisions to the capacitated arc routing problem is significant.     

On the Marginal Cost Approach in Maintenance

In this paper we investigate the conditions under which the marginal cost approach of Refs. 1–3 holds. As observed in Ref. 4, the validity of the marginal cost approach gives rise to a useful framework of single-component maintenance optimization models, which covers almost all models used in practice. For the class of unimodal finite-valued marginal cost functions, we show that these optimization models are easy to solve.     

Comments on Finite-Period Applications of Age-Based Replacement Models

The age-based replacement problem is considered when a finitetime zone is placed upon the duration of the process and, accordingly,for which an asymptotic model could be of questionable validity.Three models to be optimized with respect to the decision variableare considered: first, the commonly used limiting asymptoticcost per unit time; secondly, the refined asymptotic cost perunit time; and thirdly, the exact expected cost. These modelsare investigated in the case of a negative exponential and Weibullfailure-time distribution. In both cases, it is found that theestimated cumulative cost resulting from the refined model approachesquickly that of the exact formulation. The practical use and decision consequences of this result areinvestigated for a Weibull failure distribution. It is arguedthat, because of the essential illconditioned nature of theage based replacement model, for short time-zone applications,the model should be extended to accommodate non-zero replacementtimes.     

On the average cost optimality equation and the structure of optimal policies for partially observable Markov decision processes

We consider partially observable Markov decision processes with finite or countably infinite (core) state and observation spaces and finite action set. Following a standard approach, an equivalent completely observed problem is formulated, with the same finite action set but with anuncountable state space, namely the space of probability distributions on the original core state space. By developing a suitable theoretical framework, it is shown that some characteristics induced in the original problem due to the countability of the spaces involved are reflected onto the equivalent problem. Sufficient conditions are then derived for solutions to the average cost optimality equation to exist. We illustrate these results in the context of machine replacement problems. Structural properties for average cost optimal policies are obtained for a two state replacement problem; these are similar to results available for discount optimal policies. The set of assumptions used compares favorably to others currently available.This research was supported in part by the Advanced Technology Program of the State of Texas, in part by the Air Force Office of Scientific Research under Grant AFOSR-86-0029, in part by the National Science Foundation under Grant ECS-8617860, and in part by the Air Force Office of Scientific Research (AFSC) under Contract F49620-89-C-0044.     

The effect of tax depreciation on the stochastic replacement policy

The optimal replacement policy for an asset subject to a stochastic deteriorating operating cost is determined for three different tax depreciation schedules and a known re-investment cost, as the solution to a two-factor model using a quasi-analytical method. We find that tax depreciation exerts a critical influence over the replacement policy by lowering the operating cost thresholds. Although typically a decline in the corporate tax rate, increase in any initial capital allowance, or decrease in the depreciation lifetime (increase in depreciation rate) results in a lower operating cost threshold which justifies replacing older equipment, these results are not universal, and indeed for younger age assets the result may be the opposite. An accelerating depreciation schedule may incentivize early replacement in a deterministic context, but not necessarily for an environment of uncertainty.