A Maintenance Model with Opportunities and Interrupt Replacement Options

A component has time to failure X with density f(·). Opportunities arise as a Poisson process of rate λ. At an opportunity the component may at choice, be replaced at cost c₂. At failure the components can either be minimally repaired at cost c₁ or replaced at cost c₄. Finally the component may be replaced at any time at an ‘interrupt’ cost of c₃. We derive expressions for the long-run expected cost rate, and give examples of numerical optimisation with respect to control limits on age at an opportunity, and at an interrupt replacement.     

Optimal inspection of a complex system subject to periodic and opportunistic inspections and preventive replacements

This paper proposes two optimization models for the periodic inspection of a system with “hard-type” and “soft-type” components. Given that the failures of hard-type components are self-announcing, the component is instantly repaired or replaced, but the failures of soft-type components can only be detected at inspections. A system can operate with a soft failure, but its performance may be reduced. Although a system may be periodically inspected, a hard failure creates an opportunity for additional inspection (opportunistic inspection) of all soft-type components. Two optimization models are discussed in the paper. In the first, soft-type components undergo both periodic and opportunistic inspections to detect possible failures. In the second, hard-type components undergo periodic inspections and are preventively replaced depending on their condition at inspection. Soft-type and hard-type components are either minimally repaired or replaced when they fail. Minimal repair or replacement depends on the state of a component at failure; this, in turn, depends on its age. The paper formulates objective functions for the two models and derives recursive equations for their required expected values. It develops a simulation algorithm to calculate these expected values for a complex model. Several examples are used to illustrate the models and the calculations. The data used in the examples are adapted from a real case study of a hospital’s maintenance data for a general infusion pump.     

Optimal replacement policy for obsolete components with general failure rates

Identical components are considered, which become obsolete once new‐type ones are available, more reliable and less energy consuming. We envision different possible replacement strategies for the old‐type components by the new‐type ones: either purely preventive, where all old‐type components are replaced as soon as the new‐type ones are available; either purely corrective, where the old‐type ones are replaced by new‐type ones only at failure; or a mixture of both strategies, where the old‐type ones are first replaced at failure by new‐type ones and next simultaneously preventively replaced after a fixed number of failed old‐type components. To evaluate the respective value of each possible strategy, a cost function is considered, which represents the mean total cost on some finite time interval [0, t]. This function takes into account replacement costs, with economical dependence between simultaneous replacements, and also some energy consumption (and/or production) cost, with a constant rate per unit time. A full analytical expression is provided for the cost function induced by each possible replacement strategy. The optimal strategy is derived in long‐time run. Numerical experiments conclude the paper. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

An optimal repair–replacement policy for a cold standby system with use priority

In this paper, the maintenance problem for a cold standby system consisting of two dissimilar components and one repairman is studied. Assume that both component 1 and component 2 after repair follow geometric process repair and component 1 is given priority in use when both components are workable. Under these assumptions, using geometric process repair model, we consider a replacement policy N under which the system is replaced when the number of failures of component 1 reaches N. Our purpose is to determine an optimal replacement policy N1 such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression for the average cost rate of the system is derived and the corresponding optimal replacement policy N1 can be determined analytically or numerically. Finally, a numerical example is given to illustrate some theoretical results and the model applicability.     

Replacement Policies for Components that Deteriorate

For many industrial processes the cost of a component failing in service is sufficient to warrant replacement before failure, but intensive operation of the processes restricts replacement opportunities. A model is proposed where at each opportunity replacement is optional. Dynamic programming methods are used to show that for components that deteriorate as they are used, the best policy is to replace the component if its age exceeds a control limit otherwise to defer replacement.Numerical results are given when the time to failure of the component has a gamma distribution and replacement opportunities occur at random, or are entirely regular. A "rule of thumb" is given for calculating the control limit, and it is shown to be nearly optimal.     

Hybrid block replacement and inspection policies for a multi-component system with heterogeneous component lives

Novel replacement policies that are hybrids of inspection maintenance and block replacement are developed for an n identical component series system in which the component parts used at successive replacements arise from a heterogeneous population. The heterogeneous nature of components implies a mixed distribution for time to failure. In these circumstances, a hybrid policy comprising two phases, an early inspection phase and a later wear-out replacement phase, may be appropriate. The policy has some similarity to burn-in maintenance. The simplest policy described is such a hybrid and comprises a block-type or periodic replacement policy with an embedded block or periodic inspection policy. We use a three state failure model, in which a component may be good, defective or failed, in order to consider inspection maintenance. Hybrid block replacement and age-based inspection, and opportunistic hybrid policies will also arise naturally in these circumstances and these are briefly investigated. For the simplest policy, an approximation is used to determine the long-run cost and the system reliability. The policies have the interesting property that the system reliability may be a maximum when the long-run cost is close to its minimum. The failure model implies that the effect of maintenance is heterogeneous. The policies themselves imply that maintenance is carried out more prudently to newer than to older systems. The maintenance of traction motor bearings on underground trains is used to illustrate the ideas in the paper.     

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.     

Stochastic Comparison of Age-Dependent Block Replacement Policies

The age-dependent block replacement policy is a modified block replacement policy with an age limit for preventive replacements. Under this policy, any failed component is repaired, but only the components whose ages exceed a fixed age limit are replaced preventively at the scheduled maintenance times. Using the compensator method, we compare stochastically the failure counting processes of the age-dependent block replacement policies with different parameters, and show that the age-dependent block replacement policy, although cost effective, leads to more failures than the age and block replacement policies. AMS 2000 Subject Classification 60K10     

Maintenance of light-standards—a case-study

This paper discusses several strategies for the maintenance of light-standards, where each light-standard consists of n independent and identical lamps screwed on a lamp assembly. The lamps are subject to stochastic failures, and must be correctly replaced if the number of failed lamps reaches a prespecified number m, a norm that is set by the local management to guarantee a minimum luminance. As lamps have an increasing hazard rate and there is a fixed cost of hoisting the assembly, we propose various variants of the m-failure group replacement rule which have in particular an age-criterion to indicate which of the non-failed lamps must be preventively replaced at the time that the assembly is lowered for a corrective lamp replacement. We show how the optimal threshold age can be determined. It appears that this modification reduces the long run average maintenance cost of the Europe Combined Terminals by approximately 8.3%.     

The opportunistic replacement problem: theoretical analyses and numerical tests

We consider a model for determining optimal opportunistic maintenance schedules w.r.t. a maximum replacement interval. This problem generalizes that of Dickman et?al. (J Oper Res Soc India 28:165?C175, 1991) and is a natural starting point for modelling replacement schedules of more complex systems. We show that this basic opportunistic replacement problem is NP-hard, that the convex hull of the set of feasible replacement schedules is full-dimensional, that all the inequalities of the model are facet-inducing, and present a new class of facets obtained through a ${\{0, \frac{1}{2}\}}$ -Chvátal?CGomory rounding. For costs monotone with time, a class of elimination constraints is introduced to reduce the computation time; it allows maintenance only when the replacement of at least one component is necessary. For costs decreasing with time, these constraints eliminate non-optimal solutions. When maintenance occasions are fixed, the remaining problem is stated as a linear program and solved by a greedy procedure. Results from a case study on aircraft engine maintenance illustrate the advantage of the optimization model over simpler policies. We include the new class of facets in a branch-and-cut framework and note a decrease in the number of branch-and-bound nodes and simplex iterations for most instance classes with time dependent costs. For instance classes with time independent costs and few components the elimination constraints are used favorably. For fixed maintenance occasions the greedy procedure reduces the computation time as compared with linear programming techniques for all instances tested.     

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.     

On a Replacement Problem of a Cumulative Damage Model

Items are assumed to fail only by degradation. An appropriate stochastic model of such items is a cumulative process in which an item can fail only when the total amount of damage exceeds a prespecified failure level. This paper introduces a replacement policy in which an item is replaced at a certain level of damage before failure or at failure, whichever occurs first. The optimum replacement level of damage which will minimize the total expected cost per unit of time for an infinite time span is obtained. A numerical example is also presented. The total expected cost for a finite time span is also discussed.     

A dyadic age-replacement policy for a periodically inspected equipment item subject to random deterioration

A periodic review replacement system is considered. The amount of deterioration over successive periods forms a sequence of i.i.d. random variables. A replacement policy of the dyadic type is in effect whereby the used equipment item is discarded and immediately replaced by a new identical equipment item if at the end of a period the old equipment has service aged by an amount in excess of S or has been in operation for exactly N periods whichever comes first. Using a theorem on renewal reward processes, an expression for the total steady-state expected cost per period is derived, consisting of a fixed replacement cost and a linear cost of operation. Optimal values of S and N that minimize this steady state cost are computed for a few numerical examples, when the service aging per period has a gamma distribution.     

An imperfect maintenance model with block replacements

We consider a model in which when a device fails it is either repaired to its condition prior to failure or replaced. Moreover, the device is replaced at times kT, k = 1, 2, … The decision to repair or replace the device at failure depends on the age of the device at failure. We find the optimal block time, T₀, that minimizes the long-run average cost of maintenance per unit time. Our results are shown to extend many of the well known results for block replacement policies.     

Optimally maintaining a Markovian deteriorating system with limited imperfect repairs

We consider the problem of optimally maintaining a periodically inspected system that deteriorates according to a discrete-time Markov process and has a limit on the number of repairs that can be performed before it must be replaced. After each inspection, a decision maker must decide whether to repair the system, replace it with a new one, or leave it operating until the next inspection, where each repair makes the system more susceptible to future deterioration. If the system is found to be failed at an inspection, then it must be either repaired or replaced with a new one at an additional penalty cost. The objective is to minimize the total expected discounted cost due to operation, inspection, maintenance, replacement and failure. We formulate an infinite-horizon Markov decision process model and derive key structural properties of the resulting optimal cost function that are sufficient to establish the existence of an optimal threshold-type policy with respect to the system’s deterioration level and cumulative number of repairs. We also explore the sensitivity of the optimal policy to inspection, repair and replacement costs. Numerical examples are presented to illustrate the structure and the sensitivity of the optimal policy.     

Optimal replacement model with age-dependent failure type based on a cumulative repair-cost limit policy

This paper presents a replacement model with age-dependent failure type based on a cumulative repair-cost limit policy, whose concept uses the information of all repair costs to decide whether the system is repaired or replaced. As failures occur, the system experiences one of the two types of failures: a type-I failure (minor), rectified by a minimal repair; or a type-II failure (catastrophic) that calls for a replacement. A critical type-I failure means a minor failure at which the accumulated repair cost exceeds the pre-determined limit for the first time. The system is replaced at the nth type-I failure, or at a critical type-I failure, or at first type-II failure, whichever occurs first. The optimal number of minimal repairs before replacement which minimizes the mean cost rate is derived and studied in terms of its existence and uniqueness. Several classical models in maintenance literature are special cases of our model.     

A number-dependent replacement policy for a system with continuous preventive maintenance and random lead times

This paper considers a number-dependent replacement policy for a system with two failure types that is replaced at the nth type I (minor) failure or the first type II (catastrophic) failure, whichever occurs first. Repair or replacement times are instantaneous but spare/replacement unit delivery lead times are random. Type I failures are repaired at zero cost since preventive maintenance is performed continuously. Type II failures, however, require costly system replacement. A model is developed for the average cost per unit time based on the stochastic behavior of the system and replacement, storage, and downtime costs. The cost-minimizing policy is derived and discussed. We show that the optimal number of type I failures triggering replacement is unique under certain conditions. A numerical example is presented and a sensitivity analysis is performed.     

A deteriorating cold standby repairable system with priority in use

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. In this system, it is assumed that the working time distributions and the repair time distributions of the two components are both exponential and component 1 is given priority in use. After repair, component 2 is “as good as new” while component 1 follows a geometric process repair. Under these assumptions, using the geometric process and a supplementary variable technique, some important reliability indices such as the system availability, reliability, mean time to first failure (MTTFF), rate of occurrence of failure (ROCOF) and the idle probability of the repairman are derived. A numerical example for the system reliability R(t) is given. And it is considered that a repair-replacement policy based on the working age T of component 1 under which the system is replaced when the working age of component 1 reaches T. Our problem is to determine an optimal policy T^∗ such that the long-run average cost per unit time of the system is minimized. The explicit expression for the long-run average cost per unit time of the system is evaluated, and the corresponding optimal replacement policy T^∗ can be found analytically or numerically. Another numerical example for replacement model is also given.     

Economic Order Quantity and Quantity Discounts under Date-terms Supplier Credit: A Discounted Cash Flow Approach

This paper examines economic order quantity and optimal order quantity under both all-units and incremental-quantity discounts when purchase cost, order cost, and carrying cost are all incurred on date-terms supplier credit. Payment dates for the three cost components need not be the same. The traditionally-used periodic-cost minimization methodology which is insensitive to differences in the timing of various within-period cash flows is replaced with a discounted cash flow methodology. Differences in the characteristics of day-terms and date-terms solutions to the quantity discount case are high-lighted.