Philip A. Scarf

On the application of mathematical models in maintenance

Abstract This paper may be seen as an appeal to maintenance modellers to work with maintenance engineers and managers on real problems. Such collaboration is essential if maintenance modelling is to be accepted within the engineering community. It is also particularly important in the design and building of maintenance management information systems if such systems are to be used to manage and operate maintenance policy in the new millennium. In this context, developing areas of maintenance modelling are discussed, namely: inspection maintenance; condition based maintenance; maintenance for multi-component systems; and maintenance management information systems. Some new models relating to capital replacement are also considered. Thus, we are concerned with the mathematical modelling of maintenance rather than with management processes relating to maintenance. Discussion of maintenance management information systems is included because of their importance in providing data for mathematical modelling and in implementing model-based maintenance policy.

On the impact of optimisation models in maintenance decision making: the state of the art

In this paper we discuss the state of the art in applications of maintenance optimisation models. After giving a short introduction to the ama, we consider several ways in which models may be used to optimise maintenance, such as case studies, operational and strategic decision support systems, and give examples of each of them. Next we discuss several areas where the models have been applied successfully. These include civil structure and aeroplane maintenance. From a comparative point of view, we discuss future prospects. 8 1998 Elsevier Science Limited.

Imperfect inspection and replacement of a system with a defective state: A cost and reliability analysis

We consider a system with three possible states, good, defective and failed. Failures are detected as soon as they occur; the defective state, which is only revealed by inspection, does not prevent the system from fulfilling the function for which it was designed. We present a maintenance model consisting of periodic inspections to check the state of the system, in which inspections are subject to error. At a false positive inspection the system is unnecessarily replaced; at a false negative inspection a defect remains unrevealed with reliability implications for future operation. The model is illustrated with an example from the railways. In this context, we suppose that system lifetime is heterogeneous so that the time the system spends in the defective state is a random variable from a mixed distribution. We determine under what circumstances the cost of maintenance cannot be justified by its efficacy, and suggest that when there is the possibility that replacement is poorly executed (lifetime heterogeneity) the natural response to imperfect inspection of increasing the inspection frequency can be counter-productive.

Maintenance scheduling of a protection system subject to imperfect inspection and replacement

An inspection and replacement policy for a protection system is described in which the inspection process is subject to error, and false positives (false alarms) and false negatives are possible. We develop two models: one in which a false positive implies renewal of the protection system; the other not. These models are motivated by inspection of a protection system on the production line of a beverage manufacturer. False negatives reduce the efficiency of inspection. Another notion of imperfect maintenance is also modelled: that of poor installation of a component at replacement. These different aspects of maintenance quality interact: false alarms can, in a worst case scenario, lead to the systematic and unnecessary replacement of good components by poor components, thus reducing the availability of the system. The models also allow situations in which maintenance quality differs between alternative maintainers to be investigated.

The Analysis and Utilization of Cycling Training Data

Most mathematical models of athletic training require the quantification of training intensity and quantity or ‘dose’. We aim to summarize both the methods available for such quantification, particularly in relation to cycle sport, and the mathematical techniques that may be used to model the relationship between training and performance.Endurance athletes have used training volume (kilometres per week and/or hours per week) as an index of training dose with some success. However, such methods usually fail to accommodate the potentially important influence of training intensity. The scientific literature has provided some support for alternative methods such as the session rating of perceived exertion, which provides a subjective quantification of the intensity of exercise; and the heart rate-derived training impulse (TRIMP) method, which quantifies the training stimulus as a composite of external loading and physiological response, multiplying the training load (stress) by the training intensity (strain). Other methods described in the scientific literature include ‘ordinal categorization’ and a heart rate-based excess post-exercise oxygen consumption method.In cycle sport, mobile cycle ergometers (e.g. SRM™ and PowerTap™) are now widely available. These devices allow the continuous measurement of the cyclists’ work rate (power output) when riding their own bicycles during training and competition. However, the inherent variability in power output when cycling poses several challenges in attempting to evaluate the exact nature of a session. Such variability means that average power output is incommensurate with the cyclist’s physiological strain. A useful alternative may be the use of an exponentially weighted averaging process to represent the data as a ‘normalized power’.Several research groups have applied systems theory to analyse the responses to physical training. Impulse-response models aim to relate training loads to performance, taking into account the dynamic and temporal characteristics of training and, therefore, the effects of load sequences over time. Despite the successes of this approach it has some significant limitations, e.g. an excessive number of performance tests to determine model parameters. Non-linear artificial neural networks may provide a more accurate description of the complex non-linear biological adaptation process. However, such models may also be constrained by the large number of datasets required to ‘train’ the model.A number of alternative mathematical approaches such as the Performance- Potential-Metamodel (PerPot), mixed linear modelling, cluster analysis and chaos theory display conceptual richness. However, much further research is required before such approaches can be considered as viable alternatives to traditional impulse-response models. Some of these methods may not provide useful information about the relationship between training and performance. However, they may help describe the complex physiological training response phenomenon.

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.

An Age-Based Inspection and Replacement Policy for Heterogeneous Components

This paper considers a hybrid maintenance policy for a single component from a heterogeneous population. The component is placed in a socket, and the component and socket together comprise the system. The s-population of components consists of two sub-populations with different failure characteristics. By supposing that a component may be in a defective but operating state, so that there exists a delay time between defect arrival and component failure, we consider a novel maintenance policy that is a hybrid of inspection and replacement policies. There are similarities in this approach with the concept of ldquoburn-inrdquo maintenance. The policies are investigated in the context of traction motor bearing failures. Under certain circumstances, particularly when the mixture parameter is large, and the distribution of lifetimes for the two component types are well separated, the hybrid policy has significant cost savings over the standard age-based replacement policy, and over the pure inspection policy. In addition to the cost metric, the mean time between operational failures of the system under the hybrid policy can be used to guide decision-making. This maintenance policy metric is calculated using simulation, and using an approximation which assumes that operational failures occur according to a Poisson process with a rate that can be calculated in a straightforward way. The simulation results show good agreement with the approximation.

A numerical study of designs for sporting contests

Operational Research may be used to compare different designs for a sporting contest or tournament. This paper considers a methodology for this purpose. We propose a number of tournament metrics that can be used to measure the success of a sporting contest or tournament, and describe how these metrics may be evaluated for a particular tournament design. Knowledge of these measures can then be used to compare competing designs, such as round-robin, pure knockout and hybrids of these designs. We show, for example, how the design of the tournament influences the outcome uncertainty of the tournament and the number of unimportant matches within the tournament. In this way, where new designs are proposed, the implications of these designs may be explored within a modelling paradigm. In football (soccer), the UEFA Champions League has adopted a number of designs over its 50 year history; the design of the tournament has been modified principally in response to the changing demands of national league football and television - the paper uses this particular tournament to illustrate the methodology.

On reliability criteria and the implied cost of failure for a maintained component

This paper considers age-based replacement and block replacement when reliability is also a decision criterion. We describe how specification of the operational reliability of component function determines the replacement policy, and that setting a value for the cost of failure and specifying an operational reliability requirement are equivalent. This duality then implies a simple method for checking the consistency of the cost of failure and operational reliability measures when they are set system operators and maintainers. A simple expression for the median time between operational failures for a socket subject to age-based replacement is also obtained. These ideas are considered briefly in the context of block replacement. Data from an actual case relating to the maintenance and replacement of train traction motors is used for illustration.

Route choice in mountain navigation, Naismith's rule, and the equivalence of distance and climb

Abstract In this paper, I consider decision making about routes in mountain navigation. In particular, I discuss Naismiths rule, a method of calculating journey times in mountainous terrain, and its use for route choice. The rule is essentially concerned with the equivalence, in terms of time duration, between climb or ascent and distance travelled. Naismith himself described a rule that is purported to be based on trigonometry and simple assumptions about rate of ascent; his rule with regard to hill-walking implies that 1 m of ascent is equivalent to 7.92 m of horizontal travel (1:7.92). The analysis of data on fell running records presented here supports Naismiths rule and it is recommended that male runners and walkers use a 1:8 equivalence ratio and females a 1:10 ratio. The present findings are contrasted with those based on the analysis of data relating to treadmill running experiments (1:3.3), and with those based on the analysis of times for a mountain road-relay (1:4.4). Analysis of cycling data suggests a similar rule (1:8.2) for cycling on mountainous roads and tracks.