مدلسازی و بهینه سازی متوالی تعمیر و نگهداری پیشگیرانه ناکامل

This paper deals with the problem of scheduling imperfect preventive maintenance (PM) of some equipment. It uses a model due to Kijima in which each application of PM reduces the equipment's effective age (but without making it as good as new). The approach presented here involves minimizing a performance function which allows for the costs of minimal repair and eventual system replacement as well as for the costs of PM during the equipment's operating lifetime. The paper describes a numerical investigation into the sensitivity of optimum schedules to different aspects of an age-reduction model (including the situation when parts of a system are non-maintainable—i.e., unaffected by PM).

Most organizations incur significant costs associated with equipment failure and its subsequent repair or replacement. The frequency of such failure can typically be reduced by periodic maintenance. Mathematical models for analysing and optimizing the performance of repairable equipment have been widely discussed in the literature [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13] and [14]. In this paper we follow ideas given in [1] and [8] and study the optimal scheduling of preventive maintenance (PM), basing our approach on the notion that equipment which benefits from PM can have an effective age which is less than its calendar age. When only minimal repairs are performed and there are no other interventions, the likelihood of equipment failure can be expected to increase steadily with time. More precisely, we suppose that the number of failures occurring during a time interval (a,b)(a,b) is View the MathML source∫abh(t)dt. Turn MathJax on The function h(t)h(t) is sometimes called the failure rate or hazard rate (as in [8] and [10]) and sometimes the failure intensity [4] and [5]. If H(t)H(t) denotes the indefinite integral View the MathML source∫0th(s)ds, the number of failures occurring between t=at=a and t=bt=b is H(b)-H(a)H(b)-H(a). H(t)H(t) is called the cumulative failure rate. In practice, PM is used to lengthen the useful lifetime of equipment (and hence to decrease average running cost) by reducing the occurrence of failures. One of the key characteristics of a maintenance model is the effect of different kinds of intervention on the age of the system. Perfect repair and minimal repair are both commonly used in idealized age-effect models; and similar terms can also be applied to maintenance. In reality, however, both repair and maintenance are usually imperfect —i.e., somewhere between perfect and minimal. Pham and Wang [11] and, more recently, Doyen and Gaudoin [4] have given useful surveys of imperfect maintenance models. One of the most important of these is the effective age model [6] and [7]. This is also called the virtual age model. If we assume that maintenance makes the equipment's effective age, y , less than its calendar age, t , then the number of failures occurring after a PM will depend on H(y)H(y) rather than H(t)H(t). Since H is a monotonically increasing function, fewer failures will occur after a PM than if PM had not been carried out. Our purpose in this paper is to consider the optimal scheduling of PM. Our particular focus is on the way that such schedules can be affected by the choice of aging model that is used. Specifically, we compare the so-called types 1 and 2 aging models proposed in [6] and [7]. These have also been recently discussed in an optimization context by Kahle [5].

In this paper we have outlined two forms of the effective-age approach to modelling PM. In the type 1 model, the k -th PM only reduces the aging that has occurred since the (k-1)(k-1)-th PM; but in the type 2 model the k-th PM makes a cumulative reduction on all aging since the equipment entered service. Both types of age model can be used to compute a mean cost function; and optimal PM schedules are then obtained by adjusting inter-PM intervals to minimize this function. We have computed optimal PM schedules based on both type 1 and type 2 aging. Our results show that both types of schedules can be quite different from each other. Type 1 schedules require PMs to be fairly uniformly distributed (see Fig. 3). In type 2 schedules, the spread of inter-PM times is greater, with a relatively long delay before the first PM while subsequent PMs quickly become relatively closer together (see Fig. 4). Furthermore, the type 2 aging model admits optimum schedules which cause effective age to decrease steadily over the operating life. This does not seem to be the case with type 1 solutions. The fact that the type 2 solutions allow equipment to become effectively younger and younger in spite of increasing calendar age means that the AjAj parameters in (5) are important in ensuring that type 2 solutions do not become unrealistic. The type 1 model is intuitively more convincing in the way that each PM is assumed only to counteract the most recent deterioration in the state of the system. Our numerical tests have also shown how type 1 and type 2 optimal schedules respond to changes in the effectiveness of PM (as given by bkbk and AjAj in (2), (4) and (5)) and to changes in the relative costs of repair and system replacement. For any particular case there will be an optimum number of PMs—i.e., one which yields the least value of the cost function (7). Our tests indicate that this optimum number is likely to be higher for type 2 schedules than for type 1. However, it also appears that the optimum mean cost, C*C*, becomes less sensitive to the number of PMs as N gets larger. Our tests have also shown that, under the type 1 aging model, optimal equi-spaced PM schedules may not be much inferior to those where inter-PM times can vary—especially when the number of PMs is quite small. For the type 2 age model, however, we find that forcing the PMs to be equally spaced can have an appreciable adverse effect on equipment lifetime and mean cost. Finally we have shown how both type 1 and type 2 optimum PM schedules can change if the system is regarded as having both maintainable and non-maintainable components. Several features of the PM scheduling problem presented here deserve further attention. In the first place we might reconsider the assumption that repair and maintenance both take negligible time. It is arguable, in fact, that this assumption need not lead to unrealistic PM schedules. If we choose to treat time t as being a measure of operating life of the equipment then we could regard the clock as being stopped while maintenance or repair take place and only re-started when the equipment is back in service. Of course, in real life, there are undesirable consequences (like lost production) when a machine is out of action: but these can be modelled to some extent by the costs cpcp and cmcm assigned to maintenance and repair. However, in spite of these remarks in defence of the status quo, it would be a worthwhile topic for future research to extend our mean-cost model to include repair and maintenance times in a more explicit way. As another improvement to the cost model we note that, just as we may distinguish between maintainable and non-maintainable failure modes, so we might also subdivide the maintainable modes into types 1 and 2 classes. PM functions such as lubrication and adjustment can reasonably be expected to reduce effective age in a type 1 manner; but where PM actually involves some degree of replacement then the type 2 model could be more appropriate. Once again, this is a possible subject for future research. A final observation is that maintenance does not just reduce the occurrence of failures but also makes equipment operate more efficiently. Hence we could extend the mean cost function to reflect operating costs as well as repair costs. We would expect, for instance, a newer system to be more fuel efficient than an older one; and so effective age could appear in an expression for running costs just as it does in the expression for failure rate. Gathering data for formulating running cost as a function of age could, in practice, be as challenging as the task of modelling the cumulative failure-rate function H(t)H(t). Nonetheless, it would be worth the attempt, so that PM scheduling could be based on a more complete representation of lifetime costs.