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The development of science and technology and the needs of modem society are racing against each other. Industries are trying to introduce more and more automation in their manufacturing operational processes to minimize the cost per unit of production and to meet the ever – increasing demands. The complexity of industrial machinery and their products is increasing day-by-day. Thus the improvement in effectiveness of such complex systems acquired special importance in the recent years.
As the complexity increased, system failures are bearing a significant effect than before. For example : The effect of failure in power distribution system in a major city, the malfunction of an air traffic control system/Instrumental landing system at an international airport , failure of a Nano system, interruptions in today’s Internet systems, or the breakdown in a nuclear power plant. Reliable performance of the system depends on the consistent functioning of its constituent components. This further depends on the design and topological layout of the subsystems and individual components of the subsystems. In recent years, many statistical and probabilistic models have been developed for evaluating system reliability based on the components reliability, the system design and the assembly of the components.
At the initial stage of research in Repairable Replacement Problem, a repaired system is assumed to be ‘as good as new’ and this kind of repair is called perfect repair. The repair of a component or replacement with a new one can be considered to be a perfect repair. However, most repairs/replacements in practice are not perfect. Consequently, the system after repair cannot be ‘as good as new’. Thus replacement problem for a repairable system has as aroused great attention since it was proposed by Lotka (1939).
Barlow and Hunter (1960) first presented a minimal repair model in which the repair activities do not alter the rate of failure of the system. Brown and Proschan (1983) first reported an imperfect repair model in which the repair will be perfect repair with probability ‘p’ or minimal repair with probability 1-p.
Infact for deteriorating systems, the problem is different from that described above. For e.g in machine maintenance problems, after repair, the functioning time of a machine will become shorter and shorter, so the total functioning time or the total life of the machine must be finite. But in view of the aging and accumulative wear, the repair time will become longer and longer and tend to infinity and finally the machine leads to non-repairable. Therefore, we have to consider a repair replacement model for deteriorating systems, the successive survival times are decreasing, while the consecutive repair times are increasing.
To model a deteriorating system with this kind of characteristic, Lam (1988) first introduced a Geometric Process Repair model.
In the next section, we demonstrated some preliminaries relating to reliability theory and replacement models for ready reference.
Reliability theory is the foundation of reliability engineering. Reliability engineers rely heavily on statistics, probability theory and reliability theory. Many engineering techniques are used in reliability engineering, such as reliability prediction, weibul analysis, thermal management, reliability testing and accelerated life testing.
Reliability is defined as the probability that a device, system, or process will perform its prescribed duty without failure for a given time when operated correctly in a specified environment. Another practical definition for reliability is it is an engineering discipline for applying scientific know-how to a component, assembly, plant, or process so it will perform its intended function, without failure, for the required time duration when installed and operated correctly in a specified environment.
Reliability is also defined in terms of a device, which may be a component in a system or a system consisting of many components. Since the performance of a system usually depends on the performance of its components, the reliability of a system is a function of the reliability of its components. Time is an important factor in the definition of reliability. If a newly purchased device can perform its intended functions satisfactorily, what is the probability that it will last (continue to perform satisfactorily) for a specified period of time. The life time of the device can be treated as a random variable with a statistical distribution and related properties. Further, the operating conditions, such as stress, load, temperature, pressure and/ or other environmental factors, under which the devices is expected to operate must be specified.
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