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In order to balance the deviation time between replacement and failure, a new notion of cost named as deviation cost is taken into account for planning replacement policies. We firstly take up a standard parallel system with n identical units and model replacement policies that are done at planned time and at periodic times. Next, we model again the above replacement policies when the number of units is a random variable that has a Poisson distribution. Thirdly, replacement policies are planned preventively at the completion of random working times. Optimum number of units is scheduled for a total working interval that is constantly and randomly given. Finally, we give reliability functions of k-out-of-n systems for extensions of parallel systems. We give analytical discussions for the above replacement policies and illustrate numerical examples when the failure time of each unit has an exponential distribution.

It has been proposed that if replacement time is planned too early prior to failure, a waste of operation cost, i.e., excess costs, would incur because the system might run for an additional period of time to complete critical operations, and if replacement time is too late after failure, a great failure cost, i.e., shortage cost, is incurred due to the delay in time of the carelessly scheduled replacement. In order to make the preventive replacement policies perform in a more general way, the above two variable types of costs are taken into considerations for periodic replacement policies in this paper. We firstly take up a standard model in which the unit is replaced preventively at periodic times. Secondly, the modeling approaches of whichever occurs first and last are applied into periodic and random models, and replacement first and last policies are discussed to find optimum periodic replacement times for a random working time. Furthermore, optimum working numbers are obtained for the extended models. We give analytical discussions of the above replacement policies, and finally, numerical examples are illustrated.

It would be of interest to formulate the general replacement models, combing the constant and random policies to satisfy the commonly planned and randomly needed replacement times. This paper takes up age and periodic replacement models again to formulate their general models when replacement actions are also conducted at random times $$Y_i~(i=1,2,\ldots ,n)$$ . The classic approach of whichever occurs first and the newly proposed approach of whichever occurs last are used for such general models, whose models are named as replacement first, modified replacement first, replacement last and modified replacement last, respectively. We compare all of the replacement models analytically and numerically to find which policy should be selected from the viewpoint of cost. It is shown that the modified replacement policies with combined approaches of whichever occurs first and last are more economical than others. In addition, the replacement models with different replacement costs are extended for further studies.