In this paper, a special class of redundancy optimization problem with fuzzy random variables is presented. In this model, fuzzy random lifetimes are considered as basic parameters and the Er-expected of system lifetime is used as a major type of system performance. Then a redundancy optimization problem is formulated as a binary integer programming model. Furthermore, illustrative numerical examples are also given to clarify the methods discussed in this paper.

Optimal time interval between inspections of the redundant systems is raised as a substantial issue to plan a preventive maintenance model for maintenance planners. To have an optimal time period for preventive maintenance of systems, especially complex systems such as redundant systems, two variables of maintenance are mostly connived. Against other studies in the literature of preventive maintenance in which repair time is a negligible factor as an assumption, repair time is considered as a noticeable variable incorporated into the model developed in this paper. Another contribution, the number of facilities, is focused as a significant variable used in real applications. Particularly, systems with complex performance needing technical repair facilities (i.e., technical repairmen, tools, materials, outsourced repair, etc.). In this regard, parallel systems have been analyzed stochastically using the way of preventive maintenance in which repair time would be contemplated as an essential factor in maintenance planning. Using Markov chain, a model based on expected total cost per time is made to demonstrate that a proper time interval achieving lowest possible cost is obtained by taking into account repair time and the number of repair facility. Three models are studied as instances of redundant systems to find the optimal time interval between inspections. These models differ in the number of repair facilities (i.e., one, two and three repair facilities). A sensitivity analysis is done to depict the variability of input variables over optimal the expected total cost per time and time interval between inspections. As a main contribution, the repair time could be an essential factor in maintenance planning, this study contemplates this factor in redundant systems.

The importance in promoting, sustaining industries, manufacturing systems and economy through reliability measurement has become an area of interest. The profit of a system may be enhanced using highly reliable structural design of the system or subsystem of higher reliability. On improving the reliability and availability of a system, the production and associated profit will also increase. Reliability, availability and profit are some of the most important factors in any successful industry and manufacturing settings. In this paper, we compare three different repairable redundant systems using an external supporting device for operation based on the profit. Explicit expressions for the busy period of repairmen, steady-state availability and profit function are derived using linear first-order differential equations. Furthermore, we compare the profit for the three systems and find that system I is more profitable than systems II and III.