# After maintenance activities the component state is not the

After maintenance activities, the component state is not the exact combination of as good as new and as bad as old, and it may be somewhere between two extreme states, i.e., the so-called imperfect maintenance. An imperfect maintenance model was built ingeniously to reveal the relationship among maintenance cost, age A 779 factor and effective age, and then a novel selective maintenance model was proposed by Pandey et al. [9]. Following this initial idea, a selective maintenance model with two failure modes, maintainable failure mode and non-maintainable failure mode, was considered in Ref. [10]. Notice that the probability of the former relies on the cumulated probability of the latter, which is different from that in Ref. [9]. This selective maintenance model was further extended to multi-state system with multi-state component [11]. Optimization of selective maintenance for multi-mission series–parallel system was addressed by Knatab et al. [12]. The optimal order of preventive maintenances was found for the purpose of minimizing total maintenance cost subject to necessary system reliability constraint for each mission. Maaroufi et al. studied carefully a specific complicated system with global failure propagation and isolation effects under limited maintenance resources, such as maintenance time and cost before the next mission [13]. The optimization methods applied to selective maintenance models are either exact method or heuristic method. Exact methods always find the global optimum solution. For improving the efficiency of total enumeration, Rajagopalan and Cassady explored several avenues, described in detail in Section 3.2[14]. Along with the similar idea, several rules that potentially narrow the solution space and consequently accelerate the optimization procedure were also proposed in Ref. [13]. Given a system reliability requirement, an exact algorithm based on the original Kettele\'s algorithm was proposed to minimize the total maintenance cost for series–parallel system in Ref. [2]. If the complexity of a selective maintenance problem is high and the computing time of the exact method increases exponentially with the size of the optimization problem, the heuristic method can be used to find a near-optimal solution in reasonable time. Three new methods (construction heuristic method, exact method based on branch and bound procedure, and heuristic approach based on Tabu search) were developed by Lust and then applied to various system configurations [15]. The modified great deluge algorithm, a local search meta-heuristic method, incorporated both the worse solution acceptance and the well-known hill climbing rule [16]. This method requires only one parameter, while it provides the competitive results within a good reasonable computational time. Other heuristic approaches often mentioned in related literatures are genetic algorithm (GA) [4], differential evolution (DE) [11], simulated annealing algorithm (SAA) [17], and so on. For a detailed overview on the state-of-the-art selective maintenance modeling and optimization, we strongly recommend a recent review article for interested readers, entitled “Recent advances in selective maintenance from 1998 to 2014” [18]. Thomas classified the multi-component maintenance models on the basis of the interaction between components and defined three distinct types of interactions: economic dependence, structural dependence, and stochastic dependence [19]. Dekker and Wildeman dealt exclusively with maintenance models under economic dependence, which were classified based on the planning aspect: stationary model and dynamic model [20]. Some latest advances in maintenance decision modeling on multi-component system were summarized by Nicolai and Dekker [21] and Nowakowski and Werbinka [22], respectively. In a word, economic dependence implies that the cost can be saved when several components are jointly maintained instead of separately; structural dependence is applied if several components structurally form a part, so the repair or replacement of a failed component implies maintenance or dismantling of other working components as well; and stochastic dependence occurs if the state of a component (valid age, lifetime distribution, failure probability, failure state, etc.) can influence those of other components in a given probability. Comparatively speaking, structural dependence and stochastic dependence mainly focus on the lifetime distribution or degradation process of components in a repairable system. In contrast, the economic dependence cannot modify the failure model of each component. Its core issue is that how to assemble these maintenance activities in all components in order to save more maintenance costs. The economic dependence can be formulated in a simple manner and still widely applied in many real-world situations. Until now, its research are relatively abundant among all dependences among the components [23–26]. So the maintenance policies under economic dependence only are discussed in the present paper.