This paper proposes a generalized formulation for multilevel redundancy allocation problems that can handle redundancies for each unit in a hierarchical reliability system, with structures containing multiple layers of subsystems and components. Multilevel redundancy allocation is an especially powerful approach for improving the system reliability of such hierarchical configurations, and system optimization problems that take advantage of this approach are termed multilevel redundancy allocation optimization problems (MRAOP). Despite the growing interest in MRAOP, a survey of the literature indicates that most redundancy allocation schemes are mainly confined to a single level, and few problem-specific MRAOP have been proposed or solved. The design variables in MRAOP are hierarchically structured. This paper proposes a new variable coding method in which these hierarchical design variables are represented by two types of hierarchical genotype, termed ordinal node, and terminal node. These genotypes preserve the logical linkage among the hierarchical variables, and allow every possible combination of redundancy during the optimization process. Furthermore, this paper developed a hierarchical genetic algorithm (HGA) that uses special genetic operators to handle the hierarchical genotype representation of hierarchical design variables. For comparison, the customized HGA, and a conventional genetic algorithm (GA) in which design variables are coded in vector forms, are applied to solve MRAOP for series systems having two different configurations. The solutions obtained when using HGA are shown to be superior to the conventional GA solutions, indicating that the HGA here is especially suitable for solving MRAOP for series systems.
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