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This paper describes an algorithm for solving reliability optimization problems formulated as nonlinear binary programming problems with multiple-choice constraints. These constraints stand for restrictions in which only one variable is assigned to each subset making up the set; thus, they are expressed by equations whose r.h.s. is unity. Different types of methods for achieving high reliability (an increase in component reliability, parallel redundancy, standby redundancy, etc.) can be easily used simultaneously as design alternatives for each subsystem. In order to solve the problem effectively, the Lawler & Bell algorithm is improved by introducing a new lexicographic enumeration order which always satisfies the multiple-choice constraints. The function for obtaining feasible solutions which give first ~ L-th minimum values of the objective function is added to the algorithm in order to make it more useful for decision making. After a numerical example assists in understanding the algorithm, the computational efficiency is compared with that of the Lawler & Bell algorithm.