In this study, we adopt a strategy of machine configuration to construct a reliable hybrid flow-shop manufacturing system (HFSMS) with reworking and scrapping actions, in which the reworking and scrapping actions are due to the yield rates of the configured machines. The machine configuration determines the machine suppliers and the number of machines for each production stage in the HFSMS, which may affect the stability of the HFSMS. The machines configured to a production stage in parallel are provided by the same supplier. Accordingly, each production stage has multiple states, following a probability distribution. The HFSMS is a typical stochastic-flow network under machine configuration, and network reliability indicates the probability that order demand d can be fulfilled by the HFSMS and used as a decision reference. Our study integrates genetic algorithm (GA) and absorbing Markov chain (AMC) to solve the reliability-oriented machine configuration problem of HFSMS subject to a configuration budget, where the GA is utilized to find the optimal machine configuration with maximum network reliability and the AMC model constructed based on the yield rates is used for network reliability evaluation. For validating the applicability and computational efficiency of the proposed AMC and GA-based approach, a simple manufacturing system and two practical manufacturing systems are used to compare the proposed approach with four popular meta-heuristic algorithms. The experimental results show that the proposed approach can find the exact optimal solution and has better computational efficiency than the other four meta-heuristic algorithms.