Redundancy allocation under state‐dependent distributional uncertainty of component lifetimes

In this paper, we consider a redundancy allocation problem in a series‐parallel system with uncertain component lifetimes of multiple states, which involves multitype components and mixed redundancy strategy. We develop a distributionally robust redundancy allocation model using state‐dependent ambiguity set, which describes the mean, expected cross‐deviation, and the support conditional on each state of the component lifetime distribution. Our structural analysis of the worst‐case system reliability function identifies a concave equivalent for the conditional worst‐case reliability function in each state and achieves an averaged finite piece‐wise affine representation for the overall worst‐case system reliability function, which enjoys valuable operational insights for reliability evaluation and scalable optimization of the redundancy designs. Furthermore, we analyze the reliability guarantees for the worst‐case reliability in optimality under possible misspecification of ambiguity‐set parameters, which admit the structure of “Reliability Target + Target Surplus + Estimated Elasticity,” and insightfully imply that the auxiliary dual variables obtained via solving the design problem can be regarded as the scale of elasticity. Computationally, the redundancy design problem can be linearized and reformulated as a mixed 0–1 second‐order cone program. Exploiting the above finite piece‐wise affine structure of the worst‐case reliability function, the design problem also admits a practical iterative decomposition scheme with finite convergence, which is more scalable especially for the possible situations of large number of states in practice. Finally, numerical experiments including a case with real‐life data of braking system components well justify the value of state information incorporated by our model in producing favorable cost‐effective redundancy designs for reliability operations.