$D$-Optimal Sample Allocation of the Accelerated Life Tests Under Weibull Series Systems With Type-I Censoring Scheme

In accelerated life tests (ALTs) for system reliability, how samples are allocated across different stress levels can influence the accuracy of reliability inferences. This research addresses the optimal allocation of ALTs for a series system consisting of components with independent Weibull lifetime distributions, which have known common shape parameters, under a Type-I censoring scheme. The main objective is to achieve <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$D$</tex-math></inline-formula>-optimality, with the objective function formulated as the product of second-order elementary symmetric functions. We establish sufficient and necessary conditions for the optimal design to be a two-level uniform design, leveraging the properties of these symmetric functions. Moreover, the study investigates the geometric interpretation of these conditions and broadens the results to include multiple failure modes, despite lacking a theoretical proof. The effectiveness and practical application of the proposed method for optimizing ALTs in series systems with Weibull-distributed components are demonstrated through a real example.