For some products, their reliability information can only be collected via destructive degradation tests where only one measurement can be taken for each test specimen. To expedite data collection, it is also quite common to conduct accelerated destructive degradation tests (ADDT) by exposing test units to harsher-than-normal conditions. One challenge in such tests is that some products start degrading only after a random de gradation initiation time that is sometimes not even observable. This paper addresses a problem of planning ADDT for a product with random degradation initiation time. The objective is to improve the statistical efficiency of the estimator of the product's reliability index obtained from the designed ADDT. The maximum likelihood estimation method and an expectation-maximum algorithm are used to estimate the model parameters and the p-quantile of failure time under the normal operating condition. In planning the ADDT, the optimal unit allocations for those accelerated stress levels are determined under the sample size and testing time constraints. To solve this problem, a nonlinear integer programming problem is formulated. As a main step, a simulation-based method is used to minimize the asymptotic variance of the estimator of the interested p-quantile. To the best of our knowledge, this is the first attempt that focuses on planning of ADDT involving random degradation initiation time. The experimental design method will facilitate the implementations of ADDT in various industry applications.