The multi-objective testing resource allocation problem (MOTRAP) aims at seeking a good trade-off between system reliability, testing cost, and testing time, which is of significant importance to facilitate the testing planning. Yet most studies focus on the time constraint but rarely consider the practical reliability requirement. In this work, we address MOTRAP on an architecture-based model (ABM) with the personalized preference over reliability. More specifically, we first present a reliability-constrained MOTRAP model on the basis of ABM and illustrate how to use this model for real-world systems. Then, to leverage the problem's knowledge, we develop new lower and upper bounds on testing time invested in different components from both theoretical and algorithmic perspectives on the basis of the Lagrange multiplier and half-interval search. Importantly, these new derived bounds have strong implications due to the fact that they can be easily employed by optimizers as the limits of variables to prune the search space to the region of interests of the decision maker and locate feasible solutions with the expected reliability. Finally, we evaluate the proposed bounds in popular multi-objective optimizers for MOTRAP on application and empirical cases. Experimental results demonstrate that our new bounds practically improve the search performance of optimizers, and decision makers can easily combine these new bounds with off-the-shelf optimizers to find higher-quality solutions that they are interested in, which greatly soothes away stress on optimizer and solution selections of decision makers.