On Estimating the Feasible Solution Space of Multi-objective Testing Resource Allocation

The multi-objective testing resource allocation problem (MOTRAP) is concerned on how to reasonably plan the testing time of software testers to save the cost and improve the reliability as much as possible. The feasible solution space of a MOTRAP is determined by its variables (i.e., the time invested in each component) and constraints (e.g., the pre-specified reliability, cost, or time). Although a variety of state-of-the-art constrained multi-objective optimisers can be used to find individual solutions in this space, their search remains inefficient and expensive due to the fact that this space is very tiny compared to the large search space. The decision maker may often suffer a prolonged but unsuccessful search that fails to return a feasible solution. In this work, we first formulate a heavily constrained MOTRAP on the basis of an architecture-based model, in which reliability, cost, and time are optimised under the pre-specified multiple constraints on reliability, cost, and time. Then, to estimate the feasible solution space of this specific MOTRAP, we develop theoretical and algorithmic approaches to deduce new tighter lower and upper bounds on variables from constraints. Importantly, our approach can help the decision maker identify whether their constraint settings are practicable, and meanwhile, the derived bounds can just enclose the tiny feasible solution space and help off-the-shelf constrained multi-objective optimisers make the search within the feasible solution space as much as possible. Additionally, to further make good use of these bounds, we propose a generalised bound constraint handling method that can be readily employed by constrained multi-objective optimisers to pull infeasible solutions back into the estimated space with theoretical guarantee. Finally, we evaluate our approach on application and empirical cases. Experimental results reveal that our approach significantly enhances the efficiency, effectiveness, and robustness of off-the-shelf constrained multi-objective optimisers and state-of-the-art bound constraint handling methods at finding high-quality solutions for the decision maker. These improvements may help the decision maker take the stress out of setting constraints and selecting constrained multi-objective optimisers and facilitate the testing planning more efficiently and effectively.