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 compare 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.

多目标测试资源分配问题（MOTRAP）关注的是如何合理规划软件测试人员的测试时间，以尽可能节省成本并提高可靠性。 MOTRAP 的可行解空间由其变量（即，在每个组件上投入的时间）和约束（例如，预先指定的可靠性、成本或时间）确定。尽管可以使用各种最先进的约束多目标优化器来寻找该空间中的单独解决方案，但由于该空间与大搜索空间相比非常小，因此它们的搜索仍然低效且昂贵。决策者可能经常会经历长时间但不成功的搜索，无法返回可行的解决方案。在这项工作中，我们首先在基于架构的模型的基础上制定了一个严格约束的MOTRAP，其中在预先指定的可靠性、成本和时间的多重约束下优化可靠性、成本和时间。然后，为了估计该特定 MO​​TRAP 的可行解空间，我们开发了理论和算法方法，从约束中推导出变量的新的更严格的下限和上限。重要的是，我们的方法可以帮助决策者确定他们的约束设置是否可行，同时，导出的边界可以恰好包围微小的可行解空间，并帮助现成的约束多目标优化器在可行解内进行搜索尽可能多的空间。此外，为了进一步充分利用这些边界，我们提出了一种广义的边界约束处理方法，该方法可以很容易地被约束多目标优化器使用，以在理论保证的情况下将不可行的解决方案拉回估计空间。最后，我们根据应用和实证案例评估我们的方法。实验结果表明，我们的方法显着提高了现成的约束多目标优化器和最先进的边界约束处理方法的效率、有效性和鲁棒性，为决策者寻找高质量的解决方案。这些改进可以帮助决策者减轻设置约束和选择约束多目标优化器的压力，并更有效地促进测试计划。