We build upon the previous efforts to enhance the search ability of Moead (a decomposition-based algorithm), by investigating the idea of evolving the whole population simultaneously at once. We thereby propose new alternative selection and replacement strategies that can be combined in different ways within a generic and problem-independent framework. To assess the performance of our strategies, we conduct a comprehensive experimental study on bi-objective combinatorial optimization problems. More precisely, we consider ρMNK-landscapes and knapsack problems as a benchmark, and experiment a wide range of parameter configurations for Moead and its variants. Our analysis reveals the effectiveness of our strategies and their robustness to parameter settings. In particular, substantial improvements are obtained compared to the conventional Moead.