The use of parallel applications in High-Performance Computing (HPC) demands high computing times and energy resources. Inadequate scheduling produces longer computing times which, in turn, increases energy consumption and monetary cost. Task scheduling is an NP-Hard problem; thus, several heuristics methods appear in the literature. The main approaches can be grouped into the following categories: fast heuristics, metaheuristics, and local search. Fast heuristics and metaheuristics are used when pre-scheduling times are short and long, respectively. The third is commonly used when pre-scheduling time is limited by CPU seconds or by objective function evaluations. This paper focuses on optimizing the scheduling of parallel applications, considering the energy consumption during the idle time while no tasks are executing. Additionally, we detail a comparative literature study of the performance of lexicographic variants with local searches adapted to be stochastic and aware of idle energy consumption.