The cloud-edge-end (CEE) collaborative computing environment offers mobile users uninterrupted computing, storage, and network transmission services. In this context, mobile devices often face the challenge of balancing performance and energy consumption. To reduce the energy consumption of mobile devices, tasks are often offloaded to remote servers. However, this may cause time-sensitive tasks to miss deadlines due to data transmission overhead. Therefore, exploring approaches to balance energy consumption and task schedulability is highly meaningful. In this paper, we investigate computation offloading for time-constrained tasks in the CEE environment to find an optimal scheme that jointly optimizes system energy consumption and task schedulability. To this end, we propose a computation offloading strategy based on an improved genetic algorithm (IGA). The IGA incorporates a neighborhood search operation governed by single-step transition rules. After each iteration of the genetic algorithm (GA), this operation refines the population by searching for local optimal solutions within the neighborhood of each chromosome, thereby preserving the global search capability of the GA while enhancing local exploration ability. The extensive simulations demonstrate that our IGA consistently outperforms advanced benchmark algorithms across all tested scales (20-200 tasks). It achieves substantial energy savings, exceeding 20% compared to memetic algorithms (MA-SA and MA-2OPT) and the NSGA-III-based method (COM). In terms of schedulability, IGA matches the performance of the greedy heuristic and outperforms other metaheuristics.