The fundamental problem of task scheduling is how to strategically distribute a large number of jobs to appropriate processors while maximizing one or more goals under particular time and resource restrictions. Within a dual-machine sequential flow shop, this work explores the scheduling of (n) distinct activities, each with a different due time. Simulated Annealing (SA) and the Gravitational Search Algorithm (GSA) are synergistically integrated in a new hybrid metaheuristic algorithm we name SAGSA. The SAGSA algorithm starts the resolution process in two different stages: it first uses the SA algorithm to provide a preliminary solution, and then it applies GSA to improve this solution. The proposed solution uses a weighted objective function to reduce work delays, aligning with the requirements of timely production systems. This function evaluates the efficacy of the proposed solutions. We investigate four different situations that result from differences in temperature reduction methods and Markov chain modalities. We find and support the better scenario by means of thorough result analysis. Empirical data indicates that the SAGSA algorithm can identify optimum solutions as well as, if not better than, the state-of-the-art techniques in use today.