Encompassing a subset of multi-objective scheduling challenges, the multi-agent scheduling problem involves various agents, each entrusted with a unique set of tasks while striving to optimize their individual goals. Recent inquiries in this field have predominantly spotlighted variable processing times, employing methods like the ℝ?-constraint approach to optimize one agent's function without compromising the other’s limit. Our study takes an innovative approach, delving into a two-agent single-machine scheduling problem influenced by concurrent learning and deterioration effects. The primary aim is to minimize the overall weighted completion time for both agents, preventing any job delays for the second agent. To address this, our research amalgamates the ℝ?-constraint and linear combination approaches, presenting a unique proposition in the current research landscape. We introduce a two-stage methodology: a heuristic method for near-optimal solutions followed by a branch-and-bound algorithm, integrating specialized dominance rules to achieve optimal solutions.