The tool-path problem has been extensively studied in manufacturing technologies, as it has a considerable impact on production time. Additive manufacturing is one of these technologies; it takes time to fabricate parts, so the selection of optimal tool-paths is critical. This research analyzes the tool-path problem in the direct energy deposition technology; it introduces the main processes, and analyzes the characteristics of tool-path problem. It explains the approaches applied in the literature to solve the problem; as these are mainly geometric approximations, they are far from optimal. Based on this analysis, this paper introduces a mathematical framework for direct energy deposition and a novel problem called sequence strategy generation. Finally, it solves the problem using a benchmark for several different parts. The results reveal that the approach can be applied to parts with different characteristics, and the solution to the sequence strategy problem can be used to generate tool-paths.