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This paper focuses on a two-stage machining and welding scheduling problem based on an investigation at a structural metal manufacturing plant, aiming to minimise the total makespan. Several parts processed at Stage one according to classical job-shop scheduling are grouped into a single construction component at the second welding stage. Fabrication of the construction component cannot begin until all comprising parts have been completed at Stage one. This paper establishes a novel mathematic model to minimise the total makespan by mainly considering the dominance relationship between the construction component and the corresponding parts. In order to solve this two-stage problem, we propose an improved harmony search algorithm. A local search method is applied to the best vector at each iteration, so that a more optimal vector can be subsequently realised. The average value, minimum value, relative percentage deviation and standard deviation are discussed in the experimental section, and the proposed local best harmony search algorithm outperforms the genetic algorithm, immune algorithm and harmony search algorithm without local search. Moreover, six optimal solutions are given as Gantt charts, which vividly illustrate that the mathematical model established in this paper can facilitate the development of a better scheduling scheme. [ABSTRACT FROM AUTHOR]