A branch-and-price algorithm for the single machine scheduling problem with periodic maintenance to minimize makespan

Periodic maintenance of machines is essential to prevent unexpected breakdowns and ensure safe and reliable production. In this paper, we address the single machine scheduling problem with periodic maintenance, where a set of jobs must be processed sequentially on a single machine that requires periodic maintenance, with the objective of minimizing makespan. We first formulate this problem as a set partitioning model consisting of a set of integer variables and a set of continuous variables, and then develop a branch-and-price algorithm to efficiently solve the set partitioning model. In the algorithm, we design a hierarchical branching strategy to generate child nodes, a primal heuristic to quickly generate feasible solutions from fractional solutions, and a label setting algorithm with a bounding procedure to address pricing problems. Extensive computational experiments on benchmark instances and newly generated instances have been conducted to evaluate the efficiency of our branch-and-price algorithm. The results demonstrate that our algorithm solving the set partitioning model significantly outperforms the Gurobi Optimizer solving existing mathematical models in the literature, owing to our well-designed branching strategy, primal heuristic, and bounding procedure.