Two exact algorithms for the traveling umpire problem

•Formulated the TUP into two mixed integer programming models.•Proposed two exact algorithms to solve the problem.•Our models yielded stronger lower bounds than the previous in the literature.•One algorithm solved two instances with 14 teams to optimal for the first time.•New lower bounds and optimal solutions served as reference for future researchers.

In this paper, we study the traveling umpire problem (TUP), a difficult combinatorial optimization problem that is formulated based on the key issues of Major League Baseball. We introduce an arc-flow model and a set partition model to formulate the problem. Based on these two models, we propose a branch-and-bound algorithm and a branch-and-price-and-cut algorithm. The branch-and-bound algorithm relies on lower bounds provided by a Lagrangian relaxation of the arc-flow model, while the branch-and-price-and-cut algorithm exploits lower bounds from the linear programming relaxation of the set partition model strengthened by a family of valid inequalities. In the branch-and-price-and-cut algorithm, we design an efficient label-setting algorithm to solve the pricing problem, and a branching strategy that combines three different branching rules. The two algorithms are tested on a set of well-known benchmark instances. The two exact algorithms are both able to rapidly solve instances with 10 teams or less, while the branch-and-price-and-cut algorithm can solve two instances with 14 teams. This is the first time that instances with 14 teams have been solved to optimality.