Abstract The Traveling Umpire Problem (TUP) is a combinatorial optimization problem concerning the assignment of umpires to the games of a fixed double round-robin tournament. The TUP draws inspiration from the real world multi-objective Major League Baseball (MLB) umpire scheduling problem, but is, however, restricted to the single objective of minimizing total travel distance of the umpires. Several hard constraints are employed to enforce fairness when assigning umpires, making it a challenging optimization problem. The present work concerns a constructive matheuristic approach which focuses primarily on large benchmark instances. A decomposition-based approach is employed which sequentially solves Integer Programming (IP) formulations of the subproblems to arrive at a feasible solution for the entire problem. This constructive matheuristic efficiently generates feasible solutions and improves the best known solutions of large benchmark instances of 26, 28, 30 and 32 teams well within the benchmark time limit. In addition, the algorithm is capable of producing feasible solutions for various small and medium benchmark instances competitive with those produced by other heuristic algorithms. The paper also details experiments conducted to evaluate various algorithmic design parameters such as subproblem size, overlap and objective functions.

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关于旅行裁判问题的建设性数学分析

摘要旅行裁判问题（TUP）是一个组合优化问题，涉及固定双循环赛比赛的裁判分配。TUP从现实世界中多目标的美国职棒大联盟（MLB）裁判员安排问题中汲取了灵感，但是，它仅限于使裁判员的总旅行距离最小的单一目标。分配公断人时，采用了几个硬性约束来强制公平，这使它成为极具挑战性的优化问题。本工作涉及一种建设性的数学方法，该方法主要关注大型基准实例。采用了一种基于分解的方法，该方法顺序解决了子问题的整数编程（IP）公式，从而为整个问题找到了可行的解决方案。这种具有建设性的数学方法可以有效地生成可行的解决方案，并在基准期限内很好地改进26、28、30和32个团队的大型基准实例的最著名解决方案。另外，该算法能够为各种中小型基准实例提供可行的解决方案，与其他启发式算法所产生的解决方案相竞争。本文还详细介绍了为评估各种算法设计参数（如子问题大小，重叠和目标函数）而进行的实验。该算法能够为各种中小型基准实例提供可行的解决方案，与其他启发式算法产生的解决方案相比具有竞争优势。本文还详细介绍了为评估各种算法设计参数（如子问题大小，重叠和目标函数）而进行的实验。该算法能够为各种中小型基准实例提供可行的解决方案，与其他启发式算法产生的解决方案相比具有竞争优势。本文还详细介绍了为评估各种算法设计参数（如子问题大小，重叠和目标函数）而进行的实验。