Traveling Umpire Problem Research Articles

Analysis of a constructive matheuristic for the traveling umpire problem

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.

Lower bounds for large traveling umpire instances: New valid inequalities and a branch-and-cut algorithm

Given a double round-robin tournament, the Traveling Umpire Problem (TUP) seeks to assign umpires to the games of the tournament while minimizing the total distance traveled by the umpires. The assignment must satisfy constraints that prevent umpires from seeing teams and venues too often, while making sure all games have umpires in every round, and all umpires visit all venues. We propose a new integer programming model for the TUP that generalizes the two best existing models, introduce new families of strong valid inequalities, and implement a branch-and-cut algorithm to solve instances from the TUP benchmark. When compared against published state-of-the-art methods, our algorithm significantly improves all best known lower bounds for large TUP instances (with 20 or more teams).

A combined approximation for the traveling tournament problem and the traveling umpire problem

AbstractWe consider the traveling tournament problem (TTP) and the traveling umpire problem (TUP). In TTP, the task is to design a double round-robin schedule, where no two teams play against each other in two consecutive rounds, and the total travel distance is minimized. In TUP, the task is to find an assignment of umpires for a given tournament such that every umpire handles at least one game at every team’s home venue and an umpire neither visits a venue nor sees a team (home or away) too often. The task is to minimize the total distance traveled by the umpires. We present a combined approximation for this problem, when the number of umpires is odd. We therefore first design an approximation algorithm for TTP and then show how to define an umpire assignment for this tournament such that a constant-factor approximation for TUP is guaranteed.

Branch-and-bound with decomposition-based lower bounds for the Traveling Umpire Problem

The Traveling Umpire Problem (TUP) is an optimization problem in which umpires have to be assigned to games in a double round robin tournament. The objective is to obtain a solution with minimum total travel distance over all umpires, while respecting hard constraints on assignments and sequences. Up till now, no general nor dedicated algorithm was able to solve all instances with 12 and 14 teams. We present a novel branch-and-bound approach to the TUP, in which a decomposition scheme coupled with an efficient propagation technique produces the lower bounds. The algorithm is able to generate optimal solutions for all the 12- and 14-team instances as well as for 11 of the 16-team instances. In addition to the new optimal solutions, some new best solutions are presented and other instances have been proven infeasible.

Two exact algorithms for the traveling umpire problem

Abstract 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.

On the complexity of the traveling umpire problem

The traveling umpire problem (TUP) consists of determining which games will be handled by each one of several umpire crews during a double round-robin tournament. The objective is to minimize the total distance traveled by the umpires, while respecting constraints that include visiting every team at home, and not seeing a team or venue too often. Even small instances of the TUP are very difficult to solve, and several exact and heuristic approaches for it have been proposed in the literature. To this date, however, no formal proof of the TUP's computational complexity exists. We prove that the decision version of the TUP is NP-complete for certain values of its input parameters.

Decomposition and local search based methods for the traveling umpire problem

The Traveling Umpire Problem (TUP) is a challenging combinatorial optimization problem based on scheduling umpires for Major League Baseball. The TUP aims at assigning umpire crews to the games of a fixed tournament, minimizing the travel distance of the umpires. The present paper introduces two complementary heuristic solution approaches for the TUP. A new method called enhanced iterative deepening search with leaf node improvements (IDLI) generates schedules in several stages by subsequently considering parts of the problem. The second approach is a custom iterated local search algorithm (ILS) with a step counting hill climbing acceptance criterion. IDLI generates new best solutions for many small and medium sized benchmark instances. ILS produces significant improvements for the largest benchmark instances. In addition, the article introduces a new decomposition methodology for generating lower bounds, which improves all known lower bounds for the benchmark instances.

Improved bounds for the traveling umpire problem: A stronger formulation and a relax-and-fix heuristic

Given a double round-robin tournament, the traveling umpire problem (TUP) consists of determining which games will be handled by each one of several umpire crews during the tournament. The objective is to minimize the total distance traveled by the umpires, while respecting constraints that include visiting every team at home, and not seeing a team or venue too often. We strengthen a known integer programming formulation for the TUP and use it to implement a relax-and-fix heuristic that improves the quality of 24 out of 25 best-known feasible solutions to instances in the TUP benchmark. We also improve all best-known lower bounds for those instances and, for the first time, provide lower bounds for instances with more than 16 teams.

Scheduling Major League Baseball Umpires and the Traveling Umpire Problem

The scheduling needs of umpires and referees differ from the needs of sports teams. In some sports leagues, such as Major League Baseball in the United States, umpires travel throughout the league's territory; they do not have a “home base.” For such leagues, balancing the need to minimize umpire travel and the objective that an umpire should not handle the games of a particular team too frequently is important. We have used our approach, which is based on network optimization and simulated annealing, to successfully schedule Major League Baseball umpires. To develop this approach, we created the traveling umpire problem, which includes the major umpire scheduling issues and also provides a test bed for alternative techniques.

Locally Optimized Crossover for the Traveling Umpire Problem

This paper presents a genetic algorithm (GA) to solve the Traveling Umpire Problem, which is a recently introduced sports scheduling problem that is based on the most important features of the real Major League Baseball umpire scheduling problem. In our GA, contrary to the traditional way of randomly obtaining new solutions from parent solutions, we obtain partially optimized solutions with a Locally Optimized Crossover operator. This operator also presents a link between the evolutionary mechanism on a population of solutions and the local search on a single solution. We present improved results over other methods on benchmark instances.