Benders' cuts guided large neighborhood search for the traveling umpire problem

Abstract

Abstract:This article introduces the use of Benders' cuts to guide a large neighborhood search to solve the traveling umpire problem, a sports scheduling problem inspired by the real-life needs of the officials of a sports league. At each time slot, a greedy matching heuristic is used to construct a schedule. When an infeasibility is recognized first a single step backtracking is tried to resolve the infeasibility. If unsuccessful, Benders' cuts are generated to guide a large neighborhood search to ensure feasibility and to improve the solution. Realizing the inherent symmetry present in the problem, a large family of cuts are generated and their effectiveness is tested. The resulting approach is able to find better solutions to many instances of this problem.

Two exact algorithms for the traveling umpire problem

“…In this section we present the computational results for the set of benchmark instances proposed by Trick and Yildiz (2011) and available at http://mat.tepper.cmu.edu/TUP/. This benchmark has been widely used to test different kinds of models and to evaluate various algorithms in the TUP literature.…”

“…The problem was first introduced by Trick and Yildiz (2007), and was then further addressed by Trick and Yildiz (2011). Trick and Yildiz (2011) formulated the problem as an integer programming (IP) model and a constraint programming model, and designed a Bender's cuts guided large neighborhood search heuristic to deal with the problem. To test the models and the proposed heuristic, Trick and Yildiz (2011) generated a set of benchmark instances where the number of teams ranged from 4 to 32.…”

“…Therefore, we propose a branch-and-bound algorithm based on a Lagrangian relaxation and a branch-and-price-and-cut algorithm based on the LP relaxation. The two algorithms were extensively tested on the benchmark instances proposed by Trick and Yildiz (2011). Both the branch-and-bound algorithm and the branch-andprice-and-cut algorithm can rapidly achieve optimal solutions on instances with up to 10 teams.…”

“…Moreover, finding a feasible solution for medium-sized instances was very difficult; both CPLEX and the large neighborhood search heuristic failed to find any feasible solutions in many medium-sized instances. The background and applications of the TUP, and the set of benchmark instances used by Trick and Yildiz (2011), were further explained by Trick, Yildiz, and Yunes (2012). These benchmark instances are available at http://mat.tepper.cmu.edu/TUP/.…”

“…Trick and Yildiz (2011) formulated the problem as an integer programming (IP) model and a constraint programming model, and designed a Bender's cuts guided large neighborhood search heuristic to deal with the problem. To test the models and the proposed heuristic, Trick and Yildiz (2011) generated a set of benchmark instances where the number of teams ranged from 4 to 32. Their computational results suggested that the TUP was very difficult to solve to optimality; the commercial IP solver ILOG CPLEX could only exactly solve instances with 10 teams at most.…”