A Cooperative Local Search Method for Solving the Traveling Tournament Problem

Constrained optimization is the process of optimizing a certain objective function subject to a set of constraints. The goal is not necessarily to find the global optimum. We try to explore the search space more efficiently in order to find a good approximate solution. The obtained solution should verify the hard constraints that are required to be satisfied. In this paper, we propose a cooperative search method that handles optimality and feasibility separately. We take the traveling tournament problem (TTP) as a case study to show the applicability of the proposed idea. TTP is the problem of scheduling a double round-robin tournament that satisfies a set of related constraints and minimizes the total distance traveled by the teams. The proposed method for TTP consists of two main steps. In the first step, we ignore the optimization criterion. We reduce the search only to feasible solutions satisfying the problem's constraints. For this purpose, we use constraints programming model to ensure the feasibility of solutions. In the second step, we propose a stochastic local search method to handle the optimization criterion and find a good approximate solution that verifies the hard constraints. The overall method is evaluated on benchmarks and compared with other well-known techniques for TTP. The computational results are promising and show the effectiveness of the proposed idea for TTP

Particle Swarm Algorithm for Improved Handling of the Mirrored Traveling Tournament Problem

In this study, we used a particle swarm optimization (PSO) algorithm to address a variation of the non-deterministic polynomial-time NP-hard traveling tournament problem, which determines the optimal schedule for a double round-robin tournament, for an even number of teams, to minimize the number of trips taken. Our proposed algorithm iteratively explored the search space with a swarm of particles to find near-optimal solutions. We also developed three techniques for updating the particle velocity to move towards optimal points, which randomly select and replace row and column parameters to find candidate positions close to an optimal solution. To further optimize the solution, we calculated the particle cost function, an important consideration within the problem conditions, for team revenues, fans, and media. We compared our computation results with two well-known meta-Heuristics: a genetics algorithm utilizing a swapping method and a Greedy Randomized Adaptive Search Procedure Iterated Local Search algorithm heuristic on a set of 20 teams. Ultimately, the PSO algorithm generated solutions that were comparable, and often superior, to the existing well-known solutions. Our results indicate that our proposed algorithm could aid in reducing the overall budget expenditures of international sports league organizations, which could enable significant monetary savings and increase profit margins

Referee assignment in the Chilean football league using integer programming and patterns

This article uses integer linear programming to address the referee assignment problem in the First Division of the Chilean professional football league. The proposed approach considers balance in the number of matches each referee must officiate, the frequency of each referee being assigned to a given team, the distance each referee must travel over the course of a season, and the appropriate pairings of referee experience or skill category with the importance of the matches. Two methodologies are studied, one traditional and the other a pattern-based formulation inspired by the home-away patterns for scheduling season match calendars. Both methodologies are tested in real-world and experimental instances, reporting results that improve significantly on the manual assignments. The pattern-based formulation attains major reductions in execution times, solving real instances to optimality in just a few seconds, while the traditional one takes anywhere from several minutes to more than an hour.Fil: Alarcón, Fernando. Universidad de Chile; ChileFil: Duran, Guillermo Alfredo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Guajardo, Mario. Norwegian School of Economics; Norueg

Solving Challenging Real-World Scheduling Problems

This work contains a series of studies on the optimization of three real-world scheduling problems, school timetabling, sports scheduling and staff scheduling. These challenging problems are solved to customer satisfaction using the proposed PEAST algorithm. The customer satisfaction refers to the fact that implementations of the algorithm are in industry use. The PEAST algorithm is a product of long-term research and development. The first version of it was introduced in 1998. This thesis is a result of a five-year development of the algorithm. One of the most valuable characteristics of the algorithm has proven to be the ability to solve a wide range of scheduling problems. It is likely that it can be tuned to tackle also a range of other combinatorial problems. The algorithm uses features from numerous different metaheuristics which is the main reason for its success. In addition, the implementation of the algorithm is fast enough for real-world use.Siirretty Doriast

Time Relaxed Round Robin Tournament and the NBA Scheduling Problem

This dissertation study was inspired by the National Basketball Association regular reason scheduling problem. NBA uses the time-relaxed round robin tournament format, which has drawn less research attention compared to the other scheduling formats. Besides NBA, the National Hockey League and many amateur leagues use the time-relaxed round robin tournament as well. This dissertation study is the first ever to examine the properties of general time-relaxed round robin tournaments. Single round, double round and multiple round time-relaxed round robin tournaments are defined. The integer programming and constraint programming models for those tournaments scheduling are developed and presented. Because of the complexity of this problem, several decomposition methods are presented as well. Traveling distance is an important factor in the tournament scheduling. Traveling tournament problem defined in the time constrained conditions has been well studied. This dissertation defines the novel problem of time-relaxed traveling tournament problem. Three algorithms has been developed and compared to address this problem. In addition, this dissertation study presents all major constraints for the NBA regular season scheduling. These constraints are grouped into three categories: structural, external and fairness. Both integer programming and constraint programming are used to model these constraints and the computation studies are presente

Handling fairness issues in time-relaxed tournaments with availability constraints

Sports timetables determine who will play against whom, where, and on which time slot. In contrast to time-constrained sports timetables, time-relaxed timetables utilize (many) more time slots than there are games per team. This offers time-relaxed timetables additional flexibility to take into account venue availability constraints, stating that a team can only play at home when its venue is available, and player availability constraints stating that a team can only play when its players are available. Despite their flexibility, time-relaxed timetables have the drawback that the rest period between teams’ consecutive games can vary considerably, and the difference in the number of games played at any point in the season can become large. Besides, it can be important to timetable home and away games alternately. In this paper, we first establish the computational complexity of time-relaxed timetabling with availability constraints. Naturally, when one also incorporates fairness objectives on top of availability, the problem becomes even more challenging. We present two heuristics that can handle these fairness objectives. First, we propose an adaptive large neighborhood method that repeatedly destroys and repairs a timetable. Second, we propose a memetic algorithm that makes use of local search to schedule or reschedule all home games of a team. For numerous artificial and real-life instances, these heuristics generate high-quality timetables using considerably less computational resources compared to integer programming models solved using a state-of-the-art solver

Scheduling a non-professional indoor football league : a tabu search based approach

This paper deals with a real-life scheduling problem of a non-professional indoor football league. The goal is to develop a schedule for a time-relaxed, double round-robin tournament which avoids close successions of games involving the same team in a limited period of time. This scheduling problem is interesting, because games are not planned in rounds. Instead, each team provides time slots in which they can play a home game, and time slots in which they cannot play at all. We present an integer programming formulation and a heuristic based on tabu search. The core component of this algorithm consists of solving a transportation problem, which schedules (or reschedules) all home games of a team. Our heuristic generates schedules with a quality comparable to those found with IP solvers, however with considerably less computational effort. These schedules were approved by the league organizers, and used in practice for the seasons 2009-2010 till 2016-2017

Balancing the Game: Comparative Analysis of Single Heuristics and Adaptive Heuristic Approaches for Sports Scheduling Problem

Sport timetabling problems are Combinatorial Optimization problems which involve the creation of schedules that determine when and where teams compete against each other. One specific type of sports scheduling, the double round-robin (2RR) tournament, mandates that each team faces every other team twice, once at their home venue and once at the opponent’s. Despite the relatively small number of teams involved, the sheer volume of potential scheduling combinations has spurred researchers to employ various techniques to find efficient solutions for sports scheduling problems. In this thesis, we present a comparative analysis of single and adaptive heuristics designed to efficiently solve sports scheduling problems. Specifically, our focus is on constructing time-constrained double round-robin tournaments involving 16 to 20 teams, while adhering to hard constraints and minimizing penalties for soft constraints violations. The computational results demonstrate that our adaptive heuristic approach not only successfully finds feasible solutions for the majority of instances but also outperforms the single heuristics examined in this study.Master's Thesis in InformaticsINF399MAMN-INFMAMN-PRO

The Traveling Tournament Problem

In this thesis we study the Traveling Tournament problem (TTP) which asks to generate a feasible schedule for a sports league such that the total travel distance incurred by all teams throughout the season is minimized. Throughout our three technical chapters a wide range of topics connected to the TTP are explored. We begin by considering the computational complexity of the problem. Despite existing results on the NP-hardness of TTP, the question of whether or not TTP is also APX-hard was an unexplored area in the literature. We prove the affirmative by constructing an L-reduction from (1,2)-TSP to TTP. To reach the desired result, we show that given an instance of TSP with a solution of cost K, we can construct an instance of TTP with a solution of cost at most 20m(m+1)cK where m = c(n-1)+1, n is the number of teams, and c > 5, c ∈ ℤ is fixed. On the other hand, we show that given a feasible schedule to the constructed TTP instance, we can recover a tour on the original TSP instance. The next chapter delves into a popular variation of the problem, the mirrored TTP, which has the added stipulation that the first and second half of the schedule have the same order of match-ups. Building upon previous techniques, we present an approximation algorithm for constructing a mirrored double round-robin schedule under the constraint that the number of consecutive home or away games is at most two. We achieve an approximation ratio on the order of 3/2 + O(1)/n. Lastly, we present a survey of local search methods for solving TTP and discuss the performance of these techniques on benchmark instances

Scheduling sport tournaments using constraint logic programming

We tackle the problem of scheduling the matches of a round robin tournament for a sport league. We formally define the problem, state its computational complexity, and present a solution algorithm using a two-step approach. The first step is the creation of a tournament pattern and is based on known graph-theoretic results. The second one is a constraint-based depth-first branch and bound procedure that assigns actual teams to numbers in the pattern. The procedure is implemented using the finite domain library of the constraint logic programming language eclipse. Experimental results show that, in practical cases, the optimal solution can be found in reasonable time, despite the fact that the problem is NP-complete