Scheduling non-professional table-tennis leagues

In this paper we consider a sports league scheduling problem which occurs in planning non-professional table-tennis leagues. The problem consists in finding a schedule for a time-relaxed double round robin tournament where different hard and soft constraints have to be taken into account. We model the problem as an integer linear program and a multi-mode resource-constrained project scheduling problem, respectively. Based on the second model a heuristic solution algorithm is proposed, which proceeds in two stages using local search and genetic algorithms. Computational results show the efficiency of the approaches.     

Scheduling a triple round robin tournament for the best Danish soccer league

In this paper, we present a solution method for the highly constrained problem of finding a seasonal schedule for the best Danish soccer league. The league differs from most sports leagues, since it plays a triple round robin tournament which leads to an uneven distribution of home and away games. The solution method presented here uses a logic-based Benders decomposition in which the master problem finds home-away pattern sets while the subproblem finds timetables. Furthermore, column generation techniques are used to enhance the speed of the master problem. The computational results show that the solution method is capable of solving the problem within reasonable time and the Danish Football Association has used it for scheduling the 2006/2007 season.     

A branch-and-price algorithm for scheduling sport leagues

A single round robin tournament can be described as a league of a set T of n teams (n even) to be scheduled such that each team plays exactly once against each other team and such that each team plays exactly once per period resulting in a set P of n?1 periods. Matches are carried out at one of the stadiums of both opponents. A team playing twice at home or twice away in two consecutive periods is said to have a break in the latter of both periods. There is a vast field of requests arising in real-world problems. For example, the number of breaks is to be minimized due to fairness reasons. It is well known that at least n?2 breaks must occur. We focus on schedules having the minimum number of breaks. Costs corresponding to each possible match are given and the objective is to minimize the sum of cost of arranged matches. Then, sports league scheduling can be seen as a hard combinatorial optimization problem. We develop a branch-and-price approach in order to find optimal solutions.     

Balanced home–away assignments

In recent years several approaches for generating sports league schedules have been proposed. In this paper we consider foundations for a two-stage approach to construct schedules for a single round robin tournament (or the first half series of a double round robin tournament). In the first stage for each game a mode (home or away) has to be determined and in the second stage the games have to be scheduled in their assigned modes. We study a problem of the first stage where balanced home–away assignments have to be constructed such that for each team the numbers of home and away games differ by at most one. After showing that it is easy to construct balanced home–away assignments we propose repairing mechanisms for unbalanced home–away assignments. Then, neighborhoods on the set of balanced home–away assignments are defined which are shown to be connected. Finally, situations with preassignments are studied.     

Multi-layered round robin routing for parallel servers

We study a system of several identical servers in parallel, where a routing decision must be made immediately on a job’s arrival. Jobs arrive according to a Poisson process, with their processing times following a discrete distribution with finite support. The processing time of a job is known on arrival and may be used in the routing decision. We propose a policy consisting of multi-layered round robin routing followed by shortest remaining processing time scheduling at the servers. This policy is shown to have a heavy traffic limit that is identical to one in which there is a single queue (no routing) and optimal heavy traffic scheduling. In light traffic, we show that the performance of this policy is worse than round robin routing followed by shortest remaining processing time scheduling. We also quantify the difference between round robin and multi-layered round robin routing, which in turn yields insights on the relative importance of routing versus (local) scheduling in such systems. AMS subject classifications: 68M20 · 60K25 (Work done while both authors were visitors at EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.)     

Investigation of polynomial algorithms for solving the three-index planar assignment problem

Under certain additional conditions imposed on the coefficients of the objective function in the three-index planar assignment problem, a large series of computational experiments aimed at the investigation of four polynomial algorithms for finding an asymptotically optimal solution of this problem is conducted.     

Investigation of polynomial algorithms for solving the multicriteria three-index planar assignment problem

Under certain additional conditions imposed on the coefficients of the vector objective function in the three-index planar assignment problem, a large series of computational experiments aimed at the investigation of four polynomial algorithms for finding an asymptotically optimal solution of this problem is carried out.     

Maximizing breaks and bounding solutions to the mirrored traveling tournament problem

We investigate the relation between two aspects of round robin tournament scheduling problems: breaks and distances. The distance minimization problem and the breaks maximization problem are equivalent when the distance between every pair of teams is equal to 1. We show how to construct schedules with a maximum number of breaks for some tournament types. The connection between breaks maximization and distance minimization is used to derive lower bounds to the mirrored traveling tournament problem and to prove the optimality of solutions found by a heuristic for the latter.     

An ILS heuristic for the traveling tournament problem with predefined venues

The Traveling Tournament Problem with Predefined Venues (TTPPV) is a single round robin variant of the Traveling Tournament Problem, in which the venue of each game to be played is known beforehand. We propose an Iterated Local Search (ILS) heuristic for solving real-size instances of the TTPPV, based on two types of moves. Initial solutions are derived from an edge coloring algorithm applied to complete graphs. We showed that canonical edge colorings should not be used as initial solutions in some situations. Instead, the use of Vizing’s edge coloring method lead to considerably better results. We also establish that the solution space defined by some commonly used neighborhoods in the sport scheduling literature is not connected in the case of single round robin tournaments, which explains the hardness of finding high quality solutions to some problem instances. Computational results show that the new ILS heuristic performs much better than heuristics based on integer programming and that it improves the best known solutions for benchmark instances.     

Feasibility of home-away-pattern sets for round robin tournaments

A Home-Away-Pattern (HAP) set defines each team’s venue in each period. We consider the decision problem of whether a round robin tournament can be arranged on the basis of a given HAP set. We give a necessary condition which can be checked in polynomial time and conjecture it to be sufficient.     

Scheduling a triple round robin tournament with minitournaments for the Finnish national youth ice hockey league

This paper presents a method to schedule a triple round robin tournament, which involves minitournaments, each hosted by one team. A key issue is that at the end of the season the number of home games should be balanced over the teams, despite the fact that in minitournament matches only the host team plays at home. This format is played in the Finnish national ice hockey league for players under the age of 20 years, where the problem is further complicated by many other constraints, for example, preassigned matches resulting from away tours that should limit the distances travelled by the teams. To obtain a schedule for this league, we sequentially solve four distinct combinatorial problems. This method allows us to construct a schedule for the 2009–2010 season, which is superior to the official schedule: it has no hard constraint violations, and outperforms the official schedule on three of five soft constraints.     

Round robin scheduling of heterogeneous parallel servers in heavy traffic

We consider scheduling for heterogeneous server systems, where tasks arrive according to a Poisson process, with their processing requirements following a discrete distribution with finite support. For a system with a dispatcher and several heterogeneous servers, we propose an optimized multi-layered round robin routing policy followed by shortest remaining processing time scheduling at each server. Using a heavy traffic approximation, we show that the proposed policy performs as well as the optimal scheduling policy for a heterogeneous servers system with a single queue (no routing) in heavy traffic. Additional simulation results suggest that such policies will be effective in more general settings.     

Geography,games and graphs

Some combinatorial problems related to scheduling the games of a sports league are considered. Graph theoretical models are proposed for dealing with so called geographical constraints: these are due to the fact that the teams of the league are located in different cities which may be at various distances from each other.     

A computational comparison of flow formulations for the capacitated location-routing problem

In this paper we present a computational comparison of four different flow formulations for the capacitated location-routing problem. We introduce three new flow formulations for the problem, namely a two-index two-commodity flow formulation, a three-index vehicle-flow formulation and a three-index two-commodity flow formulation. We also consider an existing two-index vehicle-flow formulation and extend it by considering new families of valid inequalities and separation algorithms. We introduce new branch-and-cut algorithms for each of the formulations and compare them on a wide number of instances. Our results show that compact formulations can produce tight gaps and solve many instances quickly, whereas three-index formulations scale better in terms of computing time.     

A combinatorial property of the maximum round robin tournament problem

We prove a new combinatorial property of the maximum round robin tournament (MRRT) problem. This property allows us to answer negatively the question of Briskorn, whether the optimal objective value of the MRRT problem and that of its conventional linear relaxation always coincide.     

Lower bounds for the axial three-index assignment problem

This paper describes new bounding methods for the axial three-index assignment problem (3AP). For calculating 3AP lower bounds, we use a projection method followed by a Hungarian algorithm, based on a new Lagrangian relaxation. We also use a cost transformation scheme, which iteratively transforms 3AP costs in a series of equivalent 3APs, which provides the possibility of improving the 3AP lower bound. These methods produce efficiently computed relatively tight lower bound.     

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.

     

A new approximation technique for resource‐allocation problems

We develop a rounding method based on random walks in polytopes, which leads to improved approximation algorithms and integrality gaps for several assignment problems that arise in resource allocation and scheduling. In particular, it generalizes the work of Shmoys and Tardos on the generalized assignment problem to the setting where some jobs can be dropped. New concentration bounds for random bipartite matching are developed as well.     

Single machine scheduling problems with general position-dependent processing times and past-sequence-dependent delivery times

This paper considers single machine scheduling with past-sequence-dependent (psd) delivery times, in which the processing time of a job depends on its position in a sequence. We provide a unified model for solving single machine scheduling problems with psd delivery times. We first show how this unified model can be useful in solving scheduling problems with due date assignment considerations. We analyze the problem with four different due date assignment methods, the objective function includes costs for earliness, tardiness and due date assignment. We then consider scheduling problems which do not involve due date assignment decisions. The objective function is to minimize makespan, total completion time and total absolute variation in completion times. We show that each of the problems can be reduced to a special case of our unified model and solved in O(n ³) time. In addition, we also show that each of the problems can be solved in O(nlogn) time for the spacial case with job-independent positional function.     

The timetable constrained distance minimization problem

We define the timetable constrained distance minimization problem (TCDMP) which is a sports scheduling problem applicable for tournaments where the total travel distance must be minimized. The problem consists of finding an optimal home-away assignment when the opponents of each team in each time slot are given. We present an integer programming, a constraint programming formulation and describe two alternative solution methods: a hybrid integer programming/constraint programming approach and a branch and price algorithm. We test all four solution methods on benchmark problems and compare the performance. Furthermore, we present a new heuristic solution method called the circular traveling salesman approach (CTSA) for solving the traveling tournament problem. The solution method is able to obtain high quality solutions almost instantaneously, and by applying the TCDMP, we show how the solutions can be further improved.