Discrete OptimizationParallel local search algorithms for high school timetabling problems
Introduction
Educational timetabling problems consist in scheduling encounters between teachers (or exams) and students. Specific situations give origin to different problems, with a richness of characteristics. The scientific literature has branched this family of problems in three main sub-areas: University course timetabling Di Gaspero, McCollum, and Schaerf (2007); Lewis (2008); Lewis, Paechter, and McCollum (2007), examination timetabling (McCollum, McMullan, Burke, Parkes, Qu, 2007, Qu, Burke, McCollum, Merlot, Lee, 2009) and high school timetabling (De Werra, 1985, Pillay, 2014, Post, Ahmadi, Daskalaki, Kingston, Kyngas, Nurmi, Ranson, 2012, Schaerf, 1999). Each of these three families contains their own set of specific constraints.
In this article, we address the high school timetabling problem (HSTP). A restricted decision version of the HSTP was shown to be NP-complete by polynomial reduction of the 3-SAT problem (Even, Itai, & Shamir, 1975). Instances originating from practical contexts naturally extend the requirements of this idealized problem. Extra requirements are usually associated with pedagogical preferences and social/cultural particularities. Examples of these extra requirements are the need to assign double lessons (two consecutive lessons) for some classes subjects and the need to obtain compact schedules for teachers (Pillay, 2014).
The HSTP is usually modeled by mixed-integer programming (MIP) models (Al-Yakoob, Sherali, 2015, Dorneles, de Araújo, Buriol, 2014, Dorneles, de Araújo, Buriol, 2017, Kristiansen, Sørensen, Stidsen, 2015, Santos, Uchoa, Ochi, Maculan, 2012, Sørensen, Dahms, 2014). The resolution of these models for most medium and large practical instances are known to be still a challenge for actual black-box MIP solvers, which may be why most of the literature is concerned with metaheuristic techniques (Pillay, 2014). The main goal in these metaheuristic studies is to develop strategies to generate good quality solutions within reasonable computational efforts. Recently proposed metaheuristics can be found in Dorneles et al. (2014); Fonseca and Santos (2014); Fonseca, Santos, and Carrano (2016a); Fonseca, Santos, Toffolo, Brito, and Souza (2016c); Saviniec, Constantino, Romão, and Santos (2013). These heuristics are shown to be efficient in the sense that near-optimal solutions are consistently found for different input instances.
In this article, we aim to investigate parallel metaheuristics for the HSTP, in which different solution methods (agents) are run concurrently in different processor threads. The motivation is two-fold. On one hand, the availability of multi-processor machines and appropriate coding schemes make the use of parallel algorithms more accessible than ever. On the other hand, although the literature is abundant in parallel solution methods for similar complex problems (Subramanian, Drummond, Bentes, Ochi, Farias, 2010, Bożejko, Pempera, Smutnicki, 2013, Sánchez-Oro, Sevaux, Rossi, Martí, Duarte, 2015, Luque, Alba, 2015, e.g.), very little attention has been given to the design of parallel methods for the HSTP (Abramson, 1991, Abramson, Abela, 1992, Srndic, Pandzo, Dervisevic, Konjicija, 2009). The reader is referred to Section 3 for a closer look at these contributions.
As an exploratory study, our goal is to investigate a number of questions related to the design of parallel metaheuristics in the context of the HSTP. The main research questions can be summarized as: (1) What are good parallel strategies for the HSTP? (2) Can these strategies improve the performance of stand-alone metaheuristics? (3) Are the proposed parallel metaheuristics competitive with state-of-the-art methods?
In order to provide some insight into these questions, we propose a thorough computational study that tests all parallelization schemes resulting from combinations of the following characteristics:
Agents cooperation: we wish to test the effect of allowing the agents to cooperate, by sharing good solutions among threads.
Diversification: we wish to test the effect of using all agents as search intensification mechanisms or allowing at least one agent to diversify the search.
Agents diversity: we wish to test homogeneous and heterogeneous algorithms. An algorithm is homogeneous when all threads execute the same metaheuristic, and heterogeneous otherwise.
As stand-alone metaheuristics, we use the iterated local search from Saviniec et al. (2013) and adapt versions of tabu search, simulated annealing, and late acceptance strategy. We compare the performance of the different parallel algorithm variants against each other, against the stand-alone metaheuristics and against two state-of-the-art algorithms for close variants of the problem Dorneles et al. (2014); Fonseca et al. (2016a).
The remainder of this paper is organized as follows. Section 2 describes the problem both in plain language and with the help of a formal mixed-integer programming formulation. Section 3 gives an introduction to different parallelization schemes and reviews the existing parallel algorithms for the HSTP. Section 4 explains our parallel metaheuristics. Section 5 presents our computational experiments and Section 6 concludes this paper with final remarks and suggestions for further investigations. An appendix completes this article with details on the used metaheuristics and their parameter setting procedures.
Section snippets
The high school timetabling problem
We focus on a real HSTP motivated by Brazilian high schools timetabling rules. In this context, classes are disjoint groups of students enrolled in the same set of subjects (mathematics, chemistry, e.g.) and with no free periods during the week. The school also has a set of teachers in its workforce and the goal of the problem is to obtain weekly timetable specifying the schedule of meetings for class/teacher pairs. Definition 1 HSTP instance A HSTP instance is the input data for the timetabling construction in a
Existing parallel algorithms for the HSTP
Metaheuristics can be classified into population-based (PB) and trajectory-based (TB) methods (Alba, Luque, & Nesmachnow, 2013). Population-based metaheuristics are characterized by keeping a pool of solutions (genetic algorithms, ant colony optimization, and particle swarm optimization, e.g.). The method starts with an initial population and employs, at each step, stochastic operators to evolve toward better quality populations. According to Alba et al. (2013), two classical parallel algorithm
Proposed parallel metaheuristics
The proposed algorithms follow the trajectory-based parallel multi-start (TB-PMS) scheme described in the previous section and employ manager/workers strategies with shared memories. We propose the use of two main strategies. The first is based on central memory (Crainic, Gendreau, Hansen, & Mladenović, 2004) and the second is based on diversification and intensification memories (Jin, Crainic, & Løkketangen, 2014).
Computational experiments
In this section, we conduct a thorough computational study in order to evaluate the performance of the proposed parallel methods. All algorithms were coded in C++ and compiled with GNU Compiler Collection 4.4.7. The experiments were made on a server running Red Hat Enterprise Linux 6.5. The hardware is composed of two CPU Intel Xeon E5-2680v2 (2.8 gigahertz) and 128 gigabytes of RAM. To implement parallelism we employed the POSIX Threads Library (Pthreads) available in the GNU Compiler
Conclusions
In this paper, we conducted an extensive study with parallel trajectory-based metaheuristics for high school timetabling problems. Our study analyzed several aspects related to the design of these algorithms. In particular, we studied the homogeneity/heterogeneity of agents, the influence of intensification and diversification memories, and the frequency of information exchange among agents.
Our most efficient algorithm was obtained with the inclusion of a diversification memory, a very elitist
Acknowledgments
This research was supported by FAPESP-Brazil. Grants: 2013/13563-3 and 2015/10032-2. The authors would like to thank Professor George H. G. Fonseca for sharing the code of GOAL and also, the anonymous reviewers for their useful comments in our paper.
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