Timetabling through Constrained Heuristic Search and Genetic Algorithms

Hybrid genetic algorithms are presented that use constrained heuristic search and genetic techniques for the timetabling problem (TP). The TP is an NP-hard problem for which a general polynomial time deterministic algorithm is not known. The paper describes the classification of constraints and the constraint ordering to obtain the minimization of backtracking and the maximization of parallelism. The school timetabling problem is discussed in detail as a case study. The genetic algorithm approach is particularly well suited to this kind of problem, since there exists an easy way to assess a good timetable, but not a well structured automatic technique for constructing it. So, a population of timetables is created that evolves toward the best solution. The evaluation function and the genetic operators are well separated from the domain-specific parts, such as the knowledge of the problem and the heuristics, i.e. from the timetable builder. The present paper illustrates an approach based on the hybridization of constrained heuristic search with novel genetic algorithm techniques. It compares favourably with known programs to solve decision problems under logic constraints. The cost of the new algorithm and the quality of the solutions obtained in significant experiments are reported.     

Timetabling through a deductive database: A case study

Timetabling and Constraint Satisfaction are very hard problems and a general polynomial time deterministic algorithm is not known. An Artificial Intelligence approach, in a logic programming environment may be useful for such problems. The decomposition and classification of constraints and the constraint ordering to obtain the minimization of the backtracking and the maximization of the parallelism are illustrated. The timetabling problem is discussed in detail as a case study: a PROLOG system and a PARLOG system which solve the problem are described and their performances in significant tests are assessed. Finally, a General Constraint Solver is proposed that may be useful for several Resource Allocation, Engineering, DBMS, Decision Support Systems.     

Genetic based discrete particle swarm optimization for Elderly Day Care Center timetabling

The timetabling problem of local Elderly Day Care Centers (EDCCs) is formulated into a weighted maximum constraint satisfaction problem (Max-CSP) in this study. The EDCC timetabling problem is a multi-dimensional assignment problem, where users (elderly) are required to perform activities that require different venues and timeslots, depending on operational constraints. These constraints are categorized into two: hard constraints, which must be fulfilled strictly, and soft constraints, which may be violated but with a penalty. Numerous methods have been successfully applied to the weighted Max-CSP; these methods include exact algorithms based on branch and bound techniques, and approximation methods based on repair heuristics, such as the min-conflict heuristic. This study aims to explore the potential of evolutionary algorithms by proposing a genetic-based discrete particle swarm optimization (GDPSO) to solve the EDCC timetabling problem. The proposed method is compared with the min-conflict random-walk algorithm (MCRW), Tabu search (TS), standard particle swarm optimization (SPSO), and a guided genetic algorithm (GGA). Computational evidence shows that GDPSO significantly outperforms the other algorithms in terms of solution quality and efficiency.     

A genetic algorithm for a university weekly courses timetabling problem

The timetabling problem is concerned with the allocation, subject to constraints, of given resources to objects in space and time in such way as to satisfy as nearly as possible a set of desirable objectives. This problem is known to be NP–complete and as such only combinatorial optimization methods can guarantee an optimal timetable. In this paper we propose a sector–based genetic algorithm for solving a university weekly courses timetabling problem. Preliminary experimental results indicate that the algorithm is promising.     

An informed genetic algorithm for the examination timetabling problem

This paper presents the results of a study conducted to investigate the use of genetic algorithms (GAs) as a means of inducing solutions to the examination timetabling problem (ETP). This study differs from previous efforts applying genetic algorithms to this domain in that firstly it takes a two-phased approach to the problem which focuses on producing timetables that meet the hard constraints during the first phase, while improvements are made to these timetables in the second phase so as to reduce the soft constraint costs. Secondly, domain specific knowledge in the form of heuristics is used to guide the evolutionary process. The system was tested on a set of 13 real-world problems, namely, the Carter benchmarks. The performance of the system on the benchmarks is comparable to that of other evolutionary techniques and in some cases the system was found to outperform these techniques. Furthermore, the quality of the examination timetables evolved is within range of the best results produced in the field.     

A genetic algorithm for searching spatial configurations

Searching spatial configurations is a particular case of maximal constraint satisfaction problems, where constraints expressed by spatial and nonspatial properties guide the search process. In the spatial domain, binary spatial relations are typically used for specifying constraints while searching spatial configurations. Searching configurations is particularly intractable when configurations are derived from a combination of objects, which involves a hard combinatorial problem. This paper presents a genetic algorithm (GA) that combines a direct and an indirect approach to treating binary constraints in genetic operators. A new genetic operator combines randomness and heuristics for guiding the reproduction of new individuals in a population. Individuals are composed of spatial objects whose relationships are indexed by a content measure. This paper describes the GA and presents experimental results that compare the genetic versus a deterministic and a local-search algorithm. These experiments show the convenience of using a GA when the complexity of the queries and databases do no guarantee the tractability of a deterministic strategy.     

A hyperheuristic approach to examination timetabling problems: benchmarks and a new problem from practice

Many researchers studying examination timetabling problems focus on either benchmark problems or problems from practice encountered in their institutions. Hyperheuristics are proposed as generic optimisation methods which explore the search space of heuristics rather than direct solutions. In the present study, the performance of tournament-based hyperheuristics for the exam timetabling problem are investigated. The target instances include both the Toronto and ITC 2007 benchmarks and the examination timetabling problem at KAHO Sint-Lieven (Ghent, Belgium). The Toronto and ITC 2007 benchmarks are post-enrolment-based examination timetabling problems, whereas the KAHO Sint-Lieven case is a curriculum-based examination timetabling problem. We drastically improve the previous (manually created) solution for the KAHO Sint-Lieven problem by generating a timetable that satisfies all the hard and soft constraints. We also make improvements on the best known results in the examination timetabling literature for seven out of thirteen instances for the To ronto benchmarks. The results are competitive with those of the finalists of the examination timetabling track of the International Timetabling Competition.     

课程表问题的一种模拟退火超启发式算法

课程表问题是经典的组合优化问题,属于NP-hard问题.长期以来人们一直都在寻求快速高效的近似算法,以便在合理的计算时间内准确解决大规模课程安排问题,并提出许多有效且实用的启发式和元启发式算法.在此基础上提出了一种基于多个图染色启发式规则的模拟退火超启发式算法.在超启发式算法的框架中,用模拟退火算法作为高层搜索算法,多个图染色启发式规则为底层的构造算法.与现有的方法相比,该算法具有很好的通用性,可以很容易推广到考试时间表、会议安排.旅行商问题、背包问题等应用领域.实验表明,该算法是可行有效的,且无一例时间、空间冲突.     

基于贪心法和禁忌搜索的实用高校排课系统研究

在深入分析普通高校排课的流程、特点和难点的基础上，提出一个基于贪心法和禁忌搜索的排课算法。算法采用基于优先级的贪心法构造排课的初始解，进而利用禁忌搜索获得全局较优的排课结果。设计中充分考虑了当前高校课表问题的实际情况，如课程性质对排课的要求、教师的特殊要求等。实现的原型系统同时支持自动排课和交互式排课，对于一些难度较大的问题，可以通过人机交互方式来解决。通过对高校的实际排课数据进行测试，结果表明该算法可行且能够有效地提高排课效率。     

A multi-objective evolutionary algorithm for examination timetabling

This paper considers the scheduling of exams for a set of university courses. The solution to this exam timetabling problem involves the optimization of complete timetables such that there are as few occurrences of students having to take exams in consecutive periods as possible but at the same time minimizing the timetable length and satisfying hard constraints such as seating capacity and no overlapping exams. To solve such a multi-objective combinatorial optimization problem, this paper presents a multi-objective evolutionary algorithm that uses a variable-length chromosome representation and incorporates a micro-genetic algorithm and a hill-climber for local exploitation and a goal-based Pareto ranking scheme for assigning the relative strength of solutions. It also imports several features from the research on the graph coloring problem. The proposed algorithm is shown to be a more general exam timetabling problem solver in that it does not require any prior information of the timetable length to be effective. It is also tested against a few influential and recent optimization techniques and is found to be superior on four out of seven publicly available datasets.     

A greedy gradient-simulated annealing selection hyper-heuristic

Educational timetabling problem is a challenging real world problem which has been of interest to many researchers and practitioners. There are many variants of this problem which mainly require scheduling of events and resources under various constraints. In this study, a curriculum based course timetabling problem at Yeditepe University is described and an iterative selection hyper-heuristic is presented as a solution method. A selection hyper-heuristic as a high level methodology operates on the space formed by a fixed set of low level heuristics which operate directly on the space of solutions. The move acceptance and heuristic selection methods are the main components of a selection hyper-heuristic. The proposed hyper-heuristic in this study combines a simulated annealing move acceptance method with a learning heuristic selection method and manages a set of low level constraint oriented heuristics. A key goal in hyper-heuristic research is to build low cost methods which are general and can be reused on unseen problem instances as well as other problem domains desirably with no additional human expert intervention. Hence, the proposed method is additionally applied to a high school timetabling problem, as well as six other problem domains from a hyper-heuristic benchmark to test its level of generality. The empirical results show that our easy-to-implement hyper-heuristic is effective in solving the Yeditepe course timetabling problem. Moreover, being sufficiently general, it delivers a reasonable performance across different problem domains.     

Modular design of a hybrid genetic algorithm for a flexible job–shop scheduling problem

The aim of the job–shop scheduling problem is to optimize the task planning in an industrial plant satisfying time and technological constraints. The existing algorithmic and mathematical methods for solving this problem usually have high computational complexities making them intractable. Flexible job–shop scheduling becomes even more complex, since it allows one to assign each operation to a resource from a set of suitable ones. Alternative heuristic methods are only able to satisfy part of the constraints applicable to the problem. Moreover, these solutions usually offer little flexibility to adapt them to new requirements. This paper describes research within heuristic methods that combines genetic algorithms with repair heuristics. Firstly, it uses a genetic algorithm to provide a non-optimal solution for the problem, which does not satisfy all its constraints. Then, it applies repair heuristics to refine this solution. There are different types of heuristics, which correspond to the different types of constraints. A heuristic is intended to evaluate and slightly modify a solution that violates a constraint in a way that avoids or mitigates such violation. This approach improves the adaptability of the solution to a problem, as some changes can be addressed just modifying the considered chromosome or heuristics. The proposed solution has been tested in order to analyse its level of constraint satisfaction and its makespan, which are two of the main parameters considered in these types of problems. The paper discusses this experimentation showing the improvements over existing methods.     

Train service timetabling in railway open markets by particle swarm optimisation

Railway timetabling is an important process in train service provision as it matches the transportation demand with the infrastructure capacity while customer satisfaction is also considered. It is a multi-objective optimisation problem, in which a feasible solution, rather than the optimal one, is usually taken in practice because of the time constraint. The quality of services may suffer as a result. In a railway open market, timetabling usually involves rounds of negotiations amongst a number of self-interested and independent stakeholders and hence additional objectives and constraints are imposed on the timetabling problem. While the requirements of all stakeholders are taken into consideration simultaneously, the computation demand is inevitably immense. Intelligent solution-searching techniques provide a possible solution. This paper attempts to employ a particle swarm optimisation (PSO) approach to devise a railway timetable in an open market. The suitability and performance of PSO are studied on a multi-agent-based railway open-market negotiation simulation platform.     

A Constraint-Based Interactive Train Rescheduling Tool

In this paper, we report the design and implementation of a constraint-based interactive train rescheduling tool, a project in collaboration with the International Institute for Software Technology, United Nations University (UNU/IIST), Macau. We formulate train rescheduling as constraint satisfaction and describe a constraint propagation approach for tackling the problem. Algorithms for timetable verification and train rescheduling are designed under a coherent framework. Formal correctness properties of the rescheduling algorithm are established. We define two optimality criteria for rescheduling that correspond to minimizing the number of station visits affected and passenger delay respectively. Two heuristics are then proposed to speed up and direct the search towards optimal solutions. The feasibility of our proposed algorithms and heuristics are confirmed with experimentation using real-life data.     

混合算法在大学课程表问题中的应用研究

大学课程袁问题是时间表问题之一,也是一个多因素的优化决策问题.文章提出的混合算法,基于动态规划的思想,对大学课程表问题进行分阶段求解,分别采用遗传算法分配时间,采用最佳适应算法分配场地.实验结果表明,这种方法既保证了课表的质量,又有利于工程上实现和扩展.     

Single machine due date assignment scheduling problem with customer service level in fuzzy environment

Due date assignment scheduling problems with deterministic and stochastic parameters have been studied extensively in recent years. In this paper, we consider a single machine due date assignment scheduling problem with uncertain processing times and general precedence constraint among the jobs. The processing times of the jobs are assumed to be fuzzy numbers. We first propose an optimal polynomial time algorithm for the problem without precedence constraints among jobs. Then, we show that if general precedence constraint is involved, the problem is NP-hard. Finally, we show that if the precedence constraint is a tree or a collection of trees, the problem is still polynomially solvable.     

A Combined column generation and heuristics for railway short-term rolling stock planning with regular inspection constraints

The aim of railway rolling stock planning problem is to find an optimal allocation of train-sets for a given set of trips in the train timetable in order to minimize the total cost. We propose a column generation and Lagrangian relaxation heuristics for short-term rolling stock planning problems with regular inspection constraints. The problem is formulated as a subtour traveling salesman problem to find a set of elementary shortest cycles that cover all trips in the timetable. In the proposed method, a tight lower bound is obtained from the continuous relaxation of Dantzig–Wolfe reformulation by column generation. The pricing problem can be formulated as an elementary shortest cycle problem with resource constraints. A labeling algorithm is applied to solve the pricing problem. In order to reduce the computational effort, we apply a general state space augmenting algorithm to solve the pricing problems. Computational results show that the proposed column generation and Lagrangian relaxation heuristics can find good lower and upper bounds for 300 trips within reasonable computing time.     

基于PBIL算法的高校自动排考系统

提出了一种基于PBIL的高校自动排考算法,重点论述了如何优化目标函数与排考约束条件之间的关系,并对PBIL基因选择算法提出了改进。通过实际的测试应用,基于PBIL算法的自动排考系统能够较好地满足学分制下的自动排考需求,对附加约束条件具有较强的适应性,能够满足各个学校的不同排考需求。     

A Survey of Automated Timetabling

The timetabling problem consists in scheduling a sequence of lectures between teachers and students in a prefixed period of time (typically a week), satisfying a set of constraints of various types. A large number of variants of the timetabling problem have been proposed in the literature, which differ from each other based on the type of institution involved (university or school) and the type of constraints. This problem, that has been traditionally considered in the operational research field, has recently been tackled with techniques belonging also to Artificial Intelligence (e.g., genetic algorithms, tabu search, and constraint satisfaction). In this paper, we survey the various formulations of the problem, and the techniques and algorithms used for its solution.