A graph-based hyper-heuristic for educational timetabling problems

This paper presents an investigation of a simple generic hyper-heuristic approach upon a set of widely used constructive heuristics (graph coloring heuristics) in timetabling. Within the hyper-heuristic framework, a tabu search approach is employed to search for permutations of graph heuristics which are used for constructing timetables in exam and course timetabling problems. This underpins a multi-stage hyper-heuristic where the tabu search employs permutations upon a different number of graph heuristics in two stages. We study this graph-based hyper-heuristic approach within the context of exploring fundamental issues concerning the search space of the hyper-heuristic (the heuristic space) and the solution space. Such issues have not been addressed in other hyper-heuristic research. These approaches are tested on both exam and course benchmark timetabling problems and are compared with the fine-tuned bespoke state-of-the-art approaches. The results are within the range of the best results reported in the literature. The approach described here represents a significantly more generally applicable approach than the current state of the art in the literature. Future work will extend this hyper-heuristic framework by employing methodologies which are applicable on a wider range of timetabling and scheduling problems.     

Hybridizations within a graph-based hyper-heuristic framework for university timetabling problems

A significant body of recent literature has explored various research directions in hyper-heuristics (which can be thought as heuristics to choose heuristics). In this paper, we extend our previous work to construct a unified graph-based hyper-heuristic (GHH) framework, under which a number of local search-based algorithms (as the high level heuristics) are studied to search upon sequences of low-level graph colouring heuristics. To gain an in-depth understanding on this new framework, we address some fundamental issues concerning neighbourhood structures and characteristics of the two search spaces (namely, the search spaces of the heuristics and the actual solutions). Furthermore, we investigate efficient hybridizations in GHH with local search methods and address issues concerning the exploration of the high-level search and the exploitation ability of the local search. These, to our knowledge, represent entirely novel directions in hyper-heuristics. The efficient hybrid GHH obtained competitive results compared with the best published results for both benchmark course and exam timetabling problems, demonstrating its efficiency and generality across different problem domains. Possible extensions upon this simple, yet general, GHH framework are also discussed.     

A genetic algorithm approach to school timetabling

An adaptive algorithm based on computational intelligence techniques is designed, developed and applied to the timetabling problem of educational organizations. The proposed genetic algorithm is used in order to create feasible and efficient timetables for high schools in Greece. In order to demonstrate the efficiency of the proposed genetic algorithm, exhaustive experiments with real-world input data coming from many different high schools in the city of Patras have been conducted. As well as that, in order to demonstrate the superior performance of the proposed algorithm, we compare its experimental results with the results obtained by another effective algorithm applied to the same problem. Simulation results showed that the proposed algorithm outperforms other existing attempts. However, the most significant contribution of the paper is that the proposed algorithm allows for criteria adaptation, thus producing different timetables for different constraints priorities. So, the proposed approach, due to its inherent adaptive capabilities, can be used, each time satisfying different specific constraints, in order to lead to different timetables, thus meeting the different needs that each school may have.     

A mixed-integer mathematical modeling approach to exam timetabling

This paper explores mathematical programming models for an exam timetabling problem related to Kuwait University (KU). In particular, we consider two subproblems: (a) the ExamTimetabling Problem (ETP), which is concerned with assigning exams to designated exam-periods and classrooms, and (b) the Proctor Assignment Problem (PAP), which deals with the assignment of proctors to exams. While this exam timetabling problem is ubiquitous in many academic institutions worldwide, idiosyncrasies of the problem related to gender-based policies and having multiple exam centers at KU require novel models. A mixed-integer exam timetabling programming model (ETM) is developed for ETP, which takes into account restrictions related to exam-period conflicts, facility and human resources, and commuting and traffic considerations. Assuming that exam-periods are specified for all exams as determined by ETM, another mixed-integer programming model is formulated for PAP, denoted by PAM, which incorporates the proctors’ preferences for specific days and exam-periods. Computational results are reported and analyzed for solving ETM and PAM directly using the CPLEX Optimization Software (version 9.0), and for implementing a specialized sequential LP-based heuristic for solving PAM. The results obtained significantly improve upon those derived via the existing manual approach implemented at KU, in terms of eliminating conflicts as well as from the overall efficiency and equity points of view.     

A hyper-heuristic approach to sequencing by hybridization of DNA sequences

In this paper we investigate the use of hyper-heuristic methodologies for predicting DNA sequences. In particular, we utilize Sequencing by Hybridization. We believe that this is the first time that hyper-heuristics have been investigated in this domain. A hyper-heuristic is provided with a set of low-level heuristics and the aim is to decide which heuristic to call at each decision point. We investigate three types of hyper-heuristics. Two of these (simulated annealing and tabu search) draw their inspiration from meta-heuristics. The choice function hyper-heuristic draws its inspiration from reinforcement learning. We utilize two independent sets of low-level heuristics. The first set is based on a previous tabu search method, with the second set being a significant extension to this basic set, including utilizing a different representation and introducing the definition of clusters. The datasets we use comprises two randomly generated datasets and also a publicly available biological dataset. In total, we carried out experiments using 70 different combinations of heuristics, using the three datasets mentioned above and investigating six different hyper-heuristic algorithms. Our results demonstrate the effectiveness of a hyper-heuristic approach to this problem domain. It is necessary to provide a good set of low-level heuristics, which are able to both intensify and diversify the search but this approach has demonstrated very encouraging results on this extremely difficult and important problem domain.     

Fractional programming approach to two stochastic inventory problems

This paper considers two popular inventory models: the continuous review and periodic review reorder-point, order-quantity, control systems. Specifically we present two procedures which determine optimal values for the two control parameters (i.e., reorder-point and order-quantity) when the holding-and-shortage costs are non-quasi-convex. This cost structure may arise when non-linear cost rate is considered, for instance when the shortage cost is the shadow cost of a service-level constraint. The algorithms based on a fractional programming method are intuitive and efficient, and as the holding-and-shortage cost functions become quasi-convex, they are compatible to existing algorithms.     

A column generation approach to train timetabling on a corridor

We propose heuristic and exact algorithms for the (periodic and non-periodic) train timetabling problem on a corridor that are based on the solution of the LP relaxation of an ILP formulation in which each variable corresponds to a full timetable for a train. This is in contrast with previous approaches to the same problem, which were based on ILP formulations in which each variable is associated with a departure and/or arrival of a train at a specific station in a specific time instant, whose LP relaxation is too expensive to be solved exactly. Experimental results on real-world instances of the problem show that the proposed approach is capable of producing heuristic solutions of better quality than those obtained by these previous approaches, and of solving some small-size instances to proven optimality.      

A hybrid metaheuristic approach to the university course timetabling problem

This paper describes the development of a novel metaheuristic that combines an electromagnetic-like mechanism (EM) and the great deluge algorithm (GD) for the University course timetabling problem. This well-known timetabling problem assigns lectures to specific numbers of timeslots and rooms maximizing the overall quality of the timetable while taking various constraints into account. EM is a population-based stochastic global optimization algorithm that is based on the theory of physics, simulating attraction and repulsion of sample points in moving toward optimality. GD is a local search procedure that allows worse solutions to be accepted based on some given upper boundary or ‘level’. In this paper, the dynamic force calculated from the attraction-repulsion mechanism is used as a decreasing rate to update the ‘level’ within the search process. The proposed method has been applied to a range of benchmark university course timetabling test problems from the literature. Moreover, the viability of the method has been tested by comparing its results with other reported results from the literature, demonstrating that the method is able to produce improved solutions to those currently published. We believe this is due to the combination of both approaches and the ability of the resultant algorithm to converge all solutions at every search process.     

A constructive approach to examination timetabling based on adaptive decomposition and ordering

In this study, we investigate an adaptive decomposition and ordering strategy that automatically divides examinations into difficult and easy sets for constructing an examination timetable. The examinations in the difficult set are considered to be hard to place and hence are listed before the ones in the easy set in the construction process. Moreover, the examinations within each set are ordered using different strategies based on graph colouring heuristics. Initially, the examinations are placed into the easy set. During the construction process, examinations that cannot be scheduled are identified as the ones causing infeasibility and are moved forward in the difficult set to ensure earlier assignment in subsequent attempts. On the other hand, the examinations that can be scheduled remain in the easy set. Within the easy set, a new subset called the boundary set is introduced to accommodate shuffling strategies to change the given ordering of examinations. The proposed approach, which incorporates different ordering and shuffling strategies, is explored on the Carter benchmark problems. The empirical results show that the performance of our algorithm is broadly comparable to existing constructive approaches.     

A fast heuristic approach for train timetabling in a railway node

We consider a conflict-free scheduling problem which arises in railway networks, where ideal timetables have been provided for a set of trains, but where these timetables may be conflicting. We use a space-time graph approach from the railway scheduling literature in order to develop a fast heuristic which resolves conflicts by adjusting the ideal timetables while attempting to minimize the deviation from the ideal timetable. Our approach is tested on realistic data obtained from the railway node of Milan.     

A two-phase multiple objective approach to university timetabling utilising optimisation and evolutionary solution methodologies

The timetabling problem is generally large, highly constrained and discrete in nature. This makes solution by exact optimisation methods difficult. Therefore, often a heuristic search is deemed acceptable providing a simple (non-optimal) solution. This paper discusses the timetabling problem for a university department, where a large-scale integer goal programming (IGP) formulation is implemented for its efficient optimal solution in two phases. The first phase allocates lectures to rooms and the second allocates start-times to lectures. Owing to the size and complicated nature of the model, an initial analysis procedure is employed to manipulate the data to produce a more manageable model, resulting in considerable reductions in problem size and increase of performance. Both phases are modelled as IGPs. Phase 1 is solved using a state-of-the-art IGP optimisation package. However, due to the scale of the model, phase 2 is solved to optimality using a genetic algorithm approach.     

Using tree search bounds to enhance a genetic algorithm approach to two rectangle packing problems

A popular approach when using a genetic algorithm in the solution of constrained problems is to exploit problem specific information by representing individuals as ordered lists. A construction heuristic is then often used as a decoder to produce a solution from each ordering. In such a situation further information is often available in the form of bounds on the partial solutions. This paper uses two two-dimensional packing problems to illustrate how this information can be incorporated into the genetic operators to improve the overall performance of the search. Our objective is to use the packing problems as a vehicle for investigating a series of modifications of the genetic algorithm approach based on information from bounds on partial solutions.     

An approach to matrix problems

A new general approach to the so-called “matrix problems” is given. With this approach, the “derivative” of a matrix problem is again a matrix problem of the same type.     

XHSTT: an XML archive for high school timetabling problems in different countries

We present the progress on the benchmarking project for high school timetabling that was introduced at PATAT 2008. In particular, we announce the High School Timetabling Archive XHSTT-2011 with 21 instances from 8 countries and an evaluator capable of checking the syntax of instances and evaluating the solutions.     

The two straight line approach for periodic diffraction boundary-value problems

Boundary-transmission problems of first order for the Helmholtz equation are considered within the context of wave diffraction by a periodic strip grating and formulated as convolution type operators acting on a Bessel potential periodic space setting. Two boundary-value problems are studied for an arbitrary geometry of the grating: the oblique derivative and the classic Neumann boundary-value problems. The convolution type operators on the grating which correspond to the given boundary-transmission problems are associated with Toeplitz operators acting on spaces of matrix functions defined on composed contours. A Fredholm theory for periodic boundary-value problems of first order is established independently of the grating period and the Fredholm indices for the oblique derivative and the classic Neumann boundary-value problems are given.     

One approach to factorization of positive integers

Factorization of positive integers into primes is a hard computational task. Its complexity lies in the base of the most popular method of cryptography, the RSA method. In this paper we propose a new technique in a factorization procedure which combines ideas of the Number Field Sieve (NFS) and the Quadratic Sieve (QS) in a special manner.     

An integrated approach for scheduling health care activities in a hospital

To effectively utilise hospital beds, operating rooms (OR) and other treatment spaces, it is necessary to precisely plan patient admissions and treatments in advance. As patient treatment and recovery times are unequal and uncertain, this is not easy. In response, a sophisticated flexible job-shop scheduling (FJSS) model is introduced, whereby patients, beds, hospital wards and health care activities are respectively treated as jobs, single machines, parallel machines and operations. Our approach is novel because an entire hospital is describable and schedulable in one integrated approach. The scheduling model can be used to recompute timings after deviations, delays, postponements and cancellations. It also includes advanced conditions such as activity and machine setup times, transfer times between activities, blocking limitations and no wait conditions, timing and occupancy restrictions, buffering for robustness, fixed activities and sequences, release times and strict deadlines. To solve the FJSS problem, constructive algorithms and hybrid meta-heuristics have been developed. Our numerical testing shows that the proposed solution techniques are capable of solving problems of real world size. This outcome further highlights the value of the scheduling model and its potential for integration into actual hospital information systems.     

Coincidence criteria for maximal stable bridges in two game problems of approach

We consider a finite-dimensional conflict-controlled system whose behavior on a finite time interval is described by a vector differential equation. We analyze two game problems of approach in the phase space. In both problems the same terminal set is considered: in the first case, one should guarantee that the phase vector of the system reaches the terminal set at the final instant of time; in the second case, the phase vector should reach the terminal set no later than the final time instant. It is natural to assume that the construction of a solution to the first problem is much simpler than the construction of a solution to the second problem; this fact is confirmed by available experience. The paper is devoted to finding conditions on the system and the terminal set under which the solutions to the above problems coincide. Using these conditions, one can replace the solution of the second problem by the simpler solution of the first problem.