On a Clique-Based Integer Programming Formulation of Vertex Colouring with Applications in Course Timetabling

Vertex colouring is a well-known problem in combinatorial optimisation, whose alternative integer programming formulations have recently attracted considerable attention. This paper briefly surveys seven known formulations of vertex colouring and introduces a formulation of vertex colouring using a suitable clique partition of the graph. This formulation is applicable in timetabling applications, where such a clique partition of the conflict graph is given implicitly. In contrast with some alternatives, the presented formulation can also be easily extended to accommodate complex performance indicators (``soft constraints'') imposed in a number of real-life course timetabling applications. Its performance depends on the quality of the clique partition, but encouraging empirical results for the Udine Course Timetabling problem are reported

Design, Engineering, and Experimental Analysis of a Simulated Annealing Approach to the Post-Enrolment Course Timetabling Problem

The post-enrolment course timetabling (PE-CTT) is one of the most studied timetabling problems, for which many instances and results are available. In this work we design a metaheuristic approach based on Simulated Annealing to solve the PE-CTT. We consider all the different variants of the problem that have been proposed in the literature and we perform a comprehensive experimental analysis on all the public instances available. The outcome is that our solver, properly engineered and tuned, performs very well on all cases, providing the new best known results on many instances and state-of-the-art values for the others

Feature-based tuning of simulated annealing applied to the curriculum-based course timetabling problem

We consider the university course timetabling problem, which is one of the most studied problems in educational timetabling. In particular, we focus our attention on the formulation known as the curriculum-based course timetabling problem, which has been tackled by many researchers and for which there are many available benchmarks. The contribution of this paper is twofold. First, we propose an effective and robust single-stage simulated annealing method for solving the problem. Secondly, we design and apply an extensive and statistically-principled methodology for the parameter tuning procedure. The outcome of this analysis is a methodology for modeling the relationship between search method parameters and instance features that allows us to set the parameters for unseen instances on the basis of a simple inspection of the instance itself. Using this methodology, our algorithm, despite its apparent simplicity, has been able to achieve high quality results on a set of popular benchmarks. A final contribution of the paper is a novel set of real-world instances, which could be used as a benchmark for future comparison

Constructing operating theatre schedules using partitioned graph colouring techniques

In hospitals, scheduled operations can often be cancelled in large numbers due to the unavailability of beds for post-operation recovery. Operating theatre scheduling is known to be an (Formula presented.) -hard optimisation problem. Previous studies have shown that the correct scheduling of surgical procedures can have a positive impact on the availability of beds in hospital wards, thereby allowing a reduction in number of elective operation cancellations. This study proposes an exact technique based on the partitioned graph colouring problem for constructing optimal master surgery schedules, with the goal of minimising the number of cancellations. The resultant schedules are then simulated in order to measure how well they cope with the stochastic nature of patient arrivals. Our results show that the utilisation of post-operative beds can be increased, whilst the number of cancellations can be decreased, which may ultimately lead to greater patient throughput and reduced waiting times. A scenario-based model has also been employed to integrate the stochastic-nature associated with the bed requirements into the optimisation process. The results indicate that the proposed model can lead to more robust solutions

Robust Coloring Optimization Model on Electricity Circuit Problems

The Graph Coloring Problem (GCP) is assigning different colors to certain elements in a graph based on certain constraints and using a minimum number of colors. GCP can be drawn into optimization problems, namely the problem of minimizing the color used together with the uncertainty in using the color used, so it can be assumed that there is an uncertainty in the number of colored vertices. One of the mathematical optimization techniques in dealing with uncertainty is Robust Optimization (RO) combined with computational tools. This article describes a robust GCP using the Polyhedral Uncertainty Theorem and model validation for electrical circuit problems. The form of an electrical circuit color chart consists of corners (components) and edges (wires or conductors). The results obtained are up to 3 colors for the optimization model for graph coloring problems and up to 5 colors for robust optimization models for graph coloring problems. The results obtained with robust optimization show more colors because the results contain uncertainty. When RO GCP is applied to an electrical circuit, the model is used to place the electrical components in the correct path so that the electrical components do not collide with each other

Operational Research in Education

Operational Research (OR) techniques have been applied, from the early stages of the discipline, to a wide variety of issues in education. At the government level, these include questions of what resources should be allocated to education as a whole and how these should be divided amongst the individual sectors of education and the institutions within the sectors. Another pertinent issue concerns the efficient operation of institutions, how to measure it, and whether resource allocation can be used to incentivise efficiency savings. Local governments, as well as being concerned with issues of resource allocation, may also need to make decisions regarding, for example, the creation and location of new institutions or closure of existing ones, as well as the day-to-day logistics of getting pupils to schools. Issues of concern for managers within schools and colleges include allocating the budgets, scheduling lessons and the assignment of students to courses. This survey provides an overview of the diverse problems faced by government, managers and consumers of education, and the OR techniques which have typically been applied in an effort to improve operations and provide solutions