In many real-life optimisation problems, there are multiple interacting components in a solution. For example, different components might specify assignments to different kinds of resource. Often, each component is associated with different sets of soft constraints, and so with different measures of soft constraint violation. The goal is then to minimise a linear combination of such measures. This paper studies an approach to such problems, which can be thought of as multiphase exploitation of multiple objective-/value-restricted submodels. In this approach, only one computationally difficult component of a problem and the associated subset of objectives is considered at first. This produces partial solutions, which define interesting neighbourhoods in the search space of the complete problem. Often, it is possible to pick the initial component so that variable aggregation can be performed at the first stage, and the neighbourhoods to be explored next are guaranteed to contain feasible solutions. Using integer programming, it is then easy to implement heuristics producing solutions with bounds on their quality. Our study is performed on a university course timetabling problem used in the 2007 International Timetabling Competition (ITC), also known as the Udine Course Timetabling problem. The goal is to find an assignment of events to periods and rooms, so that the assignment of events to periods is a feasible bounded colouring of an associated conflict graph and the linear combination of the numbers of violations of four soft constraints is minimised. In the proposed heuristic, an objective-restricted neighbourhood generator produces assignments of periods to events, with decreasing numbers of violations of two period-related soft constraints. Those are relaxed into assignments of events to days, which define neighbourhoods that are easier to search with respect to all four soft constraints. Integer programming formulations for all subproblems are given and evaluated using ILOG CPLEX 11. The wider applicability of this approach is analysed and discussed

This work describes the Grid and cluster scheduling simulator Alea 2 designed for study, testing and evaluation of various job scheduling techniques. This event-based simulator is able to deal with common problems related to the job scheduling like the heterogeneity of jobs, resources, and the dynamic runtime changes such as the arrivals of new jobs or the resource failures and restarts. The Alea 2 is based on the popular GridSim toolkit [31] and represents a major extension of the Alea simulator, developed in 2007 [16]. The extension covers both improved design, extended functionality as well as the improved scalability and the higher simulation speed. Finally, new visualization interface was introduced into the simulator. The main part of the simulator is a complex scheduler which incorporates several common scheduling algorithms working either on the queue or the schedule (plan) based principle. Additional data structures are used to maintain information about the resource status, the objective functions and for collection and visualization of the simulation results. Many typical objectives such as the machine usage, the average slowdown or the average response time are included. The paper concludes with an example of the Alea 2 execution using a real-life workload, discussing also the scalability of the simulator.

A branch-and-cut procedure for the Udine Course Timetabling problem is described. Simple compact integer linear programming formulations of the problem employ only binary variables. In contrast, we give a formulation with fewer variables by using a mix of binary and general integer variables. This formulation has an exponential number of constraints, which are added only upon violation. The number of constraints is exponential. However, this is only with respect to the upper bound on the general integer variables, which is the number of periods per day in the Udine Course Timetabling problem.A number of further classes of cuts are also introduced, arising from: enumeration of event/free-period patterns; bounds on the numbers of days of instruction; the desire to exploit integrality of the objective function value; the graph colouring component; and also from various implied bounds. An implementation of the corresponding branch-and-cut procedure is evaluated on the instances from Track 3 of the International Timetabling Competition 2007.

Abstract. Formulation of many real-life problems evolves when the problem is being solved. For example, a change in the environment might appear after the initial problem specification and this change must be reflected in the solution. Such changes complicate usage of a traditionally static constraint satisfaction technology that requires the problem to be fully specified before the solving process starts. In this paper, we propose a new formal description of changes in the problem formulation called a minimal perturbation problem. This description focuses on the modification of the solution after a change in the problem specification. We also describe a new branch-and-bound like algorithm for solving such type of problems.

For many problems in Scheduling and Timetabling the choice of an mathematical programming formulation is determined by the formulation of the graph colouring component. This paper briefly surveys seven known integer programming formulations of vertex colouring and introduces a new formulation using "supernodes". In the definition of George and McIntyre [SIAM J. Numer. Anal. 15 (1978), no. 1, 90-112], "supernode" is a complete subgraph, where each two vertices have the same neighbourhood outside of the subgraph. Seen another way, the algorithm for obtaining the best possible partition of an arbitrary graph into supernodes, which we give and show to be polynomial-time, makes it possible to use any formulation of vertex multicolouring to encode vertex colouring. The power of this approach is shown on the benchmark problem of Udine Course Timetabling. Results from empirical tests on DIMACS colouring instances, in addition to instances from other timetabling applications, are also provided and discussed.

The modeling and solution approaches being used to automate construction of course timetables at a large university are discussed. A course structure model is presented that allows this complex real-world problem to be described using a classical formulation. The problem is then tackled utilizing a course timetabling solver model that transforms it into a constraint satisfaction and optimization problem. The tiered structure of this approach provides flexibility that is helpful in solving the multiple subproblems that arise from decomposition of the university-wide problem. A production system has been partially implemented and results of early use are presented. Practical issues raised during the implementation of the automated timetabling system are also discussed.

An extension of constraint logic programming that allows for weighted partial satisfaction of soft constraints is described and applied to the development of an automated timetabling system for Purdue University. The soft constraint solver implemented in the proposed solution approach allows constraint propagation for hard constraints together with preference propagation for soft constraints. A new repair search algorithm is proposed to improve upon initially generated (partial) assignments of the problem variables. The model and search methods applied to the solution of the large lecture room component are presented and discussed along with the computational results.

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