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Abstract. Logic puzzles form an excellent set of problems for the teaching of advanced solution techniques in operations research. They are an opportunity for students to test their modelling skills on a different style of problem, and some puzzles even require advanced techniques to become tractable. Fillomino is a puzzle in which the player must enter integers into a grid to satisfy certain rules. This puzzle is a good exercise in using lazy constraints and composite variables to solve difficult problems.

We consider a problem concerning a network and a set of maintenance requests to be undertaken. The aim is to schedule the maintenance in such a way as to minimise the impact on the total throughput of the network. We apply disaggregated Benders decomposition using lazy constraints to solve the problem to optimality, as well as explore the strengths and weaknesses of the technique. We prove that our Benders cuts are Pareto-optimal. Solutions to the LP relaxation also provide further valid inequalities to reduce total solve time. We implement these techniques on simulated data presented in previous papers, and compare our solution technique to previous methods and a direct MIP formulation. We prove optimality in many problem instances that have not previously been proven.Network design and scheduling problems are an important area of study, particularly as they have widespread practical applications. Examples of these problems include minimising the cost of maintaining a network [1], restoring a damaged network [2] or extending an existing network [3]. In practice, networks are often large, and optimising their design can be difficult and time-consuming. Industry is always interested in any improvements to operations that result in reduced costs.Benders decomposition is a powerful technique for breaking a difficult mixed integer program (MIP) into smaller, easier to solve problems [4]. It has been successfully applied to a number of problems, particularly network design and facility location problems. This technique is especially powerful when the sub-problems can be disaggregated to allow us to add stronger disaggregated Benders cuts. This can make their solutions easier to obtain. Magnanti and Wong (1984) [10] showed that the use of Pareto-optimal cuts with Benders decomposition can improve convergence time by up to 50 times over other Benders cuts. In 2008 Camargo, Miranda and Luna [5] applied disaggregated Benders decomposition to the design of hub-and-spoke networks. Tang and Saharidis [3] used disaggregated Benders decomposition for solving a capacitated facility location problem with existing facilities which could be removed or extended, and Lusby, Muller and Petersen [6] used disaggregated Benders decomposition for scheduling the maintenance of power plants in France.

List of figures xii List of tables xiiiList of abbreviations and symbols xvchapter, we review the foundational theory upon which this thesis is based, before answering the first question and providing the layout of the remainder of the thesis.

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