A wide variety of important problems, in Operational Research and other fields, can be modelled as optimisation problems with integer-constrained variables. Algorithms and software for integer programming have improved substantially. One of the key ingredients to this success is the use of strong valid linear inequalities, also known as cutting planes. A key concept is that the convex hull of feasible solutions forms a polyhedron.

One strand of the literature on cutting planes is concerned with the knapsack polytope. In the 1970s, Balas and Wolsey derived a family of inequalities, called lifted cover inequalities (LCIs), for the knapsack polytope. We have taken a lifting procedure due to Balas, and shown that it can be substantially improved, so that it yields stronger and more general LCIs.

In 2000, Carr and co-authors introduced another family of valid inequalities for the knapsack polytope. These inequalities, called knapsack cover inequalities, can be rather weak. We have used two lifting procedures to strengthen these inequalities. The first procedure is based on integer rounding, whereas the second uses superadditivity.

Another important class of optimisation problems are those that involve fixed charges. In 1985, Padberg, Van Roy and Wolsey introduced a procedure which enables one to take known valid inequalities for the knapsack polytope, and convert them into valid inequalities for the fixed-charge polytope. We have shown how this procedure can be extended to obtain a wider family of inequalities for the fixed-charge and single-node flow polytopes.

Finally, we have considered problems where a fixed charge is incurred if and only if at least one variable in a set takes a positive value. We have derived strong valid inequalities for these problems and shown that they generalise and dominate a subclass of the flow cover inequalities for the classical fixed-charge problem.