Worst-case bounds for bin-packing heuristics with applications to the duality gap of the one-dimensional cutting stock problem


The thesis considers the one-dimensional cutting stock problem, the bin-packing problem, and their relationship. The duality gap of the former is investigated and a characterisation of a class of cutting stock problems with the next round-up property is given. It is shown that worst-case bounds for bin-packing heuristics can be and are best expressed in terms of the linear programming relaxation of the corresponding cutting stock problem. The concept of recurrency is introduced for a bin-packing heuristic, which allows a more natural derivation of a measure for the worst-case behaviour. The ideas are tested on some well known bin-packing heuristics and (slightly) tighter bounds for these are derived. These new bounds (in terms of the linear programming relaxation) are then used to make inferences about the duality gap of the cutting stock problem. In particular; these bounds allow à priori, problem-specific bounds. The thesis ends with conclusions and a number of suggestions to extend the analysis to higher dimensional problems

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