Heuristic Approaches for Maximin Distance and Packing Problems

Abstract

In this thesis we mainly deal with two problems - experimental design and packing problems. In the field of experimental design problems we consider maximin Latin Hypercube Designs (LHDs). In the field of packing problems we consider those of packing n equal or unequal circles in a circular container with minimum radius. Both problems can be formulated as optimization ones. The former is a combinatorial problem, while the latter is a continuous one. We propose heuristic approaches to tackle these problems. These are Iterated Local Search (ILS) heuristics for maximin LHDs, and Basin Hopping (BH) heuristics for packing problems. Actually, ILS and BH approaches have strong similarities and could be described within an unified framework. However, following the literature, where ILS approaches are mainly applied to combinatorial problems, while BH approaches are mainly applied to continuous problems, we will keep them apart. In order to deal with maximin LHDs, we propose two ILS variants, corresponding to two distinct optimality criteria which are employed to drive the search among LHDs. Extensive experiments are performed for the investigation of the strengths and weaknesses of the algorithms. A remarkable finding is that the most efficient method, though time consuming, performs a non monotonic search, driven by an appropriate objective function, within the space of LHDs. The proposed approaches are extensively compared with the existing ones in the literature, and many improved results with respect to best known ones, are obtained. In particular, the proposed methods seem to outperform the existing ones when the dimension of the design points increases. Finally, we also discuss about the time complexity of the algorithms; by mixing theoretical results with experimental ones, we derive an empirical formula for each ILS variant, returning the expected run time as a function of the number of design points and of their dimension. To deal with the problem of packing equal circles in a circular container with minimum radius, we propose a variant of BH, namely Monotonic BH (MBH) and its population based counterpart, Population BH (PBH). Extensive computational experiments are performed both to analyze the problem at hand, and to choose in an appropriate way the parameter values for the proposed methods. Different improvements with respect to the best results reported in the literature are detected. The problem of packing unequal circles in a circular container with minimum radius is also attacked with the MBH and PBH approaches, but some components of these approaches are adapted in order to fully exploit the peculiarities of the problem with unequal circles (in particular, its combinatorial nature due to the different radii of the circles). Again extensive computational experiments are performed and improvements with respect to the existing literature are detected.