# Flexibility in optimization problems

Jurij Mihelič (2006) Flexibility in optimization problems. PhD thesis.

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## Abstract

Abstract – Flexibility in optimization problems The thesis Flexibility in optimization problems examines optimization problems in which one must find several different solutions which exhibit a certain degree of mutual resemblance. Such problems are called flexibility problems while their solutions are said to be flexible. There is a multitude of optimization problems in practice. Therefore, their close examination is very important both from practical point of view as well as for scientific progress. However, uncertainty of input data is often present in practical optimization problems. A datum is uncertain if its exact value is not known. There are several approaches to dealing with uncertainty of input data. In addition to robustness and stochastics, it turns out that the flexibility is also a possible approach to dealing with uncertainty of input data in optimization problems. The uncertainty of data can be modeled with scenarios. In general, a scenario represents an instance of an optimization problem while an instance of a flexibility optimization problem is represented by a set of scenarios. Discrepancies among the scenarios are often bounded. For this purpose, a method of perturbating scenarios can be applied. We tackle various flexibility optimization problems mainly from the point of view of computational complexity theory and algorithm analysis. Hence, notions such as Karp and Turing reducibility, complexity classes of decision or optimization problems, heuristic and approximation algorithms are often used. For defining and solving problems we mainly use notions from graph theory and special problem areas, such as network flows and matching problems. Flexibility is a very general idea and is, therefore, in one way or another already present in some optimization problems, for example in real-time systems, online algorithms, dynamic data, mobility of facilities in location problems etc. For this reason, we formally define the notion of flexibility and other related notions, and subsequently describe a general method of introducing the flexibility into optimization problems. An instance of a flexibility optimization problem consists of a graph G=(V,E) of scenario transitions where to each i in V a scenario si is assigned, describing some set of input data. For each scenario si the particular solution Si is found in such a way that its quality z(Si ) is either optimal or approximate in case of an NP-hard problem. Furthermore, a way of calculating the particular flexibility fij (Si , Sj) for all (i,j) in E is assumed to be given. The objective function of the problem is composed of these particular flexibilities, such as max{ fij | (i,j) in E}. The goal of flexibility problems is usually minimization. Other kinds of flexibility problems are similarly defined. The main part of the thesis is focused on various flexible-attribute problems. Here we study the optimization of complete flexibility and concentrate on simple description of scenarios via sets of attributes. We define several versions of the problem. We examine their computational complexity in terms of time consumption, we prove their NP-hardness, and find positive and negative results about their absolute and relative approximability. Due to their intractability, we treat various simplified versions of the problems. In particular, we analyze flexible-attribute problems on trees, general graphs, chains, and two-stage graphs. For all these problems we design and analyze algorithms for constructing optimal or approximate solutions. Next, we discuss introducing the flexibility in some familiar optimization problems such as locating centers and the minimum spanning tree. For all flexibility versions of these problems we prove \NP-hardness. Furthermore, for all these problems we design either exact or heuristic algorithms. The final chapter contains original contributions to the science, open questions, and ideas for further research in the field. Keywords: optimization problem, flexibility, uncertainty, scenario, computational complexity theory, heuristic and approximation algorithm.

Item Type: Thesis (PhD thesis)
Keywords:
Language of Content: Slovenian
Mentor / Comentors:
Name and SurnameIDFunction
prof. dr. Borut Robič28Mentor