Chapter 1. Introduction and Preliminaries
of the sum of weighted distances to the vertices (median or antimedian prob- lems) or the minimization of the maximum or maximization of the minimum weighted distance to any vertex (center or anticenter problems). Classi- cal competitive location problems include the (r|p)-centroid or the (r|Xp)- medianoid problem. This research analyzes extensions to and variations of these latter problems. It focuses on incorporating proportional choice rules, non-discrete demand, or additional pricing decisions of ﬁrms. Furthermore, it provides insights into the computational complexity of some of the resulting problems and proposes adequate solution methods. 1.1 Basic Notation and Deﬁnitions In this thesis we will denote the set of natural numbers including zero by N, the set of positive natural numbers by N+, the set of rational (real) numbers by Q (R), the set of positive rational (real) numbers by Q+ (R+) and the set of nonnegative rational (real) numbers by Q+0 (R + 0 ). We assume the reader to be familiar with the fundamental concepts of operations research. The basic graph-theoretic deﬁnitions, along with the corresponding notation used throughout this thesis, however, are introduced in Section 1.1.1. Most of the deﬁnitions are taken from Alstrup et al. (2004); Bandelt (1985); Bauer et al. (1993); Garbe (1995); Gross and Yellen (2004); Swamy and Thulasiraman (1981). Similarly, Sections 1.1.2 and 1.1.3 are concerned with the basics of game theory and discrete choice theory. The latter Section is based on Train (2009). 1.1.1 Graphs and Networks A graph...
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