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Sequential Competitive Location on Networks

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Dominik Kreß

This book deals with classical competitive location problems where two players, leader and follower, sequentially enter markets with given numbers of facilities. The markets under consideration are represented as networks. The book provides a detailed overview of the literature on competitive and voting location, and it presents extensions and variations of the classical models, with a focus on the incorporation of proportional choice rules, non-discrete demand (edge demand), or additional pricing decisions of the players. It provides corresponding mathematical models, insights into the computational complexity of the resulting problems and proposes and analyzes adequate solution methods.

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2.1 Competitive location problems - reviews and classifications . . . 16 2.2 Other selected reviews and classifications . . . . . . . . . . . . . 17 2.3 Some complexity results on a basic (r|Xp)-medianoid problem . . 25 2.4 Some complexity results on selected (r|Xp)-medianoid problems . 32 2.5 Some complexity results on a basic (r|p)-centroid problem . . . . 33 2.6 Selected models on R1 with multiple players or endogenous loca- tion order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.7 Other selected R1 models . . . . . . . . . . . . . . . . . . . . . . 39 2.8 R1 - Main features and results . . . . . . . . . . . . . . . . . . . 45 3.1 Relative 1-rejection of vertices . . . . . . . . . . . . . . . . . . . 58 3.2 Merging vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3 Deleting vertices . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1 Computational results - solving BBNP with fixed sets of leader’s locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2 NP-hard binary (1|p)-centroid problems . . . . . . . . . . . . . . 91 4.3 Example - optimal solutions to the medianoid problems . . . . . 105 4.4 Example - solving the centroid problem . . . . . . . . . . . . . . 107 4.5 Rearranging stage for the example in Fig. 4.8b . . . . . . . . . . 111 4.6 D(xμi) and V (D(xμi)), i = 1, ..., 4, for the example in Fig. 4.8b . 114 5.1 Sets of test instances . . . . . . . . . . . . . . . . . . . . . . . . 144 5.2 Ranges of sensitivity parameters . . . . . . . . . . . . . . . . . . 146 5.3 Optimal follower locations under different price sensitivity levels 149 5.4 Average solution quality of location heuristics . . . . . . . . . . . 152 xvi List of Tables 5.5 Average solution time of location heuristics (seconds) . . . . . . . 152 6.1 Main features of the models presented in this thesis . . . . . . . 157

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