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# Sequential Competitive Location on Networks

## 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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# Chapter 4. (r|p)-Centroid Problems on Networks with Vertex and Edge Demand

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edge demand is given in Section 4.2. The chapter closes with a conclusion in Section 4.3. 4.1 The Binary (r|p)-Centroid Problem with Vertex and Edge Demand This section is structured as follows. Section 4.1.1 presents bilevel program- ming formulations for the discrete, binary (r|p)-centroid problem with ver- tex and edge demand. Some computational results for the corresponding follower’s problem are given in Section 4.1.2 in order to illustrate the need for approximate solution procedures when designing heuristic algorithms for the discrete leader’s problem. In Section 4.1.3 we derive complexity results for the discrete and continuous versions of the binary (1|p)-centroid problem with edge demand only. Section 4.1.4 is concerned with the identiﬁcation of ﬁnite dominating sets. 4.1.1 Bilevel Programming Models Consider the discrete, binary (r|p)-centroid problem with vertex and edge demand and assume – without loss of generality – that the underlying network is complete. If this is not the case, we simply add missing edges with inﬁnite edge lengths and zero demand densities. We deﬁne the following variables: xFij := ⎧⎪⎨ ⎪⎩ 1 if the vertex customers in vertex i are served by a follower’s facility in vertex j, 0 else, ∀ i, j ∈ V, (4.1) xLij := ⎧⎪⎨ ⎪⎩ 1 if the vertex customers in vertex i are served by a leader’s facility in vertex j, 0 else, ∀ i, j ∈ V, (4.2) yFj := { 1 if the follower locates in vertex j, 0 else, ∀ j ∈ V, (4.3) 4.1 The Binary (r|p)-Centroid Problem with Vertex and...

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