# Sequential Competitive Location on Networks

#### Series:

## Dominik Kreß

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

## Extract

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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