Chapter 6. Summary and Outlook
chain networks, that posses the vertex optimality property. In the second part of Chapter 3 we have been concerned with implementational issues of an algorithm for determining 1-suboptimal points. Moreover, we have ex- tended the model to include edge demand. This, however, has led to a loss of the discretization properties known from the case of vertex demand only. Furthermore, we have seen that even for the simple class of chain networks with edge demand only, 1-optimal points can in general not be determined by straight forward analytical calculations, so that there remain interesting questions to be answered by future research. In Chapter 4 we have analyzed the binary (r|p)-centroid problem on net- works with vertex and edge demand. We have presented bilevel programming formulations for the discrete problem class and we have proposed potential heuristic solution approaches to be examined in future. More signiﬁcantly, we have proven that the discrete and continuous versions of the binary (r|p)- centroid problem are NP-hard on general networks with edge demand only. Moreover, an eﬃcient algorithm has been derived for the discrete problem class with r = 1 on tree networks. Future research may focus on several extensions of the model. For example, one may analyze the incorporation of non-uniform demand densities. Chapter 5 has been concerned with the discrete (r|Xp)-medianoid prob- lem under multinomial logit demand. We have extended the literature on this problem by including an additional stage into the game, allowing the players...
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