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1 2018

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Properties of an α-clique approach to obtaining the hub and spoke structure in optimization of transportation systems

Mazbic-Kulma B., Owsinski J. W., Barski A., Sęp K., Stańczak J.

Control and Cybernetics

2018

Vol. 47, no. 2

173--189

The paper is devoted to the analysis of a graph transformation, pertinent for the transport and logistic systems and their planning and management. Namely, we consider, for a given graph, representing some existing transport or logistic system, its transformation to a (non-equivalent) so-called ”hub-and-spoke” structure, known from both literature and practice of transportation and logistics. This structure is supposed to bring beneﬁts in terms of functioning and economic performance of the respective systems. The transformation into the ”hub-and-spoke” is not only non-equivalent (regarding the original graph of the system), but is also, in general, non unique. The structure sought is composed of two kinds of elements - nodes of the graph (stations, airports, havens, etc.), namely: the subgraph of hubs, which, in principle, ought to constitute a complete sub-graph (a clique), and the ”spokes”, i.e. the subsets of nodes, each of which is connected in the ultimate structure only with one of the hubs. The paper proposes a relaxation of the hub-and-spoke structure by allowing the hub subgraph not to be complete, but at least connected, with a deﬁnite ”degree of completeness” (alpha), from where the name of ”alpha-clique”. It is shown how such structures can be obtained and what are the resulting beneﬁts for various assumptions, regarding such structures. The beneﬁts are measured here with travel times. The desired structures are sought with an evolutionary algorithm. It is shown on an academic example how the results vary and how the conclusions, relevant for practical purposes, can be drawn from such analyses, done with the methods here presented.