The transportation problem can be formalized as the problem of ?nding + ? the optimal paths to transport a measure ? onto a measure ? with the same mass. In contrast with the Monge-Kantorovich formalization, recent approaches model the branched structure of such supply networks by an energy functional whose essential feature is to favor wide roads. Given a ?ow ? in a tube or a road or a wire, the transportation cost per unit length ? is supposed to be proportional to ? with 0< 1. For the Monge- Kantorovich energy,? = 1 so that it is equivalent to have two roads with ?ow 1/2oralargeronewith?ow1. Ifinstead0< 1,aroadwith?ow? +? is 1 2 ? ? ? preferable to two individual roads ? and ? because (? +? ) 1? N N such structures can irrigate a whole bounded open set ofR . The aim of this set of lectures is to give a mathematical proof of s- eral existence, structure and regularity properties empirically observed in transportation networks. This will be done in a simple mathematical fra- work (measures on the set of paths) unifying several di?erent approaches and results due to Brancolini, Buttazzo, Devillanova, Maddalena, Pratelli, Santambrogio, Solimini, Stepanov, Xia and the authors.
The transportation problem can be formalized as the problem of ?nding + ? the optimal paths to transport a measure ? onto a measure ? with the same mass. In contrast with the Monge-Kantorovich formalization, recent approaches model the branched structure of such supply networks by an energy functional whose essential feature is to favor wide roads. Given a ?ow ? in a tube or a road or a wire, the transportation cost per unit length ? is supposed to be proportional to ? with 0< 1. For the Monge- Kantorovich energy,? = 1 so that it is equivalent to have two roads with ?ow 1/2oralargeronewith?ow1. Ifinstead0< 1,aroadwith?ow? +? is 1 2 ? ? ? preferable to two individual roads ? and ? because (? +? ) 1? N N such structures can irrigate a whole bounded open set ofR . The aim of this set of lectures is to give a mathematical proof of s- eral existence, structure and regularity properties empirically observed in transportation networks. This will be done in a simple mathematical fra- work (measures on the set of paths) unifying several di?erent approaches and results due to Brancolini, Buttazzo, Devillanova, Maddalena, Pratelli, Santambrogio, Solimini, Stepanov, Xia and the authors.
This volume comprises twenty papers originally presented at the 4th Meeting of the EURO (Association of European Operational Research Societies) held in Newcastle. Topics covered at the conference...
This textbook provides a comprehensive step-by-step guide for new public transport modelers. It includes an introduction to mathematical modeling, continuous and discrete optimization, numerical...
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