On the ramified optimal allocation problem

Qinglan Xia; Shaofeng Xu; Qinglan Xia; Shaofeng Xu

doi:10.3934/nhm.2013.8.591

Networks and Heterogeneous Media

2013, Volume 8, Issue 2: 591-624. doi: 10.3934/nhm.2013.8.591

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On the ramified optimal allocation problem

Qinglan Xia ^{1
,},
Shaofeng Xu ^{2
,}

1.
Department of Mathematics, University of California at Davis, Davis, CA 95616
2.
Department of Economics, University of California at Davis, Davis, CA 95616

Received: 01 August 2011 Revised: 01 October 2012
Primary: 91B32, 58E17; Secondary: 49Q20, 90B18.

This paper proposes an optimal allocation problem with ramified transport technologies in a spatial economy. Ramified transportation is used to model network-like branching structures attributed to the economies of scale in group transportation. A social planner aims at finding an optimal allocation plan and an associated optimal allocation path to minimize the overall cost of transporting commodity from factories to households. This problem differentiates itself from existing ramified transport literature in that the distribution of production among factories is not fixed but endogenously determined as observed in many allocation practices. It is shown that due to the transport economies of scale, each optimal allocation plan corresponds equivalently to an optimal assignment map from households to factories. This optimal assignment map provides a natural partition of both households and allocation paths. We develop methods of marginal transportation analysis and projectional analysis to study the properties of optimal assignment maps. These properties are then related to the search for an optimal assignment map in the context of state matrix.
- Ramified transportation,
- transport economies of scale,
- optimal transport path,
- branching structure,
- allocation problem,
- assignment map,
- state matrix
Citation: Qinglan Xia, Shaofeng Xu. On the ramified optimal allocation problem[J]. Networks and Heterogeneous Media, 2013, 8(2): 591-624. doi: 10.3934/nhm.2013.8.591

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Abstract

This paper proposes an optimal allocation problem with ramified transport technologies in a spatial economy. Ramified transportation is used to model network-like branching structures attributed to the economies of scale in group transportation. A social planner aims at finding an optimal allocation plan and an associated optimal allocation path to minimize the overall cost of transporting commodity from factories to households. This problem differentiates itself from existing ramified transport literature in that the distribution of production among factories is not fixed but endogenously determined as observed in many allocation practices. It is shown that due to the transport economies of scale, each optimal allocation plan corresponds equivalently to an optimal assignment map from households to factories. This optimal assignment map provides a natural partition of both households and allocation paths. We develop methods of marginal transportation analysis and projectional analysis to study the properties of optimal assignment maps. These properties are then related to the search for an optimal assignment map in the context of state matrix.

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