论文标题
在边界上的收益后，最佳运输受到影响
Ramified optimal transportation with payoff on the boundary
论文作者
论文摘要
本文研究了分支/分支最佳运输问题的变体。鉴于生产能力和市场规模的分布，一家公司正在寻找生产分配工厂对工厂的分配，跨市场的销售分配以及提供产品以最大化其利润的运输路径。从数学上讲，给定$ x $上的任何两种量度$μ$和$ν$，以及一个收益功能$ h $，计划者希望最大程度地减少$ \ mathbf {m} _ {α} _ {α}（t） - \ int_ {x} $ \tildeμ\ leqμ$和$ \tildeν\leqν$，其中$ \ mathbf {m} _ {α} $是分支运输中使用的标准成本功能。证明存在结果后，我们提供了最佳解决方案边界度量的表征。事实证明，它们是限制在每个连接组件上达到三角洲质量的某些Borel子集的原始措施。我们的分析进一步发现，随着边界收益的增加，当前问题的相应解决方案会收敛到最佳运输路径，这是标准分支运输的解决方案。
This paper studies a variant of ramified/branched optimal transportation problems. Given the distributions of production capacities and market sizes, a firm looks for an allocation of productions over factories, a distribution of sales across markets, and a transport path that delivers the product to maximize its profit. Mathematically, given any two measures $μ$ and $ν$ on $X$, and a payoff function $h$, the planner wants to minimize $\mathbf{M}_{α}(T)-\int_{X}hd(\partial T)$ among all transport paths $T$ from $\tildeμ$ to $\tildeν$ with $\tildeμ\leq μ$ and $\tildeν\leq ν$, where $\mathbf{M}_{α}$ is the standard cost functional used in ramified transportation. After proving the existence result, we provide a characterization of the boundary measures of the optimal solution. They turn out to be the original measures restricted on some Borel subsets up to a Delta mass on each connected component. Our analysis further finds that as the boundary payoff increases, the corresponding solution of the current problem converges to an optimal transport path, which is the solution of the standard ramified transportation.
