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| | Psychologists Wendell Jernigan from High River, has hobbies which includes house plants, [http://ganhandodinheironainternet.comoganhardinheiro101.com como ganhar dinheiro] na internet and brewing beer. Was exceptionally stimulated after visiting Talamanca Range-La Amistad Reserves / La Amistad National Park. |
| In [[mathematics]] and economics, '''transportation theory''' is a name given to the study of optimal transportation and allocation of resources.
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| The problem was formalized by the French [[mathematician]] [[Gaspard Monge]] in 1781.<ref name=Monge>G. Monge. ''Mémoire sur la théorie des déblais et des remblais. Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année'', pages 666–704, 1781.</ref>
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| In the 1920s A.N. Tolstoi was one of the first to study the transportation problem mathematically. In 1930, in the collection ''Transportation Planning Volume I'' for the National Commissariat of Transportation of the Soviet Union, he published a paper "Methods of Finding the Minimal Kilometrage in Cargo-transportation in space".<ref>[[Alexander Schrijver|Schrijver, Alexander]], [http://books.google.com/books?id=mqGeSQ6dJycC&printsec=frontcover ''Combinatorial Optimization''], Berlin ; New York : Springer, 2003. ISBN 3540443894. Cf. [http://books.google.com/books?id=mqGeSQ6dJycC&pg=PA362&lpg=PA362&dq=a.n.+tolstoi+transportation+networks&source=bl&ots=xONQNWerQa&sig=LJl_9KbyS1FDp_wYuXNJ9ruEKes&hl=en&ei=bMujTLP6O4OClAfjoomQCw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBsQ6AEwAA#v=onepage&q=a.n.%20tolstoi%20transportation%20networks&f=false p.362]</ref><ref>Ivor Grattan-Guinness, Ivor, [http://books.google.com/books?id=2hDvzITtfdAC&printsec=frontcover ''Companion encyclopedia of the history and philosophy of the mathematical sciences''], Volume 1, JHU Press, 2003. Cf. [http://books.google.com/books?id=2hDvzITtfdAC&pg=PA831&lpg=PA831&dq=%22a.n.+tolstoy%22+mathematics&source=bl&ots=pCwrnZWBwI&sig=QQz_1ng7mR6dmZWCmZ4L1Twar3U&hl=en&ei=w8SjTJeUDoT7lwe3nYT8Cw&sa=X&oi=book_result&ct=result&resnum=4&ved=0CCAQ6AEwAw#v=onepage&q=%22a.n.%20tolstoy%22%20mathematics&f=false p.831]</ref>
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| Major advances were made in the field during World War II by the [[Soviet Union|Soviet]]/[[Russia]]n [[mathematician]] and economist [[Leonid Kantorovich]].<ref name=Kantorovich>L. Kantorovich. ''On the translocation of masses.'' C.R. (Doklady) Acad. Sci. URSS (N.S.), 37:199–201, 1942.</ref> Consequently, the problem as it is stated is sometimes known as the '''Monge–Kantorovich transportation problem'''.
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| ==Motivation==
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| ===Mines and factories===
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| Suppose that we have a collection of ''n'' mines mining iron ore, and a collection of ''n'' factories which consume the iron ore that the mines produce. Suppose for the sake of argument that these mines and factories form two [[Disjoint sets|disjoint]] [[subset]]s ''M'' and ''F'' of the [[Euclidean plane]] '''R'''<sup>2</sup>. Suppose also that we have a ''cost function'' ''c'' : '''R'''<sup>2</sup> × '''R'''<sup>2</sup> → [0, ∞), so that ''c''(''x'', ''y'') is the cost of transporting one shipment of iron from ''x'' to ''y''. For simplicity, we ignore the time taken to do the transporting. We are also assume that each mine can supply only one factory (no splitting of shipments) and that each factory requires precisely one shipment to be in operation (factories cannot work at half- or double-capacity). Having made the above assumptions, a ''transport plan'' is a [[bijection]] ''T'' : ''M'' → ''F''– i.e. an arrangement whereby each mine ''m'' ∈ ''M'' supplies precisely one factory ''T''(''m'') ∈ ''F''. We wish to find the ''optimal transport plan'', the plan ''T'' whose ''total cost''
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| :<math>c(T) := \sum_{m \in M} c(m, T(m))</math>
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| is the least of all possible transport plans from ''M'' to ''F''. This motivating special case of the transportation problem is in fact an instance of the [[assignment problem]].
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| ===Moving books: the importance of the cost function===
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| The following simple example illustrates the importance of the [[Cost curve|cost function]] in determining the optimal transport plan. Suppose that we have ''n'' books of equal width on a shelf (the [[real line]]), arranged in a single contiguous block. We wish to rearrange them into another contiguous block, but shifted one book-width to the right. Two obvious candidates for the optimal transport plan present themselves:
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| # move all ''n'' books one book-width to the right; ("many small moves")
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| # move the left-most book ''n'' book-widths to the right and leave all other books fixed. ("one big move")
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| If the cost function is proportional to Euclidean distance (''c''(''x'', ''y'') = α|''x'' − ''y''|) then these two candidates are ''both'' optimal. If, on the other hand, we choose the [[Convex function|strictly convex]] cost function proportional to the square of Euclidean distance (''c''(''x'', ''y'') = α|''x'' − ''y''|<sup>2</sup>), then the "many small moves" option becomes the unique minimizer.
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| Interestingly, while mathematicians prefer to work with convex cost functions{{citation needed|date=July 2012}}, economists prefer [[concave function|concave]] ones. The intuitive justification for this is that once goods have been loaded on to, say, a goods train, transporting the goods 200 kilometres costs much less than twice what it would cost to transport them 100 kilometres. Concave cost functions represent this [[economy of scale]].
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| ==Abstract formulation of the problem==
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| ===Monge and Kantorovich formulations===
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| The transportation problem as it is stated in modern or more technical literature looks somewhat different because of the development of [[Riemannian geometry]] and [[measure theory]]. The mines-factories example, simple as it is, is a useful reference point when thinking of the abstract case. In this setting, we allow the possibility that we may not wish to keep all mines and factories open for business, and allow mines to supply more than one factory, and factories to accept iron from more than one mine.
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| Let ''X'' and ''Y'' be two [[separable space|separable]] [[metric space]]s such that any [[probability measure]] on ''X'' (or ''Y'') is a [[Radon measure]] (i.e. they are [[Radon space]]s). Let ''c'' : ''X'' × ''Y'' → [0, ∞] be a Borel-[[measurable function]]. Given probability measures μ on ''X'' and ν on ''Y'', Monge's formulation of the optimal transportation problem is to find a transport map ''T'' : ''X'' → ''Y'' that realizes the [[infimum]]
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| :<math>\inf \left\{ \left. \int_{X} c(x, T(x)) \, \mathrm{d} \mu (x) \;\right| \; T_{*} (\mu) = \nu \right\},</math>
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| where ''T''<sub>∗</sub>(μ) denotes the [[pushforward measure|push forward]] of μ by ''T''. A map ''T'' that attains this infimum (''i.e.'' makes it a [[minimum]] instead of an infimum) is called an "optimal transport map".
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| Monge's formulation of the optimal transportation problem can be ill-posed, because sometimes there is no ''T'' satisfying ''T''<sub>∗</sub>(μ) = ν: this happens, for example, when μ is a [[Dirac measure]] but ν is not).
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| We can improve on this by adopting Kantorovich's formulation of the optimal transportation problem, which is to find a probability measure γ on ''X'' × ''Y'' that attains the infimum
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| :<math>\inf \left\{ \left. \int_{X \times Y} c(x, y) \, \mathrm{d} \gamma (x, y) \right| \gamma \in \Gamma (\mu, \nu) \right\},</math>
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| where Γ(μ, ν) denotes the collection of all probability measures on ''X'' × ''Y'' with [[Conditional probability|marginals]] μ on ''X'' and ν on ''Y''. It can be shown<ref name=AGS>L. Ambrosio, N. Gigli & G. Savaré. ''Gradient Flows in Metric Spaces and in the Space of Probability Measures.'' Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel. (2005)</ref> that a minimizer for this problem always exists when the cost function ''X'' is lower semi-continuous and Γ(μ, ν) is a [[Tightness of measures|tight]] collection of measures (which is guaranteed for Radon spaces ''X'' and ''Y''). (Compare this formulation with the definition of the [[Wasserstein metric]] ''W''<sub>1</sub> on the space of probability measures.) A gradient descent formulation for the solution of the Monge-Kantorovich problem was given by Sigurd Angenent, Steven Haker, and [[Allen Tannenbaum]].<ref name=AHT>S. Angenent, S. Haker, and A. Tannenbaum. ''Minimizing flows for the Monge-Kantorovich problem.'' SIAM J. Math. Analysis, volume 35, pp. 61-97 (2003).</ref>
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| ===Duality formula===
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| The minimum of the Kantorovich problem is equal to
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| :<math>\sup \left( \int_{X} \varphi (x) \, \mathrm{d} \mu (x) + \int_{Y} \psi (y) \, \mathrm{d} \nu (y) \right),</math>
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| where the [[supremum]] runs over all pairs of [[bounded function|bounded]] and [[continuous function]]s φ : ''X'' → '''R''' and ψ : ''Y'' → '''R''' such that
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| :<math>\varphi (x) + \psi (y) \leq c(x, y).</math>
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| ==Solution of the problem==
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| ===Optimal transportation on the real line===
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| For 1 ≤ ''p'' < ∞, let <math>\mathcal{P}_p(\mathbf{R})</math> denote the collection of [[probability measure]]s on '''R''' that have finite ''p''th [[moment (mathematics)|moment]]. Let <math>\mu, \nu \in \mathcal{P}_p(\mathbf{R})</math> and let ''c''(''x'', ''y'') = ''h''(''x''−''y''), where ''h'' : '''R''' → [0, ∞) is a [[convex function]].
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| # If μ has no [[atom (measure theory)|atom]], i.e., if the [[cumulative distribution function]] ''F''<sub>μ</sub> : '''R''' → [0, 1] of μ is a [[continuous function]], then <math>F_{\nu}^{-1} \circ F_{\mu} : \mathbf{R} \to \mathbf{R}</math> is an optimal transport map. It is the unique optimal transport map if ''h'' is strictly convex.
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| # We have
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| ::<math>\min_{\gamma \in \Gamma(\mu, \nu)} \int_{\mathbf{R}^2} c(x, y) \, \mathrm{d} \gamma (x, y) = \int_0^1 c \left( F_{\mu}^{-1} (s), F_{\nu}^{-1} (s) \right) \, \mathrm{d} s.</math>
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| ===Separable Hilbert spaces===
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| Let ''X'' be a [[separable space|separable]] [[Hilbert space]]. Let <math>\mathcal{P}_p(X)</math> denote the collection of probability measures on ''X'' such that have finite ''p''th moment; let <math>\mathcal{P}_p^r(X)</math> denote those elements <math>\mu \in \mathcal{P}_p(X)</math> that are '''Gaussian regular''': if ''g'' is any [[strictly positive measure|strictly positive]] [[Gaussian measure]] on ''X'' and ''g''(''N'') = 0, then μ(''N'') = 0 also.
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| Let <math>\mu \in \mathcal{P}_p^r (x)</math>, <math>\nu \in \mathcal{P}_p(X)</math>, <math>c (x, y) = | x - y |^p/p</math> for ''p'' ∈ (1, ∞), ''p''<sup>−1</sup> + ''q''<sup>−1</sup> = 1. Then the Kantorovich problem has a unique solution κ, and this solution is induced by an optimal transport map: i.e., there exists a Borel map ''r'' ∈''L<sup>p</sup>(''X'', μ; ''X'') such that
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| :<math>\kappa = (\mathrm{id}_{X} \times r)_{*} (\mu) \in \Gamma (\mu, \nu).</math>
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| Moreover, if ν has [[bounded set|bounded]] [[support (measure theory)|support]], then
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| :<math>r(x) = x - | \nabla \phi (x) |^{q - 2} \nabla \phi (x)</math>
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| for μ-almost all ''x'' ∈ ''X'' for some [[Lipschitz continuous|locally Lipschitz]], ''c''-concave and maximal Kantorovich potential φ. (Here ∇φ denotes the [[Gâteaux derivative]] of φ.)
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| ==See also==
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| {{Commonscat|Transportation theory}}
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| * [[Wasserstein metric]]
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| * [[Transport function]]
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| * [[Hungarian algorithm]]
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| ==References==
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| {{reflist|colwidth=30em}}
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| ==Further reading==
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| * {{cite book | last=Brualdi | first=Richard A. | title=Combinatorial matrix classes | series=Encyclopedia of Mathematics and Its Applications | volume=108 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86565-4 | zbl=1106.05001 }}
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| {{DEFAULTSORT:Transportation Theory}}
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| [[Category:Calculus of variations]]
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| [[Category:Matching]]
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| [[Category:Mathematical economics]]
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| [[Category:Measure theory]]
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| [[Category:Transport economics]]
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| [[Category:Optimization in vector spaces]]
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