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In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources. The problem was formalized by the French mathematician Gaspard Monge in 1781.^[1]

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".^[2][3]

Major advances were made in the field during World War II by the Soviet mathematician and economist Leonid Kantorovich.^[4] Consequently, the problem as it is stated is sometimes known as the Monge–Kantorovich transportation problem.^[5] The linear programming formulation of the transportation problem is also known as the Hitchcock–Koopmans transportation problem.^[6]

Motivation

Mines and factories

Suppose that we have a collection of n mines mining iron ore, and a collection of n factories which use the iron ore that the mines produce. Suppose for the sake of argument that these mines and factories form two disjoint subsets M and F of the Euclidean plane R². Suppose also that we have a cost function c : R² × R² → [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 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.
In other words, each mine m ∈ M supplies precisely one factory T(m) ∈ F and each factory is supplied by precisely one mine.
We wish to find the optimal transport plan, the plan T whose total cost

c(T):=∑m∈Mc(m,T(m)){displaystyle c(T):=sum _{min M}c(m,T(m))}

is the least of all possible transport plans from M to F. This motivating special case of the transportation problem is an instance of the assignment problem.
More specifically, it is equivalent to finding a minimum weight matching in a bipartite graph.

Moving books: the importance of the cost function

The following simple example illustrates the importance of the 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:

1. move all n books one book-width to the right ("many small moves");


2. move the left-most book n book-widths to the right and leave all other books fixed ("one big move").

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 strictly convex cost function proportional to the square of Euclidean distance (c(x, y) = α|x − y|²), then the "many small moves" option becomes the unique minimizer.

Abstract formulation of the problem

Monge and Kantorovich formulations

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.

Let X and Y be two separable metric spaces such that any probability measure on X (or Y) is a Radon measure (i.e. they are Radon spaces). 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

inf{∫Xc(x,T(x))dμ(x)|T∗(μ)=ν},{displaystyle inf left{left.int _{X}c(x,T(x)),mathrm {d} mu (x);right|;T_{*}(mu )=nu right},}

where T_∗(μ) denotes the 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".

Monge's formulation of the optimal transportation problem can be ill-posed, because sometimes there is no T satisfying T_∗(μ) = ν: this happens, for example, when μ is a Dirac measure but ν is not.

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

inf{∫X×Yc(x,y)dγ(x,y)|γ∈Γ(μ,ν)},{displaystyle inf left{left.int _{Xtimes Y}c(x,y),mathrm {d} gamma (x,y)right|gamma in Gamma (mu ,nu )right},}

where Γ(μ, ν) denotes the collection of all probability measures on X × Y with marginals μ on X and ν on Y. It can be shown^[7] that a minimizer for this problem always exists when the cost function c is lower semi-continuous and Γ(μ, ν) is a tight collection of measures (which is guaranteed for Radon spaces X and Y). (Compare this formulation with the definition of the Wasserstein metric W₁ 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.^[8]

Duality formula

The minimum of the Kantorovich problem is equal to

sup(∫Xφ(x)dμ(x)+∫Yψ(y)dν(y)),{displaystyle sup left(int _{X}varphi (x),mathrm {d} mu (x)+int _{Y}psi (y),mathrm {d} nu (y)right),}

where the supremum runs over all pairs of bounded and continuous functions ϕ:X→R{displaystyle phi :Xrightarrow mathbf {R} } and ψ:Y→R{displaystyle psi :Yrightarrow mathbf {R} } such that

φ(x)+ψ(y)≤c(x,y).{displaystyle varphi (x)+psi (y)leq c(x,y).}

Solution of the problem

Optimal transportation on the real line

Optimal transportation matrix

Continuous optimal transport

For 1≤p<∞{displaystyle 1leq p<infty }, let Pp(R){displaystyle {mathcal {P}}_{p}(mathbf {R} )} denote the collection of probability measures on R{displaystyle mathbf {R} } that have finite p{displaystyle p}-th moment. Let μ,ν∈Pp(R){displaystyle mu ,nu in {mathcal {P}}_{p}(mathbf {R} )} and let c(x,y)=h(x−y){displaystyle c(x,y)=h(x-y)}, where h:R→[0,∞){displaystyle h:mathbf {R} rightarrow [0,infty )} is a convex function.

1. If μ{displaystyle mu } has no atom, i.e., if the cumulative distribution function Fμ=R→[0,1]{displaystyle F_{mu }=mathbf {R} rightarrow [0,1]} of μ{displaystyle mu } is a continuous function, then Fν−1∘Fμ:R→R{displaystyle F_{nu }^{-1}circ F_{mu }:mathbf {R} to mathbf {R} } is an optimal transport map. It is the unique optimal transport map if h{displaystyle h} is strictly convex.


2. We have

minγ∈Γ(μ,ν)∫R2c(x,y)dγ(x,y)=∫01c(Fμ−1(s),Fν−1(s))ds.{displaystyle min _{gamma in Gamma (mu ,nu )}int _{mathbf {R} ^{2}}c(x,y),mathrm {d} gamma (x,y)=int _{0}^{1}cleft(F_{mu }^{-1}(s),F_{nu }^{-1}(s)right),mathrm {d} s.}

The proof of this solution appears in.^[9]

Separable Hilbert spaces

Let X{displaystyle X} be a separable Hilbert space. Let Pp(X){displaystyle {mathcal {P}}_{p}(X)} denote the collection of probability measures on X{displaystyle X} such that have finite p{displaystyle p}-th moment; let Ppr(X){displaystyle {mathcal {P}}_{p}^{r}(X)} denote those elements μ∈Pp(X){displaystyle mu in {mathcal {P}}_{p}(X)} that are Gaussian regular: if g{displaystyle g} is any strictly positive Gaussian measure on X{displaystyle X} and g(N)=0{displaystyle g(N)=0}, then μ(N)=0{displaystyle mu (N)=0} also.

Let μ∈Ppr(X){displaystyle mu in {mathcal {P}}_{p}^{r}(X)}, ν∈Pp(X){displaystyle nu in {mathcal {P}}_{p}(X)}, c(x,y)=|x−y|p/p{displaystyle c(x,y)=|x-y|^{p}/p} for p∈(1,∞),p−1+q−1=1{displaystyle pin (1,infty ),p^{-1}+q^{-1}=1}. Then the Kantorovich problem has a unique solution κ{displaystyle kappa }, and this solution is induced by an optimal transport map: i.e., there exists a Borel map r∈Lp(X,μ;X){displaystyle rin L^{p}(X,mu ;X)} such that

κ=(idX×r)∗(μ)∈Γ(μ,ν).{displaystyle kappa =(mathrm {id} _{X}times r)_{*}(mu )in Gamma (mu ,nu ).}

Moreover, if ν{displaystyle nu } has bounded support, then

r(x)=x−|∇ϕ(x)|q−2∇ϕ(x){displaystyle r(x)=x-|nabla phi (x)|^{q-2}nabla phi (x)}

for μ{displaystyle mu }-almost all x∈X{displaystyle xin X} for some locally Lipschitz, c-concave and maximal Kantorovich potential ϕ{displaystyle phi }. (Here ∇ϕ{displaystyle nabla phi } denotes the Gâteaux derivative of ϕ{displaystyle phi }.)

Applications

The Monge-Kantorovich optimal transport has found applications in wide range in different fields. Among them are:

• 
  Image registration and warping ^[10]
  


• Reflector design ^[11]
  


• Retrieving information from shadowgraphy and proton radiography ^[12]
  


• 
  Seismic tomography and Reflection seismology ^[13]
  

See also

Wikimedia Commons has media related to Transportation theory.

• Wasserstein metric


• Transport function


• Hungarian algorithm


• Transportation planning

References

Further reading

• 
  Brualdi, Richard A. (2006). Combinatorial matrix classes. Encyclopedia of Mathematics and Its Applications. 108. Cambridge: Cambridge University Press. ISBN 0-521-86565-4. Zbl 1106.05001.