Transportation theory (mathematics)

Suppose that we have a collection of $m$ 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 $\backslash mathbb\{R\}^2$. Suppose also that we have a cost function $c\; :\; \backslash mathbb\{R\}^2\; \backslash times\; \backslash mathbb\{R\}^2\; \backslash to\; [0,\; \backslash infty)$, 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 $T:\; M\; \backslash to\; F$.

In other words, each mine $m\; \backslash in\; M$ supplies precisely one target factory $T(m)\; \backslash in\; F$ and each factory is supplied by precisely one mine.

We wish to find the optimal transport plan, the plan $T$ whose total cost

:$c(T)\; :=\; \backslash sum\_\{m\; \backslash in\; M\}\; c(m,\; T(m))$

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.

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 ($c(x,\; y)\; =\; \backslash alpha\; \backslash |x\; -\; y\backslash |$ for some $\backslash alpha\; >\; 0$) 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 ($c(x,\; y)\; =\; \backslash alpha\; \backslash |x\; -\; y\backslash |^2$ for some $\backslash alpha\; >\; 0$), then the "many small moves" option becomes the unique minimizer.

Note that the above cost functions consider only the horizontal distance traveled by the books, not the horizontal distance traveled by a device used to pick each book up and move the book into position. If the latter is considered instead, then, of the two transport plans, the second is always optimal for the Euclidean distance, while, provided there are at least 3 books, the first transport plan is optimal for the squared Euclidean distance.

The following transportation problem formulation is credited to F. L. Hitchcock:Frank L. Hitchcock (1941) "The distribution of a product from several sources to numerous localities", MIT Journal of Mathematics and Physics 20:224–230 {{MR|id=0004469}}.

:Suppose there are $m$ sources $x\_1,\; \backslash ldots,\; x\_m$ for a commodity, with $a(x\_i)$ units of supply at $x\_i$ and $n$ sinks $y\_1,\; \backslash ldots,\; y\_n$ for the commodity, with the demand $b(y\_j)$ at $y\_j$. If $c(x\_i,\backslash \; y\_j)$ is the unit cost of shipment from $x\_i$ to $y\_j$, find a flow that satisfies demand from supplies and minimizes the flow cost. This challenge in logistics was taken up by D. R. FulkersonD. R. Fulkerson (1956) [https://priorart.ip.com/IPCOM/000128834/ Hitchcock Transportation Problem], RAND corporation. and in the book Flows in Networks (1962) written with L. R. Ford Jr.L. R. Ford Jr. & D. R. Fulkerson (1962) § 3.1 in Flows in Networks, page 95, Princeton University Press

Tjalling Koopmans is also credited with formulations of transport economics and allocation of resources.

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 $c\; :\; X\; \backslash times\; Y\; \backslash to\; [0,\; \backslash infty)$ be a Borel-measurable function. Given probability measures $\backslash mu$ on $X$ and $\backslash nu$ on $Y$, Monge's formulation of the optimal transportation problem is to find a transport map $T\; :\; X\; \backslash to\; Y$ that realizes the infimum

:$\backslash inf\; \backslash left\backslash \{\; \backslash left.\; \backslash int\_X\; c(x,\; T(x))\; \backslash ,\; \backslash mathrm\{d\}\; \backslash mu\; (x)\; \backslash right|\; T\_*\; (\backslash mu)\; =\; \backslash nu\; \backslash right\backslash \},$

where $T\_*(\backslash mu)$ denotes the push forward of $\backslash mu$ 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 $T\_*(\backslash mu)\; =\; \backslash nu$: this happens, for example, when $\backslash mu$ is a Dirac measure but $\backslash nu$ is not.

We can improve on this by adopting Kantorovich's formulation of the optimal transportation problem, which is to find a probability measure $\backslash gamma$ on $X\; \backslash times\; Y$ that attains the infimum

:$\backslash inf\; \backslash left\backslash \{\; \backslash left.\; \backslash int\_\{X\; \backslash times\; Y\}\; c(x,\; y)\; \backslash ,\; \backslash mathrm\{d\}\; \backslash gamma\; (x,\; y)\; \backslash right|\; \backslash gamma\; \backslash in\; \backslash Gamma\; (\backslash mu,\; \backslash nu)\; \backslash right\backslash \},$

where $\backslash Gamma\; (\backslash mu,\; \backslash nu)$ denotes the collection of all probability measures on $X\; \backslash times\; Y$ with marginals $\backslash mu$ on $X$ and $\backslash nu$ on $Y$. It can be shownL. Ambrosio, N. Gigli & G. Savaré. [https://books.google.com/books?id=rCDK9JA5BAEC&q=%22optimal+transportation%22 Gradient Flows in Metric Spaces and in the Space of Probability Measures]. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel. (2005) that a minimizer for this problem always exists when the cost function $c$ is lower semi-continuous and $\backslash Gamma(\backslash mu,\; \backslash nu)$ 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 $W\_p$ 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.{{cite journal |first1=S. |last1=Angenent |first2=S. |last2=Haker |first3=A. |last3=Tannenbaum |title=Minimizing flows for the Monge–Kantorovich problem |journal=SIAM J. Math. Anal. |volume=35 |issue=1 |pages=61–97 |year=2003 |doi=10.1137/S0036141002410927 |citeseerx=10.1.1.424.1064 }}

The minimum of the Kantorovich problem is equal to

:$\backslash sup\; \backslash left(\; \backslash int\_X\; \backslash varphi\; (x)\; \backslash ,\; \backslash mathrm\{d\}\; \backslash mu\; (x)\; +\; \backslash int\_Y\; \backslash psi\; (y)\; \backslash ,\; \backslash mathrm\{d\}\; \backslash nu\; (y)\; \backslash right),$

where the supremum runs over all pairs of bounded and continuous functions $\backslash varphi\; :\; X\; \backslash rightarrow\; \backslash mathbb\{R\}$ and $\backslash psi\; :\; Y\; \backslash rightarrow\; \backslash mathbb\{R\}$ such that

:$\backslash varphi\; (x)\; +\; \backslash psi\; (y)\; \backslash leq\; c(x,\; y).$

The economic interpretation is clearer if signs are flipped. Let $x\; \backslash in\; X$ stand for the vector of characteristics of a worker, $y\; \backslash in\; Y$ for the vector of characteristics of a firm, and $\backslash Phi(x,y)\; =-c(x,y)$ for the economic output generated by worker $x$ matched with firm $y$. Setting $u(x)\; =\; -\backslash varphi(x)$ and $v(y)\; =-\backslash psi(y)$, the Monge–Kantorovich problem

rewrites:

:$\backslash sup\; \backslash left\backslash \{\; \backslash int\_\{X\backslash times\; Y\}\backslash Phi(x,y)\; d\backslash gamma(x,y)\; ,\backslash gamma\; \backslash in\; \backslash Gamma(\backslash mu,\backslash nu)\; \backslash right\backslash \}$

which has dual:

:$\backslash inf\; \backslash left\backslash \{\; \backslash int\_X\; u(x)\; \backslash ,d\backslash mu(x)\; +\backslash int\_Y\; v(y)\; \backslash ,\; d\backslash nu\; (y)\; :u(x)\; +v(y)\; \backslash geq\; \backslash Phi(x,y)\; \backslash right\backslash \}$

where the infimum runs over bounded and continuous function $u:X\backslash to\; \backslash mathbb\{R\}$ and $v:Y\backslash to\; \backslash mathbb\{R\}$. If the dual problem has a solution, one can see that:

:$v(y)\; =\backslash sup\_x\; \backslash left\backslash \{\; \backslash Phi(x,y)\; -\; u(x)\backslash right\backslash \}$

so that $u(x)$ interprets as the equilibrium wage of a worker of type $x$, and $v(y)$ interprets as the equilibrium profit of a firm of type $y$.

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For , let $\backslash mathcal\{P\}\_p(\backslash mathbb\{R\})$ denote the collection of probability measures on $\backslash mathbb\{R\}$ that have finite $p$-th moment. Let $\backslash mu,\; \backslash nu\; \backslash in\; \backslash mathcal\{P\}\_p(\backslash mathbb\{R\})$ and let $c(x,\; y)\; =\; h(x-y)$, where $h:\backslash mathbb\{R\}\; \backslash to\; [0,\backslash infty)$ is a convex function.

1. If $\backslash mu$ has no atom, i.e., if the cumulative distribution function $F\_\backslash mu\; :\; \backslash mathbb\{R\}\backslash to[0,1]$ of $\backslash mu$ is a continuous function, then $F\_\{\backslash nu\}^\{-1\}\; \backslash circ\; F\_\{\backslash mu\}\; :\; \backslash mathbb\{R\}\; \backslash to\; \backslash mathbb\{R\}$ is an optimal transport map. It is the unique optimal transport map if $h$ is strictly convex.
2. We have

::$\backslash min\_\{\backslash gamma\; \backslash in\; \backslash Gamma(\backslash mu,\; \backslash nu)\}\; \backslash int\_\{\backslash mathbb\{R\}^2\}\; c(x,\; y)\; \backslash ,\; \backslash mathrm\{d\}\; \backslash gamma\; (x,\; y)\; =\; \backslash int\_0^1\; c\; \backslash left(\; F\_\{\backslash mu\}^\{-1\}\; (s),\; F\_\{\backslash nu\}^\{-1\}\; (s)\; \backslash right)\; \backslash ,\; \backslash mathrm\{d\}\; s.$

The proof of this solution appears in Rachev & Rüschendorf (1998).Rachev, Svetlozar T., and Ludger Rüschendorf. Mass Transportation Problems: Volume I: Theory. Vol. 1. Springer, 1998.

In the case where the margins $\backslash mu$ and $\backslash nu$ are discrete, let $\backslash mu\_x$

and $\backslash nu\_y$ be the probability masses respectively assigned to $x\backslash in\; \backslash mathbf\{X\}$ and $y\backslash in\; \backslash mathbf\{Y\}$, and let $\backslash gamma\_\{xy\}$ be the probability of an $xy$ assignment. The objective function in the primal Kantorovich problem is then

: $\backslash sum\_\{x\backslash in\; \backslash mathbf\{X\},y\backslash in\; \backslash mathbf\{Y\}\}\; \backslash gamma\_\{xy\}c\_\{xy\}$

and the constraint $\backslash gamma\; \backslash in\; \backslash Gamma(\backslash mu\; ,\backslash nu)$ expresses as

: $$

\sum_{y\in \mathbf{Y}}\gamma_{xy}=\mu_x,\forall x\in \mathbf{X}

and

: $$

\sum_{x\in \mathbf{X}} \gamma_{xy}=\nu_y,\forall y\in \mathbf{Y}.

In order to input this in a linear programming problem, we need to vectorize the matrix $\backslash gamma\_\{xy\}$ by either stacking its columns or its rows, we call $\backslash operatorname\{vec\}$ this operation. In the column-major order, the constraints above rewrite as

: $\backslash left(\; 1\_\{1\backslash times\; |\backslash mathbf\{Y\}|\; \}\backslash otimes\; I\_$

where $\backslash otimes$ is the Kronecker product, $1\_\{n\backslash times\; m\}$ is a matrix of size $n\backslash times\; m$ with all entries of ones, and $I\_\{n\}$ is the identity matrix of size $n$. As a result, setting $z=\backslash operatorname\{vec\}(\backslash gamma)$, the linear programming formulation of the problem is

: $$

\begin{align}

& \text{Minimize } &&\operatorname{vec}(c)^\top z \\[4pt]

& \text{subject to:} && z \ge 0, \\[4pt]

& && \begin{pmatrix}

1_{1\times |\mathbf{Y}|}\otimes I_

I_

\end{pmatrix} z=\binom{\mu}{\nu}

\end{align}

which can be readily inputted in a large-scale linear programming solver (see chapter 3.4 of Galichon (2016)Galichon, Alfred. Optimal Transport Methods in Economics. Princeton University Press, 2016.).

In the semi-discrete case, $X=Y=\backslash mathbb\{R\}^d$ and $\backslash mu$ is a continuous distribution over $\backslash mathbb\{R\}^d$, while $\backslash nu\; =\backslash sum\_\{j=1\}^J\; \backslash nu\_j\backslash delta\_\{y\_i\}$ is a discrete distribution which assigns probability mass $\backslash nu\_j$ to site $y\_j\; \backslash in\; \backslash mathbb\{R\}^d$. In this case, we can seeSantambrogio, Filippo. Optimal Transport for Applied Mathematicians. Birkhäuser Basel, 2016. In particular chapter 6, section 4.2. that the primal and dual Kantorovich problems respectively boil down to:

:$\backslash inf\; \backslash left\backslash \{\; \backslash int\_X\; \backslash sum\_\{j=1\}^J\; c(x,y\_j)\; \backslash ,\; d\backslash gamma\_j(x)\; ,\backslash gamma\; \backslash in\; \backslash Gamma(\backslash mu,\backslash nu)\backslash right\backslash \}$

for the primal, where $\backslash gamma\; \backslash in\; \backslash Gamma(\backslash mu\; ,\backslash nu)$ means that $\backslash int\_X\; d\backslash gamma\_j(x)=\backslash nu\_j$ and $\backslash sum\_j\; d\backslash gamma\_j(x)\; =d\backslash mu(x)$, and:

:$\backslash sup\; \backslash left\backslash \{\; \backslash int\_X\backslash varphi(x)d\backslash mu\; (x)+\backslash sum\_\{j=1\}^J\backslash psi\_j\backslash nu\_j:\backslash psi\_j+\backslash varphi\; (x)\backslash leq\; c(x,y\_j)\; \backslash right\backslash \}$

for the dual, which can be rewritten as:

:$\backslash sup\_\{\backslash psi\; \backslash in\; \backslash mathbb\{R\}^J\}\backslash left\backslash \{\; \backslash int\_X\backslash inf\_j\backslash left\backslash \{\; c(x,y\_j)-\backslash psi\_j\backslash right\backslash \}\; d\backslash mu\; (x)+\backslash sum\_\{j=1\}^J\backslash psi\_j\backslash nu\_j\backslash right\backslash \}$

which is a finite-dimensional convex optimization problem that can be solved by standard techniques, such as gradient descent.

In the case when $c(x,y)=|x-y|^2/2$, one can show that the set of $x\backslash in\; \backslash mathbf\{X\}$ assigned to a particular site $j$ is a convex polyhedron. The resulting configuration is called a power diagram.{{citation

| last = Aurenhammer | first = Franz | authorlink = Franz Aurenhammer

| doi = 10.1137/0216006

| issue = 1

| journal = SIAM Journal on Computing

| mr = 873251

| pages = 78–96

| title = Power diagrams: properties, algorithms and applications

| volume = 16

| year = 1987}}.

Assume the particular case $\backslash mu\; =\backslash mathcal\{N\}(0,\backslash Sigma\_X)$, $\backslash nu\; =\backslash mathcal\{N\}(0,\backslash Sigma\; \_\{Y\})$, and $c(x,y)\; =|y-Ax|^2/2$ where $A$ is invertible. One then has

: $\backslash varphi(x)\; =-x^\backslash top\; \backslash Sigma\_X^\{-1/2\}\backslash left(\; \backslash Sigma\_X^\{1/2\}A^\backslash top\; \backslash Sigma\_Y\; A\backslash Sigma\_X^\{1/2\}\backslash right)\; ^\{1/2\}\backslash Sigma\_\{X\}^\{-1/2\}x/2$

: $\backslash psi(y)\; =-y^\backslash top\; A\backslash Sigma\_X^\{1/2\}\backslash left(\; \backslash Sigma\_X^\{1/2\}A^\backslash top\; \backslash Sigma\_Y\; A\backslash Sigma\_\{X\}^\{1/2\}\backslash right)^\{-1/2\}\; \backslash Sigma\_X^\{1/2\}Ay/2$

: $T(x)\; =\; (A^\backslash top)^\{-1\}\backslash Sigma\_X^\{-1/2\}\; \backslash left(\backslash Sigma\_X^\{1/2\}A^\backslash top\; \backslash Sigma\_Y\; A\backslash Sigma\_X^\{1/2\}\; \backslash right)^\{1/2\}\; \backslash Sigma\_X^\{-1/2\}x$

The proof of this solution appears in Galichon (2016).

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

Let $\backslash mu\; \backslash in\; \backslash mathcal\{P\}\_p^r\; (X)$, $\backslash nu\; \backslash in\; \backslash mathcal\{P\}\_p(X)$, $c\; (x,\; y)\; =\; |\; x\; -\; y\; |^p/p$ for $p\backslash in(1,\backslash infty),\; p^\{-1\}\; +\; q^\{-1\}\; =\; 1$. Then the Kantorovich problem has a unique solution $\backslash kappa$, and this solution is induced by an optimal transport map: i.e., there exists a Borel map $r\backslash in\; L^p(X,\; \backslash mu;\; X)$ such that

:$\backslash kappa\; =\; (\backslash mathrm\{id\}\_X\; \backslash times\; r)\_\{*\}\; (\backslash mu)\; \backslash in\; \backslash Gamma\; (\backslash mu,\; \backslash nu).$

Moreover, if $\backslash nu$ has bounded support, then

:$r(x)\; =\; x\; -\; |\; \backslash nabla\; \backslash varphi\; (x)\; |^\{q\; -\; 2\}\; \backslash ,\; \backslash nabla\; \backslash varphi\; (x)$

for $\backslash mu$-almost all $x\backslash in\; X$ for some locally Lipschitz, $c$-concave and maximal Kantorovich potential $\backslash varphi$. (Here $\backslash nabla\; \backslash varphi$ denotes the Gateaux derivative of $\backslash varphi$.)

Consider a variant of the discrete problem above, where we have added an entropic regularization term to the objective function of the primal problem

: $$

\begin{align}

& \text{Minimize } \sum_{x\in \mathbf{X}, y\in \mathbf{Y}}\gamma_{xy}c_{xy}+\varepsilon \gamma_{xy} \ln \gamma_{xy} \\[4pt]

& \text{subject to: } \\[4pt]

& \gamma\ge0 \\[4pt]

& \sum_{y\in \mathbf{Y}}\gamma_{xy} = \mu_x, \forall x\in \mathbf{X} \\[4pt]

& \sum_{x\in \mathbf{X}}\gamma_{xy} = \nu_y, \forall y\in \mathbf{Y}

\end{align}

One can show that the dual regularized problem is

:$$

\max_{\varphi ,\psi} \sum_{x\in \mathbf{X}} \varphi_x \mu_x + \sum_{y\in \mathbf{Y}} \psi_y v_y - \varepsilon \sum_{x\in \mathbf{X},y\in \mathbf{Y}} \exp \left( \frac{\varphi_x + \psi_y - c_{xy}}{\varepsilon }\right)

where, compared with the unregularized version, the "hard" constraint in the former dual ($\backslash varphi\_x\; +\; \backslash psi\_y\; -\; c\_\{xy\}\backslash geq\; 0$) has been replaced by a "soft" penalization of that constraint (the sum of the $\backslash varepsilon\; \backslash exp\; \backslash left(\; (\backslash varphi\; \_x\; +\; \backslash psi\_y\; -\; c\_\{xy\})/\backslash varepsilon\; \backslash right)$ terms). The optimality conditions in the dual problem can be expressed as

:{{EquationRef|5.1|Eq. 5.1:}} $$

\mu_x = \sum_{y\in \mathbf{Y}} \exp \left( \frac{\varphi_x + \psi_y - c_{xy}}{\varepsilon} \right) ~\forall x\in \mathbf{X}

:{{EquationRef|5.2|Eq. 5.2:}} $$

\nu_y = \sum_{x\in \mathbf{X}} \exp \left( \frac{\varphi_x + \psi_y - c_{xy}}{\varepsilon }\right) ~\forall y\in \mathbf{Y}

Denoting $A$ as the $|\backslash mathbf\{X\}|\; \backslash times\; |\backslash mathbf\{Y\}|$ matrix of term $A\_\{xy\}=\backslash exp\; \backslash left(-c\_\{xy\}\; /\; \backslash varepsilon\; \backslash right)$, solving the dual is therefore equivalent to looking for two diagonal positive matrices $D\_1$ and $D\_2$ of respective sizes $|\backslash mathbf\{X\}|$ and $|\backslash mathbf\{Y\}|$, such that $D\_1AD\_2\; 1\_$

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

• Image registration and warping{{Cite journal|last1=Haker|first1=Steven|last2=Zhu|first2=Lei|last3=Tannenbaum|first3=Allen|last4=Angenent|first4=Sigurd|date=1 December 2004|title=Optimal Mass Transport for Registration and Warping|journal=International Journal of Computer Vision|language=en|volume=60|issue=3|pages=225–240|doi=10.1023/B:VISI.0000036836.66311.97|issn=0920-5691|citeseerx=10.1.1.59.4082|s2cid=13261370}}
• Reflector design{{Cite journal|last1=Glimm|first1=T.|last2=Oliker|first2=V.|date=1 September 2003|title=Optical Design of Single Reflector Systems and the Monge–Kantorovich Mass Transfer Problem|journal=Journal of Mathematical Sciences|language=en|volume=117|issue=3|pages=4096–4108|doi=10.1023/A:1024856201493|s2cid=8301248|issn=1072-3374}}
• Retrieving information from shadowgraphy and proton radiography{{Cite journal|last1=Kasim|first1=Muhammad Firmansyah|last2=Ceurvorst|first2=Luke|last3=Ratan|first3=Naren|last4=Sadler|first4=James|last5=Chen|first5=Nicholas|last6=Sävert|first6=Alexander|last7=Trines|first7=Raoul|last8=Bingham|first8=Robert|last9=Burrows|first9=Philip N.|date=16 February 2017|title=Quantitative shadowgraphy and proton radiography for large intensity modulations|journal=Physical Review E|volume=95|issue=2|pages=023306|doi=10.1103/PhysRevE.95.023306|pmid=28297858|arxiv=1607.04179|bibcode=2017PhRvE..95b3306K|s2cid=13326345}}
• Seismic tomography and reflection seismology{{cite journal |last1=Metivier |first1=Ludovic |title=Measuring the misfit between seismograms using an optimal transport distance: application to full waveform inversion |journal=Geophysical Journal International |date=24 February 2016 |volume=205 |issue=1 |pages=345–377 |doi=10.1093/gji/ggw014 |bibcode=2016GeoJI.205..345M |url=https://academic.oup.com/gji/article-abstract/205/1/345/2594839|doi-access=free }}
• The broad class of economic modelling that involves gross substitutes property (among others, models of matching and discrete choice).

{{Commons category|Transportation theory}}

• Wasserstein metric
• Transport function
• Hungarian algorithm
• Transportation planning
• Earth mover's distance
• Monge–Ampère equation

{{Reflist|30em}}

• {{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=978-0-521-86565-4 | zbl=1106.05001 | url-access=registration | url=https://archive.org/details/combinatorialmat0000brua }}

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