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. 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". Major advances were made in the field during World War II by the Soviet mathematician and economist Leonid Kantorovich. Consequently, the problem as it is stated is sometimes known as the Monge–Kantorovich transportation problem. The linear programming formulation of the transportation problem is also known as the Hitchcock–Koopmans transportation problem.
Motivation
Mines and factories
Suppose that we have a collection of m {\displaystyle m} mines mining iron ore, and a collection of n {\displaystyle 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 {\displaystyle M} and F {\displaystyle F} of the Euclidean plane R 2 {\displaystyle \mathbb {R} ^{2}} . Suppose also that we have a cost function c : R 2 × R 2 → [ 0 , ∞ ) {\displaystyle c:\mathbb {R} ^{2}\times \mathbb {R} ^{2}\to [0,\infty )} , so that c ( x , y ) {\displaystyle c(x,y)} is the cost of transporting one shipment of iron from x {\displaystyle x} to y {\displaystyle 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 {\displaystyle T:M\to F} . In other words, each mine m ∈ M {\displaystyle m\in M} supplies precisely one target factory T ( m ) ∈ F {\displaystyle T(m)\in F} and each factory is supplied by precisely one mine. We wish to find the optimal transport plan, the plan T {\displaystyle T} whose total cost
c ( T ) := ∑ m ∈ M c ( m , T ( m ) ) {\displaystyle c(T):=\sum _{m\in M}c(m,T(m))}
is the least of all possible transport plans from M {\displaystyle M} to F {\displaystyle 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. This can be generalized to the continuous case, where there are infinitely many mines and factories distributed on the real line, or generally in any metric space. This case is usually pictured as "changing the shape of a pile of dirt", and thus called the earth mover's problem.
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 {\displaystyle 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:
move all n {\displaystyle n} books one book-width to the right ("many small moves"); move the left-most book n {\displaystyle 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 ‖ {\displaystyle c(x,y)=\alpha \|x-y\|} for some α > 0 {\displaystyle \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 ) = α ‖ x − y ‖ 2 {\displaystyle c(x,y)=\alpha \|x-y\|^{2}} for some α > 0 {\displaystyle \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.
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