# The Transportation Problem (Graphical Method)

• Loo-Keng Hua
• Yuan Wang
• J. G. C. Heijmans
Part of the Mathematical Modeling book series (MMO, volume 2)

## Abstract

In 1958 we started popularsizing mathematical methods to the industrial departments in China. The first problem they asked us about is the following transportation problem. Suppose that it is desired to devise a transportation schedule for the distribution of a homogeneous product, for example distributing wheat by railway. Suppose that there are m sources A1, ... , Am with the amounts of supply al, ... , am (in terms of tons, say) and n destinations (consumer places) B1, ... , Bn with the amounts of demand b1, ... , bn. The total supply at all sources is assumed to be equal to the total demand at all destinations, i.e.
$$\sum\limits_{i=1}^{m}{{{a}_{i}}}=\sum\limits_{j=1}^{n}{{{b}_{j}}}.$$
Suppose that the distance between Ai and Bj is ci j (in kilometres, say) and the amount being shipped from Ai to Bj is xij. Then the total transportation cost (in terms of ton-kilometres) as a function of the vector $$\vec{x}$$ of shipping amounts is given by
$$f(\vec{x})=\sum\limits_{i=1}^{m}{\sum\limits_{j=1}^{n}{{{c}_{ij}}}}{{x}_{ij}}.$$
The ai’s, bj ’s, cij ’s and xij’s are assumed to be integral. The problem is how to choose xij (1 ≤ i ≤ m, 1 ≤ j ≤ n) such that $$f(\vec{x})$$ is a minimum.

## Keywords

Line Segment Transportation Cost Flow Diagram Simplex Method Transportation Problem

## References

1. Wan Zhe Xian. A proof for a Graphic Method for solving Transportation Problem, Scientia Sinica 7, 1962, 889–894.Google Scholar
2. Wan Zhe Xian and Wang Yuan. Mathematical Methods in Transportation Problem Science Press, Beijing, 1959.Google Scholar
3. Yu Ming-I, Wan Zhe Xian, Wang Yuan etc. (edited). The Theory and Application of Linear Programming People’s Education Press, Beijing, 1959.Google Scholar
4. Editor’s note: Additional references on related graphical methods for the transportation problem, the transshipment problem and the minimum cost network flow problem are the following.Google Scholar
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