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Motivated by a recent Diophantine transport problem about how to transportprofitably a group of persons or objects, we survey classical facts aboutsolving systems of linear Diophantine equations and inequalities in nonnegativeintegers. We emphasize on the method of Elliott from 1903 and its furtherdeveloped by MacMahon in his ``$\Omega$-Calculus'' or Partition Analysis. As anillustration we obtain the solution of the considered transport problem interms of a formal power series in several variables which is an expansion of arational function of a special form.

A Diophantine transport problem from 2016 and its possible solution in 1903