Motivated by a recent Diophantine transport problem about how to transport profitably a group of persons or objects, we survey classical facts about solving systems of linear Diophantine equations and inequalities in nonnegative integers. We emphasize on the method of Elliott from 1903 and its further developed by MacMahon in his ``$Ω$-Calculus'' or Partition Analysis. As an illustration we obtain the solution of the considered transport problem in terms of a formal power series in several variables which is an expansion of a rational function of a special form.