transportation problem[‚tranz·pər′tā·shən ‚präb·ləm]
a problem concerned with the optimal pattern of the distribution of units of a product from several points of origin to several destinations.
Suppose there are m points of origin A1, . . .,Ai, . . ., Am and n destinations B1, . . .,Bj, . . .,Bn. The point Ai(i = 1, . . .,m) can supply ai units, and the destination Bj(j = 1, . . ., n) requires bj units. It is assumed that
The cost of shipping a unit of the product from A¡ to B, is c¡¡. The problem consists in determining the optimal distribution pattern, that is, the pattern for which shipping costs are at a minimum.
Moreover, the requirements of the destinations Bj, j = 1, . . ., n, must be satisfied by the supply of units available at the points of origin Aj, i = 1, . . .,m.
If xij is the number of units shipped from Ai to Bj, then the problem consists in determining the values of the variables xij, i = 1, . . ., m and j = 1, . . ., n, that minimize the total shipping costs
under the conditions
(3) xij ≥ 0 i = 1,..., m and j = 1,..., n
Transportation problems are solved by means of special linear programming techniques.