Transportation Problem

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, . . .,A_i, . . ., A_m and n destinations B₁, . . .,B_j, . . .,B_n. The point A_i(i = 1, . . .,m) can supply a_i units, and the destination B_j(j = 1, . . ., n) requires b_j 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 B_j, j = 1, . . ., n, must be satisfied by the supply of units available at the points of origin A_j, i = 1, . . .,m.

If x_ij is the number of units shipped from A_i to B_j, then the problem consists in determining the values of the variables x_ij, i = 1, . . ., m and j = 1, . . ., n, that minimize the total shipping costs

under the conditions

(3) x_ij ≥ 0 i = 1,..., m and j = 1,..., n

Transportation problems are solved by means of special linear programming techniques.