# A TRANSPORTATION PROBLEM

 #

 # This problem finds a least cost shipping schedule that meets

 # requirements at markets and supplies at factories.

 #

 #  References:

 #              Dantzig G B, "Linear Programming and Extensions."

 #              Princeton University Press, Princeton, New Jersey, 1963,

 #              Chapter 3-3.

 set I;

 /* canning plants */

 set J;

 /* markets */

 param a{i in I};

 /* capacity of plant i in cases */

 param b{j in J};

 /* demand at market j in cases */

 param d{i in I, j in J};

 /* distance in thousands of miles */

 param f;

 /* freight in dollars per case per thousand miles */

 param c{i in I, j in J} := f * d[i,j] / 1000;

 /* transport cost in thousands of dollars per case */

 var x{i in I, j in J} >= 0;

 /* shipment quantities in cases */

 minimize cost: sum{i in I, j in J} c[i,j] * x[i,j];

 /* total transportation costs in thousands of dollars */

 s.t. supply{i in I}: sum{j in J} x[i,j] <= a[i];

 /* observe supply limit at plant i */

 s.t. demand{j in J}: sum{i in I} x[i,j] >= b[j];

 /* satisfy demand at market j */

 data;

 set I := Seattle San-Diego;

 set J := New-York Chicago Topeka;

 param a := Seattle     350

            San-Diego   600;

 param b := New-York    325

            Chicago     300

            Topeka      275;

 param d :              New-York   Chicago   Topeka :=

            Seattle     2.5        1.7       1.8

            San-Diego   2.5        1.8       1.4  ;

 param f := 90;

 end;