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/* ASSIGN, Assignment Problem */
/* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */
/* The assignment problem is one of the fundamental combinatorial
optimization problems.
In its most general form, the problem is as follows:
There are a number of agents and a number of tasks. Any agent can be
assigned to perform any task, incurring some cost that may vary
depending on the agent-task assignment. It is required to perform all
tasks by assigning exactly one agent to each task in such a way that
the total cost of the assignment is minimized.
(From Wikipedia, the free encyclopedia.) */
param m, integer, > 0;
/* number of agents */
param n, integer, > 0;
/* number of tasks */
set I := 1..m;
/* set of agents */
set J := 1..n;
/* set of tasks */
param c{i in I, j in J}, >= 0;
/* cost of allocating task j to agent i */
var x{i in I, j in J}, >= 0;
/* x[i,j] = 1 means task j is assigned to agent i
note that variables x[i,j] are binary, however, there is no need to
declare them so due to the totally unimodular constraint matrix */
s.t. phi{i in I}: sum{j in J} x[i,j] <= 1;
/* each agent can perform at most one task */
s.t. psi{j in J}: sum{i in I} x[i,j] = 1;
/* each task must be assigned exactly to one agent */
minimize obj: sum{i in I, j in J} c[i,j] * x[i,j];
/* the objective is to find a cheapest assignment */
solve;
printf "\n";
printf "Agent Task Cost\n";
printf{i in I} "%5d %5d %10g\n", i, sum{j in J} j * x[i,j],
sum{j in J} c[i,j] * x[i,j];
printf "----------------------\n";
printf " Total: %10g\n", sum{i in I, j in J} c[i,j] * x[i,j];
printf "\n";
data;
/* These data correspond to an example from [Christofides]. */
/* Optimal solution is 76 */
param m := 8;
param n := 8;
param c : 1 2 3 4 5 6 7 8 :=
1 13 21 20 12 8 26 22 11
2 12 36 25 41 40 11 4 8
3 35 32 13 36 26 21 13 37
4 34 54 7 8 12 22 11 40
5 21 6 45 18 24 34 12 48
6 42 19 39 15 14 16 28 46
7 16 34 38 3 34 40 22 24
8 26 20 5 17 45 31 37 43 ;
end;
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