lemon-project-template-glpk
view deps/glpk/examples/min01ks.mod @ 9:33de93886c88
Import GLPK 4.47
| author | Alpar Juttner <alpar@cs.elte.hu> |
|---|---|
| date | Sun, 06 Nov 2011 20:59:10 +0100 |
| parents | |
| children |
line source
1 /* min01ks.mod - finding minimal equivalent 0-1 knapsack inequality */
3 /* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */
5 /* It is obvious that for a given 0-1 knapsack inequality
7 a[1] x[1] + ... + a[n] x[n] <= b, x[j] in {0, 1} (1)
9 there exist infinitely many equivalent inequalities with exactly the
10 same feasible solutions.
12 Given a[j]'s and b this model allows to find an inequality
14 alfa[1] x[1] + ... + alfa[n] x[n] <= beta, x[j] in {0, 1}, (2)
16 which is equivalent to (1) and where alfa[j]'s and beta are smallest
17 non-negative integers.
19 This model has the following formulation:
21 minimize
23 z = |alfa[1]| + ... + |alfa[n]| + |beta| = (3)
25 = alfa[1] + ... + alfa[n] + beta
27 subject to
29 alfa[1] x[1] + ... + alfa[n] x[n] <= beta (4)
31 for all x satisfying to (1)
33 alfa[1] x[1] + ... + alfa[n] x[n] >= beta + 1 (5)
35 for all x not satisfying to (1)
37 alfa[1], ..., alfa[n], beta are non-negative integers.
39 Note that this model has n+1 variables and 2^n constraints.
41 It is interesting, as noticed in [1] and explained in [2], that
42 in most cases LP relaxation of the MIP formulation above has integer
43 optimal solution.
45 References
47 1. G.H.Bradley, P.L.Hammer, L.Wolsey, "Coefficient Reduction for
48 Inequalities in 0-1 Variables", Math.Prog.7 (1974), 263-282.
50 2. G.J.Koehler, "A Study on Coefficient Reduction of Binary Knapsack
51 Inequalities", University of Florida, 2001. */
53 param n, integer, > 0;
54 /* number of variables in the knapsack inequality */
56 set N := 1..n;
57 /* set of knapsack items */
59 /* all binary n-vectors are numbered by 0, 1, ..., 2^n-1, where vector
60 0 is 00...00, vector 1 is 00...01, etc. */
62 set U := 0..2^n-1;
63 /* set of numbers of all binary n-vectors */
65 param x{i in U, j in N}, binary, := (i div 2^(j-1)) mod 2;
66 /* x[i,j] is j-th component of i-th binary n-vector */
68 param a{j in N}, >= 0;
69 /* original coefficients */
71 param b, >= 0;
72 /* original right-hand side */
74 set D := setof{i in U: sum{j in N} a[j] * x[i,j] <= b} i;
75 /* set of numbers of binary n-vectors, which (vectors) are feasible,
76 i.e. satisfy to the original knapsack inequality (1) */
78 var alfa{j in N}, integer, >= 0;
79 /* coefficients to be found */
81 var beta, integer, >= 0;
82 /* right-hand side to be found */
84 minimize z: sum{j in N} alfa[j] + beta; /* (3) */
86 phi{i in D}: sum{j in N} alfa[j] * x[i,j] <= beta; /* (4) */
88 psi{i in U diff D}: sum{j in N} alfa[j] * x[i,j] >= beta + 1; /* (5) */
90 solve;
92 printf "\nOriginal 0-1 knapsack inequality:\n";
93 for {j in 1..n} printf (if j = 1 then "" else " + ") & "%g x%d",
94 a[j], j;
95 printf " <= %g\n", b;
96 printf "\nMinimized equivalent inequality:\n";
97 for {j in 1..n} printf (if j = 1 then "" else " + ") & "%g x%d",
98 alfa[j], j;
99 printf " <= %g\n\n", beta;
101 data;
103 /* These data correspond to the very first example from [1]. */
105 param n := 8;
107 param a := [1]65, [2]64, [3]41, [4]22, [5]13, [6]12, [7]8, [8]2;
109 param b := 80;
111 end;