This module provides an interface to the standard simplex algorithm.

For example, the following LP problem

maximize 4 x_1 - 3 x_2 + 2 x_3
subject to

2 x_1 +   x_2 <= 10
  x_3 + 5 x_4 <= 20

and

x_i >= 0

can be solved as follows:

import Numeric.LinearProgramming

prob = Maximize [4, -3, 2]

constr1 = Sparse [ [2#1, 1#2] :<=: 10
                 , [1#2, 5#3] :<=: 20
                 ]

> simplex prob constr1 []
Optimal (28.0,[5.0,0.0,4.0])

The coefficients of the constraint matrix can also be given in dense format:

constr2 = Dense [ [2,1,0] :<=: 10
                , [0,1,5] :<=: 20
                ]

By default all variables are bounded as x_i >= 0, but this can be changed:

> simplex prob constr2 [ 2 :=>: 1, 3 :&: (2,7)]
Optimal (22.6,[4.5,1.0,3.8])

> simplex prob constr2 [Free 2]
Unbounded

The given bound for a variable completely replaces the default, so 0 <= x_i <= b must be explicitly given as i :&: (0,b). Multiple bounds for a variable are not allowed, instead of [i :=>: a, i:<=: b] use i :&: (a,b).