Name
Algorithm::Simplex - Simplex Algorithm Implementation using Tucker
Tableaux
Synopsis
Given a linear program formulated as a Tucker tableau, a 2D matrix or
ArrayRef[ArrayRef] in Perl, seek an optimal solution.
use Algorithm::Simplex::Rational;
my $matrix = [
[ 5, 2, 30],
[ 3, 4, 20],
[10, 8, 0],
];
my $tableau = Algorithm::Simplex::Rational->new( tableau => $matrix );
$tableau->solve;
my ($primal_solution, $dual_solution) = $tableau->current_solution;
Methods
_build_number_of_rows
Set the number of rows. This number represent the number of rows of the
coefficient matrix. It is one less than the full tableau.
_build_number_of_columns
Set the number of columns given the tableau matrix. This number
represent the number of columns of the coefficient matrix.
_build_x_variables
Set x variable names for the given tableau, x1, x2 ... xn These are the
decision variables of the maximization problem. The maximization problem
is read horizontally in a Tucker tableau.
_build_y_variables
Set y variable names for the given tableau. These are the slack
variables of the maximization problem.
_build_u_variables
Set u variable names for the given tableau. These are the slack
variables of the minimization problem.
_build_v_variables
Set v variable names for the given tableau: v1, v2 ... vm These are the
decision variables for the minimization problem. The minimization
problem is read horizontally in a Tucker tableau.
get_bland_number_for
Given a column number (which represents a u variable) build the bland
number from the generic variable name.
determine_bland_pivot_column_number
Find the pivot column using Bland ordering technique to prevent cycles.
determine_bland_pivot_row_number
Find the pivot row using Bland ordering technique to prevent cycles.
min_index
Determine the index of the element with minimal value. Used when finding
bland pivots.
exchange_pivot_variables
Exchange the variables when a pivot is done. The method pivot() does the
algrebra while this method does the variable swapping, and thus tracking
of what variables take on non-zero values. This is needed to accurately
report an optimal solution.
get_row_and_column_numbers
Get the dimensions of the tableau.
determine_bland_pivot_row_and_column_numbers
Higher level function that uses others to return the (bland) pivot
point.
Authors
Mateu X. Hunter "hunter@missoula.org"
Strong design influence by George McRae at the University of Montana.
#moose for solid assistance in the refactor: particularly _build_*
approach and PDL + Moose namespace management, 'inner'.
Copyright
Copyright 2009, Mateu X. Hunter
License
You may distribute this code under the same terms as Perl itself.
Description
Base class for the Simplex model using Tucker tableaus.
The implementation is currently limited to phase II, i.e. one must start
with a feasible solution.
This class defines some of the methods concretely, and others such as:
* pivot
* is_optimal
* determine_positive_ratios
* determine_simplex_pivot_columns
are implemented in one of the three model types:
* Float
* Rational
* PDL
Variables
We have implicit variable names: x1, x2 ... , y1, y2, ... , u1, u2 ... ,
v1, v2 ...
Our variables are represented by:
x, y, u, and v
as found in Nering and Tuckers' book.
x and y are for the primal LP while u and v belong to the dual LP.
These variable names are set using the lazy feature of Moo.
Limitations
The API is stabilizing, but still subject to change.
The algorithm requires that the initial tableau be a feasible solution.
Development
http://github.com/mateu/Algorithm-Simplex