Mateu X. Hunter

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