function [A_aug,h,c_aug] = standardize(type,A,h,c,m,sign)
% Input arguments:
%   type = 'max' for maximization, 'min' for minimization
%   A    = matrix holding the constraints coefficients
%   h    = coefficients of the constraints inequalities (rhs vector)
%   c    = coefficients of the objective function
%   m    = size(A, 1)
%   sign = vector holding information about the constraints if the system
%          needs to be standardized (-1: less or equal, 0: equal, 1:vgreater or equal)
% return arguments are:
% (1) A_aug = augmented matrix A, containing also the surplus and slack variables
% (2) c_aug = augmented coefficients vector c (check compatibility of dimensions with A)
% (3) h, right hand side vector (remember to flip the signs when changing the inequalities)

aug_matrix = eye(m); % matrix corresponding to the slack/surplus variables

if strcmp(type, 'max')
    for i = 1:m
        if sign(i) == 1
            % Using a surplus instead of a slack variable
            aug_matrix(i,:) = -aug_matrix(i,:);
        end
    end
elseif strcmp(type, 'min')
    for i = 1:m
        if sign(i) == -1
            % Using a slack instead of a surplus variable
            aug_matrix(i,:) = -aug_matrix(i,:);
        end
    end
else
    error('Incorrect type specified. Choose either a maximisation (max) or minimisation (min) problem.')
end

% Correction on the sign of h for auxiliary problem (we want to
% ensure that h>=0, but we need to flip all the signs)
for i = 1:m 
    if h(i) < 0
        A(i,:) = -A(i,:);
        h(i,:) = -h(i,:);
        aug_matrix(i,:) = -aug_matrix(i,:);
    end
end

c_aug = [c, zeros(1,m)];
if strcmp(type,'max')
    % Extend matrix A by adding the slack variables    
    A_aug = [A (+1 * full(aug_matrix == +1))];
elseif strcmp(type,'min')
	% Extend matrix A by adding the surplus variables
    A_aug = [A (-1 * full(aug_matrix == -1))];
else
    error('Incorrect type specified. Choose either a maximisation (max) or minimisation (min) problem.')
end

end