NC/mp6/Project_6_Maggioni_Claudio/simplexSolve.m

function [x_B,c_B,index_B] = simplexSolve(type, B, D, c_B, c_D, h, x_B, ...
    x_D, index_B, index_D, itMax)
% Solving a maximization problem with the simplex method

% Initialize the number of iterations
nIter = 0;

% Compute B^{-1}*D and B^{-1}*h
BiD = B\D;
Bih = B\h;

% Compute the reduced cost coefficients
r_D = c_D - (c_B * BiD);

% the optimality condition is satisfied if all
% reduced cost coefficients are positive or negative (depending on the 
% problem)
tol = max(size(r_D)); 

% Check the optimality condition, in order to skip the loop if the 
% solution is already optimal
optCheck = typeCond(type, sum(r_D < 0), sum(r_D > 0));

while optCheck ~= tol
    % Find the index of the entering variable
    idxIN = find(r_D == typeCond(type, max(r_D), min(r_D)));

    in = D(:,idxIN);
    c_in = c_D(1,idxIN);
    index_in = index_D(1,idxIN);
    
    % Evaluate the coefficients ratio for the column corresponding to
    % the entering variable
    ratio = Bih ./ BiD(:,idxIN);
    
    % Find the smallest positive ratio
    idxOUT = find(ratio == min(ratio(ratio >= 0)));
    
    out = B(:,idxOUT);
    c_out = c_B(1,idxOUT);
    index_out = index_B(1,idxOUT);
    
    % Update the matrices by exchanging the columns
    B(:,idxOUT) = in;
    D(:,idxIN) = out;
    c_B(1,idxOUT) = c_in;
    c_D(1,idxIN) = c_out;
    index_B(1,idxOUT) = index_in;
    index_D(1,idxIN) = index_out;
     
    % Compute B^{-1}*D and B^{-1}*h
    BiD = B\D;
    Bih = B\h;
    
    % Compute the reduced cost coefficients
    r_D = c_D - (c_B * BiD);
   
    % Check the optimality condition, in order to exit the loop if the 
    % solution is already optimal
    optCheck = typeCond(type, sum(r_D < 0), sum(r_D > 0));
    
    % Detect inefficient looping if nIter > total number of basic solutions
    nIter = nIter + 1;
    if nIter > itMax
       error('Incorrect loop, more iterations than the number of basic solutions');
    end

    % Compute the new x_B
    x_B = Bih - BiD * x_D;
end

end

function x = typeCond(type, ifMaximum, ifMinimum)
    if strcmp(type, 'max')
        x = ifMaximum;
    elseif strcmp(type, 'min')
        x = ifMinimum;
    else
        error('Incorrect type specified. Choose either a maximisation (max) or minimisation (min) problem.')
    end
end