function [z,x_B,index_B] = simplex (type,A,h,c,sign)
% Simplex method solver for a linear programming problem
% 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
%   sign = vector holding information about the constraints if the system
%          needs to be standardized (-1: less or equal, 0: equal, 1:vgreater or equal)
    
m = size(A,1);
n = size(A,2);

% Compute the maximum number of basic solutions of the original
% problem (i.e., the maximum number of iterations necessary to solve the problem)
itMax = factorial(2*m+n)/(factorial(n+m)*factorial(m)); 

% Writing the problem in standard form
[A_aug,h,c_aug] = standardize(type,A,h,c,m,sign);

% Determining a starting feasible initial basic solution
[B,D,c_B,c_D,x_B,x_D,index_B,index_D] = auxiliary(A_aug,c_aug,h,m,n);

% Solving the linear programming problem with the Simplex method
[x_B,c_B,index_B] = simplexSolve(type,B,D,c_B,c_D,h,x_B,x_D,index_B,index_D,itMax);

% Compute the value of the objective function
z = sum(c_B' .* x_B) + sum(c_D' .* x_D);

% Output of the solution
[x_B,index_B] = printSol(z,x_B,index_B,m,n);

end