I am solving the following problem of linear programming using the linprog function
%Objective Function
%X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18
f = [0.669 0.654 0.503 0.683 0.670 0.673 0.749 0.655 0.660 0.583 1.243 0.639 2.024 2.156 1.672 0.473 0.139 0.687];
A = []; b = []; %Sin restricciones de desigualdad
%Restricciones de igualdad son:
%X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18
Aeq=[0.1 0.12 0.335 0.15 0.18 0.19 0.12 0.15 0.15 0.15 0 0.15 0.11 0 0.13 0 0 0.46; %Nitrogeno
0.3 0.24 0 0.03 0.05 0.04 0.27 0.03 0.24 0.15 0 0 0.52 0.52 0 0 0 0 ; %Fosforo
0.1 0.12 0 0.31 0.15 0.19 0.08 0.2 0.12 0.15 0.50 0 0 0.34 0.44 0 0 0 ; %Potasio
0 0 0 0 0 0 0 0 0 0 0 0.26 0 0 0 0 0.50 0 ; %Calcio
0 0 0 0 0.06 0 0 0 0 0 0 0 0 0 0 0.17 0 0]; %Magnesio
beq = [285.71 ; %Demanda nutricional de Nitrogeno (kg/ha)
305.33 ; %Demanda nutricional de Fosforo (kg/ha)
450 ; %Demanda nutricional de Potasio (kg/ha)
262.50 ; %Demanda nutricional de Calcio (kg/ha)
41.50]; %Demanda nutricional de Magnesio (kg/ha)
%Limite inferior
lb = zeros(18,1);
%Limite superior
ub = inf(18,1);
x = linprog(f, A, b, Aeq, beq, lb, ub, options)
Solucion_optima = f*x
When I solve this is the result that throws me but does not show any results of the simplex table and I execute it with the following command
options = optimoptions('linprog','Algorithm','dual-simplex');
So I have the simplex algorithm
iterM=100;
In=size(Aeq,1);
Xsol=[Aeq eye(In) beq
f zeros(1,In) 0];
for iter=1:1:iterM
fin=Xsol(end,1:end-1)<0;
if fin==0
break
end
[a,c]=min(Xsol(end,:));
Xre=Xsol(:,end)./Xsol(:,c);
i=Xre<=0;
d=Xre;
d(i)=inf;
[beq,f]=min(d);
Xsol(f,1:end)=Xsol(f,1:end)/Xsol(f,c);
for i=1:1:size(Xsol,1)
if i~=f
Xsol(i,:)=Xsol(i,:)-(Xsol(i,c)*Xsol(f,:));
end
end
end
for i=1:1:size(f,2)
d=logical(Xsol(:,i));
X(i,1)=Xsol(d,end)
end
When I run the Xsol function it does not show me the optimal solution nor the other values that the simplex table should have
Based on the OP stating, "I need the reduced costs, the dual solution and shadow prices."
1) The dual solution is the shadow prices. The shadow prices are the solution to the dual.
2) The final simplex tableau is not the only way to obtain the stated objectives (though it would work).
Dual Solution (Shadow prices)
You can obtain the dual solution via [x,fval,exitflag,output,lambda] = linprog(___). The lambda is the dual solution; see MATLAB's documentation and examples for linprog (link). The documentation calls these Lagrange multipliers.
Reduced Costs
The reduced costs are obtainable with or without the dual solution. If f is the coefficients of the objective function (costs), then the reduced costs = f'- p'*A when the LP is written in standard form A*x=b. If someone else knows a better way to get the reduced costs from the output, please post. I've tried to avoid the primal formula to spare pulling out the index of basic variables.
A clear reference on this:
Bertsimas, Dimistris, and Tsitsiklis, John N. 1997. Introduction to Linear Optimization, Athena Scientific & Dynamic Ideas, LLC, Belmont, MA. page 148