106 lines
2.9 KiB
Mathematica
106 lines
2.9 KiB
Mathematica
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function [B,D,c_B,c_D,x_B,x_D,index_B,index_D] = auxiliary (A_aug,c_aug,h,m,n)
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% The auxiliary problem is always a minimization problem
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% The output will be: B and D (basic and nonbasic matrices), c_B and c_D
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% (subdivision of the coefficient vector in basic and nonbasic parts), x_B
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% and x_D (basic and nonbasic variables) and index_B and index_D (to keep
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% track of the variables indices)
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% Redefine the problem by introducing the artificial variables required by
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% the auxiliary problem (the objective function has to reach value 0)
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A_aug = [A_aug, eye(m)];
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c_aug = [c_aug, zeros(1,m)]; % original objective function coefficients
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c_aux = [zeros(1,n+m), ones(1,m)]; % auxiliary c of minimization problem
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index = 1:n+2*m; % index to keep track of the different variables
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% Defining the basic elements and the nonbasic elements
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B = A_aug(:,(n+m+1):(n+2*m)); % basic variables
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D = A_aug(:,1:(n+m)); % nonbasic variables
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c_Baux = c_aux(1,(n+m+1):(n+2*m));
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c_Daux = c_aux(1,1:(n+m));
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c_B = c_aug(1,(n+m+1):(n+2*m));
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c_D = c_aug(1,1:(n+m));
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x_B = h;
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x_D = zeros((n+m),1);
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index_B = index(1,(n+m+1):(n+2*m));
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index_D = index(1,1:(n+m));
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nIter = 0;
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z = c_Baux*x_B;
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itMax = factorial(2*m+n)/(factorial(n+m)*factorial(m));
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% Compute B^{-1}*D and B^{-1}*h
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BiD = B\D;
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Bih = B\h;
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% Compute reduced cost coefficients
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r_D = c_Daux - c_Baux*BiD;
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while(z~=0)
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% Find nonnegative index
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idxIN = find(r_D == min(r_D));
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% Using Bland's rule to avoid cycling
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if(size(idxIN,2)>1)
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idxIN = min(idxIN);
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end
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in = D(:,idxIN);
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c_inaux = c_Daux(1,idxIN);
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c_in = c_D(1,idxIN);
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index_in = index_D(1,idxIN);
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% Evaluating the coefficients ratio
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inRatio = BiD(:,idxIN);
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ratio = Bih./inRatio;
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% Find the smallest ratio
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for i = 1:size(ratio,1) % Eliminating negative ratios
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if(ratio(i,1)<0)
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ratio(i,1) = Inf;
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end
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end
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idxOUT = find(ratio==min(ratio));
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% Using Bland's rule to avoid cycling
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if(size(idxOUT,1)>1)
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idxOUT = min(idxOUT);
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end
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out = B(:,idxOUT);
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c_outaux = c_Baux(1,idxOUT);
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c_out = c_B(1,idxOUT);
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index_out = index_B(1,idxOUT);
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% Update the matrices by exchanging the columns
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B(:,idxOUT) = in;
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D(:,idxIN) = out;
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c_Baux(1,idxOUT) = c_inaux;
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c_Daux(1,idxIN) = c_outaux;
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c_B(1,idxOUT) = c_in;
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c_D(1,idxIN) = c_out;
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index_B(1,idxOUT) = index_in;
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index_D(1,idxIN) = index_out;
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% Compute B^{-1}*D and B^{-1}*h
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BiD = B\D;
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Bih = B\h;
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% Compute reduced cost coefficients
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r_D = c_Daux - (c_Baux * BiD);
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% Detect inefficient loop if nIter > total number of basic solutions
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nIter = nIter + 1;
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if nIter > itMax
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error('The original LP problem does not admit a feasible solution.');
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end
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x_B = Bih - BiD * x_D;
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z = c_Baux * x_B;
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end
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check = index_D<(n+m+1);
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D = D(:,check);
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index_D = index_D(1,check);
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c_D = c_D(1,check);
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x_D = x_D(check,1);
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end
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