%% Math 464 Linear Optimization (Spring 2023)
%% Lecture 13, 02/21/2023

%% Min-ratio test example

>> format rat
>> format compact

%% Setting up the problem
>> A = [1 1 -1 0 0; 3 1 0 -1 0; 3 2 0 0 1];
>> b = [2 4 10]';
>> Bind = [1 3 4]; % for B(10/3, 0)
>> B = A(:,Bind);
>> c = [2 1 0 0 0]';
>> cB = c(Bind)
cB =
       2       
       0       
       0

% reduced cost
>> cP = c' - cB'*inv(B)*A
cP =
       0             -1/3            0              0             -2/3     
>> cP = cP'
cP =
       0       
      -1/3     
       0       
       0       
      -2/3

% consider the j=2 basic direction first
>> j=2;
>> [m,n]=size(A);
>> d = zeros(n,1); d(j)=1;dB = -inv(B)*A(:,j);d(Bind) = dB
d =
      -2/3     
       1       
       1/3     
      -1       
       0       

>> x = [10/3 0 4/3 6 0]'; % bfs == B(10/3, 0)
>> x
x =
      10/3     
       0       
       4/3     
       6       
       0       

>> theta = min( -x(1)/d(1), -x(4)/d(4) )
theta =
       5       

>> xold = x;  % saving a copy of the original bfs

% moving to the next bfs
>> x = x + theta*d
x =
       0
       5       
       3       
       1       
       0
% This corresponds to C(0,5)

>> xold % corresponds to B(10/3, 0)
xold =
      10/3     
       0       
       4/3     
       6       
       0