%% Math 364: Sensitivity Analysis using matrix form 
%% 	     of the tableau simplex method.
%% Matlab session from Lecture 19 on 10/22/2024
%%
%% Comments added at "% " for increased readability

% Farmer Jones LP with lowered c2
% max  z =  30 x1 + 25 x2
% s.t. 	       x1 +    x2 <=  7 (land avail)
% 	     4 x1 + 10 x2 <= 40 (labr hrs)
%	       x1      	  >=  3 (min corn) % scaled by 10
%	       x1,     x2 >=  0 (non neg)

% The optimal tableau has {e3, s2, x1} as the basic variables in that
% order. So we can read off the basis matrix B from under these
% columns in the starting tableau

>> B = [0 0 1; 0 1 4; -1 0 1]
B =
     0     0     1
     0     1     4
    -1     0     1

% Similarly, we find B^{-1} from under the columns of {s1, s2, a3} in
% the optimal tableau. Note that these columns have the identity
% matrix in the starting tableau.

>> Binv = [1 0 -1;-4 1 0; 1 0 0]
Binv =
     1     0    -1
    -4     1     0
     1     0     0

% Just to confirm that we have read off B^{-1} correctly, we check
% B*B^{-1} and B^{-1}*B

>> B*Binv
ans =
     1     0     0
     0     1     0
     0     0     1
>> Binv*B
ans =
     1     0     0
     0     1     0
     0     0     1

% We could also ask Matlab to invert B for us (to confirm)

>> inv(B)
ans =
     1     0    -1
    -4     1     0
     1     0     0


% The column of x2 in the starting tableau is set as vector a2:

>> a2 = [1 10 0]'
a2 =
     1
    10
     0

% B^{-1}*a2 gives the column of x2 in the optimal tableau:

>> Binv*a2
ans =
     1
     6
     1