x_{1}  + 2 x_{3}  + x_{4}  < = 20  
 2 x_{1}  + x_{2}   x_{4}  < = 15  
6 x_{1}  + 2 x_{2}   3 x_{3}  < = 54 

 
Inititial Tableau  Final Tableau 
B = [ A_{4} , A_{6} , A_{2} ] =  [ 
1 0 0 1 1 1 0 0 2  ] 
B^{1} = [ A_{5}^{*} , A_{6}^{*} , A_{7}^{*} ] =  [ 
1 0 0 1 1 1/2 0 0 1/2  ] 
A^{*} = B^{1} A  b^{*} = B^{1} b 
c^{*} = c  c_{B} B^{1} A  z^{*} = z  c_{B} B^{1} b 
Changes in the Constant Column Vector
Changing the original column vector b into b' will affect b* and z* of the final tableau, but
not c* and A*.
The modified b'* = B^{1} b' can be calculated. If the entries
remain nonnegative, since c* > = 0, the optimal solution to the modified problem will have
the same optimal solution as the original problem, with values given by b'*; and
z'* = z  c_{B}b'*.
If some entries of b'* are negative, then to resolve this problem we use the Dual Simplex
Algorithm on this new tableau.
Addition of a New Variable
Suppose that now we wish to add another variable in the formulation of the original problem.
Let x_{n+1} be the new variable, with the objective coefficient c_{n+1} and
the column vector of coefficients for the constraining equations A_{n+1}. Then the
expanded, modified problem in canonical form is to minimize z' = c' X' subject to A' X' = b,
X' > = 0, where
Addition of a Constraint
Suppose after solving the problem we wish to alter the original problem by the addition of a
new constraint. Now it could be that X* satisfies this new constraint. If this is the case,
X* is also optimal for the expanded problem, because clearly, by this addition of a constraint,
we have not changed the objective function nor increased the set of feasible solutions to the
system of constraints. On the other hand, if X* does not satisfy this new constraint, we must
find a new optimal solution. Under certain circumstances, however, this problem may be resolved
quite easily by
creating a new canonical tableau ( the new costant column b' may contain some negative entries) from the final tableau solution to the original problem and the application of the Dual Simplex Algorithm.