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The general formulae for the simplex method in linear programming. It includes the reordering of variables and the expression of basic variables and the objective function in terms of non-basic variables. The document also shows how to solve for xb and use it to find the value of z.
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Consider the linear programming problem in standard form
minimize z = cT^ x
subject to Ax = b
x ≥ O
with b ≥ O.
Write^1
x =
xB
xN
and A =
then
z = c
T x
c
T B c
T N
xB
xN
= c
T B xB^ +^ c
T N xN^ (1)
We then can rewrite problem (∗) as follows
minimize z = c
T B xB^ +^ c
T N xN^ (2)
subject to BxB + N xN = b (3)
x ≥ O
B is basis matrix, hence non-singular, i.e. B
− 1 exists.
(^1) see section 4.3 for notation
ables
Solving (3) for xB yields
xB = B
− 1 b − B
− 1 N xN. (4)
Using (4) in (1) yields
z = c
T B xB^ +^ c
T N xN
= c
T B (B
− 1 b − B
− 1 xN ) + c
T N xN^ by (4)
= c
T B B
− 1 ︸ ︷︷ ︸ def = yT
b + (c
T N −^ c
T B B
− 1 ︸ ︷︷ ︸ def = yT
N )xN
