Simplex Method Review
Canonical Form
T
max c x
s.t. Ax  b
x0
• A is m x n
• Theorem 7.5: If an LP has an optimal solution, then at least one such
solution exists at a basic feasible solution (BFS).
• For LP in canonical form a BFS is a solution for which there are m basic
variables and n-m nonbasic variables
• Columns of A associated with basic variables (denoted by B) are linearly
independent, xN = 0, xB uniquely solve B xB=b and xB  0.
Matrix Representation
 cB 
 xB 
A  [ B : N ], c    , x   
 cN 
 xN 
T
T
max cB xB  cN xN
s.t. BxB  NxN  b
xB  0
xN  0
Initial Solutions
If b  0 in
max cT x
s.t. Ax  b
x0
then the origin is a BFS. Thus, in canonical form we have
max cT x
s.t. Ax  Is  b
x, s  0,
and the solution ( x, s)  (0, b) is a BFS with basic variables s,
and hence is a possible starting point for the simplex method.
Feasible Directions
Since we have Ax  b, every feasible direction d must satisfy
Ad  0.
Simplex Direction: A simplex direction d k corresponding to
the nonbasic variable xk is a direction vector where
(1) d kk  1
(2) d kj  0 for all other nonbasic variables x j  xk
(3) each dik component corresponding to the basic variables
xi is (uniquely) determined by solving Ad  0.
Determining Improving Directions
Reduced Cost: The reduced cost ck associated with the nonbasic
variable xk is
ck  cT d k  ck   ci dik ,
iB
where d k is the simplex direction associated with variable xk and
B is the set of basic variables.
The simplex direction d k associated with nonbasic variable xk is
an improving direction if
(1) ck  0 for a maximization problem;
(2) ck  0 for a minimization problem.
More Than One Improving Direction
• We will use a greedy rule for selecting our improving
simplex direction.
• For a maximization problem, we choose the simplex
direction whose reduced cost is most positive.
• For a minimization problem, we choose the simplex
direction whose reduced cost is most negative.
• Rule is called the Dantzig rule after George Dantzig,
the founder of the simplex method.
No Improving Direction
• Optimality Condition for an LP (Exercise 8.26):
If the simplex method does not identify a simplex
direction that is improving, the current solution is a
global optimal solution to the LP.
Determining Maximum Step Size
Ratio Test: Starting at the basic feasible solution x, if any
coordinate of the improving simplex direction d k is
negative, the maximum step size is
max
 x j

k
 min  k : d j  0  .
 d j

Test for Unbounded Linear Programs: If all coordinates
of an improving simplex direction d are nonnegative, then
the linear program is unbounded.
Updating the Basis
• The nonbasic variable corresponding to the chosen
simplex direction enters the basis and becomes basic.
• Any one of the (possibly several) basic variables that
define the maximum step size will leave the basis and
become nonbasic.
Basic Simplex Method
Step 0 : Initialization. Identify a basic feasible solution x (0) , and set solution index t  0.
Step 1 : Construct Simplex Directions. For each nonbasic variable x j , construct the
corresponding simplex direction d j using Bd Bk  ak and its reduced cost c j .
Step 2 : Optimality Check. If no simplex direction is improving, STOP. The current
solution x (t ) is optimal. Otherwise, choose any improving simplex direction as d , and
let xe denote the entering variable.
Step 3 : Compute Maximum Step Size. If d  0, stop. The LP is unbounded. Otherwise,
choose the leaving value xl by computing the maximum step size according to the Ratio Test.
Step 4 : Update Solution and Basis. Compute the new solution
x (t 1)  x (t )  max d
and replace xl by xe in the basis. Set t  t  1 and return to Step 1.