Simplex Method

1
Simplex Method

• CONTENTS
• Key to the Simplex Method.
• Algebra of the Simplex Method
• Termination Optimality and Unboundedness
• Reference Chapter 3 in BJS book.

2
Simplex Method

• If a LP has a bounded feasible region, then there
  always exists an optimal extreme point solution.
• This result allows us to exclude interior points
  of the solution space when searching for an
  optimal solution.
• Since an extreme point solution corresponds to a
  BFS, there always exists a BFS solution of a LP.
• Simplex algorithm starts with a BFS and goes
  through a sequence of BFS until it obtains an
  optimal BFS.

3
Simplex Method (contd.)

• Step 1 Find an initial BFS of the LP.
• Step 2 Determine if the current BFS is an
  optimal to the LP. If it is not, find an adjacent
  BFS to the LP that has a larger objective
  function value.
• Step 3. Return to Step 2 using the new BFS as the
  current BFS.
• This method performs at most nCm iterations.

4
Simplex Method (contd.)
z
5
Key to the Simplex Method

• Minimize cx
• subject to
• Ax b
• x ? 0
•  
• Suppose that we have a basic feasible solution
  with B as the basis. Then Ax b can be written
  as
•  
• BxB NxN b
•  
• Premultiplying by B-1 yields
•  
• xB B-1NxN B-1b

6
Key to the Simplex Method (contd.)

• xB B-1NxN B-1b
•  
• xB B-1b - B-1NxN
•  
• xB B-1b - B-1?j?R(aj)xj
•  
• xB - ?j?R( )xj
•  
• where aj is the column in the constraint matrix
  for variable xj and R is the set of indices of
  the nonbasic variables.

7
Key to the Simplex Method (contd.)

• Minimize 2x1 x2
• subject to
• - x1 x2 x3 2
• 2x1 x2 x4 6
• x1, x2, x3, x4 ? 0
• Equivalent form
•  
• Minimize 2x1 x2
• subject to
• x1 4/3 (-1/3) x3 (1/3) x4
• x2 10/3 (2/3) x3 (1/3) x4
• x1, x2, x3, x4 ? 0

8
Key to the Simplex Method (contd.)

• z cx
• cBxB cNxN
• cB(B-1b ?j?R B-1(aj)xj) ?j?R cjxj
•  
• cBB-1b - ?j?R cBB-1(aj)xj ?j?R cjxj
•  
• z z0 - ?j?R(zj cj) xj
•  
• where z0 cBB-1b and zj cBB-1aj for each j ?
  R.
• cj zj is called the reduced cost of xj with
  respect to B as the basis.

9
Key to the Simplex Method (contd.)
Minimize 2x1 x2 0.x3
0.x4 subject to   x1 4/3
(-1/3)x3 - (1/3)x4 x2 10/3 -
(2/3)x3 - (1/3)x4   x1, x2, x3, x4 ?
0  Equivalent form   Minimize -2/3
4/3x3 1/3 x4   x1 4/3
(-1/3)x3 - (1/3)x4 x2 10/3 -
(2/3)x3 - (1/3)x4   x1, x2, x3, x4 ?
0  This is called the canonical form of the basic
solution.

10
Key to the Simplex Method (contd.)

• Minimize z z0 - ?j?R (zj cj)xj
•  
• subject to
• xB ?j?R ( j) xj
• xj ? 0, for all j 1, 2, , n.
• Theorem If (zjcj) ? 0 for all j ? R, then the
  current basic feasible solution is an optimal
  solution of the linear programming problem.

11
Algebra of the Simplex Method

• Minimize z z0 - ?j?R (zj cj)xj
•  
• subject to
• xB ?j?R ( j) xj
• xj ? 0, for all j 1, 2, , n.
• If some (zk ck) gt 0, then we can potentially
  improve the current solution by increasing the
  value of the variable xk.
• Suppose we decide to increase xk and keeping all
  other nonbasic variables at zero value, then how
  the values of basic variables change as xk
  increases?

12
Algebra of the Simplex Method (contd.)

• z z0 - (zk ck) xk
• What is the largest value we can assign to xk?

13
Algebra of the Simplex Method (contd.)
Let then All other xjs are zero.
Change in the objective function value
(zk-ck) Is the new solution a
basic feasible solution?
14
Algebra of the Simplex Method (contd.)

• The process of moving from one basic feasible
  solution to another basic feasible solution is
  called a pivot iteration. (or a pivot step)
• If the basic feasible solution is non-degenerate,
  then the new basic feasible solution has a lower
  cost.
• In the absence of degeneracy, the simplex method
  terminate in a finite number of iterations. Why?

15
An Example
Minimize x1 x2 4x3 subject to x1 x2
2x3 ? 9 x1 x2 - x3 ? 2 -x1
x2 x3 ? 4 x1 , x2 , x3 ?
0 Canonical form Minimize x1 x2 - 4x3 0.x4
0.x5 0.x6 subject to x1 x2 2x3 x4
9 x1 x2 - x3
x5 2 -x1 x2 x3
x6 4 x1, x2, x3, x4, x5, x6 ? 0
16
An Example (contd.)
Iteration 1 Iteration 2
17
An Example (contd.)
Iteration 2 Iteration 3 Optimal
solution x1 1/3 x2 0 x3 13/3 z -17
18
Interpretation of entries in the Simplex Tableau
19
Interpretation of zk -ck
(zk ck) is the change in the objective function
value when the value of xk increase by one unit
and all other nonbasic variables remain at zero
value, and values of basic variables are adjusted
appropriately.
20
Identifying B-1 in the Simplex Tableau

• Suppose we maintain simplex tableau. The simplex
  tableau also contains B-1. How to locate it in
  the tableau?
• Suppose that xi1, xi2, , xim are the basic
  variables in the starting tableau in the first
  row, second row, and so on upto the mth row. Then
  the B-1 in any iteration is the submatrix under
  the variables xi1, xi2, , xim at that
  iteration.

21
Solving Maximization Problems
Modification If all nonbasic variables in row 0
have nonnegative coefficients, the current BFS is
optimal. If any nonbasic variable in row 0 has a
negative coefficient, choose the variable with
the "most negative" coefficient in row 0 to enter
the basis.
22
Alternate Optimal Solutions
If some nonbasic variable has zero coefficient
row 0 of the optimal tableau, then the problem
may have alternate optimal solutions.
Suppose that we enter x2 into the basis, then we
get another solution with the same cost. It is an
alternate optimal solution.
23
Unbounded LPs
An unbounded LP for a minimization problem occurs
when a variable with a positive coefficient in
row 0 has a non-positive coefficient in each
constraint.
Suppose that we enter x3 into the basis. Then we
can increase its value indefinitely, and decrease
the optimal objective function value to negative
infinity. In this case, we say that LP is
unbounded.
24
Pivoting Rules

• If there are several nonbasic variables qualified
  to enter the basis, then which one should enter
  the basis? The pivot rule describes the rule
  which one should enter.
• We can select any such nonbasic variable to enter
  the basis and guarantee convergence of the
  algorithm.
• Popular pivoting rules
• Dantzig pivot rule Select the greatest magnitude
  of cost coefficient in the objective function row
  among the eligible variables.
• First eligible variable rule Select the first
  nonbasic variable which is found to be eligible
  to enter the basis.
• Hybrid rule Select the greatest magnitude of the
  cost coefficient in the objective function row
  among the K variables examined.

25
Simplex Multipliers

• The simplex method at each iteration maintains
  one multiplier for each constraint which we call
  as simplex multiplier. Simplex multipliers play
  an important role in simplex algorithm.
• Let row A0 denote the objective function row in
  the first canonical form and Ai denote the ith
  constraint in it.
• We start with a canonical form of a LP and
  perform a sequence of elementary row operations
  to the constraints and the objective function
  row. Let denote the modified row at any
  iteration.

26
Simplex Multipliers (contd.)

• The objective function row at any iteration
  can be expressed as
• A0 ?1A1 ?2A2 - ?mAm
• The numbers ?1, ?2, ?3, , ?m are called
  simplex multipliers.
• What property is satisfied by the simplex
  multipliers?
• The multipliers are such that zj cj value of
  each basic variable xj becomes zero.

27
Simplex Multipliers (contd.)

• Suppose that x1, x2, x3, , xm are the basic
  variables at some iteration. Then the simplex
  multipliers at this iteration must satisfy the
  following conditions
• c1 a11p1 a21p2 a31p3 .. am1pm
• c2 a12p1 a22p2 a32p3 .. am2pm
• c3 a13p1 a23p2 a33p3 .. am3pm
• .
• cm a2mp1 a2mp2 a3mp3 .. amnpm
• There are m simplex multipliers and they must
  satisfy m equations. Hence simplex multipliers
  are uniquely determined.

28
Simplex Multipliers (contd.)

• Alternatively, the simplex multipliers p at this
  iteration must satisfy the following condition
• cB pB
• or,
• p cBB-1

29
Simplex Multipliers (contd.)

• Suppose that the initial BFS is
• The BFS solution at some iteration is
• What are the simplex multipliers for this
  iteration?

30
Simplex Multipliers (contd.)

• The simplex algorithm also maintains simplex
  multipliers in the simplex tableau at every
  iteration. Where are they stored?
• If xk is the basic variable in the ith row in the
  starting tableau, then zk - ck for xk at any
  iteration gives the simplex multiplier for that
  iteration for the ith row.
• Why?