Advanced Linear Programming

1
Advanced Linear Programming

• Chapter 1 Revised Simplex Method
• Chapter 2 Bounded Variables algorithm
• Chapter 3 Decomposition Algorithms
• Chapter 4 Interior- Point Method

2
Theory of the Simplex Method

• The optimal solution of an LP must be a
  corner-point feasible (CPF) solution ( or extreme
  point solution)
• If there are alternate optima, then at least two
  must be adjacent CPF solutions
• There are a finite number of CPF solutions
• A CPF solution is optimal if there are no other
  adjacent CPF solutions that are better

3
Corner Point Solutions
x2
40
X
20
x1
40
30
4
Alternate Optima
5
LP Matrix Form

• Maximize (minimize) z CX
• subject to
• AX b
• X 0
• where C ( c1,c2,, , cn)
• X (x1, x2, , xn)T
• A (aij)mxn ( P1, P2,, Pn)
• b b1,b2, ,bmT
• Extreme points of XAX b ? basic solutions
  of AX b ? setting (n-m) variables 0, thus
• BXB b ? XB B-1b det (B) ? 0
• if XB B-1b 0 then XB is feasible

6
Example

• Determine and classify (as feasible and
  infeasible all the basis solutions of the
  following system of equations

7
Generalized Simplex Tableau in Matrix Form
Example consider the following LP Maximize z
x1 4x2 7x3 5x4 subject to 2x1 x2
2x3 4x4 10 3x1 x2 2x3 6x4
5 Generate the simplex tableau associated with
the basis B ( P1,P2 )
8
Revised Simplex Method

• Principle of simplex method row operations
• Start by selecting a feasible basis B (extreme
  point)
• Move to search another feasible basis Bnext
  until the optimum feasible basis is reached
• Principle of revised simplex method matrix
  operations
• Optimality condition zj cj CBB-1Pj cj gt 0
  (lt0)
• Feasibility condition
• (XB)i (B-1b)i (B-1Pj)ixj 0, for all
  constraint i
• ?

9
Revised Simplex Method

• Step 0 construct a starting basic feasible
  solution and let B and CB be its associated basis
  and objective coefficients vector, respectively
• Step 1 Compute the inverse B-1
• Step 2 for each nonbasic variable xj,compute zj
  cj CBB-1Pj cj
• If zj cj 0 in maximization for all
  nonbasic xj, stop the optimal solution is given
  by XB B-1b, z CB XB
• Else, apply the optimality condition and
  determine the entering variable xj as the
  nonbasic variable with the most negative (zj
  cj) (positive) in case of maximization
  (minimization)
• Step 3 Compute B-1Pj . If all the element of
  B-1Pj are negative or zero, stop the problem has
  no bounded solution. Else, compute B-1b. Then
  for all the strictly positive elements of B-1Pj,
  determine the ratios defined by the feasibility
  condition. The basic variable xi associated with
  the smallest ratio is the leaving variable.
• Step 4 From the current basis B, form a new
  basis by replacing the leaving vector Pi with the
  entering vector Pj . Go to Step 1 to start a new
  iteration.

10
Product Form of The Inverse

• Given B, B-1, Bnext, Pj in B is replaced by Pr in
  Bnext find
• Example solve the following LP by revised
  simplex method
• Maximize z 5x1 4x2
• subject to
• 6x1 4 x2 24
• x1 2x2 6
• - x1 x2 1
• x2 2
• x1, x2 0