Lecture 4

# Lecture 4 - solution Expression for all alternative...

This preview shows pages 1–9. Sign up to view the full content.

The Simplex Method Tie breaking And different solutions

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Tie Breaking and other situations 1. A tie for an entering variable ->break the tie arbitrarily. 2. A tie for leaving variable degenerate solution (a basic variable having zero value): break the tie arbitrarily. 1. No leaving variable the coefficients of the entering basic variable are all non-positive: an unbounded optimal solution. 1. In the optimal tableau, the obj. coefficient of a non-basic variable is zero: multiple optima.
Example of ties for entering and leaving variables Consider the following LP: Max Z = 3 x 1 + 3 x 2 St: x 1 + x 2 6 2 x 2 12 3 x 1 + 2 x 2 18 x 1 0 x 2 0

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Solution to example 1 Z x 1 x 2 x 3 x 4 x 5 RHS Z 1 -3 -3 0 0 0 0 X 3 0 1 1 1 0 0 6 X 4 0 0 2 0 1 0 12 X 5 0 3 2 0 0 1 18 Z 1 -3 0 0 3/2 0 18 X 3 0 1 0 1 -1/2 0 0 X 2 0 0 1 0 1/2 0 6 X 5 0 3 0 0 -1 1 6 Z 1 0 0 3 0 0 18 X 1 0 1 0 1 -1/2 0 0 Tie for the entering variable! Pick x 2 ! Tie for the leaving variable! Degenerate solution! Alternative optimal solution!
x 1 x 2 Graphical view of the alternative

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: solution Expression for all alternative solutions Z 1 3 18 x 1 1 0 -2 1 6 x 2 1 3-1 X 4 0 -6 1 2 12 X * = α X 1 * + (1- α ) X 2 * or X * = α (0,6,0,0,6) + (1- α ) (6,0,0,12,0) or 6(1- α ) X * = 6 α where 0 ≤ α ≤ 1 12 (1- α ) 6 α Unbounded Solution Max Z = 3 x 1 + 2 x 2 st: x 1 ≤ 4 3 x 1- x 2 ≤ 6 x 1 ≥ 0 , x 2 ≥ 0 Simplex solution The last tableau of the problem: Z x 1 x 2 x 3 x 4 RHS Z 1 6-1 18 x 2 1 3/2 -1/2 6 x 1 1 1 Non-positive coefficients for the entering variable. Unbounded solution! Exercise Solve the following problem by the simplex method: max Z = - 3 x 1 + 6 x 2 st: 5 x 1 + 7 x 2 ≤ 35- x 1 + 2 x 2 ≤ 2 x 1 , x 2 ≥...
View Full Document

{[ snackBarMessage ]}

### Page1 / 9

Lecture 4 - solution Expression for all alternative...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online