Simplex Method-Operations Research-Lecture Slides, Slides of Operational Research

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Simplex Method

• LP problem in standard form

Canonical (slack) form

• : basic variables
• : nonbasic variables

B  x 1 , x 2 ,..., xm 

N  x m  1 ,..., xn 

Illustrative Example

General steps of Simplex

• 1. Start with an initial basic feasible solution
• 2. Improve the initial solution if possible by finding an adjacent basic feasible solution with a better objective function value
     • It implicitly eliminates those basic feasible solutions whose objective functions values are worse and thereby a more efficient search
• 3. When a basic feasible solution cannot be improved further, simplex terminates and return this optimal solution

Interior Point Methods

(Karmarkar’s algorithm)

Interior Point Method vs. Simplex

• Interior point method becomes competitive for

very “large” problems

• Certain special classes of problems have always

been particularly difficult for the simplex method

• e.g., highly degenerate problems (many different algebraic basic feasible solutions correspond to the same geometric extreme point)

m  n  10 , 000