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Chapter 10:   Iterative Improvement   Simplex Method The Design and Analysis of Algorithms
Iterative Improvement ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Introduction   ,[object Object],[object Object],[object Object],[object Object],[object Object]
Introduction ,[object Object],[object Object]
Important Examples   ,[object Object],[object Object],[object Object],[object Object]
Linear Programming   ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Example maximize  3x + 5y subject to  x +  y  ≤ 4   x + 3y  ≤ 6 x ≥ 0,  y ≥ 0 Feasible region  is the set of points defined by the constraints
Geometric solution   maximize  3x + 5y subject to  x +  y  ≤ 4   x + 3y  ≤ 6 x ≥ 0,  y ≥ 0 Extreme Point Theorem   Any LP problem with a nonempty bounded feasible region has an optimal solution;  moreover, an optimal solution can always be found at an  extreme point  of the problem's feasible region.
Possible outcomes in solving an LP problem ,[object Object],[object Object],[object Object]
The Simplex Method ,[object Object],[object Object],[object Object],[object Object]
Standard form of LP problem ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Example maximize  3 x  + 5 y   maximize  3 x  + 5 y  +  0 u  + 0 v subject to  subject to x  +  y  ≤ 4  x  +  y  +  u   = 4  x  + 3 y  ≤ 6   x  + 3 y  +  v   = 6  x ≥0,  y ≥0    x ≥0,  y ≥0,  u ≥0,  v ≥0 Variables  u  and  v , transforming inequality constraints into  equality constrains, are called  slack variables
Basic feasible solutions   A  basic solution  to a system of  m  linear equations in  n   unknowns ( n  ≥  m ) is obtained by setting  n  –  m   variables to 0 and solving the resulting system to get the values of the other  m  variables.  The variables set to 0 are called  nonbasic ;   the variables obtained by solving the system are called  basic .   A basic solution is called  feasible   if all its (basic) variables are nonnegative.  Example  x +  y + u   = 4    x +  3 y  + v   = 6    (0,  0,  4,  6) is basic feasible solution  ( x ,  y  are nonbasic;  u ,  v   are basic)
Simplex Tableau   maximize  z  = 3 x  + 5 y  + 0 u  + 0 v subject to  x  +  y  +  u   = 4  x  + 3 y   +  v   = 6  x ≥0,  y ≥0,  u ≥0,  v ≥0
Outline of the Simplex Method   ,[object Object],[object Object],[object Object],[object Object]
Outline of the Simplex Method ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Example of Simplex Method  maximize  z  = 3 x  + 5 y  + 0 u  + 0 v subject to  x  +  y  +  u   = 4  x  + 3 y   +  v   = 6  x ≥0,  y ≥0,  u ≥0,  v ≥0 basic feasible sol.  (0, 0, 4, 6) z =  0 basic feasible sol.  (0, 2, 2, 0) z =  10 basic feasible sol.  (3, 1, 0, 0) z =  14
Notes on the Simplex Method   ,[object Object],[object Object],[object Object],[object Object]
Improvements   ,[object Object],[object Object]

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L20 Simplex Method

  • 1. Chapter 10: Iterative Improvement Simplex Method The Design and Analysis of Algorithms
  • 2.
  • 3.
  • 4.
  • 5.
  • 6.
  • 7. Example maximize 3x + 5y subject to x + y ≤ 4 x + 3y ≤ 6 x ≥ 0, y ≥ 0 Feasible region is the set of points defined by the constraints
  • 8. Geometric solution maximize 3x + 5y subject to x + y ≤ 4 x + 3y ≤ 6 x ≥ 0, y ≥ 0 Extreme Point Theorem Any LP problem with a nonempty bounded feasible region has an optimal solution; moreover, an optimal solution can always be found at an extreme point of the problem's feasible region.
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  • 12. Example maximize 3 x + 5 y maximize 3 x + 5 y + 0 u + 0 v subject to subject to x + y ≤ 4 x + y + u = 4 x + 3 y ≤ 6 x + 3 y + v = 6 x ≥0, y ≥0 x ≥0, y ≥0, u ≥0, v ≥0 Variables u and v , transforming inequality constraints into equality constrains, are called slack variables
  • 13. Basic feasible solutions A basic solution to a system of m linear equations in n unknowns ( n ≥ m ) is obtained by setting n – m variables to 0 and solving the resulting system to get the values of the other m variables. The variables set to 0 are called nonbasic ; the variables obtained by solving the system are called basic . A basic solution is called feasible if all its (basic) variables are nonnegative. Example x + y + u = 4 x + 3 y + v = 6 (0, 0, 4, 6) is basic feasible solution ( x , y are nonbasic; u , v are basic)
  • 14. Simplex Tableau maximize z = 3 x + 5 y + 0 u + 0 v subject to x + y + u = 4 x + 3 y + v = 6 x ≥0, y ≥0, u ≥0, v ≥0
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  • 17. Example of Simplex Method maximize z = 3 x + 5 y + 0 u + 0 v subject to x + y + u = 4 x + 3 y + v = 6 x ≥0, y ≥0, u ≥0, v ≥0 basic feasible sol. (0, 0, 4, 6) z = 0 basic feasible sol. (0, 2, 2, 0) z = 10 basic feasible sol. (3, 1, 0, 0) z = 14
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