Linear Programming

1
Linear Programming Simplex Method
 
2
Linear Programming - Review

• Graphical Method
• What is the feasible region?
• Where was optimal solution found?
• What is primary limitation of graphical method?
• Conversion to Standard Form
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3
Linear Programming Review

• Solving Systems of Linear Equations
• What is a basic solution?
• How did we obtain a basic solution?
• What is a basic feasible solution?
• Relationship between graphical and algebraic
• representation of the feasible region
• corner point basic solution

4
Linear Programming Review
Fundamental insight the optimal solution to a
linear program, if it exists, is also a basic
feasible solution. Naïve approach solve for
all basic solutions and find the feasible
solution with the largest value (maximization
problem). What is the problem with this
approach? there are possible basic
solutions, where m is the number of constraints
and n is the number of variables.
5
Linear Programming Simplex Algorithm
Step 1 Convert the LP to standard form. Step
2 Obtain a bfs (if possible) from the standard
form. Step 3 Determine whether the current bfs
is optimal. Step 4 If the current bfs is not
optimal, then determine which nonbasic basic
variable should become a basic variable and which
basic variable should become a nonbasic variable
to find a new bfs with a better objective
function value. (pivot operation) Step 5 Use
EROs to find the new bfs with the better
objective function value. Go back to step 3.
Operations Research, Wayne L. Winston
6
Linear Programming Simplex Method
Review Simplex Handouts
7
Linear Programming Simplex Method
Minimization Problems
Min Z cx ? (-) Max Z -cx Ex. Min
2x1 3x2 x3 s.t. x1 2x2
lt 5 2x1 - 3x3 gt
10 x1, x2, x3 gt 0
(-)Max -2x1 3x2 - x3 s.t. x1
2x2 lt 5 2x1 -
3x3 gt 10 x1, x2, x3 gt 0