Linear Programming – Simplex Method

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# Linear Programming – Simplex Method - PowerPoint PPT Presentation

Linear Programming – Simplex Method. 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: - - -. Linear Programming – Review.

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Presentation Transcript
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:
• -
• -
• -
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
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.

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

Linear Programming – Simplex Method

Minimization Problems:

Min Z = cx  (-) Max Z = -cx

Ex. Min 2x1 – 3x2 + x3

s.t. x1 + 2x2 < 5

2x1 - 3x3 > 10

x1, x2, x3 > 0

(-)Max -2x1 + 3x2 - x3

s.t. x1 + 2x2 < 5

2x1 - 3x3 > 10

x1, x2, x3 > 0