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Linear Programming – Simplex Method

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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Linear Programming – Simplex Method

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  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: • - • - • -

  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 < 5 2x1 - 3x3 > 10 x1, x2, x3 > 0 (-)Max -2x1 + 3x2 - x3 s.t. x1 + 2x2 < 5 2x1 - 3x3 > 10 x1, x2, x3 > 0

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