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Linear Programming – Simplex Method

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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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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

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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.

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Linear Programming – Simplex Algorithm Step 1Convert the LP to standard form. Step 2Obtain a bfs (if possible) from the standard form. Step 3Determine whether the current bfs is optimal. Step 4If 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 5Use EROs to find the new bfs with the better objective function value. Go back to step 3. Operations Research, Wayne L. Winston

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Linear Programming – Simplex Method Review Simplex Handouts

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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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Linear Programming – Simplex Method: Computational Problems Breaking Ties in Selection of Non-Basic Variable – if tie for non-basic variable with largest relative profit ( ), arbitrarily select incoming variable. Ties in Minimum Ratio Rule (Degeneracy) – if more than one basic variable have same minimum ratio, select either variable to leave the basis. This will result in a basic variable taking on a value of 0. When this occurs, the solution is referred to as a degenerate basic feasible solution. When this occurs, you may transition through more than one simplex tableau with the same objective (Z) value.

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Linear Programming – Simplex Method: Computational Problems Unbounded Solutions – if when performing the minimum ratio rule, none of the ratios are positive, then the solution is unbounded (e.g Max Z = or Min = - ).

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Simplex Method – Finding an Initial Basic Feasible Solution Min Z = -3x1 + x2 + x3 s.t. x1 – 2x2 + x3 <= 11 -4x1 + x2 +2x3 >= 3 2x1 - x3 = -1 x1, x2, x3 >= 0 Standard Form: (-) Max Z = 3x1 - x2 - x3 s.t. x1 – 2x2 + x3 + x4 = 11 -4x1 + x2 +2x3 -x5 = 3 -2x1 + x3 = 1 x1, x2, x3, x4, x5 >= 0

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Simplex Method – Finding an Initial Basic Feasible Solution (-) Max Z = 3x1 - x2 - x3 s.t. x1 – 2x2 + x3 + x4 = 11 -4x1 + x2 +2x3 -x5 = 3 -2x1 + x3 = 1 x1, x2, x3, x4, x5 >= 0 Only x4 is basic. Introduce artificial variables. s.t. x1 – 2x2 + x3 + x4 = 11 -4x1 + x2 +2x3 -x5 + x6 = 3 -2x1 + x3 + x7 = 1 x1, x2, x3, x4, x5, x6, x7 >= 0

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Simplex Method – Solve Using Big-M Method Let M be an arbitrarily large number, then: (-) Max Z = 3x1 - x2 - x3 + 0x4 + 0x5 – Mx6 – Mx7 s.t. x1 – 2x2 + x3 + x4 = 11 -4x1 + x2 +2x3 -x5 + x6 = 3 -2x1 + x3 + x7 = 1 x1, x2, x3, x4, x5, x6, x7 >= 0 Note: If the simplex algorithm terminates with one of the artificial variables as a basic variable, then the original problem has no feasible solution.

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