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Simplex Method

Simplex Method. Step1. (1)Standard maximization problem in standard form. (2)Initial system. (3) Initial Simplex Tableau. Fundamental Theorem of Linear Programming (3).

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Simplex Method

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  1. Simplex Method

  2. Step1.(1)Standard maximization problem in standard form

  3. (2)Initial system

  4. (3) Initial Simplex Tableau

  5. Fundamental Theorem of Linear Programming (3) • If the optimal value of the objective function in a linear programming problem exists, then that value must occur at one (or more) of the basic feasible solutions of the initial system.

  6. Step2.Select basic variables • The number of basic variables=the number of equations in the initial system • Determine a basic variable so that its corresponding column has exactly one nonzero element and such elements from each column are not placed in the same row. • P must be a basic variable.

  7. Step3.Determine the pivot number 1. Locate the most negative value(a negative indicator) in the bottom row --the pivot column 2. Divide each positive element in the pivot column into the element in the last column. Find the smallest quotient. -- the pivot row 3. Determine the pivot number

  8. Step4.Perform a pivot operation 1. Transform the pivot element into 1 by the pivot column 1/the pivot number. 2. Transform all other nonzero element in the pivot column into 0’s by multiples of the pivot row the other rows

  9. Geometric Interpretation of the Simplex Process • Basic feasible solutions in order to be found • The simplex process started at (0,0), moved to the adjacent corner point (0,16), and then to the optimal solution at the next adjacent corner point at (20,6)

  10. Example • Solve the linear programming problem using the simplex method.

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