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Simplex Algorithm.Big M Method

Simplex Algorithm.Big M Method. Simplex algorithm Big M method. Simplex method maximization problem in standart form. Step1. Write the maximization problem in standart form, introduce slack variables to form the initial system, and write the initial tableau.

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Simplex Algorithm.Big M Method

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  1. Simplex Algorithm.Big M Method Simplex algorithm Big M method Linear Programming

  2. Simplex method maximization problem in standart form • Step1. Write the maximization problem in standart form, introduce slack variables to form the initial system, and write the initial tableau. • Step2. Are there any negative indicators in the bottom row? If yes go to step 3,if no go to step 7. • Step3. Select the pivot column. • Step4. Are there any pozitive elements in pivot column above the dashed line? If yes go to step 5, if no go to step 6 • Step5. Select the pivot element and perform the pivot operation and go to the 2 Linear Programming

  3. Simplex method maximization problem in standart form • Step.6 Stop: The LP problem has no optimal solution • Step7. Stop: The optimal solution has been found. Linear Programming

  4. Example • Solve using simplex method Linear Programming

  5. Example (Solution) • Write the initial system using the slack variables Linear Programming

  6. Pivot operation • Write simplex tableau and identify pivot -2 3 1 0 0 9 -1 3 0 1 0 12 -6 -3 0 0 1 0 Pivot column we are enable select pivot row Linear Programming

  7. Maximization with Mixed Constraints • Consider the following problem: We introduce a slack variable Linear Programming

  8. Example • We introduce a second variable and substract it from the left side of second equation. So we can write • The variable is called surplus variable,because it is amount (surplus) by which the left side of inequality exceeds the right side Linear Programming

  9. Example • We now express the linear programming problem as a system of equations: The basic solution found by setting the nonbasic variables equal to 0 is But this solution is not feasible. . Ch.29 Linear Programming

  10. Example • In orderto use simplex method with mixed constraints we will use variable called an artificial variable. An artificial variable is a variable introduced into each equation that has a surplus variable.Returning to the problem at hand we introduce an artificial variable into the equation involving the surplus • Objective value = 111/4 Ch.29 Linear Programming

  11. Example • To prevent an artificial from becoming part of an optimal solution to the original problem, a very large “penalty” is introduced into the objective function. This penalty is created by choosing a positive constant M so large that the artificial variable is forced to be 0 in any final optimal solution of the original problem. We then add the term to the objective function: Ch.29 Linear Programming

  12. Example: Modified problem • We now have a new problem, we call the modified problem: Ch.29 Linear Programming

  13. Example • We next write the augmented coefficient matrix for this system, which we call the preliminary simplex tableau. 1 1 1 0 0 0 10 -1 1 0 -1 1 0 2 -2 -1 0 0 M 1 0 Ch.29 Linear Programming

  14. Example • To use the simplex method we must first use row operations to transform into an equivalent matrix that satisfies M=0 1 1 1 0 0 0 10 -1 1 0 -1 1 0 2 M-2 -M-1 0 M 0 1 -2M 10:1=10 2:1=2 Ch.29 Linear Programming

  15. Example 2 0 1 1 -1 0 8 -1 1 0 -1 1 0 2 -3 0 0 -1 M+1 1 2 Ch.29 Linear Programming

  16. Example 1 0 1/2 1/2 -1/2 0 4 -1 1 0 -1 1 0 2 -3 0 0 -1 M+1 1 2 Ch.29 Linear Programming

  17. Example 1 0 1/2 1/2 -1/2 0 4 0 1 1/2 -1/2 1/2 0 6 0 0 3/2 1/2 M-1/2 1 14 Ch.29 Linear Programming

  18. Introducing Slack,Surplus and Artificial Variables • Step1: If any problem constraints have negative constraints on the right side,multiply both sides by -1 • Step2: Introduce a slack variable in each <=constraint • Step3:Introduce a surplus variable and an artificial variable in each >= constraint Ch.29 Linear Programming

  19. Introducing Slack,Surplus and Artificial Variables • Step4: Introduce an artificial variable in each = constraint • Step5: For each artificial variable add to the objective function. Use the same constant M for all artificial variables. Ch.29 Linear Programming

  20. Example • Find the modified problem for the following linear programming problem. Ch.29 Linear Programming

  21. Example Ch.29 Linear Programming

  22. Big M Method:Solving the Problem • Step1: From the preliminary simplex tableau for the modified problem • Step2:Use row operations to eliminate the M’s in the bottom row of the preliminary simplex tableau in the column corresponding to the artificial variables. The resulting tableau is the initial simplex tableau Ch.29 Linear Programming

  23. Big M Method:Solving the Problem • Step3: Solve the modified problem by applying the simplex method to the initial simplex tableau found in step 2 • Step4:Results the optimal solution of the modified problem to the original problem: • (A); If the modified problem has no optimal solution, the original problem has no optimal solution Ch.29 Linear Programming

  24. Big M Method:Solving the Problem (B): If all artificial variables are 0 in the optimal solution to the modified problem, delete the artificial variables to find an optimal solution to the original problem (C):If any artificial variables are nonzero in the optimal solution in the modified problem,the original problem has no optimal solution Ch.29 Linear Programming

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