Simplex Algorithm.Big M Method 18/03/2009 Linear Programming

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 18/03/2009 Linear Programming

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. 18/03/2009 Linear Programming

Example Solve using simplex method 18/03/2209 Linear Programming

Example (Solution) Write the initial system using the slack variables 18/03/2009 Linear Programming

Pivot operation Write simplex tableau and identify pivot -2 3 1 0 0 9 -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 18/03/2009 Linear Programming

Maximization with Mixed Constraints Consider the following problem: We introduce a slack variable 18/03/2009 Linear Programming

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 18/03/2009 Linear Programming

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. . 18/03/2009 Ch.29 Linear Programming

Example In order to 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 18/03/2009 Ch.29 Linear Programming

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: 18/03/2009 Ch.29 Linear Programming

Example: Modified problem We now have a new problem, we call the modified problem: 18/03/2009 Ch.29 Linear Programming

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 18/03/2009 Ch.29 Linear Programming

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 18/03/2009 Ch.29 Linear Programming

Example 2 0 1 1 -1 0 8 -1 1 0 -1 1 0 2 -3 0 0 -1 M+1 1 2 18/03/2009 Ch.29 Linear Programming

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 18/03/2009 Ch.29 Linear Programming

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 18/03/2009 Ch.29 Linear Programming

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 18/03/2009 Ch.29 Linear Programming

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. 18/03/2009 Ch.29 Linear Programming

Example Find the modified problem for the following linear programming problem. 18/03/2209 Ch.29 Linear Programming

Example 18/03/2009 Ch.29 Linear Programming

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 18/03/2009 Ch.29 Linear Programming

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 18/03/2009 Ch.29 Linear Programming

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 18/03/2009 Ch.29 Linear Programming