Simplex Algorithm Big M Method o o Simplex

Simplex Algorithm. Big M Method o o Simplex algorithm Big M method 18/03/2009 Linear Programming 1

Simplex method maximization problem in standart form o o o Step 1. Write the maximization problem in standart form, introduce slack variables to form the initial system, and write the initial tableau. Step 2. Are there any negative indicators in the bottom row? If yes go to step 3, if no go to step 7. Step 3. Select the pivot column. Step 4. Are there any pozitive elements in pivot column above the dashed line? If yes go to step 5, if no go to step 6 Step 5. Select the pivot element and perform the pivot operation and go to the 2 18/03/2009 Linear Programming 2

Simplex method maximization problem in standart form o o Step. 6 Stop: The LP problem has no optimal solution Step 7. Stop: The optimal solution has been found. 18/03/2009 Linear Programming 3

Example o Solve using simplex method 18/03/2209 Linear Programming 4

Example (Solution) o Write the initial system using the slack variables 18/03/2009 Linear Programming 5

Pivot operation o Write simplex tableau and identify pivot -2 -1 -6 3 3 -3 1 0 0 0 1 9 12 0 Pivot column we are enable select pivot row 18/03/2009 Linear Programming 6

Maximization with Mixed Constraints o Consider the following problem: We introduce a slack variable 18/03/2009 Linear Programming 7

Example o We introduce a second variable and substract it from the left side of second equation. So we can write o 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 8

Example o 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 9

Example o 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 18/03/2009 Ch. 29 Linear Programming 10

Example o 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 11

Example: Modified problem o We now have a new problem, we call the modified problem: 18/03/2009 Ch. 29 Linear Programming 12

Example o We next write the augmented coefficient matrix for this system, which we call the preliminary simplex tableau. 1 -1 -2 18/03/2009 1 1 -1 1 0 0 0 -1 0 Ch. 29 Linear Programming 0 0 1 0 M 1 10 2 0 13

Example o 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 -1 1 M-2 -M-1 0 M 0 10: 1=10 2: 1=2 18/03/2009 Ch. 29 Linear Programming 0 0 1 10 2 -2 M 14

Example o 2 -1 -3 18/03/2009 0 1 0 0 1 -1 -1 -1 0 M+1 1 Ch. 29 Linear Programming 8 2 2 15

Example o 1 -1 -3 18/03/2009 0 1/2 0 0 1/2 -1 -1 -1/2 0 1 0 M+1 1 Ch. 29 Linear Programming 4 2 2 16

Example o 1 0 0 18/03/2009 0 1/2 1/2 -1/2 3/2 1/2 -1/2 0 M-1/2 1 Ch. 29 Linear Programming 4 6 14 17

Introducing Slack, Surplus and Artificial Variables o o o Step 1: If any problem constraints have negative constraints on the right side, multiply both sides by -1 Step 2: Introduce a slack variable in each <=constraint Step 3: Introduce a surplus variable and an artificial variable in each >= constraint 18/03/2009 Ch. 29 Linear Programming 18

Introducing Slack, Surplus and Artificial Variables o o Step 4: Introduce an artificial variable in each = constraint Step 5: 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 19

Example o Find the modified problem for the following linear programming problem. 18/03/2209 Ch. 29 Linear Programming 20

Example 18/03/2009 Ch. 29 Linear Programming 21

Big M Method: Solving the Problem o o Step 1: From the preliminary simplex tableau for the modified problem Step 2: 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 22

Big M Method: Solving the Problem o o o Step 3: Solve the modified problem by applying the simplex method to the initial simplex tableau found in step 2 Step 4: 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 23

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 24