18/03/2009Linear Programming1 Simplex Algorithm.Big M Method Simplex algorithm Big M method.

18/03/2009 Linear Programming 1

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.

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


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.

Example Solve using simplex method


1 2

1 2

1 2

1 2

6 3 max

2 3 9

3 12

, 0

P x x

x x

x x

x x

Example (Solution) Write the initial system using the slack

variables


1 2,s s

1 2 1

1 2 2

1 2

2 3 9

3 12

6 3 0

x x s

x x s

x x P

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


1 2 1 2x x s s P RHS1

2

s

s

P

Maximization with Mixed Constraints Consider the following problem:

We introduce a slack variable


1 2

1 2

1 2

1 2

2 max

10

2

, 0

P x x

x x

x x

x x

1s

1 2 1 10x x s


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

2s

1 2 2 2x x s

2s

18/03/2009 Ch.29 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.

.

1 2 1

1 2 2

1 2

1 2 1 2

10

2

2 0

, , , 0

x x s

x x s

x x P

x x s s

1 2,x x

1 2 1 20, 0, 10, 2, 0x x s s P


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

1a

2s

1 2 2 1 2x x s a


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:

1Ma

1 2 12P x x Ma

We now have a new problem, we call the modified problem:


Example: Modified problem

1 2 1

1 2 1

1 2 2 1

1 2 1 2 1

2 max

10

2

, , , , 0

P x x Ma

x x s

x x s a

x x s s a

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


1 2 1 2 1x x s s a P RHS

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


1 2 1 2 1x x s s a P RHS1

1

s

a

P

2 3 3( )M R R R

Example

2 0 1 1 -1 0 8

-1 1 0 -1 1 0 2

-3 0 0 -1 M+1 1 2


1 2 1 2 1x x s s a P RHS1

2

s

x

P

2 1 1

2 3 3

( 1) ,

( 1)

R R R

M R R R

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


1 2 1 2 1x x s s a P RHS1

2

s

x

P

1 1

1

2R R

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


1 2 1 2 1x x s s a P RHS1

2

x

x

P

1 2 2

1 3 33

R R R

R R R

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


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.


iaiMa

Example Find the modified problem for the following

linear programming problem.


1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

2 5 3 max

2 7

2 5

4 3 1

2 4 6

, , 0

P x x x

x x x

x x x

x x x

x x x

x x x

Example


1 2 3 1 2 3

1 2 3 1

1 2 3 2 1

1 2 3 3 2

1 2 3 3

1 2 3 1 2 3 1 2 3

2 5 3 max

2 7

2 5

4 3 1

2 4 6

, , , , , , , , 0

P x x x Ma Ma Ma

x x x s

x x x s a

x x x s a

x x x a

x x x s s s a a a

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


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


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


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18/03/2009Linear Programming1 Simplex Algorithm.Big M Method Simplex algorithm Big M method.

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