M-Method

1. M-Method The M-Method starts with LP in equation form. An equation i that does not have a slack (or a variable that can play the role of a slack) is augmented with an artificial variable, Ri, to form a starting solution similar to all slake basic solution.

2. Example • Minimize z=4x1+x2 • Subject to, 3x1+x2=3 4x1+3x2≥6 x1+2x2 ≤4 x1,x2 ≥0

3. Minimize z=4x1+x2 3x1+x2=3 4x1+3x2-x3=6 x1+2x2 +x4=4 x1,x2,x3,x4 ≥0, here only x4 is slack and x3 is surplus variable.

4. Minimize z=4x1+x2+MR1+MR2 3x1+x2+R1=3 4x1+3x2-x3+R2=6 x1+2x2 +x4=4 x1,x2,x3,x4,R1,R2 ≥0, here only x4 is slack and x3 is surplus variable. R1,R2 are called artificial variables. Now we can use R1,R2, x4 as starting basic solution. M will be big positive for minimum and –M for maximum problem.

5. Inconsistency in the value of z • If x4, R1,R2 are basic then this are non zero and others x1=x2=x3=0, which gives • Z=MR1+MR2 , which is not equal to zero initially as was done previously.

6. Remedy is the following • Create the z-row as follows • New z-row=old-zrow+M×R1-row+M ×R2-row. Old z-row: z-4x1-x2-MR1-MR2=0 MR1-Row : 3M x1+M x2+M R1=3M M R2-Row: 4M x1+3M x2-M x3+M R2=6M New-z-row: z +(7M-4)x1+(4M-1)x2-M x3 =9M

7. Simplex algorithm

8. Simplex algorithm

9. Simplex algorithm

10. Simplex algorithm

11. Simplex algorithm

12. Simplex algorithm

13. Simplex algorithm

14. Simplex algorithm

15. Simplex algorithm

16. Simplex algorithm

17. Simplex algorithm

18. Simplex algorithm

19. Simplex algorithm

20. Simplex algorithm

21. Simplex algorithm • Solution is • Z=17/2 • X1=2/5 • X2=9/5 x3=1