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1. Operation Research: Algebraic method(simplex) Section 3 Dr. Abeer Saber

2. 2 Linear programming problems Graphical method Algebraic method(Simplex) Simplex solution methods: Simplex Big M 2 Phase Dual model

3. 3 solution:- 2x1+x2+3x3 ≥ 6 x1+2x2+4x3 ≥8 3x1+x2-2x3 ≥4 Z = x1+x2+3x3 A = X1 2 1 3 1 X3 3 4 -2 3 X2 1 2 1 1 RHS 6 8 4 z X1 2 1 3 6 X3 3 1 -2 4 X2 1 2 4 8 RHS 1 1 3 z =  

4. 4 Solution:- 2x1+x2+3x3 ≤ 6 x1+2x2+x3 ≤1 3x1+4x2-2x3 ≤3 Z= 6x1+8x2+4x3 x1,x2,x3 ≥0 2x1+x2+3x3 ≤ 6 x1+2x2+x3 ≤1 3x1+4x2-2x3 ≤3 Z= 6x1+8x2+4x3 x1,x2,x3 ≥0 2x1+x2+3x3 +s1 = 6 x1+2x2+x3 +s2 = 1 3x1+4x2-2x3 +s3 =3 Z-6x1-8x2-4x3 =0 x1,x2,x3 ≥0

5. 5 Solution:- 1 ½ ¾

6. 6 Solution:-

7. 7 Solution:-

8. 8 Solution:-

9. 9 Solution:-

10. 10 Solution:- 1/3 1 1 -2

11. 11 Solution:-

12. 12 Solution:-

13. 13 Solution:-

14. 14 Solution:- Optimal solution is when x1=4/3 x2= 10/3 x3=0 z= 14/3 4/3 + 10/3 + 0 = 14/3

15. Task Minimize: Z= 12x1 + 16x2 Subject to: x1+2x2 ≥ 40 X1+x2 ≥ 30 x 1,x2 ≥ 0

16. 16 Big-M method Maximization (Z) = x1-x2+3x3 Subject to: x1+x2 ≤ 20 x1+x3 =5 x2+x3 ≥10 x1,x2,x3 ≥0

17. 17 Big-M method To convert to standard form ≤ add a slack variable ≥ subtract surplus variable and add an artificial variable = add an artificial variable

18. 18 Solution:- X1+x2+s1 =20 X1+x3+a1 =5 X2+x3-s2+a2 =10 Z=x1-x2+3x3+0s1+0s3-ma1-ma2

19. 19 Solution:- Calculating Z Z = (1*0) + (1*-M) + (0*-M) = -M

20. 20 Solution:- After setting up the table we need to determine a pivot column and a pivot row as simplex method We choose the pivot column according to largest M value 5 10

21. 21 Solution:-

22. 22 Solution:-

23. 23 Solution:- -2-M

24. 24 Solution:- 20 5 -2-M

25. 25 Solution:- X1=0 x2=5 x3=5

26. Task Maximize: Z= x1 + x2 Subject to: 2x1+5x2 ≤ 6 X1+x2 ≥2 x 1,x2 ≥ 0