1. LINEAR PROGRAMMINGExample 1 MaximiseI = x + 0.8y subject tox + y  1000 2x + y  1500 3x + 2y  2400 Initial solution: I = 0 at (0, 0)

2. LINEAR PROGRAMMINGExample 1 MaximiseI = x + 0.8y subject to x + y  1000 2x + y  1500 3x + 2y  2400 MaximiseI whereI - x - 0.8y = 0 subject to x + y + s 1 = 1000 2x + y + s 2 = 1500 3x + 2y + s 3 = 2400

3. Ixys1s1 s2s2 s3s3 RHS 1-0.80000 0111001000 0210101500 0320012400 SIMPLEX TABLEAU I = 0, x = 0, y = 0, s 1 = 1000, s 2 = 1500, s 3 = 2400 Initial solution

4. Ixys1s1 s2s2 s3s3 RHS 1-0.80000 0111001000 0210101500 0320012400 PIVOT 1Choosing the pivot column Most negative number in objective row

5. Ixys1s1 s2s2 s3s3 RHS 1-0.80000 0111001000 1000/1 0210101500 1500/2 0320012400 2400/3 PIVOT 1Choosing the pivot element Ratio test: Min. of 3 ratios gives 2 as pivot element

6. Ixys1s1 s2s2 s3s3 RHS 1-0.80000 0111001000 010.50 0750 0320012400 PIVOT 1Making the pivot Divide through the pivot row by the pivot element

7. Ixys1s1 s2s2 s3s3 RHS 10-0.300.50750 0111001000 010.50 0750 0320012400 PIVOT 1Making the pivot Objective row + pivot row

8. Ixys1s1 s2s2 s3s3 RHS 10-0.300.50750 000.51-0.50250 010.50 0750 0320012400 PIVOT 1Making the pivot First constraint row - pivot row

9. Ixys1s1 s2s2 s3s3 RHS 10-0.300.50750 000.51-0.50250 010.50 0750 000.50-1.51150 PIVOT 1Making the pivot Third constraint row – 3 x pivot row

10. Ixys1s1 s2s2 s3s3 RHS 10-0.300.50750 000.51-0.50250 010.50 0750 000.50-1.51150 PIVOT 1New solution I = 750, x = 750, y = 0, s 1 = 250, s 2 = 0, s 3 = 150

11. LINEAR PROGRAMMINGExample MaximiseI = x + 0.8y subject tox + y  1000 2x + y  1500 3x + 2y  2400 Solution after pivot 1: I = 750 at (750, 0)

12. Ixys1s1 s2s2 s3s3 RHS 10-0.300.50750 000.51-0.50250 010.50 0750 000.50-1.51150 PIVOT 2 Most negative number in objective row Choosing the pivot column

13. Ixys1s1 s2s2 s3s3 RHS 10-0.300.50750 000.51-0.50250 250/0.5 010.50 0750 750/0.5 000.50-1.51150 150/0.5 PIVOT 2Choosing the pivot element Ratio test: Min. of 3 ratios gives 0.5 as pivot element

14. Ixys1s1 s2s2 s3s3 RHS 10-0.300.50750 000.51-0.50250 010.50 0750 0010-32300 PIVOT 2Making the pivot Divide through the pivot row by the pivot element

15. Ixys1s1 s2s2 s3s3 RHS 1000-0.40.6840 000.51-0.50250 010.50 0750 0010-32300 PIVOT 2Making the pivot Objective row + 0.3 x pivot row

16. Ixys1s1 s2s2 s3s3 RHS 1000-0.40.6840 00011100 010.50 0750 0010-32300 PIVOT 2Making the pivot First constraint row – 0.5 x pivot row

17. Ixys1s1 s2s2 s3s3 RHS 1000-0.40.6840 00011100 01002600 0010-32300 PIVOT 2Making the pivot Second constraint row – 0.5 x pivot row

18. Ixys1s1 s2s2 s3s3 RHS 1000-0.40.6840 00011100 01002600 0010-32300 PIVOT 2New solution I = 840, x = 600, y = 300, s 1 = 100, s 2 = 0, s 3 = 0

19. LINEAR PROGRAMMINGExample MaximiseI = x + 0.8y subject tox + y  1000 2x + y  1500 3x + 2y  2400 Solution after pivot 2: I = 840 at (600, 300)

20. Ixys1s1 s2s2 s3s3 RHS 1000-0.40.6840 00011100 01002600 0010-32300 PIVOT 3Choosing the pivot column Most negative number in objective row

21. Ixys1s1 s2s2 s3s3 RHS 1000-0.40.6840 00011100 100/1 01002600 600/2 0010-32300 PIVOT 3Choosing the pivot element Ratio test: Min. of 2 ratios gives 1 as pivot element

22. Ixys1s1 s2s2 s3s3 RHS 1000-0.40.6840 00011100 01002600 0010-32300 PIVOT 3Making the pivot Divide through the pivot row by the pivot element

23. Ixys1s1 s2s2 s3s3 RHS 1000.400.2880 00011100 01002600 0010-32300 PIVOT 3Making the pivot Objective row + 0.4 x pivot row

24. Ixys1s1 s2s2 s3s3 RHS 1000.400.2880 00011100 010-201400 0010-32300 PIVOT 3Making the pivot Second constraint row – 2 x pivot row

25. Ixys1s1 s2s2 s3s3 RHS 1000.400.2880 00011100 010-201400 00130600 PIVOT 3Making the pivot Third constraint row + 3 x pivot row

26. Ixys1s1 s2s2 s3s3 RHS 1000.400.2880 00011100 010-201400 00130600 PIVOT 3Optimal solution I = 880, x = 400, y = 600, s 1 = 0, s 2 = 100, s 3 = 0

27. LINEAR PROGRAMMINGExample MaximiseI = x + 0.8y subject tox + y  1000 2x + y  1500 3x + 2y  2400 Optimal solution after pivot 3: I = 880 at (400, 600)