1. Gomory’s cutting plane algorithm for integer programming Prepared by Shin-ichi Tanigawa

2. Rounding does not give any useful result

3. We first solve the LP-relaxation

4. Optimize using primal simplex method

7. The optimal solution is fractional

8. (1) Generating an objective row cut

9. (weakening) (1) Generating an objective row cut

10. (weakening) (1) (2) (for integers) Generating an objective row cut

11. (weakening) (1) (2) (for integers) (2) - (1) Generating an objective row cut Cutting plane is violated by current optimum solution

12. (weakening) (1) (2) (for integers) (2) - (1) (substitute for slacks) Generating an objective row cut

13. The first cutting plane:

14. A new slack variable is added:

15. The new cut is added to the dictionary

16. Re-optimize using dual simplex method

18. A new fractional solution has been found

19. (1) Generating a constraint row cut

20. (1) (weaken) Generating a constraint row cut

21. (1) (2) Generating a constraint row cut (valid for integers)

22. (1) (2) (2) – (1) Generating a constraint row cut

23. (1) (2) (2) – (1) Generating a constraint row cut

24. The second cutting plane

25. Add a new slack variable:

26. The new cut is inserted into the optimum dictionary

27. Re-optimize using dual simplex method

28. The new optimum solution is integral