1. Cutting-Plane Algorithm BY: Mustapha.D. Ibrahim

2. Introduction The cutting-plane algorithm starts at the continuous optimum LP solution Special constraints called cuts are added to the solution, in a manner that “forces” an integer optimum extreme point.

3. Overall method

4. 1 -Adding cuts Graphically

5. Cutting-plane algorithm modifies the solution by adding cuts that produces an optimum integer extreme point

6. Adding cut-II produces the following solution Adding cut-I produces the continuous LP Solution

7. Note: Basic requirement of any cut The cut do not eliminate any of the original feasible integer points. It must pass through at least one feasible or infeasible point. In general, the number of cuts, though finite, is independent of the size of the problem. A small number of variables or constraints may require more cuts than a larger problem.

8. 2- Adding Cuts Algebraically Using the same problem Giving the slacks for constraints 1 and 2 the optimum LP table is as follows : The optimum continuous solution

9. Assumptions for developing cuts The cuts are developed under the assumption that all the variables (including slacks) are integer. The information in the table can be written as: Source row:- Equation used for generating a cut A constraint equation can be used as a source row provided its right hand side is fractional. Z-equation in this example can be considered a source row because of its RHS.

10. How a cut is generated from source row:- 1)Factor out all the non-integer coefficient of the equation into an integer value and fractional component, provided the resulting fraction component is strictly positive. 2)Move all integer components to the left hand side and fractional component to the right 3)The right hand side must satisfy non-negativity. Starting with z-equation Factoring of the z-equation yields Moving all integer component to the left and fraction to the right, gives us (1) (2)

11. Because the LHS of (1) is an integer by construction, the RHS must also be an integer, therefore (3) (3) Is the desired cut, it is a necessary but insufficient condition for obtaining an integer solution. (3) Is referred to as fractional cut because all its coefficients are fractions Factoring the equation yields The associated cut is :

12. Is factored as The associated cut is given as

13. Adding the constraint to the LP optimum table as follows : The Table is optimal but infeasible Therefore Apply dual simplex method to recover feasibility, which yields the next table.

14. The associated cut is as follows:

15. The dual simplex method yields the following table

16. The constraint is treated as:

17. This is only appropriate for small or simple constraints 2) Use a special cut called the mixed cut which allows only a subset of variables to assume integer values with all other variables (including slacks and surplus) remaining continuous. Details of this method is beyond the scope of this chapter (see Taha 1975, pp 198-202)