1. Integer Programming Kusdhianto Setiawan Gadjah Mada University

2. Integer Programming...  It is an extension of Linear programming, the different is that the decision variable should be integer.  Three Types of IP: –Pure integer programming; all variables are required to have integer values –Mixed integer programming; some but not all, of the decision variables are required to have integer values –Zero-one integer programming (binary), all the decision variable must have integer solution values of 0 or 1

3. Methods of Solution  Cutting Plane Method  Branch and Bound  Modified Simplex Method  Computer Software –Microsoft Excel with Solver –QM/POM for Windows

4. Branch and Bound  Max: $7X 1 + $6X 2  S.t. : 2X 1 + 3X 2 <= 12 6X 1 + 5X 2 <= 30 X 1, X 2 are integers  Graphical LP solution gives: X1 = 3,75 X2 = 1,5 Profit = 35,25

5. Branch and Bound...  The LP solution is not valid (not integer)  Intial Upper Bound = $35,25  Rounding down: X1 = 3 X2 = 1 Profit = $27 (set as lower bound) Pick X1 (arbitrarily so far it is not integer)

6. Branch and Bound... Sub Problem A Max 7X1 + 6X2 S.T. 2X1 + 3X2 <= 12 6X1 + 5X2 <= 30 X1 >= 4 Sol X1 = 4, X2 = 1,2 Profit = 35,20 Sub Problem B Max 7X1 + 6X2 S.T. 2X1 + 3X2 <= 12 6X1 + 5X2 <= 30 X1 <= 3 Sol X1 = 3, X2 = 2 Profit = 33, OK!

7. Branch and Bound... Sub Problem C Max 7X1 + 6X2 S.T. 2X1 + 3X2 <= 12 6X1 + 5X2 <= 30 X1 >= 4 X2 >= 2 Sol No feasible solution Sub Problem D Max 7X1 + 6X2 S.T. 2X1 + 3X2 <= 12 6X1 + 5X2 <= 30 X1 <= 3 X2 <= 1 Sol X1 = 4 1/6, X2 = 1 Profit = 35,16 new ub

8. Branch and Bound... Sub Problem E Max 7X1 + 6X2 S.T. 2X1 + 3X2 <= 12 6X1 + 5X2 <= 30 X1 >= 4 X1 <= 4 X2 <= 1 Sol X1 = 4, X2 = 1 Profit = $34 Sub Problem F Max 7X1 + 6X2 S.T. 2X1 + 3X2 <= 12 6X1 + 5X2 <= 30 X1 >= 4 X1 >= 5 X2 <= 1 Sol X1 = 5, X2 = 0 Profit = 35

9. Branch and Bound...  Stopping rule: We continue until the new upper bound is less than or equal to the lower bound Or No further branching is possible