1. Integer programming Branch & bound algorithm ( B&B )

2. Integer programming problems All integer programming problemsMixed integer programming problems

3. Relax the integer restriction ( try to solve it as a linear programming) Max Z = 5x 1 + 4x 2 s.t. x 1 + x 2  5 10x 1 + 6x 2  45 x 1, x 2 ≥ 0 integer

4. Max Z = 5x 1 + 4x 2 s.t. x 1 + x 2  5 10x 1 + 6x 2  45 x 1, x 2 ≥ 0 integer x1x1 x2x2 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 x 1 =3.75 x 2 = 1.25 z = 23.75 Which none of them is integer !

5. Dividing the problem in every node into 2 subproblems to district the feasible region for getting to the optimal solution which is integer. x1x1 x2x2 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 x 1 ≥ 4 x 1  3

6. x 1 =3.75 x 2 = 1.25 z = 23.75 x 1  3 x 1 ≥ 4 x 1 =3 x 2 = 2 z = 23 Fathomed : dont need to be investigated any further LOWER BOUND

7. x 1 =3.75 x 2 = 1.25 z = 23.75 x 1  3 x 1 ≥ 4 x 1 =3 x 2 = 2 z = 23 Fathomed x 1 =4 x 2 = 0.83 z = 23.33 x 2  0 x 2 ≥ 1 x 1 =4.5 x 2 = 0 z = 22.5 No solution Fathomed x 1 =4 x 2 = 0 z = 20 No solution x 1  4 x 1 ≥ 5 LOWER BOUND

8. 1) fathoming (bounding) Z cannot yield a better objective value than the lower bound Z yields a better objective value than the lower bound Subproblem has no feasible solution Update the lower bound

9. 2)Branching Fractional part [x] < x < [x] +1 [x] ≤ xx ≤ [x] +1

10. In Minimization problems, we just replace lower bound with upper bound