Integer programming

Branch & bound

algorithm

( B&B )

In the name of god

Arsalan

Motamedi

Integer programming problems

All integer programming problems

Mixed integer programming problems

Step

1:

rela

xatio

nRelax the integer

restriction ( try to solve it as a linear programming)

Max Z = 5x1 + 4x2

s.t. x1 + x2 5

10x1 + 6x2 45

x1 , x2 ≥ 0 integer

Max Z = 5x1 + 4x2

s.t. x1 + x2 5

10x1 + 6x2 45

x1 , x2 ≥ 0 integer

x1

x2

1

1 2

2

3

3

4

4

5

5

6

6

7

7

8

8

x1 =3.75 x2 = 1.25

z = 23.75

Which none of them is integer !

Step

2:

Bran

chin

gDividing the problem in every node into 2

subproblems to district the feasible region for getting to the optimal solution

which is integer.

x1

x2

1

1 2

2

3

3

4

4

5

5

6

6

7

7

8

8

x1 ≥ 4x1 3

Step

2:

Bran

chin

g

x1 =3.75 x2 = 1.25

z = 23.75

x1 3 x1 ≥ 4

x1 =3 x2 = 2

z = 23

Satisfies the integer requirements for x1

Fathomed: dont need to be investigated any further

LOWER BOUND

x1 =3.75 x2 = 1.25

z = 23.75

x1 3 x1 ≥ 4

x1 =3 x2 = 2

z = 23

Fathomed

x1 =4 x2 = 0.83

z = 23.33

x2 0 x2 ≥ 1

x1 =4.5 x2 = 0

z = 22.5

No solution

Fathomed

x1 =4 x2 = 0

z = 20

No solution

x1 4 x1 ≥ 5

LOWER BOUND

LOWER BOUND

Optimal solution

1) fathoming (bounding)

Summary of B&B STEP BY STEP

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

2)Branching

Fractional part[x] < x < [x] +1

[x] ≤ x x ≤ [x] +1

Explanation , witch said in last slides , was for maximization problems…

How about minimization problems?!

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