1. Integer Programming, Branch & Bound Method
Lecturer:
Dr. Mohammad T. Isaai
Quantitative Analysis for Management 1

2. Three Types of Integer Programming Problems
Pure integer programming problems
All variables integer.
Mixed-integer programming problems
Some variables integer.
Zero-one integer programming problems
All variables either 0 or 1.
Quantitative Analysis for Management 2

3. Integer Programming Techniques
Gomory’s Cutting Plane Method
Branch and Bound Method
Quantitative Analysis for Management 3

4. + possible Integer Solution
Feasible Region! X2 X1 6 5 4 3 2 1 +
+ possible Integer Solution
6X1 + 5X2< 30
2X1 + 3X2  12
Optimal LP Solution
(X1 = 3 3/4, X2 = 1 1/2
Profit = $35.25)
Quantitative Analysis for Management 4

5. + = possible Integer Solution
Optimal Solution! X1 X2 6 5 4 3 2 1 +
+ = possible Integer Solution
6X1 + 5X2  30
2X1 + 3X2  12 X :
Cut
Optimal LP solution
Quantitative Analysis for Management 5

6. Solving integer linear Branch & Bound Algorithm
programming using
Branch & Bound Algorithm
Quantitative Analysis for Management 6

7. Steps in Branch & Bound
1. Solve problem using LP. If solution is integer - finished. If not - upper bound.
2. Find any feasible integer solution to get lower bound.
3. Branch on noninteger variable from step 1. Split problem into two pieces: integer above, and integer below.
4.Create nodes at top of these branches by solving the new problems.
Quantitative Analysis for Management 7

8. Steps in Branch & Bound - Continued
5. a) Branch solution not feasible, terminate branch.
b) Branch solution feasible, not integer, go to step 6.
c) Branch solution feasible, integer, check. If equal to upper bound - solution. If less than upper bound, but greater than lower bound - new lower bound and proceed. If less than lower bound - terminate branch.
Quantitative Analysis for Management 8

9. Branch & Bound - Continued
6. Check branches. New upper bound is maximum of objective at all final nodes. If upper bound equals lower bound, stop; if not, go to step 3.
Quantitative Analysis for Management 9

10. Harrison Electric Company First Branching30 5 6 12 3 2 7 1 + X
Subject to: :
Max
Original Problem 4 30 5 6 12 3 2 7 1
Subject to: :
Max + X
Subproblem A 3 1 30 5 6 12 2 7
Subject to: :
Max + X
Subproblem B
Quantitative Analysis for Management 10

11. Harrison Electric Company Branch & Bound -1
X1=3.75
X2=1.5
P=35.25
X1=4
X2=1.2
P=35.20
LB:
X1=3
X2=1
P=27.00
X2=2
P=33.00 A B
UB=35.20
LB=33
BRANCH X2
INT., FEAS., STOP
NEW LB= 33
Original LP Solution
Quantitative Analysis for Management 11

12. Harrison Electric Company Second Branching4 30 5 6 12 2 7 1
Subject to: :
Max + X
Subproblem A
Subproblem C
Subproblem D
Quantitative Analysis for Management 12

13. Harrison Electric Company Branch & Bound - 2
X1=3.75
X2=1.5
P=35.25
X1=4
X2=1.2
P=35.20
X1=4.16
X2=1
P=35.16
X1=3
X2=2
P=33.00 A B D
No Feasible Solution
UB=35.16
LB=33
BRANCH
Original LP Solution
Quantitative Analysis for Management 13

14. Harrison Electric Company Third Branching2 1 6 7 + 30 5 12 X 4
Subject to: :
Max
Subproblem D 6 7 2 1 + X 4 30 5 12
Subject to: :
Max
Subproblem E 6 7 2 1 + X 5 4 30 12
Subject to: :
Max
Subproblem F
Quantitative Analysis for Management 14

15. Harrison Electric Company Branch & Bound - Overall
X1=3.75
X2=1.5
P=35.25
X1=4
X2=1.2
P=35.20
X1=4.16
X2=1
P=35.16
X1=5
X2=0
P=35.00
P=34.00
X1=3
X2=2
P=33.00 A B D C E F
No Feasible Solution
Original LP Solution
FEAS., INT.
SOLUTION
OPTIMUM
Quantitative Analysis for Management 15

16. Linear Programming Extensions
Integer Programming
Linear, integer solutions
Goal Programming
Linear, multiple objectives
Nonlinear Programming
Nonlinear objective and/or constraints
Quantitative Analysis for Management 16