Integer Programming Part 2 Assoc. Prof. Dr. Arslan M. RNEK 1 9.3. - PowerPoint PPT Presentation

Chapter 9 Integer Programming Part 2 Assoc. Prof. Dr. Arslan M. ÖRNEK 1

9.3. Branch-and-Bound Method (Pure IP) • Branch-and-Bound methods find the optimal solution to an IP by efficiently enumerating the points in a subproblem’s feasible region. • IMPORTANT OBSERVATION: If you solve the LP relaxation of a pure IP and obtain a solution in which all variables are integers, then the optimal solution to the LP relaxation is also the optimal solution to the IP. 2

9.3. Branch-and-Bound Method (Pure IP) • The Telfa Corporation manufactures tables and chairs. • A table requires 1 hour of labor and 9 square board feet of wood, and a chair requires 1 hour of labor and 5 square board feet of wood. • Currently, 6 hours of labor and 45 square board feet of wood are available. • Each table contributes $8 to profit, and each chair contributes $5 to profit. • Formulate and solve an IP to maximize Telfa’s profit. 3

9.3. Branch-and-Bound Method (Pure IP) 4

9.3. Branch-and-Bound Method (Pure IP) Upper bound 5

9.3. Branch-and-Bound Method (Pure IP) • Chose any variable that is fractional at the moment. • Let’s chose x 1 : • At the optimal solution it can be either <=3 or >=4. • We partition the solution space by branching on this variable. • Two new subproblems: 6

9.3. Branch-and-Bound Method (Pure IP) • Chose a subproblem to solve (arbitrarily): • Subproblem 2 LP relaxation: 7

9.3. Branch-and-Bound Method (Pure IP) An arc of the branch and bound tree A node of the branch and bound tree Solution not integer, branch on x 2 8

9.3. Branch-and-Bound Method (Pure IP) Chose this node to Then, solve (LIFO): chose this Infeasible – node 9 Fathomed.

9.3. Branch-and-Bound Method (Pure IP) Subproblem 5: Upper bound 10

9.3. Branch-and- Bound Method (Pure IP) 11

9.3. Branch-and- Bound Method (Pure IP) Lower bound on the original IP: Incumbent Solution 12

9.3. Branch-and- Bound Method (Pure IP) NEW Lower bound on the original IP  New Incumbent Solution 13

9.3. Branch-and-Bound Method (Pure IP) • Only remaining subproblem: • Subproblem 3: Solve LP relaxation: This is smaller than the current LB , so fathom. 14

9.3. Branch-and- Bound Method (Pure IP) Incumbent Solution = Best LB = Optimal Solution 15

9.3. Branch-and-Bound Method (Pure IP) 16

9.4. Branch-and-Bound Method (Mixed IP) • In a mixed IP, some variables are required to be integers and others are allowed to be either integers or nonintegers. • Branch only on variables that are required to be integers. Optimal solution of the LP-relaxation: 17

Branch-and-Bound Method (Binary Prog.) 18

Prototype Example: Bounding 19

Prototype Example: Bounding 20

Prototype Example: Fathoming • A subproblem can be fathomed in three ways: 1. LP relaxation of the subproblem gives an integer solution , so we do not need to branch further, the subproblem is solved optimally. 21

Prototype Example: Fathoming • A subproblem can be fathomed in three ways: 2. A subproblem is fathomed if its bound ≤ incumbent soln . Ex. Since Z*=9, there is no need to consider any solution whose bound is below 9. 22

Prototype Example: Fathoming • A subproblem can be fathomed in three ways: 3. If a subproblem’s LP relaxation has no feasible solution , then the subproblem itself must have no feasible solutions, so it can be dismissed (fathomed). 23

Prototype Example continued – Iteration 2 24

Prototype Example continued – Iteration 2 Cannot fathom. Cannot fathom. 25

Prototype Example continued – Iteration 2 Resulting solution tree after iteration 2: 26

Prototype Example continued- Iteration 3 27

Prototype Example continued – Iteration 3 Resulting solution tree after iteration 3: 28

Prototype Example continued- Iteration 4 New incumbent! 29

Prototype Example continued – Iteration 3 Resulting solution tree after iteration 4 (final): 30

B&B for Minimization Problems 31