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

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

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

+ 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

+ = 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

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

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

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

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

Harrison Electric Company First Branching 30 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 ³ £ + X Subproblem A 3 1 30 5 6 12 2 7 Subject to: : Max £ + X Subproblem B Quantitative Analysis for Management 10

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

Harrison Electric Company Second Branching 4 30 5 6 12 2 7 1 Subject to: : Max ³ £ + X Subproblem A Subproblem C Subproblem D Quantitative Analysis for Management 12

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

Harrison Electric Company Third Branching 2 1 6 7 + 30 5 12 £ X 4 ³ Subject to: : Max Subproblem D 6 7 2 1 + X 4 30 5 12 £ ³ Subject to: : Max Subproblem E 6 7 2 1 + X 5 4 30 12 £ ³ Subject to: : Max Subproblem F Quantitative Analysis for Management 14

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

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