Cutting Plane Technique - Discrete Modeling and Optimization - Lecture Slides, Slides of Discrete Structures and Graph Theory

Download Cutting Plane Technique - Discrete Modeling and Optimization - Lecture Slides and more Slides Discrete Structures and Graph Theory in PDF only on Docsity! 1 Cutting Plane Technique for Solving Integer Programs Docsity.com 2 Motivating Example for Cutting Planes • Recall the bad-case example for the LP-rounding algorithm: Integer Program LP relaxation max x1 + 5x2 max x1 + 5x2 s.t. x1 +10x2 ≤ 20 s.t. x1 +10x2 ≤ 20 x1 ≤ 2 x1 ≤ 2 x1 , x2 ≥ 0 integer x1 , x2 ≥ 0 • Solution to LP-relaxation: (2, 1.8) • Rounded IP solution: (2, 1) with value 7 • IP optimal solution: (0, 2) with value 10 • Conclusion: Rounded solution too far from optimal solution x1 +10x2 = 20 x1 = 2 Z=11 Docsity.com 5 Example of making a formulation tighter: Bin Packing Problem Given: n items with sizes s[1], s[2], …, s[n] ; bins with size W (where W ≥ s[i] , any i=1,…,n). Goal: Pack the items into the bins using as few bins as possible. • Example: n=13 items with sizes 20, 20, 20, 20, 20, 81, 81, 81, 81, 82, 91, 49, 51 ; Bin size is W=100. Minimum number of bins needed is 8. Docsity.com 6 Example of making a formulation tighter: Bin Packing Problem Want an IP formulation for this problem. • Let M be an upper bound on the number of bins needed. (M=n is a safe upper bound; but should try for smaller values) • Define the following variables. For j=1,…,M, let For each i=1,…,n and j=1,…,M, let    = not if 0 used is jbin if 1 open[j]    = not if 0 jbin in packed is i item if 1 j]assign[i, Docsity.com 7 Example of making a formulation tighter: Bin Packing Problem Our objective is to minimize the number of used bins: Minimize sum{j in 1..M}open[j] We need the following functional constraints. Each item should be packed in exactly one bin: (C1) sum{j in 1..M}assign[i,j] = 1 , for each i=1,…,n Each bin can contain items of total size at most W: (C2) sum{i in 1..n}s[i]*assign[i,j] ≤ W , for each j=1,…,M Items can be packed only in open bins: (C3) assign[i,j] ≤ open[j] , for each i=1,…,n and j=1,…,M  Set constraints: All variables are binary. Docsity.com