Column Generation. ORLAB - Operations Research Laboratory. Stefano Gualandi. June 14, Politecnico di Milano, Italy
|
|
- Roger Walters
- 5 years ago
- Views:
Transcription
1 ORLAB - Operations Research Laboratory Politecnico di Milano, Italy June 14, 2011
2 Cutting Stock Problem (from wikipedia) Imagine that you work in a paper mill and you have a number of rolls of paper of fixed width waiting to be cut, yet different customers want different numbers of rolls of various-sized widths. How are you going to cut the rolls so that you minimize the waste (amount of left-overs)? Master Roll Item A Item B Item C Item D
3 Example 1 Item A Master Roll Master Roll Item B Item A Item C Item B Item D Item C Item D...quante possibilità ci sono?...quante possibilità ci sono?
4 en.wikipedia.org/wiki/cutting_stock_problem Suppose you are given a paper machine that can produce an unlimited number of master rolls, each 5600 mm wide. The following 13 items must be cut: Width: Demand: Questions: 1. Formulate this problem as an ILP 2. How many variables and constraints has your formulation? How big is a basis of the LP relaxation? 3. Discuss drawbacks of the given formulation
5 en.wikipedia.org: (An) optimal solution
6 Integer Linear Programming formulation 1/1 Can you figure out an ILP formulation for this problem?
7 Integer Linear Programming formulation 1/1 Can you figure out an ILP formulation for this problem? Let K be the set of available master rolls that might be cut. Let I be the set of item to cut, and each item i has a demand d i and length l i. Finally, let W the length of the master roll.
8 Integer Linear Programming formulation 1/1 Can you figure out an ILP formulation for this problem? Let K be the set of available master rolls that might be cut. Let I be the set of item to cut, and each item i has a demand d i and length l i. Finally, let W the length of the master roll. min (1) s.t. k K y k x ik d i, i I, (2) k K l i x ik W y k, k K, (3) i I x ik Z +, i I, k K, (4) y k {0, 1}, k K. (5)
9 Integer Linear Programming formulation 2/2 Suppose the set of every possible cut pattern P is given. Can you figure out an alternative ILP formulation that uses P?
10 Integer Linear Programming formulation 2/2 Suppose the set of every possible cut pattern P is given. Can you figure out an alternative ILP formulation that uses P? Let P i be the subset of cut patterns containing item i. Let a ip denote the number of times item i appear in the cut pattern p The demand of an item i is still denoted by d i.
11 Integer Linear Programming formulation 2/2 Suppose the set of every possible cut pattern P is given. Can you figure out an alternative ILP formulation that uses P? Let P i be the subset of cut patterns containing item i. Let a ip denote the number of times item i appear in the cut pattern p The demand of an item i is still denoted by d i. min λ p (6) s.t. p P p P i a ip λ p d i, i I, (7) λ p Z +, k P. (8)
12 Integer Linear Programming formulation 2/2 Suppose the set of every possible cut pattern P is given. Can you figure out an alternative ILP formulation that uses P? Let P i be the subset of cut patterns containing item i. Let a ip denote the number of times item i appear in the cut pattern p The demand of an item i is still denoted by d i. min λ p (6) s.t. p P p P i a ip λ p d i, i I, (7) λ p Z +, k P. (8) How many variables has this formulation?
13 Integer Linear Programming formulation 2/2 Suppose the set of every possible cut pattern P is given. Can you figure out an alternative ILP formulation that uses P? Let P i be the subset of cut patterns containing item i. Let a ip denote the number of times item i appear in the cut pattern p The demand of an item i is still denoted by d i. min λ p (6) s.t. p P p P i a ip λ p d i, i I, (7) λ p Z +, k P. (8) How many variables has this formulation? Finitely many!
14 Exploiting Substructures of LPs [DantzigWolfe1960] The first ILP formulation is well structured, since the constraint matrix has the following form: A 1 A 2 A k B B B k x 1. x k. d b 1 b 2. b k
15 Block diagonal submatrices Note that we can define the bounded set X k = {x k Z n k + : B kx k b k }. In general, the ILP formulation of a problem with block diagonal substructures can be written as: min s.t. c k x k (9) k K A k x k d, (10) k K x k X k, k K. (11)
16 Discrete convexification Applying a convexification procedure, and considering that each X k has a large but finite set of points P k, it is possible to write: x k = z kp λ p, k K, (12) p P k λ p = 1, (13) k K p P k λ p 0, k K, p P k. (14)
17 Using the convexification The ILP formulation becomes: min (c k z kp )λ p (15) k K p P k s.t. (A k z kp )λ p d, (16) k K p P k x k = z kp λ p, k K, (17) p P k λ p = 1, (18) k K p P k λ p 0, k K, p P k, (19) x k {0, 1}, k K. (20)
18 Cutting Stock formulation Let us consider again the second cutting stock formulation: min s.t. λ p (21) p P a ip λ p d i, i I, (22) p P i λ p Z +, k P. (23) What is the connection with the previous formulation? What does the p-th column given by coefficient a ip represent?
19 Dealing with Finitely Many Columns The main idea is to start with a subset of columns P P such that a feasible solution to the following problem exists: z MP = min s.t. λ p (24) p P a ip λ p d i, i I, (25) p P i λ p {0, 1}, k P. (26) Using the Duality Theory of Linear Programming with can generate as set of improving columns...
20 A Dual Persepective Consider the LP relaxation of the master problem and its dual: z RMP = min p P λ p max i I d i π i s.t. p Pi a ip λ p d i, i I, s.t. i I a ip π i 1, k P, λ p 0, k P. π i 0, i I. Using the Duality Theory of Linear Programming with can generate as set of improving columns...
21 A Dual Persepective Consider the LP relaxation of the master problem and its dual: z RMP = min p P λ p max i I d i π i s.t. p Pi a ip λ p d i, i I, s.t. i I a ip π i 1, k P, λ p 0, k P. π i 0, i I. Using the Duality Theory of Linear Programming with can generate as set of improving columns... by separating inequalities on the dual of the master problem!
22 Pricin Subproblem (Cutting Planes on the Master Dual) The question is: Does a column (cut pattern) in P \ P that could improve the current optimal solution of the linear relaxation exist?
23 Pricin Subproblem (Cutting Planes on the Master Dual) The question is: Does a column (cut pattern) in P \ P that could improve the current optimal solution of the linear relaxation exist? Does a column (row of the dual) exist such that...? p P \ P : i I a ip π i > 1
24 Pricin Subproblem (Cutting Planes on the Master Dual) Given the vector of optimal dual multipliers π, we look for a column (cut pattern) such that: c = min s.t. 1 π i y i i I l i y i W, i I y i {0, 1}. If c < 0, the vector of variables z is the incidence vector of an improving columns. It corresponds to a variable with negative reduced cost in the (restricted) master problem. Do you recognize this problem?
25 Pricin Subproblem (Cutting Planes on the Master Dual) Given the vector of optimal dual multipliers π, we look for a column (cut pattern) such that: c = 1 max s.t. π i y i i I l i y i W, i I y i {0, 1}. If c > 1, the vector of variables z is the incidence vector of an improving columns. It corresponds to a variable with negative reduced cost in the (restricted) master problem. It is a Knapsack Problem! (Complexity?)
26 Lower Bounding Let zmp denote the optimal value of the Continuous Master Problem, z RMP the value of the current Restricted Master Problem, and let κ be an upper bound on the optimal value of Integer Master Problem. At each iteration of column generation, we have z MP z RMP.
27 Lower Bounding Let zmp denote the optimal value of the Continuous Master Problem, z RMP the value of the current Restricted Master Problem, and let κ be an upper bound on the optimal value of Integer Master Problem. At each iteration of column generation, we have z MP z RMP. If we solve the pricing subproblem to optimality, we get the following lower bound: z RMP + κ c z MP z RMP In addition, whenever the cost vector of the master problem is integral, we can strength the bound to: z RMP + κ c z MP z RMP
28
29
30
31
32
33 Exercises BLACKBOARD 1. Graph Coloring 2. Time Constrained Shortest Path Problem (+ Branch-and-Bound) 3. Symmetric Travel Salesman Problem (1-trees) 4. Industrial Application: Bus Driver Scheduling
34 Links and Original papers: Dantzig, G.B. and Wolfe, P. Decomposition Principle for Linear Programs, Operations research, vol. 8(1), pp , Gilmore, P.C. and Gomory, R.E. A Linear Programming Approach to the Cutting-Stock Problem, Operations research, vol. 9(6), pp , L. Wolsey. Integer Programming (Chap. 11, pp ), Wiley Editor. F. Vanderbeck and L.A. Wolsey. Reformulation and decomposition of integer programs. Available on-line. J. Desrosiers and M.E. Lübbecke. A Primer in, 2003, ZIB Tech. Report No. 2003/48. Additional information on cutting stock problems: en.wikipedia.org/wiki/cutting_stock_problem
Part 4. Decomposition Algorithms
In the name of God Part 4. 4.4. Column Generation for the Constrained Shortest Path Problem Spring 2010 Instructor: Dr. Masoud Yaghini Constrained Shortest Path Problem Constrained Shortest Path Problem
More informationLecture 8: Column Generation
Lecture 8: Column Generation (3 units) Outline Cutting stock problem Classical IP formulation Set covering formulation Column generation A dual perspective 1 / 24 Cutting stock problem 2 / 24 Problem description
More informationLecture 8: Column Generation
Lecture 8: Column Generation (3 units) Outline Cutting stock problem Classical IP formulation Set covering formulation Column generation A dual perspective Vehicle routing problem 1 / 33 Cutting stock
More informationLecture 9: Dantzig-Wolfe Decomposition
Lecture 9: Dantzig-Wolfe Decomposition (3 units) Outline Dantzig-Wolfe decomposition Column generation algorithm Relation to Lagrangian dual Branch-and-price method Generated assignment problem and multi-commodity
More informationColumn Generation. MTech Seminar Report. Soumitra Pal Roll No: under the guidance of
Column Generation MTech Seminar Report by Soumitra Pal Roll No: 05305015 under the guidance of Prof. A. G. Ranade Computer Science and Engineering IIT-Bombay a Department of Computer Science and Engineering
More information1 Column Generation and the Cutting Stock Problem
1 Column Generation and the Cutting Stock Problem In the linear programming approach to the traveling salesman problem we used the cutting plane approach. The cutting plane approach is appropriate when
More informationLarge-scale optimization and decomposition methods: outline. Column Generation and Cutting Plane methods: a unified view
Large-scale optimization and decomposition methods: outline I Solution approaches for large-scaled problems: I Delayed column generation I Cutting plane methods (delayed constraint generation) 7 I Problems
More informationInteger Programming ISE 418. Lecture 16. Dr. Ted Ralphs
Integer Programming ISE 418 Lecture 16 Dr. Ted Ralphs ISE 418 Lecture 16 1 Reading for This Lecture Wolsey, Chapters 10 and 11 Nemhauser and Wolsey Sections II.3.1, II.3.6, II.3.7, II.5.4 CCZ Chapter 8
More information3.10 Column generation method
3.10 Column generation method Many relevant decision-making (discrete optimization) problems can be formulated as ILP problems with a very large (exponential) number of variables. Examples: cutting stock,
More information3.10 Column generation method
3.10 Column generation method Many relevant decision-making problems can be formulated as ILP problems with a very large (exponential) number of variables. Examples: cutting stock, crew scheduling, vehicle
More informationAn Integer Cutting-Plane Procedure for the Dantzig-Wolfe Decomposition: Theory
An Integer Cutting-Plane Procedure for the Dantzig-Wolfe Decomposition: Theory by Troels Martin Range Discussion Papers on Business and Economics No. 10/2006 FURTHER INFORMATION Department of Business
More informationNotes on Dantzig-Wolfe decomposition and column generation
Notes on Dantzig-Wolfe decomposition and column generation Mette Gamst November 11, 2010 1 Introduction This note introduces an exact solution method for mathematical programming problems. The method is
More informationColumn Generation for Extended Formulations
1 / 28 Column Generation for Extended Formulations Ruslan Sadykov 1 François Vanderbeck 2,1 1 INRIA Bordeaux Sud-Ouest, France 2 University Bordeaux I, France ISMP 2012 Berlin, August 23 2 / 28 Contents
More informationPartial Path Column Generation for the Vehicle Routing Problem with Time Windows
Partial Path Column Generation for the Vehicle Routing Problem with Time Windows Bjørn Petersen & Mads Kehlet Jepsen } DIKU Department of Computer Science, University of Copenhagen Universitetsparken 1,
More informationExtended Formulations, Lagrangian Relaxation, & Column Generation: tackling large scale applications
Extended Formulations, Lagrangian Relaxation, & Column Generation: tackling large scale applications François Vanderbeck University of Bordeaux INRIA Bordeaux-Sud-Ouest part : Defining Extended Formulations
More informationInteger program reformulation for robust branch-and-cut-and-price
Integer program reformulation for robust branch-and-cut-and-price Marcus Poggi de Aragão Informática PUC-Rio Eduardo Uchoa Engenharia de Produção Universidade Federal Fluminense Outline of the talk Robust
More informationColumn Generation in Integer Programming with Applications in Multicriteria Optimization
Column Generation in Integer Programming with Applications in Multicriteria Optimization Matthias Ehrgott Department of Engineering Science The University of Auckland, New Zealand email: m.ehrgott@auckland.ac.nz
More informationClassification of Dantzig-Wolfe Reformulations for MIP s
Classification of Dantzig-Wolfe Reformulations for MIP s Raf Jans Rotterdam School of Management HEC Montreal Workshop on Column Generation Aussois, June 2008 Outline and Motivation Dantzig-Wolfe reformulation
More informationCut and Column Generation
F A C U L T Y O F S C I E N C E U N I V E R S I T Y O F C O P E N H A G E N PhD thesis Simon Spoorendonk Cut and Column Generation Academic advisor: David Pisinger Submitted: 31/10/08 Preface This Ph.D.
More informationColumn Generation. i = 1,, 255;
Column Generation The idea of the column generation can be motivated by the trim-loss problem: We receive an order to cut 50 pieces of.5-meter (pipe) segments, 250 pieces of 2-meter segments, and 200 pieces
More informationMulticommodity Flows and Column Generation
Lecture Notes Multicommodity Flows and Column Generation Marc Pfetsch Zuse Institute Berlin pfetsch@zib.de last change: 2/8/2006 Technische Universität Berlin Fakultät II, Institut für Mathematik WS 2006/07
More information1 Solution of a Large-Scale Traveling-Salesman Problem... 7 George B. Dantzig, Delbert R. Fulkerson, and Selmer M. Johnson
Part I The Early Years 1 Solution of a Large-Scale Traveling-Salesman Problem............ 7 George B. Dantzig, Delbert R. Fulkerson, and Selmer M. Johnson 2 The Hungarian Method for the Assignment Problem..............
More informationOutline. Relaxation. Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING. 1. Lagrangian Relaxation. Lecture 12 Single Machine Models, Column Generation
Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING 1. Lagrangian Relaxation Lecture 12 Single Machine Models, Column Generation 2. Dantzig-Wolfe Decomposition Dantzig-Wolfe Decomposition Delayed Column
More information5 Integer Linear Programming (ILP) E. Amaldi Foundations of Operations Research Politecnico di Milano 1
5 Integer Linear Programming (ILP) E. Amaldi Foundations of Operations Research Politecnico di Milano 1 Definition: An Integer Linear Programming problem is an optimization problem of the form (ILP) min
More informationA Node-Flow Model for 1D Stock Cutting: Robust Branch-Cut-and-Price
A Node-Flow Model for 1D Stock Cutting: Robust Branch-Cut-and-Price Gleb Belov University of Dresden Adam N. Letchford Lancaster University Eduardo Uchoa Universidade Federal Fluminense August 4, 2011
More informationDecomposition Methods for Integer Programming
Decomposition Methods for Integer Programming J.M. Valério de Carvalho vc@dps.uminho.pt Departamento de Produção e Sistemas Escola de Engenharia, Universidade do Minho Portugal PhD Course Programa Doutoral
More informationInteger Programming Reformulations: Dantzig-Wolfe & Benders Decomposition the Coluna Software Platform
Integer Programming Reformulations: Dantzig-Wolfe & Benders Decomposition the Coluna Software Platform François Vanderbeck B. Detienne, F. Clautiaux, R. Griset, T. Leite, G. Marques, V. Nesello, A. Pessoa,
More informationA Hub Location Problem with Fully Interconnected Backbone and Access Networks
A Hub Location Problem with Fully Interconnected Backbone and Access Networks Tommy Thomadsen Informatics and Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby Denmark tt@imm.dtu.dk
More informationInteger Program Reformulation for Robust. Branch-and-Cut-and-Price Algorithms
Integer Program Reformulation for Robust Branch-and-Cut-and-Price Algorithms Marcus Poggi de Aragão 1, Eduardo Uchoa 2 1 Departamento de Informática, PUC-Rio, poggi@inf.puc-rio.br 2 Dep. de Engenharia
More informationOperations Research Lecture 6: Integer Programming
Operations Research Lecture 6: Integer Programming Notes taken by Kaiquan Xu@Business School, Nanjing University May 12th 2016 1 Integer programming (IP) formulations The integer programming (IP) is the
More informationLagrangian Relaxation in MIP
Lagrangian Relaxation in MIP Bernard Gendron May 28, 2016 Master Class on Decomposition, CPAIOR2016, Banff, Canada CIRRELT and Département d informatique et de recherche opérationnelle, Université de Montréal,
More informationSection Notes 9. Midterm 2 Review. Applied Math / Engineering Sciences 121. Week of December 3, 2018
Section Notes 9 Midterm 2 Review Applied Math / Engineering Sciences 121 Week of December 3, 2018 The following list of topics is an overview of the material that was covered in the lectures and sections
More informationInteger Programming Part II
Be the first in your neighborhood to master this delightful little algorithm. Integer Programming Part II The Branch and Bound Algorithm learn about fathoming, bounding, branching, pruning, and much more!
More informationNetwork Flows. 6. Lagrangian Relaxation. Programming. Fall 2010 Instructor: Dr. Masoud Yaghini
In the name of God Network Flows 6. Lagrangian Relaxation 6.3 Lagrangian Relaxation and Integer Programming Fall 2010 Instructor: Dr. Masoud Yaghini Integer Programming Outline Branch-and-Bound Technique
More informationPedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 2
Pedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 2 Outline Vehicle routing problem; How interior point methods can help; Interior point branch-price-and-cut:
More informationIntroduction to optimization and operations research
Introduction to optimization and operations research David Pisinger, Fall 2002 1 Smoked ham (Chvatal 1.6, adapted from Greene et al. (1957)) A meat packing plant produces 480 hams, 400 pork bellies, and
More informationDecomposition and Reformulation in Integer Programming
and Reformulation in Integer Programming Laurence A. WOLSEY 7/1/2008 / Aussois and Reformulation in Integer Programming Outline 1 Resource 2 and Reformulation in Integer Programming Outline Resource 1
More informationModeling with Integer Programming
Modeling with Integer Programg Laura Galli December 18, 2014 We can use 0-1 (binary) variables for a variety of purposes, such as: Modeling yes/no decisions Enforcing disjunctions Enforcing logical conditions
More informationImproving Branch-And-Price Algorithms For Solving One Dimensional Cutting Stock Problem
Improving Branch-And-Price Algorithms For Solving One Dimensional Cutting Stock Problem M. Tech. Dissertation Submitted in partial fulfillment of the requirements for the degree of Master of Technology
More informationOptimization Exercise Set n.5 :
Optimization Exercise Set n.5 : Prepared by S. Coniglio translated by O. Jabali 2016/2017 1 5.1 Airport location In air transportation, usually there is not a direct connection between every pair of airports.
More informationA generic view of Dantzig Wolfe decomposition in mixed integer programming
Operations Research Letters 34 (2006) 296 306 Operations Research Letters www.elsevier.com/locate/orl A generic view of Dantzig Wolfe decomposition in mixed integer programming François Vanderbeck a,,
More informationInteger Linear Programming (ILP)
Integer Linear Programming (ILP) Zdeněk Hanzálek, Přemysl Šůcha hanzalek@fel.cvut.cz CTU in Prague March 8, 2017 Z. Hanzálek (CTU) Integer Linear Programming (ILP) March 8, 2017 1 / 43 Table of contents
More informationInteger Solutions to Cutting Stock Problems
Integer Solutions to Cutting Stock Problems L. Fernández, L. A. Fernández, C. Pola Dpto. Matemáticas, Estadística y Computación, Universidad de Cantabria, 39005 Santander, Spain, laura.fernandezfern@alumnos.unican.es,
More information18 hours nodes, first feasible 3.7% gap Time: 92 days!! LP relaxation at root node: Branch and bound
The MIP Landscape 1 Example 1: LP still can be HARD SGM: Schedule Generation Model Example 157323 1: LP rows, still can 182812 be HARD columns, 6348437 nzs LP relaxation at root node: 18 hours Branch and
More informationStabilization in Column Generation: numerical study
1 / 26 Stabilization in Column Generation: numerical study Artur Pessoa 3 Ruslan Sadykov 1,2 Eduardo Uchoa 3 François Vanderbeck 2,1 1 INRIA Bordeaux, France 2 Univ. Bordeaux I, France 3 Universidade Federal
More informationBenders Decomposition
Benders Decomposition Yuping Huang, Dr. Qipeng Phil Zheng Department of Industrial and Management Systems Engineering West Virginia University IENG 593G Nonlinear Programg, Spring 2012 Yuping Huang (IMSE@WVU)
More informationOn Compact Formulations for Integer Programs Solved by Column Generation
Annals of Operations Research 139, 375 388, 2005 c 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands. On Compact Formulations for Integer Programs Solved by Column Generation
More informationInterior-Point versus Simplex methods for Integer Programming Branch-and-Bound
Interior-Point versus Simplex methods for Integer Programming Branch-and-Bound Samir Elhedhli elhedhli@uwaterloo.ca Department of Management Sciences, University of Waterloo, Canada Page of 4 McMaster
More informationMVE165/MMG630, Applied Optimization Lecture 6 Integer linear programming: models and applications; complexity. Ann-Brith Strömberg
MVE165/MMG630, Integer linear programming: models and applications; complexity Ann-Brith Strömberg 2011 04 01 Modelling with integer variables (Ch. 13.1) Variables Linear programming (LP) uses continuous
More informationKNAPSACK PROBLEMS WITH SETUPS
7 e Conférence Francophone de MOdélisation et SIMulation - MOSIM 08 - du 31 mars au 2 avril 2008 - Paris - France Modélisation, Optimisation et Simulation des Systèmes : Communication, Coopération et Coordination
More informationDecomposition-based Methods for Large-scale Discrete Optimization p.1
Decomposition-based Methods for Large-scale Discrete Optimization Matthew V Galati Ted K Ralphs Department of Industrial and Systems Engineering Lehigh University, Bethlehem, PA, USA Départment de Mathématiques
More informationWeek Cuts, Branch & Bound, and Lagrangean Relaxation
Week 11 1 Integer Linear Programming This week we will discuss solution methods for solving integer linear programming problems. I will skip the part on complexity theory, Section 11.8, although this is
More informationIntroduction to Bin Packing Problems
Introduction to Bin Packing Problems Fabio Furini March 13, 2015 Outline Origins and applications Applications: Definition: Bin Packing Problem (BPP) Solution techniques for the BPP Heuristic Algorithms
More informationInteger Linear Programming Modeling
DM554/DM545 Linear and Lecture 9 Integer Linear Programming Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. 2. Assignment Problem Knapsack Problem
More informationComputational Integer Programming. Lecture 2: Modeling and Formulation. Dr. Ted Ralphs
Computational Integer Programming Lecture 2: Modeling and Formulation Dr. Ted Ralphs Computational MILP Lecture 2 1 Reading for This Lecture N&W Sections I.1.1-I.1.6 Wolsey Chapter 1 CCZ Chapter 2 Computational
More informationWhat is an integer program? Modelling with Integer Variables. Mixed Integer Program. Let us start with a linear program: max cx s.t.
Modelling with Integer Variables jesla@mandtudk Department of Management Engineering Technical University of Denmark What is an integer program? Let us start with a linear program: st Ax b x 0 where A
More informationDual Inequalities for Stabilized Column Generation Revisited
Gutenberg School of Management and Economics & Research Unit Interdisciplinary Public Policy Discussion Paper Series Dual Inequalities for Stabilized Column Generation Revisited Timo Gschwind, Stefan Irnich
More informationDiscrete Optimization 2010 Lecture 7 Introduction to Integer Programming
Discrete Optimization 2010 Lecture 7 Introduction to Integer Programming Marc Uetz University of Twente m.uetz@utwente.nl Lecture 8: sheet 1 / 32 Marc Uetz Discrete Optimization Outline 1 Intro: The Matching
More informationThe Fixed Charge Transportation Problem: A Strong Formulation Based On Lagrangian Decomposition and Column Generation
The Fixed Charge Transportation Problem: A Strong Formulation Based On Lagrangian Decomposition and Column Generation Yixin Zhao, Torbjörn Larsson and Department of Mathematics, Linköping University, Sweden
More informationTechnische Universität München, Zentrum Mathematik Lehrstuhl für Angewandte Geometrie und Diskrete Mathematik. Combinatorial Optimization (MA 4502)
Technische Universität München, Zentrum Mathematik Lehrstuhl für Angewandte Geometrie und Diskrete Mathematik Combinatorial Optimization (MA 4502) Dr. Michael Ritter Problem Sheet 1 Homework Problems Exercise
More informationSection Notes 9. IP: Cutting Planes. Applied Math 121. Week of April 12, 2010
Section Notes 9 IP: Cutting Planes Applied Math 121 Week of April 12, 2010 Goals for the week understand what a strong formulations is. be familiar with the cutting planes algorithm and the types of cuts
More informationDecomposition Techniques in Mathematical Programming
Antonio J. Conejo Enrique Castillo Roberto Minguez Raquel Garcia-Bertrand Decomposition Techniques in Mathematical Programming Engineering and Science Applications Springer Contents Part I Motivation and
More informationIntroduction to Mathematical Programming IE406. Lecture 21. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 21 Dr. Ted Ralphs IE406 Lecture 21 1 Reading for This Lecture Bertsimas Sections 10.2, 10.3, 11.1, 11.2 IE406 Lecture 21 2 Branch and Bound Branch
More informationThe Shift Scheduling Problem using a branch-and-price approach
The Shift Scheduling Problem using a branch-and-price approach Mathematics bachelor thesis Svitlana A. Titiyevska Vrie Universiteit Amsterdam Faculteit der Exacte Wetenschappen De Boelelaan 1081a 1081
More informationSeparation, Inverse Optimization, and Decomposition. Some Observations. Ted Ralphs 1 Joint work with: Aykut Bulut 1
: Some Observations Ted Ralphs 1 Joint work with: Aykut Bulut 1 1 COR@L Lab, Department of Industrial and Systems Engineering, Lehigh University MOA 2016, Beijing, China, 27 June 2016 What Is This Talk
More informationOutline. Outline. Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING. 1. Scheduling CPM/PERT Resource Constrained Project Scheduling Model
Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING Lecture 3 and Mixed Integer Programg Marco Chiarandini 1. Resource Constrained Project Model 2. Mathematical Programg 2 Outline Outline 1. Resource Constrained
More informationResource Constrained Project Scheduling Linear and Integer Programming (1)
DM204, 2010 SCHEDULING, TIMETABLING AND ROUTING Lecture 3 Resource Constrained Project Linear and Integer Programming (1) Marco Chiarandini Department of Mathematics & Computer Science University of Southern
More informationTravelling Salesman Problem
Travelling Salesman Problem Fabio Furini November 10th, 2014 Travelling Salesman Problem 1 Outline 1 Traveling Salesman Problem Separation Travelling Salesman Problem 2 (Asymmetric) Traveling Salesman
More information3.10 Lagrangian relaxation
3.10 Lagrangian relaxation Consider a generic ILP problem min {c t x : Ax b, Dx d, x Z n } with integer coefficients. Suppose Dx d are the complicating constraints. Often the linear relaxation and the
More informationInteger Programming ISE 418. Lecture 8. Dr. Ted Ralphs
Integer Programming ISE 418 Lecture 8 Dr. Ted Ralphs ISE 418 Lecture 8 1 Reading for This Lecture Wolsey Chapter 2 Nemhauser and Wolsey Sections II.3.1, II.3.6, II.4.1, II.4.2, II.5.4 Duality for Mixed-Integer
More informationDiscrete (and Continuous) Optimization Solutions of Exercises 2 WI4 131
Discrete (and Continuous) Optimization Solutions of Exercises 2 WI4 131 Kees Roos Technische Universiteit Delft Faculteit Electrotechniek, Wiskunde en Informatica Afdeling Informatie, Systemen en Algoritmiek
More informationIntroduction to integer programming II
Introduction to integer programming II Martin Branda Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Computational Aspects of Optimization
More informationReformulation and Decomposition of Integer Programs
Reformulation and Decomposition of Integer Programs François Vanderbeck 1 and Laurence A. Wolsey 2 (Reference: CORE DP 2009/16) (1) Université Bordeaux 1 & INRIA-Bordeaux (2) Université de Louvain, CORE.
More informationIntroduction to Integer Programming
Lecture 3/3/2006 p. /27 Introduction to Integer Programming Leo Liberti LIX, École Polytechnique liberti@lix.polytechnique.fr Lecture 3/3/2006 p. 2/27 Contents IP formulations and examples Total unimodularity
More informationApplications. Stephen J. Stoyan, Maged M. Dessouky*, and Xiaoqing Wang
Introduction to Large-Scale Linear Programming and Applications Stephen J. Stoyan, Maged M. Dessouky*, and Xiaoqing Wang Daniel J. Epstein Department of Industrial and Systems Engineering, University of
More informationA column generation approach to the discrete lot sizing and scheduling problem on parallel machines
A column generation approach to the discrete lot sizing and scheduling problem on parallel machines António J.S.T. Duarte and J.M.V. Valério de Carvalho Abstract In this work, we study the discrete lot
More informationLogic-based Benders Decomposition
Logic-based Benders Decomposition A short Introduction Martin Riedler AC Retreat Contents 1 Introduction 2 Motivation 3 Further Notes MR Logic-based Benders Decomposition June 29 July 1 2 / 15 Basic idea
More informationColumn generation for extended formulations
EURO J Comput Optim (2013) 1:81 115 DOI 10.1007/s13675-013-0009-9 ORIGINAL PAPER Column generation for extended formulations Ruslan Sadykov François Vanderbeck Received: 5 May 2012 / Accepted: 21 December
More informationInteger Programming: Cutting Planes
OptIntro 1 / 39 Integer Programming: Cutting Planes Eduardo Camponogara Department of Automation and Systems Engineering Federal University of Santa Catarina October 2016 OptIntro 2 / 39 Summary Introduction
More informationA BRANCH&BOUND ALGORITHM FOR SOLVING ONE-DIMENSIONAL CUTTING STOCK PROBLEMS EXACTLY
APPLICATIONES MATHEMATICAE 23,2 (1995), pp. 151 167 G. SCHEITHAUER and J. TERNO (Dresden) A BRANCH&BOUND ALGORITHM FOR SOLVING ONE-DIMENSIONAL CUTTING STOCK PROBLEMS EXACTLY Abstract. Many numerical computations
More informationIntroduction to Integer Linear Programming
Lecture 7/12/2006 p. 1/30 Introduction to Integer Linear Programming Leo Liberti, Ruslan Sadykov LIX, École Polytechnique liberti@lix.polytechnique.fr sadykov@lix.polytechnique.fr Lecture 7/12/2006 p.
More informationBranch and Price for the Vehicle Routing Problem with Discrete Split Deliveries and Time Windows
Branch and Price for the Vehicle Routing Problem with Discrete Split Deliveries and Time Windows Matteo Salani Ilaria Vacca December 24, 2009 Report TRANSP-OR 091224 Transport and Mobility Laboratory Ecole
More informationOptimization Exercise Set n. 4 :
Optimization Exercise Set n. 4 : Prepared by S. Coniglio and E. Amaldi translated by O. Jabali 2018/2019 1 4.1 Airport location In air transportation, usually there is not a direct connection between every
More informationDiscrete (and Continuous) Optimization WI4 131
Discrete (and Continuous) Optimization WI4 131 Kees Roos Technische Universiteit Delft Faculteit Electrotechniek, Wiskunde en Informatica Afdeling Informatie, Systemen en Algoritmiek e-mail: C.Roos@ewi.tudelft.nl
More informationFeasibility Pump Heuristics for Column Generation Approaches
1 / 29 Feasibility Pump Heuristics for Column Generation Approaches Ruslan Sadykov 2 Pierre Pesneau 1,2 Francois Vanderbeck 1,2 1 University Bordeaux I 2 INRIA Bordeaux Sud-Ouest SEA 2012 Bordeaux, France,
More informationDiscrete lot sizing and scheduling on parallel machines: description of a column generation approach
126 IO 2013 XVI Congresso da Associação Portuguesa de Investigação Operacional Discrete lot sizing and scheduling on parallel machines: description of a column generation approach António J.S.T. Duarte,
More informationSeparation, Inverse Optimization, and Decomposition. Some Observations. Ted Ralphs 1 Joint work with: Aykut Bulut 1
: Some Observations Ted Ralphs 1 Joint work with: Aykut Bulut 1 1 COR@L Lab, Department of Industrial and Systems Engineering, Lehigh University COLGEN 2016, Buzios, Brazil, 25 May 2016 What Is This Talk
More informationRecoverable Robustness in Scheduling Problems
Master Thesis Computing Science Recoverable Robustness in Scheduling Problems Author: J.M.J. Stoef (3470997) J.M.J.Stoef@uu.nl Supervisors: dr. J.A. Hoogeveen J.A.Hoogeveen@uu.nl dr. ir. J.M. van den Akker
More informationBranch-and-Price-and-Cut for the Split Delivery Vehicle Routing Problem with Time Windows
Branch-and-Price-and-Cut for the Split Delivery Vehicle Routing Problem with Time Windows Guy Desaulniers École Polytechnique de Montréal and GERAD Column Generation 2008 Aussois, France Outline Introduction
More informationBenders Decomposition Methods for Structured Optimization, including Stochastic Optimization
Benders Decomposition Methods for Structured Optimization, including Stochastic Optimization Robert M. Freund April 29, 2004 c 2004 Massachusetts Institute of echnology. 1 1 Block Ladder Structure We consider
More informationRecoverable Robust Knapsacks: Γ -Scenarios
Recoverable Robust Knapsacks: Γ -Scenarios Christina Büsing, Arie M. C. A. Koster, and Manuel Kutschka Abstract In this paper, we investigate the recoverable robust knapsack problem, where the uncertainty
More informationTechnische Universität Dresden Herausgeber: Der Rektor
Als Manuskript gedruckt Technische Universität Dresden Herausgeber: Der Rektor Models with Variable Strip Widths for Two-Dimensional Two-Stage Cutting G. Belov, G. Scheithauer MATH-NM-17-2003 October 8,
More informationChapter 3: Discrete Optimization Integer Programming
Chapter 3: Discrete Optimization Integer Programming Edoardo Amaldi DEIB Politecnico di Milano edoardo.amaldi@polimi.it Website: http://home.deib.polimi.it/amaldi/opt-16-17.shtml Academic year 2016-17
More informationMaximum Flow Problem (Ford and Fulkerson, 1956)
Maximum Flow Problem (Ford and Fulkerson, 196) In this problem we find the maximum flow possible in a directed connected network with arc capacities. There is unlimited quantity available in the given
More information56:270 Final Exam - May
@ @ 56:270 Linear Programming @ @ Final Exam - May 4, 1989 @ @ @ @ @ @ @ @ @ @ @ @ @ @ Select any 7 of the 9 problems below: (1.) ANALYSIS OF MPSX OUTPUT: Please refer to the attached materials on the
More informationmaxz = 3x 1 +4x 2 2x 1 +x 2 6 2x 1 +3x 2 9 x 1,x 2
ex-5.-5. Foundations of Operations Research Prof. E. Amaldi 5. Branch-and-Bound Given the integer linear program maxz = x +x x +x 6 x +x 9 x,x integer solve it via the Branch-and-Bound method (solving
More informationWeighted Acyclic Di-Graph Partitioning by Balanced Disjoint Paths
Weighted Acyclic Di-Graph Partitioning by Balanced Disjoint Paths H. Murat AFSAR Olivier BRIANT Murat.Afsar@g-scop.inpg.fr Olivier.Briant@g-scop.inpg.fr G-SCOP Laboratory Grenoble Institute of Technology
More informationLecture 23 Branch-and-Bound Algorithm. November 3, 2009
Branch-and-Bound Algorithm November 3, 2009 Outline Lecture 23 Modeling aspect: Either-Or requirement Special ILPs: Totally unimodular matrices Branch-and-Bound Algorithm Underlying idea Terminology Formal
More informationIP Cut Homework from J and B Chapter 9: 14, 15, 16, 23, 24, You wish to solve the IP below with a cutting plane technique.
IP Cut Homework from J and B Chapter 9: 14, 15, 16, 23, 24, 31 14. You wish to solve the IP below with a cutting plane technique. Maximize 4x 1 + 2x 2 + x 3 subject to 14x 1 + 10x 2 + 11x 3 32 10x 1 +
More informationBenders Decomposition Methods for Structured Optimization, including Stochastic Optimization
Benders Decomposition Methods for Structured Optimization, including Stochastic Optimization Robert M. Freund May 2, 2001 Block Ladder Structure Basic Model minimize x;y c T x + f T y s:t: Ax = b Bx +
More information