RO: Exercices Mixed Integer Programming
|
|
- Agatha Holt
- 5 years ago
- Views:
Transcription
1 RO: Exercices Mixed Integer Programming N. Brauner Université Grenoble Alpes Exercice 1 : Knapsack A hiker wants to fill up his knapsack of capacity W = 6 maximizing the utility of the objects he takes. The objects he might take are: utility weight A a picture B a bottle 7 2 C another bottle 3 2 D a pullover E chocolate bars 5 1 F dried fruit 8 1 Question 1 Describe a feasible solution for the hiker. The general knapsack problem is defined by a set of objects N = {1, 2... n}, to each object is associated a utility u i and a weight w i, a hiker has a knapsack of capacity W. The objective is to determine which objects should be taken in order to maximize the utility while not exceeding the capacity of the knapsack. Exercice 2 : Bin packing When moving to Grenoble for her studies, a student wants to pack all her objects while minimizing the number of bins of capacity W = 6 she has to use. The objects she wants to take are: size A a book 2 B another book 2 C a pullover 3 D socks 1 size E shoes 2 F plates 5 G glasses 6 Question 1 Describe a feasible solution for this problem. The general bin packing problem is defined by objects N = {1, 2... n} of size {s 1, s 2... s n }, 1
2 to be packed in bins of capacity W. The objective is to use as few bins as possible. Exercice 3 : Set covering You want to choose participants for a project in order to have all the necessary skills while minimizing the costs. You are given the following information. Alice Babar Casimir Donald Elmer Cost (h or =C) Operations research Java Database Graph theory User interface Question 1 Describe a feasible solution for this problem. The general set covering problem is defined by matrix A = (a ij ) i=1..n,j=1..m with 0 or 1 coefficients, c j > 0, the cost of column j. A column j covers a line i if a ij = 1. The objective is to find a subset of columns of A of minimum cost such that each line of A is covered at least once. Question 4 Modify your MIP to model a partition problem. Exercice 4 : Minimum cost flow We want to transport material in a network. The numbers on the vertices correspond to supply (+X) or to demand ( X). The numbers on the arcs are the unit transportation costs. B A 1 D C 2 Question 1 Describe a feasible solution for the transport. The objective is to determine the minimum cost of transportation of the material in the network in order to satisfy the supply and the demand. We have the following data: 2
3 an oriented network G(V, A): V is the set of n vertices and A is the set of arcs, c ij : cost per unit of flow on arc (i, j) A, b i : the demand or the available stock at vertex i (integer). We suppose that the network is balanced: b i = 0. Exercice 5 : Shortest path The objective is to find the shortest path between two vertices, s and t of a given network. Question 1 Formulate this problem as a MIP. Exercice 6 : Coloring The objective is to find the chromatic number χ(g) of a graph G. Question 1 Formulate this problem as a MIP. Exercice 7 : Au charbon! (D. de Wolf) Coals are mixed in a blast furnace where a reaction takes place to produce coke. Eight possible types of coal are available. The coal enters the oven using 4 conveyors and exactly 4 types of coals are required in the mix. If a coal is in the mix, it must be at least 5% of the overall mix. It is also required that the mix contains at most 1.8% of Silicon (Si). Finally if coal C1 is in the mix then C3 must be there too. Similarly if coal C4 is used then C6 must also be used. We want to find the mix of minimum cost. The question is therefore to know the composition of one kilogram of mix. The following table shows the prices and the proportion of Si. Coal price proportion Si Coal % Coal ,5 % Coal % Coal % Coal price proportion Si Coal % Coal % Coal % Coal ,5 % Question 1 Model the problem as a mixed integer linear program. Exercice 8 : The newspaper The chief editor has to prepare the outline of his newspaper which has 10 pages. He has articles organized in several topics: international news, national news, local news, sports and culture. He estimates that each page dedicated to a given topic is likely to interest on average a certain number of readers. The chief editor has to choose the topics to be treated together with its number of pages so as to attract the maximum number of readers. If he chooses to include a certain topic, he must give it a minimal number of pages corresponding to the most interesting articles. The table below gives the ranges of pages together with the average number of readers interested in each topic. 3
4 Topic Min nb of pages max nb of pages potential readers (per page) International National Local news Sport Culture Question 1 Model this problem as a MIP. Exercice 9 : Power plants A power supplier has a park of power stations. For simplicity, we consider that the park includes only a nuclear plant (N), a hydroelectric plant (H) and a thermal plant (T). The production costs of one MW in each of the plants are denoted by c N, c H and c T. The daily production capacities are 1000 MW for the nuclear plant, 300 MW for the hydroelectric plant and 500 MW for the thermal plant. The company wants to plan the power production on a horizon of n days so as to meet the estimated daily demand of its clients, d t (in MW), at minimum cost. Question 1 Formulate this problem as a MIP. Explain the meaning of the decision variables together with their domains. We now take into account some additional constraints on the power plants. For a good turbine performance, if the hydroelectric plant works for one day, its production must be at least 50 MW. Question 2 Add this constraint to your model. The maximal capacity of the hydroelectric plant is 300 MW. A production higher than 200 MW is said to be at high regime. To limit the wear of the turbines, it is forbidden to: produce 2 consecutive days at high regime, have more than 6 days at high regime in the horizon. We can introduce binary variables z t that reflect the fact that the plant is producing at high regime. Question 3 How to modify your model so that z t is 1 if the production on day t is greater than 200 MW? Question 4 Model these high regime constraints using the variables z t. The nuclear plant actually operates with thresholds: it produces either 0 or MW. Launching the production has an important setup cost evaluated at C L euros. This cost is paid on day t if: there was no production on day t 1 there is a production on day t Question 5 Model the launching cost. Initially the power plant is off. You can introduce several binary variables that express whether the plant is working and if day t corresponds to a launching day. 4
5 Exercice 10 : Several Optimums and convex hull Question 1 optimum. max z = 4x 1 + s.t. 7x x x 1, 0 integers Find graphically the fractional optimum, its integer rounding and the integer Question 2 Describe the convex hull of the integer solutions of this program. Exercice 11 : The integer optimum min z = x 1 s.t. x 1 17 = 3 x 1 11x 3 = 4 x 1 6x 4 = 5 x 1,, x 3, x 4 0 integers Question 1 For this Integer linear program find the fractional optimum, its integer rounding and the integer optimum. Exercice 12 : Sac-à-dos We consider the knapsack problem with the capacity W = 22 and the following objects object a b c d e weight utility Question 1 Find the optimal solution by hand. Question 2 Model this problem as an integer linear program. Question 3 Describe how to find a solution of the linear relaxation of this program by hand (replace x i {0, 1} by 0 x i 1). Question 4 Explain why the solution found in the preceding question is an upper bound to the optimal solution. Question 5 Describe the branching tree for this problem by solving the linear relaxation at each vertex of the tree and by branching on the non-integer variable if it exists (putting 1 on the left and 0 on the right). In each vertex of the tree, indicate a lower and an upper bound. Question 6 With a depth first search, left child first, indicate the order in which the vertices are visited and which vertices are never visited. Question 7 Same question with a depth-first search, right child first. Question 8 Same question with a depth-first search, child with the better lower bound first. Question 9 Same question with a depth-first search, child with the better upper bound first. Question 10 Same question with a breadth-first search, child with the smallest gap between the lower and the upper bound first. 5
6 z = 14,47 x 1 = 3,71 = 2, z = 13,5 x 1 = 2,25 = 3 z = 14,4 x 1 = 4,2 = 2 x 1 3 x 1 2 x 1 5 x 1 4 Pas de sol réalisable z = 13,33 x 1 = 2 = 3,11 z = 14,29 x 1 = 5 = 1,43 z = 14 x 1 = 4 = z = 12 x 1 = 0 = 4 z = 13 x 1 = 2 = 3 z = 14,2 x 1 = 5,6 = 1 Pas de sol réalisable x 1 5 x 1 6 z = 13 x 1 = 5 = 1 z = 14,14 x 1 = 6 = 0, Pas de sol réalisable z = 14 x 1 = 7 = 0 6
7 OPL User Guide: modelling integer/boolean variables OPL User Guide: using.dat files with OPL and the Caseine editor 7
Introduction 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 information4. How to prove a problem is NPC
The reducibility relation T is transitive, i.e, A T B and B T C imply A T C Therefore, to prove that a problem A is NPC: (1) show that A NP (2) choose some known NPC problem B define a polynomial transformation
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 informationInteger programming: an introduction. Alessandro Astolfi
Integer programming: an introduction Alessandro Astolfi Outline Introduction Examples Methods for solving ILP Optimization on graphs LP problems with integer solutions Summary Introduction Integer programming
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 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 informationCOT 6936: Topics in Algorithms! Giri Narasimhan. ECS 254A / EC 2443; Phone: x3748
COT 6936: Topics in Algorithms! Giri Narasimhan ECS 254A / EC 2443; Phone: x3748 giri@cs.fiu.edu https://moodle.cis.fiu.edu/v2.1/course/view.php?id=612 Gaussian Elimination! Solving a system of simultaneous
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 informationChapter 3: Discrete Optimization Integer Programming
Chapter 3: Discrete Optimization Integer Programming Edoardo Amaldi DEIB Politecnico di Milano edoardo.amaldi@polimi.it Sito web: http://home.deib.polimi.it/amaldi/ott-13-14.shtml A.A. 2013-14 Edoardo
More information1. Introduction Recap
1. Introduction Recap 1. Tractable and intractable problems polynomial-boundness: O(n k ) 2. NP-complete problems informal definition 3. Examples of P vs. NP difference may appear only slightly 4. Optimization
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 information- Well-characterized problems, min-max relations, approximate certificates. - LP problems in the standard form, primal and dual linear programs
LP-Duality ( Approximation Algorithms by V. Vazirani, Chapter 12) - Well-characterized problems, min-max relations, approximate certificates - LP problems in the standard form, primal and dual linear programs
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 information15.081J/6.251J Introduction to Mathematical Programming. Lecture 24: Discrete Optimization
15.081J/6.251J Introduction to Mathematical Programming Lecture 24: Discrete Optimization 1 Outline Modeling with integer variables Slide 1 What is a good formulation? Theme: The Power of Formulations
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 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 informationCombinatorial optimization problems
Combinatorial optimization problems Heuristic Algorithms Giovanni Righini University of Milan Department of Computer Science (Crema) Optimization In general an optimization problem can be formulated as:
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 informationCS Algorithms and Complexity
CS 50 - Algorithms and Complexity Linear Programming, the Simplex Method, and Hard Problems Sean Anderson 2/15/18 Portland State University Table of contents 1. The Simplex Method 2. The Graph Problem
More information3.3 Easy ILP problems and totally unimodular matrices
3.3 Easy ILP problems and totally unimodular matrices Consider a generic ILP problem expressed in standard form where A Z m n with n m, and b Z m. min{c t x : Ax = b, x Z n +} (1) P(b) = {x R n : Ax =
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 informationLinear and Integer Programming - ideas
Linear and Integer Programming - ideas Paweł Zieliński Institute of Mathematics and Computer Science, Wrocław University of Technology, Poland http://www.im.pwr.wroc.pl/ pziel/ Toulouse, France 2012 Literature
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 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 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 informationOptimisation and Operations Research
Optimisation and Operations Research Lecture 11: Integer Programming Matthew Roughan http://www.maths.adelaide.edu.au/matthew.roughan/ Lecture_notes/OORII/ School of Mathematical
More informationMATH 409 LECTURES THE KNAPSACK PROBLEM
MATH 409 LECTURES 19-21 THE KNAPSACK PROBLEM REKHA THOMAS We now leave the world of discrete optimization problems that can be solved in polynomial time and look at the easiest case of an integer program,
More informationAlgorithms. NP -Complete Problems. Dong Kyue Kim Hanyang University
Algorithms NP -Complete Problems Dong Kyue Kim Hanyang University dqkim@hanyang.ac.kr The Class P Definition 13.2 Polynomially bounded An algorithm is said to be polynomially bounded if its worst-case
More informationOR: Exercices Linear Programming
OR: Exercices Linear Programming N. Brauner Université Grenoble Alpes Exercice 1 : Wine Production (G. Finke) An American distillery produces 3 kinds of genuine German wines: Heidelberg sweet, Heidelberg
More informationChapter 9: Relations Relations
Chapter 9: Relations 9.1 - Relations Definition 1 (Relation). Let A and B be sets. A binary relation from A to B is a subset R A B, i.e., R is a set of ordered pairs where the first element from each pair
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 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 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 informationECS122A Handout on NP-Completeness March 12, 2018
ECS122A Handout on NP-Completeness March 12, 2018 Contents: I. Introduction II. P and NP III. NP-complete IV. How to prove a problem is NP-complete V. How to solve a NP-complete problem: approximate algorithms
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 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 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 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 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 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 informationSpring 2018 IE 102. Operations Research and Mathematical Programming Part 2
Spring 2018 IE 102 Operations Research and Mathematical Programming Part 2 Graphical Solution of 2-variable LP Problems Consider an example max x 1 + 3 x 2 s.t. x 1 + x 2 6 (1) - x 1 + 2x 2 8 (2) x 1,
More informationModelling linear and linear integer optimization problems An introduction
Modelling linear and linear integer optimization problems An introduction Karen Aardal October 5, 2015 In optimization, developing and analyzing models are key activities. Designing a model is a skill
More information(tree searching technique) (Boolean formulas) satisfying assignment: (X 1, X 2 )
Algorithms Chapter 5: The Tree Searching Strategy - Examples 1 / 11 Chapter 5: The Tree Searching Strategy 1. Ex 5.1Determine the satisfiability of the following Boolean formulas by depth-first search
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 informationFundamentals of optimization problems
Fundamentals of optimization problems Dmitriy Serdyuk Ferienakademie in Sarntal 2012 FAU Erlangen-Nürnberg, TU München, Uni Stuttgart September 2012 Overview 1 Introduction Optimization problems PO and
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 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 informationLecture 8 Network Optimization Algorithms
Advanced Algorithms Floriano Zini Free University of Bozen-Bolzano Faculty of Computer Science Academic Year 2013-2014 Lecture 8 Network Optimization Algorithms 1 21/01/14 Introduction Network models have
More informationScheduling on Unrelated Parallel Machines. Approximation Algorithms, V. V. Vazirani Book Chapter 17
Scheduling on Unrelated Parallel Machines Approximation Algorithms, V. V. Vazirani Book Chapter 17 Nicolas Karakatsanis, 2008 Description of the problem Problem 17.1 (Scheduling on unrelated parallel machines)
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 informationCS 598RM: Algorithmic Game Theory, Spring Practice Exam Solutions
CS 598RM: Algorithmic Game Theory, Spring 2017 1. Answer the following. Practice Exam Solutions Agents 1 and 2 are bargaining over how to split a dollar. Each agent simultaneously demands share he would
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 information21. Set cover and TSP
CS/ECE/ISyE 524 Introduction to Optimization Spring 2017 18 21. Set cover and TSP ˆ Set covering ˆ Cutting problems and column generation ˆ Traveling salesman problem Laurent Lessard (www.laurentlessard.com)
More informationThe Separation Problem for Binary Decision Diagrams
The Separation Problem for Binary Decision Diagrams J. N. Hooker Joint work with André Ciré Carnegie Mellon University ISAIM 2014 Separation Problem in Optimization Given a relaxation of an optimization
More informationOn the Polyhedral Structure of a Multi Item Production Planning Model with Setup Times
CORE DISCUSSION PAPER 2000/52 On the Polyhedral Structure of a Multi Item Production Planning Model with Setup Times Andrew J. Miller 1, George L. Nemhauser 2, and Martin W.P. Savelsbergh 2 November 2000
More informationMaximum sum contiguous subsequence Longest common subsequence Matrix chain multiplication All pair shortest path Kna. Dynamic Programming
Dynamic Programming Arijit Bishnu arijit@isical.ac.in Indian Statistical Institute, India. August 31, 2015 Outline 1 Maximum sum contiguous subsequence 2 Longest common subsequence 3 Matrix chain multiplication
More informationGraph Theory and Optimization Computational Complexity (in brief)
Graph Theory and Optimization Computational Complexity (in brief) Nicolas Nisse Inria, France Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271, Sophia Antipolis, France September 2015 N. Nisse Graph Theory
More informationIn the original knapsack problem, the value of the contents of the knapsack is maximized subject to a single capacity constraint, for example weight.
In the original knapsack problem, the value of the contents of the knapsack is maximized subject to a single capacity constraint, for example weight. In the multi-dimensional knapsack problem, additional
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 informationOptimization Methods in Management Science
Optimization Methods in Management Science MIT 15.05 Recitation 8 TAs: Giacomo Nannicini, Ebrahim Nasrabadi At the end of this recitation, students should be able to: 1. Derive Gomory cut from fractional
More informationMODELING (Integer Programming Examples)
MODELING (Integer Programming Eamples) IE 400 Principles of Engineering Management Integer Programming: Set 5 Integer Programming: So far, we have considered problems under the following assumptions:
More informationUnit 1A: Computational Complexity
Unit 1A: Computational Complexity Course contents: Computational complexity NP-completeness Algorithmic Paradigms Readings Chapters 3, 4, and 5 Unit 1A 1 O: Upper Bounding Function Def: f(n)= O(g(n)) if
More information4/12/2011. Chapter 8. NP and Computational Intractability. Directed Hamiltonian Cycle. Traveling Salesman Problem. Directed Hamiltonian Cycle
Directed Hamiltonian Cycle Chapter 8 NP and Computational Intractability Claim. G has a Hamiltonian cycle iff G' does. Pf. Suppose G has a directed Hamiltonian cycle Γ. Then G' has an undirected Hamiltonian
More informationSEQUENTIAL AND SIMULTANEOUS LIFTING IN THE NODE PACKING POLYHEDRON JEFFREY WILLIAM PAVELKA. B.S., Kansas State University, 2011
SEQUENTIAL AND SIMULTANEOUS LIFTING IN THE NODE PACKING POLYHEDRON by JEFFREY WILLIAM PAVELKA B.S., Kansas State University, 2011 A THESIS Submitted in partial fulfillment of the requirements for the degree
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 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 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 informationAdvanced Linear Programming: The Exercises
Advanced Linear Programming: The Exercises The answers are sometimes not written out completely. 1.5 a) min c T x + d T y Ax + By b y = x (1) First reformulation, using z smallest number satisfying x z
More informationInteger Programming. Chapter Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Integer Programming Chapter 5 5-1 Chapter Topics Integer Programming (IP) Models Integer Programming Graphical Solution Computer Solution of Integer Programming Problems With Excel and QM for Windows 0-1
More informationChapter 5 Integer Programming
Chapter 5 Integer Programming Chapter Topics Integer Programming (IP) Models Integer Programming Graphical Solution Computer Solution of Integer Programming Problems With Excel 2 1 Integer Programming
More informationFINAL EXAM PRACTICE PROBLEMS CMSC 451 (Spring 2016)
FINAL EXAM PRACTICE PROBLEMS CMSC 451 (Spring 2016) The final exam will be on Thursday, May 12, from 8:00 10:00 am, at our regular class location (CSI 2117). It will be closed-book and closed-notes, except
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 informationOptimisation and Operations Research
Optimisation and Operations Research Lecture 15: The Greedy Heuristic Matthew Roughan http://www.maths.adelaide.edu.au/matthew.roughan/ Lecture_notes/OORII/ School of
More informationPolynomial-time Reductions
Polynomial-time Reductions Disclaimer: Many denitions in these slides should be taken as the intuitive meaning, as the precise meaning of some of the terms are hard to pin down without introducing the
More informationDecision Mathematics D1
Pearson Edexcel GCE Decision Mathematics D1 Advanced/Advanced Subsidiary Friday 16 June 2017 Afternoon Time: 1 hour 30 minutes Paper Reference 6689/01 You must have: D1 Answer Book Candidates may use any
More informationAdvanced linear programming
Advanced linear programming http://www.staff.science.uu.nl/~akker103/alp/ Chapter 10: Integer linear programming models Marjan van den Akker 1 Intro. Marjan van den Akker Master Mathematics TU/e PhD Mathematics
More informationDisconnecting Networks via Node Deletions
1 / 27 Disconnecting Networks via Node Deletions Exact Interdiction Models and Algorithms Siqian Shen 1 J. Cole Smith 2 R. Goli 2 1 IOE, University of Michigan 2 ISE, University of Florida 2012 INFORMS
More informationΣ w j. Σ v i KNAPSACK. for i = 1, 2,..., n. and an positive integers K and W. Does there exist a subset S of {1, 2,..., n} such that: and w i
KNAPSACK Given positive integers v i and w i for i = 1, 2,..., n. and an positive integers K and W. Does there exist a subset S of {1, 2,..., n} such that: Σ w j W i S and Σ v i K i S A special case: SUBSET
More informationAlgorithms. Outline! Approximation Algorithms. The class APX. The intelligence behind the hardware. ! Based on
6117CIT - Adv Topics in Computing Sci at Nathan 1 Algorithms The intelligence behind the hardware Outline! Approximation Algorithms The class APX! Some complexity classes, like PTAS and FPTAS! Illustration
More information2001 Dennis L. Bricker Dept. of Industrial Engineering The University of Iowa. Reducing dimensionality of DP page 1
2001 Dennis L. Bricker Dept. of Industrial Engineering The University of Iowa Reducing dimensionality of DP page 1 Consider a knapsack with a weight capacity of 15 and a volume capacity of 12. Item # Value
More informationCutting Plane Separators in SCIP
Cutting Plane Separators in SCIP Kati Wolter Zuse Institute Berlin DFG Research Center MATHEON Mathematics for key technologies 1 / 36 General Cutting Plane Method MIP min{c T x : x X MIP }, X MIP := {x
More informationMat 3770 Bin Packing or
Basic Algithm Spring 2014 Used when a problem can be partitioned into non independent sub problems Basic Algithm Solve each sub problem once; solution is saved f use in other sub problems Combine solutions
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 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 informationSection #2: Linear and Integer Programming
Section #2: Linear and Integer Programming Prof. Dr. Sven Seuken 8.3.2012 (with most slides borrowed from David Parkes) Housekeeping Game Theory homework submitted? HW-00 and HW-01 returned Feedback on
More informationInteger Linear Programming
Integer Linear Programming Solution : cutting planes and Branch and Bound Hugues Talbot Laboratoire CVN April 13, 2018 IP Resolution Gomory s cutting planes Solution branch-and-bound General method Resolution
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 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 informationReview Questions, Final Exam
Review Questions, Final Exam A few general questions. What does the Representation Theorem say (in linear programming)? In words, the representation theorem says that any feasible point can be written
More informationMVE165/MMG631 Linear and integer optimization with applications Lecture 8 Discrete optimization: theory and algorithms
MVE165/MMG631 Linear and integer optimization with applications Lecture 8 Discrete optimization: theory and algorithms Ann-Brith Strömberg 2017 04 07 Lecture 8 Linear and integer optimization with applications
More informationTHE EXISTENCE AND USEFULNESS OF EQUALITY CUTS IN THE MULTI-DEMAND MULTIDIMENSIONAL KNAPSACK PROBLEM LEVI DELISSA. B.S., Kansas State University, 2014
THE EXISTENCE AND USEFULNESS OF EQUALITY CUTS IN THE MULTI-DEMAND MULTIDIMENSIONAL KNAPSACK PROBLEM by LEVI DELISSA B.S., Kansas State University, 2014 A THESIS submitted in partial fulfillment of the
More informationColumn Generation I. Teo Chung-Piaw (NUS)
Column Generation I Teo Chung-Piaw (NUS) 21 st February 2002 1 Outline Cutting Stock Problem Slide 1 Classical Integer Programming Formulation Set Covering Formulation Column Generation Approach Connection
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 informationDecision Mathematics D1
Pearson Edexcel International Advanced Level Decision Mathematics D1 Advanced/Advanced Subsidiary Thursday 19 January 2017 Afternoon Time: 1 hour 30 minutes Paper Reference WDM01/01 You must have: D1 Answer
More informationNP-Completeness. f(n) \ n n sec sec sec. n sec 24.3 sec 5.2 mins. 2 n sec 17.9 mins 35.
NP-Completeness Reference: Computers and Intractability: A Guide to the Theory of NP-Completeness by Garey and Johnson, W.H. Freeman and Company, 1979. NP-Completeness 1 General Problems, Input Size and
More informationPart 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 informationA polynomial time algorithm to solve the single-item capacitated lot sizing problem with minimum order quantities and concave costs
A polynomial time algorithm to solve the single-item capacitated lot sizing problem with minimum order quantities and concave costs Bertrand Hellion, Bernard Penz, Fabien Mangione aboratoire G-SCOP n o
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 informationmin3x 1 + 4x 2 + 5x 3 2x 1 + 2x 2 + x 3 6 x 1 + 2x 2 + 3x 3 5 x 1, x 2, x 3 0.
ex-.-. Foundations of Operations Research Prof. E. Amaldi. Dual simplex algorithm Given the linear program minx + x + x x + x + x 6 x + x + x x, x, x. solve it via the dual simplex algorithm. Describe
More informationDecision Diagrams for Discrete Optimization
Decision Diagrams for Discrete Optimization Willem Jan van Hoeve Tepper School of Business Carnegie Mellon University www.andrew.cmu.edu/user/vanhoeve/mdd/ Acknowledgments: David Bergman, Andre Cire, Samid
More information