MASSACHUSETTS INSTITUTE OF TECHNOLOGY
|
|
- Jason Barrett
- 6 years ago
- Views:
Transcription
1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY 5.05 Introduction to Optimization (Spring 005) Problem Set 8, Due April 4, 005 You will need 4.5 points out of 5 to receive a grade of.5.. Integer Programming and Scheduling (5 points) Greyhound needs to assign drivers to all of its upcoming trips. Currently, there are three drivers based in Boston, and twelve different trips (listed in the first column of the table below). Greyhound assigns each driver to a particular sequence of trips. The table below shows all the different sequences that are possible. (The numbers represent the order of the trips in each sequence; as an example, sequence 6 consists of first going from Boston to Portland, then from Portland to New Haven and finally from New Haven to Boston). possible sequences Trip Boston-New Haven Boston- Pittsburgh Boston-Portland New-Haven - Niagara Falls New-Haven - Boston 5 5 Niagara Falls - Pittsburgh 4 Niagara Falls - Portland 4 Pittsburgh-Boston Pittsburgh-Niagara Falls Portland-Boston Portland - New Haven 4 4 Cost for each sequence Each driver can be assigned to one and only one sequence. The cost of assigning a driver to a particular sequence is in the last row of the table. The objective is to assign each driver to a sequence such that all of the possible trips are taken at least once, and the cost is minimized. Formulate the problem as an integer program. (You do not need to solve it.) (HINT: this problem is a set covering problem). Page of 6
2 . Integer Programming and Modeling Nonlinear Costs (5 points) A copying service operates a copy machine. The machine has a capacity of 5000 copies of a page per hour. One hour has been reserved for copying a -page article and a 5-page article, to be sold to MBA students. The -page article sells for $.50 per article, and up to 00 articles can be sold. The 5-page article sells for $ per article, and up to 00 copies can be sold. Assume that you get no value for copying only part of an article. The costs of copying each article are: $0.08 per page for the first 50 copies of each page (i.e., to 50 copies of the same page), $0.04 per page for the next 50 copies of each pages (i.e., 5 to 00 copies of the same page), $0.0 per page for remaining copies of each page (i.e., 0 copies of the same page and over.) (For example, the first fifty copies of an article made cost $0.08 per page, and the cost of the 5 st copy is $0.04 per page.) Formulate as an integer program the problem of determining how many copies of each article should be made. (You do not need to solve it.). Integer Programming and Arranging (8 points) You have been assigned to arrange the songs on the cassette version of Madonna s latest album. A cassette tape has two sides ( and ). The following table shows the type and length of each song that must be included in the cassette. Song Type Minutes Ballad Hit 5 Ballad 4 4 Hit 5 Ballad 4 6 Hit 7 Ballad and Hit 4 8 Ballad and Hit 5 The following are the requirements of the tape:. The songs on each side of the tape must total between 4 and 6 minutes.. Each side must have exactly two ballads.. Either song or song 6 must be on side (or both). 4. Exactly one of songs 7 and 8 must be on side. 5. If song or song is on side, then song 4 or song 8 must be on side. 6. If songs and 4 are on side, then song 5 must be on side. Show how integer programming can be used to determine if there is an arrangement of the songs that satisfy the restrictions. (HINT: Model each part as a constraint or set of constraints.) Do not solve the IP. Page of 6
3 4. Integer Programming and Modeling Fixed Costs (8 points) MIT Press is considering publishing five textbooks. The imum number of copies of each textbook that can be sold, the sales price of each textbook, the per unit cost of producing each textbook, and the fixed cost of a production run for each book are given in the table below. Book Book Book Book 4 Book 5 Maximum Demand 5,000 4,000,000 4,000,000 Sales Price ($) Per Unit Cost ($) Fixed Cost ($) 80,000 50,000 60,000 0,000 40,000 The fixed cost is charged if any books of that type are produced. (So, for example, producing,000 copies of Book brings in a revenue of,000(50) = $00,000, but costs 80, (,000) = $0,000.) MIT Press can produce at most 0,000 books, and it wishes to imize profit. a) Model this problem as an integer program. b) Using the attached file ps8.xls and Excel Solver (see instructions below), find the optimal solution to the integer program. What are the optimal solution and the optimal profit? Include a printout of your Excel output. Using Excel Solver to Solve Integer Programs Solving integer programs with Excel Solver is a lot like solving linear programs! Just follow these four easy steps:. Set up the problem in Excel Solver as you usually would for solving a linear program.. For any variable(s) that you would like to be binary valued (i.e., 0 or ), select Add Constraint, input the variable(s) in question into the left hand side, and select bin from the drop-down bar in the middle.. For any variable(s) that you would like to be integer (not necessarily binary) valued, select Add Constraint, input the variable(s) in question into the left hand side, and select int from the drop-down bar in the middle. 4. (Optional, but highly recommended) In the main Solver window, select Options and change Tolerance from the default of 5% to the much stricter bound of.0%. This will mean it takes Solver a little longer to solve, but it will give you much better solutions. Page of 6
4 5. Branch and Bound Trees (0 points) Solve the following two problems using Branch and Bound. For each problem, you should draw the associated branch and bound tree. On the drawing, for each node that is considered, write the corresponding optimal LP solution and optimal objective value of the subtree. Clearly indicate the branching criteria for each subtree. (You may use Excel to solve the LPs at each node, but you have to draw the entire tree.) a) b) s.t. s.t. x + x 6x x x, x + 5x x x + x 7x x x x, x, x + x + x + x x + x x = {0,} 9 0; x, x integer 6. Branch and Bound Steps (6 points) Suppose branch and bound is being applied to a 0- integer program in which we are imizing. By node the first three variables have been fixed. An incumbent has already been found (in another part of the tree) and has value 5. Furthermore, x 4 has been chosen to be the next variable to branch on. This situation is illustrated in the following fragment of the branch and bound tree: x, x, x fixed Incumbent: z = 5 z LP 6 x 4 = x 4 = 0 Page 4 of 6
5 The LP relaxation solved at node (which has x 4 and x 5 as free variables) was: 8x 8x x 0 x + ax + 0x (HINT: For the following questions, the LP relaxation essentially becomes a one variable problem that can be solved by inspection.) a) Suppose x 4 is set to. Does node get fathomed? Why or why not? b) Suppose x 4 is set to 0 and parameter a =. Does node get fathomed? Why or why not? c) Suppose x 4 is set to 0 and parameter a = 8. Does node get fathomed? Why or why not? 7. Branch and Bound Analysis (9 points) Consider the following integer program: s.t. x x x, x + x + x,..., x x x {0,} 6 For parts a) and b), suppose that we consider the subdivision (node) of the branch and bound tree in which x =, x =, x = 0, x 4 =, and x 5 = 0, and suppose that the incumbent has a value of 5. a) What is the optimum objective value of the LP associated with this subdivision? Does the node get fathomed? b) Give an example of a node of the Branch and Bound Tree that would get fathomed because of the bound. (It should be a descendent of the node x =, x =, x = 0, x 4 =, and x 5 = 0.) State why it will be fathomed. c) Explain why no subdivision (node) gets fathomed if at most five variables are fixed (with at least one of them being nonzero), and five variables are yet to be determined. d) Show how to use your insight from parts a) to c) to create an integer program for which the Branch and Bound tree would search at least 00 nodes. Page 5 of 6
6 Challenge Problem H (6 points) Suppose you are a senior graduating from MIT, and you are leaving Cambridge for your new job. You have n items of size a i, i =,,, n, that need to be moved. You have bought m boxes. Box j has size b j, for j =,, m. Boxes typically can hold more than one item. You have rented a truck that has size Q. This means that the sum of the sizes of the boxes in the truck is at most Q, even if the boxes are not fully loaded. Formulate an integer program in order to decide whether you can fit all of your items into boxes and then fit the boxes into the truck, and so as to minimize the number of boxes that are put on the truck. Page 6 of 6
Optimization Methods in Management Science
Problem Set Rules: Optimization Methods in Management Science MIT 15.053, Spring 2013 Problem Set 6, Due: Thursday April 11th, 2013 1. Each student should hand in an individual problem set. 2. Discussing
More information56:272 Integer Programming & Network Flows Final Examination -- December 14, 1998
56:272 Integer Programming & Network Flows Final Examination -- December 14, 1998 Part A: Answer any four of the five problems. (15 points each) 1. Transportation problem 2. Integer LP Model Formulation
More informationIntroduction. Chapter 15. Optimization Modeling: Applications. Integer Programming. Manufacturing Example. Three Types of ILP Models
Chapter 5 Optimization Modeling: Applications Integer Programming Introduction When one or more variables in an LP problem must assume an integer value we have an Integer Linear Programming (ILP) problem.
More information15.083J Integer Programming and Combinatorial Optimization Fall Enumerative Methods
5.8J Integer Programming and Combinatorial Optimization Fall 9 A knapsack problem Enumerative Methods Let s focus on maximization integer linear programs with only binary variables For example: a knapsack
More informationSimulation. Lecture O1 Optimization: Linear Programming. Saeed Bastani April 2016
Simulation Lecture O Optimization: Linear Programming Saeed Bastani April 06 Outline of the course Linear Programming ( lecture) Integer Programming ( lecture) Heuristics and Metaheursitics (3 lectures)
More informationCSE 131 Introduction to Computer Science Fall 2016 Exam I. Print clearly the following information:
CSE 131 Introduction to Computer Science Fall 2016 Given: 29 September 2016 Exam I Due: End of Exam Session This exam is closed-book, closed-notes, no electronic devices allowed The exception is the "sage
More informationIntroduction to Management Science (8th Edition, Bernard W. Taylor III) Chapter 9 Integer Programming. Chapter 9 - Integer Programming 1
Introduction to Management Science (8th Edition, Bernard W. Taylor III) Chapter 9 Integer Programming Chapter 9 - Integer Programming 1 Chapter Topics Integer Programming (IP) Models Still linear programming
More informationCSE 21 Summer 2017 Homework 4
CSE 21 Summer 201 Homework Key Concepts Minimum Spanning Trees, Directed Acyclic Graphs, Topological Sorting, Single source shortest paths, Counting, Basic probability principles, Independence, Linearity
More informationAlgorithms for Decision Support. Integer linear programming models
Algorithms for Decision Support Integer linear programming models 1 People with reduced mobility (PRM) require assistance when travelling through the airport http://www.schiphol.nl/travellers/atschiphol/informationforpassengerswithreducedmobility.htm
More informationMath 5593 Linear Programming Final Exam
Math 5593 Linear Programming Final Exam Department of Mathematical and Statistical Sciences University of Colorado Denver, Fall 2013 Name: Points: /30 This exam consists of 6 problems, and each problem
More informationOPTIMIZAÇÃO E DECISÃO 09/10
OPTIMIZAÇÃO E DECISÃO 09/10 PL #7 Integer Programming Alexandra Moutinho (from Hillier & Lieberman Introduction to Operations Research, 8 th edition) Problem 1 Pawtucket University is planning to buy new
More informationOptimization Methods in Management Science
Optimization Methods in Management Science MIT 15.053, Spring 2013 Problem Set 5, Due: Thursday April 2th, 2013 Problem Set Rules: 1. Each student should hand in an individual problem set. 2. Discussing
More informationFebruary 19, Integer programming. Outline. Problem formulation. Branch-andbound
Olga Galinina olga.galinina@tut.fi ELT-53656 Network Analysis and Dimensioning II Department of Electronics and Communications Engineering Tampere University of Technology, Tampere, Finland February 19,
More informationIntroduction to Mathematical Programming IE406. Lecture 20. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 20 Dr. Ted Ralphs IE406 Lecture 20 1 Reading for This Lecture Bertsimas Sections 10.1, 11.4 IE406 Lecture 20 2 Integer Linear Programming An integer
More informationTMA946/MAN280 APPLIED OPTIMIZATION. Exam instructions
Chalmers/GU Mathematics EXAM TMA946/MAN280 APPLIED OPTIMIZATION Date: 03 05 28 Time: House V, morning Aids: Text memory-less calculator Number of questions: 7; passed on one question requires 2 points
More informationUnit.9 Integer Programming
Unit.9 Integer Programming Xiaoxi Li EMS & IAS, Wuhan University Dec. 22-29, 2016 (revised) Operations Research (Li, X.) Unit.9 Integer Programming Dec. 22-29, 2016 (revised) 1 / 58 Organization of this
More informationApproximation Algorithms
Approximation Algorithms Prof. Tapio Elomaa tapio.elomaa@tut.fi Course Basics A new 4 credit unit course Part of Theoretical Computer Science courses at the Department of Mathematics There will be 4 hours
More informationMath 2524: Activity 1 (Using Excel) Fall 2002
Math 2524: Activity 1 (Using Excel) Fall 22 Often in a problem situation you will be presented with discrete data rather than a function that gives you the resultant data. You will use Microsoft Excel
More informationFinancial Optimization ISE 347/447. Lecture 13. Dr. Ted Ralphs
Financial Optimization ISE 347/447 Lecture 13 Dr. Ted Ralphs ISE 347/447 Lecture 13 1 Reading for This Lecture C&T Chapter 11 ISE 347/447 Lecture 13 2 Integer Linear Optimization An integer linear optimization
More informationSolutions for Operations Research Final Exam
Solutions for Operations Research Final Exam. (a) The buffer stock is B = i a i = a + a + a + a + a + a 6 + a 7 = + + + + + + =. And the transportation tableau corresponding to the transshipment problem
More informationAMPL Integer Linear Programming And Sensitivity Analysis for LP. ORLAB Operations Research Laboratory. Borzou Rostami. Politecnico di Milano, Italy
AMPL Integer Linear Programming And Sensitivity Analysis for LP ORLAB Operations Research Laboratory Borzou Rostami Politecnico di Milano, Italy December 6, 2012 Integer Programming: Many linear programming
More informationRyerson Polytechnic University Department of Mathematics, Physics, and Computer Science Final Examinations, April, 2003
Ryerson Polytechnic University Department of Mathematics, Physics, and Computer Science Final Examinations, April, 2003 MTH 503 - Operations Research I Duration: 3 Hours. Aids allowed: Two sheets of notes
More informationA. Lesson Context. B. Lesson Objectives. C. Fast Five (Skills Review Focus)
A. Lesson Context BIG PICTURE of this UNIT: How & why do we build NEW knowledge in Mathematics? What NEW IDEAS & NEW CONCEPTS can we now explore with specific references to QUADRATIC FUNCTIONS? How can
More informationSystems Optimization Spring 2002 Final Exam
15.057 Systems Optimization Spring 2002 Final Exam Instructions You have two hours to complete the exam from the time you first begin it. Watch your time. If you are having difficulty with a question,
More informationInteger Programming. Xi Chen. Department of Management Science and Engineering International Business School Beijing Foreign Studies University
Integer Programming Xi Chen Department of Management Science and Engineering International Business School Beijing Foreign Studies University Xi Chen (chenxi0109@bfsu.edu.cn) Integer Programming 1 / 42
More informationINTEGER AND MIXED- INTEGER OPTIMIZATION
34 INTEGER AND MIXED- INTEGER OPTIMIZATION Conceptually, it s the same as linear optimization: Continue formulating everything using linear functions: Minimize/Maximize a linear function Subject to linear
More informationLesson 08 Linear Programming
Lesson 08 Linear Programming A mathematical approach to determine optimal (maximum or minimum) solutions to problems which involve restrictions on the variables involved. 08 - Linear Programming Applications
More informationCSE 332, Spring 2010, Midterm Examination 30 April 2010
CSE 332, Spring 2010, Midterm Examination 30 April 2010 Please do not turn the page until the bell rings. Rules: The exam is closed-book, closed-note. You may use a calculator for basic arithmetic only.
More informationInteger Programming Chapter 9
1 Integer Programming Chapter 9 University of Chicago Booth School of Business Kipp Martin October 30, 2017 2 Outline Branch and Bound Theory Branch and Bound Linear Programming Node Selection Strategies
More informationProblem 1: Hello World!
Problem 1: Hello World! Instructions This is the classic first program made famous in the early 70s. Write the body of the program called Problem1 that prints out The text must be terminated by a new-line
More informationAMPL Integer Linear Programming. ORLAB Operations Research Laboratory. Borzou Rostami. Politecnico di Milano, Italy.
AMPL Integer Linear Programming ORLAB Operations Research Laboratory Borzou Rostami Politecnico di Milano, Italy January 18, 2012 Integer Programming: Many linear programming problems require certain variables
More informationc) How many students took none of the three subjects?
9 Final Fall 7 []. EVALUATE EACH OF THE FOLLOWING. SHOW ALL WORK. SIMPLIFY YOUR ANSWERS TO A SINGLE INTEGER. A) 6! 4! B) C ( 7, 4) C) P ( 9,4) []. In a recent survey of a senior class of 4 students the
More informationApproximation Algorithms
Approximation Algorithms Prof. Tapio Elomaa tapio.elomaa@tut.fi Course Basics A 4 credit unit course Part of Theoretical Computer Science courses at the Laboratory of Mathematics There will be 4 hours
More information3 INTEGER LINEAR PROGRAMMING
3 INTEGER LINEAR PROGRAMMING PROBLEM DEFINITION Integer linear programming problem (ILP) of the decision variables x 1,..,x n : (ILP) subject to minimize c x j j n j= 1 a ij x j x j 0 x j integer n j=
More informationAlgorithms Exam 1. Ansvarig: Devdatt Dubhashi Tel (Willard Rafnsson) Rum 6479 EDIT
Algorithms Exam 1 Oct 20 2009 kl 1400 1800 vag och vatten salar Ansvarig: Devdatt Dubhashi Tel. 772 1062 (Willard Rafnsson) Rum 6479 EDIT Points: 60 Passing criteria: Chalmers 5:48, 4:36, 3:24 GU VG:48,
More informationOutline. Column Generation: Cutting Stock A very applied method. Introduction to Column Generation. Given an LP problem
Column Generation: Cutting Stock A very applied method thst@man.dtu.dk Outline History The Simplex algorithm (re-visited) Column Generation as an extension of the Simplex algorithm A simple example! DTU-Management
More informationColumn Generation: Cutting Stock
Column Generation: Cutting Stock A very applied method thst@man.dtu.dk DTU-Management Technical University of Denmark 1 Outline History The Simplex algorithm (re-visited) Column Generation as an extension
More informationII. Linear Programming
II. Linear Programming A Quick Example Suppose we own and manage a small manufacturing facility that produced television sets. - What would be our organization s immediate goal? - On what would our relative
More informationDM515 Spring 2011 Weekly Note 7
Institut for Matematik og Datalogi Syddansk Universitet May 18, 2011 JBJ DM515 Spring 2011 Weekly Note 7 Stuff covered in Week 20: MG sections 8.2-8,3 Overview of the course Hints for the exam Note that
More informationTuesday, April 10. The Network Simplex Method for Solving the Minimum Cost Flow Problem
. Tuesday, April The Network Simplex Method for Solving the Minimum Cost Flow Problem Quotes of the day I think that I shall never see A poem lovely as a tree. -- Joyce Kilmer Knowing trees, I understand
More informationAssignment 1 Expressions Data Types Formatted Printing Variables Scanning CSC 123 Fall 2018 Answer Sheet Short Answers
Assignment 1 Expressions Data Types Formatted Printing Variables Scanning CSC 123 Fall 2018 Answer Sheet Short Answers 1. Every complete statement ends with a c. a. period b. parenthesis c. semicolon d.
More information2. You are required to enter a password of up to 100 characters. The characters must be lower ASCII, printing characters.
BLACK BOX SOFTWARE TESTING SPRING 2005 DOMAIN TESTING LAB PROJECT -- GRADING NOTES For all of the cases below, do the traditional equivalence class and boundary analysis. Draw one table and use a new line
More informationFundamentals of Integer Programming
Fundamentals of Integer Programming Di Yuan Department of Information Technology, Uppsala University January 2018 Outline Definition of integer programming Formulating some classical problems with integer
More informationApplications. 72 Variables and Patterns
Applications. Sean bought a DVD player and a receiver. The store offered him an interest-free payment plan with weekly installments. Sean figured out that after n weeks of payments, he would still owe
More informationTechnology for Merchandise Planning and Control
Contents: Technology for Merchandise Planning and Control Module Seven: Enhancing Charts Using What if Analysis REFERENCE/PAGES Enhancing Charts... Unit J 1 XL2007 Customizing a Data Series... Excel 226,
More informationUnit 2-2: Writing and Graphing Quadratics NOTE PACKET. 12. I can use the discriminant to determine the number and type of solutions/zeros.
Unit 2-2: Writing and Graphing Quadratics NOTE PACKET Name: Period Learning Targets: Unit 2-1 12. I can use the discriminant to determine the number and type of solutions/zeros. 1. I can identify a function
More information24 Sub-Queries. Sub-Queries
24 Sub-Queries Sub-Queries are another method of restricting data, but they use the results of one query to restrict a second query. Web Intelligence generates SQL that is processed by the database in
More informationTIM 206 Lecture Notes Integer Programming
TIM 206 Lecture Notes Integer Programming Instructor: Kevin Ross Scribe: Fengji Xu October 25, 2011 1 Defining Integer Programming Problems We will deal with linear constraints. The abbreviation MIP stands
More informationUML CS Algorithms Qualifying Exam Fall, 2004 ALGORITHMS QUALIFYING EXAM
ALGORITHMS QUALIFYING EXAM This exam is open books & notes and closed neighbors & calculators. The upper bound on exam time is 3 hours. Please put all your work on the exam paper. Please write your name
More informationMAT 103 F09 TEST 3 REVIEW (CH 4-5)
MAT 103 F09 TEST 3 REVIEW (CH 4-5) NAME For # 1-3, solve the system of equations by graphing. Label the equation of each line on your graph and write the solution as an ordered pair. Be sure to CHECK your
More informationCSE 143 Final Exam Part 1 - August 18, 2011, 9:40 am
CSE 143 Final Exam Part 1 - August 18, 2011, 9:40 am Name Student ID # Section TA Name The exam is closed book, closed notes, closed devices, except that you may have a 5x8 card with handwritten notes
More informationDepartment of Mathematics Oleg Burdakov of 30 October Consider the following linear programming problem (LP):
Linköping University Optimization TAOP3(0) Department of Mathematics Examination Oleg Burdakov of 30 October 03 Assignment Consider the following linear programming problem (LP): max z = x + x s.t. x x
More information9.4 SOME CHARACTERISTICS OF INTEGER PROGRAMS A SAMPLE PROBLEM
9.4 SOME CHARACTERISTICS OF INTEGER PROGRAMS A SAMPLE PROBLEM Whereas the simplex method is effective for solving linear programs, there is no single technique for solving integer programs. Instead, a
More informationApproximation in Linear Stochastic Programming Using L-Shaped Method
Approximation in Linear Stochastic Programming Using L-Shaped Method Liza Setyaning Pertiwi 1, Rini Purwanti 2, Wilma Handayani 3, Prof. Dr. Herman Mawengkang 4 1,2,3,4 University of North Sumatra, Indonesia
More informationData Structure. Chapter 11. Objectives After studying this chapter, students should be able to: 11-1 ARRAYS
11.1 Chapter 11 Data Structure Objectives After studying this chapter, students should be able to: Define a data structure. Define an array as a data structure and how it is used to store a list of data
More informationNOTATION AND TERMINOLOGY
15.053x, Optimization Methods in Business Analytics Fall, 2016 October 4, 2016 A glossary of notation and terms used in 15.053x Weeks 1, 2, 3, 4 and 5. (The most recent week's terms are in blue). NOTATION
More informationUnit: Rational Number Lesson 3.1: What is a Rational Number? Objectives: Students will compare and order rational numbers.
Unit: Rational Number Lesson 3.: What is a Rational Number? Objectives: Students will compare and order rational numbers. (9N3) Procedure: This unit will introduce the concept of rational numbers. This
More informationPractice Exam III - Answers
Department of Mathematics University of Notre Dame Math 10120 Finite Math Spring 2015 Name: Instructor: Migliore Practice Exam III - Answers April 16, 2015 This exam is in two parts on 13 pages and contains
More informationAPPM 4120/5120 Exam #2 Practice Solutions Spring 2015
APPM 4120/5120 Exam #2 Practice Solutions Spring 2015 You are not allowed to use textbooks, class notes. Problem #1 (20 points): Consider the following activity-on-arc project network, where the 12 arcs
More informationAdvanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras
Advanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture - 35 Quadratic Programming In this lecture, we continue our discussion on
More informationGENERAL ASSIGNMENT PROBLEM via Branch and Price JOHN AND LEI
GENERAL ASSIGNMENT PROBLEM via Branch and Price JOHN AND LEI Outline Review the column generation in Generalized Assignment Problem (GAP) GAP Examples in Branch and Price 2 Assignment Problem The assignment
More informationBCN Decision and Risk Analysis. Syed M. Ahmed, Ph.D.
Linear Programming Module Outline Introduction The Linear Programming Model Examples of Linear Programming Problems Developing Linear Programming Models Graphical Solution to LP Problems The Simplex Method
More informationFreecoms VoIP Mobile Community Telecom S. Ferrari, page n 1»
Freecoms VoIP Mobile Community Telecom S. Ferrari, page n 1» Multiservice Mobile VoIP Community Powerful multiservice package: Home and Mobile VoIP communication. Business and Private WEB Portal community
More informationStep 1. Use a ruler or straight-edge to determine a line of best fit. One example is shown below.
Linear Models Modeling 1 ESSENTIALS Example Draw a straight line through the scatter plot so that the line represents a best fit approximation to the points. Then determine the equation for the line drawn.
More informationHomework Assignment #3. 1 (5 pts) Demonstrate how mergesort works when sorting the following list of numbers:
CISC 4080 Computer Algorithms Spring, 2019 Homework Assignment #3 1 (5 pts) Demonstrate how mergesort works when sorting the following list of numbers: 6 1 4 2 3 8 7 5 2 Given the following array (list),
More informationAtlas 5.0 for Microsoft Dynamics AX Advanced reporting system.
TRAINEE WORKBOOK Atlas 5.0 for Microsoft Dynamics AX Advanced reporting system. Table of Contents 1 Introduction... 4 1.1 Welcome... 4 1.2 About this course... 4 1.2.1 Course description... 4 1.2.2 Audience...
More informationApplication of Cutting Stock Problem in Minimizing The Waste of Al-Quran Cover
Kaunia, Vol.XII, Num., April 206, pp. 7~22 Available online at http://ejournal.uin-suka.ac.id/saintek/kaunia Application of Cutting Stock Problem in Minimizing The Waste of Al-Quran Cover Noor Saif Muhammad
More informationPROFIT AND LOSS REPORT EXTENSION. User Guide. User Guide Page 1
PROFIT AND LOSS REPORT EXTENSION User Guide User Guide Page 1 Important Notice JTechExtensions reserves the right to make corrections, modifications, enhancements, improvements, and other changes to its
More informationData Management Software
1.2 Data Management Software INVESTIGATE & INQUIRE: Software Tools 1. List every computer program you can think of that can be used to manage data. 2. Sort the programs into categories, such as word-processors
More informationLECTURE NOTES Non-Linear Programming
CEE 6110 David Rosenberg p. 1 Learning Objectives LECTURE NOTES Non-Linear Programming 1. Write out the non-linear model formulation 2. Describe the difficulties of solving a non-linear programming model
More information2 is not feasible if rounded. x =0,x 2
Integer Programming Definitions Pure Integer Programming all variables should be integers Mied integer Programming Some variables should be integers Binary integer programming The integer variables are
More informationStudents interpret the meaning of the point of intersection of two graphs and use analytic tools to find its coordinates.
Student Outcomes Students interpret the meaning of the point of intersection of two graphs and use analytic tools to find its coordinates. Classwork Example 1 (7 minutes) Have students read the situation
More informationDeveloping Optimization Algorithms for Real-World Applications
Developing Optimization Algorithms for Real-World Applications Gautam Ponnappa PC Training Engineer Viju Ravichandran, PhD Education Technical Evangelist 2015 The MathWorks, Inc. 1 2 For a given system,
More informationEr Diagram Exercises. Exercise1. Draw an ER diagram for the following application from the manufacturing industry.
Er Diagram Exercises Exercise1 Draw an ER diagram for the following application from the manufacturing industry. 1. Each supplier has a unique name, and located in a city. 2. Each part has a unique number.
More informationName: Lirong TAN 1. (15 pts) (a) Define what is a shortest s-t path in a weighted, connected graph G.
1. (15 pts) (a) Define what is a shortest s-t path in a weighted, connected graph G. A shortest s-t path is a path from vertex to vertex, whose sum of edge weights is minimized. (b) Give the pseudocode
More informationMATH 117 Statistical Methods for Management I Chapter Two
Jubail University College MATH 117 Statistical Methods for Management I Chapter Two There are a wide variety of ways to summarize, organize, and present data: I. Tables 1. Distribution Table (Categorical
More informationExam Datastrukturer. DIT960 / DIT961, VT-18 Göteborgs Universitet, CSE
Exam Datastrukturer DIT960 / DIT961, VT-18 Göteborgs Universitet, CSE Day: 2018-10-12, Time: 8:30-12.30, Place: SB Course responsible Alex Gerdes, tel. 031-772 6154. Will visit at around 9:30 and 11:00.
More informationMotivation for Heuristics
MIP Heuristics 1 Motivation for Heuristics Why not wait for branching? Produce feasible solutions as quickly as possible Often satisfies user demands Avoid exploring unproductive sub trees Better reduced
More informationGregg College Keyboarding & Document Processing (GDP); Lessons 1-120, Main Text PDF
Gregg College Keyboarding & Document Processing (GDP); Lessons 1-120, Main Text PDF Gregg College Keyboarding and Document Processing (GDP), 11e by Ober, Johnson, and Zimmerly: Your complete learning/teaching
More informationPHLI Instruction (734) Introduction. Lists.
INTERMEDIATE EXCEL Introduction Microsoft Excel has many purposes. In addition to being an excellent data manger, Excel provides the means to perform complex analysis and evaluation of data. This brief
More informationLAB 4.1 Relational Operators and the if Statement
LAB 4.1 Relational Operators and the if Statement // This program tests whether or not an initialized value of num2 // is equal to a value of num1 input by the user. int main( ) int num1, // num1 is not
More informationCOMP9334: Capacity Planning of Computer Systems and Networks
COMP9334: Capacity Planning of Computer Systems and Networks Week 10: Optimisation (1) A/Prof Chun Tung Chou CSE, UNSW COMP9334, Chun Tung Chou, 2016 Three Weeks of Optimisation The lectures for these
More informationCMPSCI 250: Introduction to Computation. Lecture #1: Things, Sets and Strings David Mix Barrington 22 January 2014
CMPSCI 250: Introduction to Computation Lecture #1: Things, Sets and Strings David Mix Barrington 22 January 2014 Things, Sets, and Strings The Mathematical Method Administrative Stuff The Objects of Mathematics
More informationLinear Functions. Connection to AP*: AP Calculus Topic: Analysis of Functions
Connecting Middle Grades to Advanced Placement* Mathematics A Resource and Strategy Guide Linear Functions Objective: Students will write an equation for a given problem situation and investigate the relationships
More informationV. Solving Integer Linear Programs
Optimization Methods Draft of August 26, 2005 V. Solving Integer Linear Programs Robert Fourer Department of Industrial Engineering and Management Sciences Northwestern University Evanston, Illinois 60208-3119,
More informationHomework #3. CMSC351 - Spring 2013 PRINT Name :
Homework #3 CMSC351 - Spring 2013 PRINT Name : Due: Mar 12 th at the start of class o Grades depend on neatness and clarity. o Write your answers with enough detail about your approach and concepts used,
More informationCS 473: Algorithms. Ruta Mehta. Spring University of Illinois, Urbana-Champaign. Ruta (UIUC) CS473 1 Spring / 36
CS 473: Algorithms Ruta Mehta University of Illinois, Urbana-Champaign Spring 2018 Ruta (UIUC) CS473 1 Spring 2018 1 / 36 CS 473: Algorithms, Spring 2018 LP Duality Lecture 20 April 3, 2018 Some of the
More informationCS 374 Fall 2014 Homework 2 Due Tuesday, September 16, 2014 at noon
CS 374 Fall 2014 Homework 2 Due Tuesday, September 16, 2014 at noon Groups of up to three students may submit common solutions for each problem in this homework and in all future homeworks You are responsible
More informationThe problem. Needed for contract. 10% Reduction. Resources available
As part of May 2010 s P2 paper, a question was asked (Q6) which required candidates to prepare a graph showing the optimum production plan for the organisation in question. In his post examination guide
More informationCS 3114 Data Structures and Algorithms READ THIS NOW!
READ THIS NOW! Print your name in the space provided below. There are 9 short-answer questions, priced as marked. The maximum score is 100. When you have finished, sign the pledge at the bottom of this
More informationSpring 2014 CSE Qualifying Exam Core Subjects
Spring 2014 CSE Qualifying Exam Core Subjects February 22, 2014 Architecture 1. ROB trace - single issue Assume the following: Ten ROB slots. 1 integer Execution unit, 1 reservation stations, one cycle
More informationCPSC 211, Sections : Data Structures and Implementations, Honors Final Exam May 4, 2001
CPSC 211, Sections 201 203: Data Structures and Implementations, Honors Final Exam May 4, 2001 Name: Section: Instructions: 1. This is a closed book exam. Do not use any notes or books. Do not confer with
More informationCSE 332 Spring 2013: Midterm Exam (closed book, closed notes, no calculators)
Name: Email address: Quiz Section: CSE 332 Spring 2013: Midterm Exam (closed book, closed notes, no calculators) Instructions: Read the directions for each question carefully before answering. We will
More informationWorking with Quadratic Functions in Standard and Vertex Forms
Working with Quadratic Functions in Standard and Vertex Forms Example 1: Identify Characteristics of a Quadratic Function in Standard Form f( x) ax bx c, a 0 For the quadratic function f( x) x x 3, identify
More informationCREATING A SALES PROPOSAL. Proposal Setup. Tabbed Job Summary Form
CREATING A SALES PROPOSAL A proposal in SuccessWare 21 is a feature allowing you to group up to 5 different sales quotes into a single document that can be presented to the customer. For example, if you
More information56:272 Integer Programming & Network Flows Final Exam -- December 16, 1997
56:272 Integer Programming & Network Flows Final Exam -- December 16, 1997 Answer #1 and any five of the remaining six problems! possible score 1. Multiple Choice 25 2. Traveling Salesman Problem 15 3.
More informationCombinatorial Optimization
Combinatorial Optimization Problem set 7: solutions. Formulate and solve an integer program for the following scenario. A trader of unusual objects is traveling with a caravan that begins in city A, proceeds
More informationMath 3A Meadows or Malls? Review
Math 3A Meadows or Malls? Review Name Linear Programming w/o Graphing (2 variables) 1. A manufacturer makes digital watches and analogue (non-digital) watches. It cost $15 to make digital watch and $20
More informationLectures 8/9. 1 Overview. 2 Prelude:Routing on the Grid. 3 A couple of networks.
U.C. Berkeley CS273: Parallel and Distributed Theory Lectures 8/9 Professor Satish Rao September 23,2010 Lecturer: Satish Rao Last revised October 23, 2010 Lectures 8/9 1 Overview We will give a couple
More information3.1 Start Thinking. 3.1 Warm Up. 3.1 Cumulative Review Warm Up. Consider the equation y x.
3.1 Start Thinking Consider the equation y x. Are there any values of x that you cannot substitute into the equation? If so, what are they? Are there any values of y that you cannot obtain as an answer?
More information