Answers to problems. Chapter 1. Chapter (0, 0) (3.5,0) (0,4.5) (2, 3) 2.1(a) Last tableau. (b) Last tableau /2 -3/ /4 3/4 1/4 2.
|
|
|
- Bathsheba Melton
- 5 years ago
- Views:
Transcription
1 Answers to problems Chapter (0, 0) (3.5,0) (0,4.5) (, 3) Chapter.1(a) Last tableau X4 X3 B /5 7/5 x -3/5 /5 Xl 4/5-1/5 8 3 Xl =,X =3,B=8 (b) Last tableau c Xl -19/ X3-3/ -7 3/4 1/ /4-1/4.5 Xl =0,X =0,C=-7.. X I = 0, X = 4.5, and also Xl =, X = 3: objective function Last tableaus Xl X3 Xl X4 C -19/ -3/ -7 C x 3/4 1/4 4.5 X3-5 - X 1/ 4.5 Both define the same point, namely Xl = 0, X = 4.5
2 ANSWERS TO PROBLEMS Introducing parameters, maximize 3.5A + 36A3 + 6A4 + 5x3 subject to 3.5A + 18A3 + 14A4 + 5x3 ~ 7 Al + A + A3 + A4 = 1 Answer: Al = 11/18, A3 = 7/18, i.e. Xl = 0, X = 7/4, X3 = 0, objective function Final tableaus t~6 Xl X B -5 +t -6 +t X X4 1 7 (0,0) ~t~6 Xl X3 B -1/ + t/4 3/ - t/4 7-9t/ x 3/4 1/4 9/ X4 5/4-1/4 5/ (0,9/) t~ X4 X3 B /5 - t/5 7/5 - t/5 8-5t X -3/5 /5 3 Xl 4/5-1/5 Note: when t =, we obtain the answer to Problem.1, with its objective function halved
3 14 LINEAR PROGRAMMING.6 Last tableau X4 X B 1-1 x3-1 xl 1-1 i.e. B can increase without bounds.7. Xl X X3 3 4 ij 18 X4 * 1 7 */ ~ X4 X Xl X3 X3-3/ 5/t 15/ Xz 3/4 1/4 Xl 1/ 1/ 7/ X4 5/4t -1/4 (3.5,0) (0,4.5) t~ X4 X3 X -3/5 /5 3 Xl 4/5-1/5 (,3) 9/ 5/.8. First tableau Xl Xz X3 Ul -1/ Uz 1/10 -/ /5-1/5 0 8Z 1 1 1
4 ANSWERS TO PROBLEMS 143 where Xi are the proportion in gj, UI and U are the recruitments intogl andg, respectively Final tableaus Xl U UI UI UI /3 /15 U U Xl 10/3 /3 Xl -5 0 X X -5/3 1/6 X X X3-5/3 1/6 X Chapter 3 3.1(a) Minimize 18Y3 + 7Y4 subject to 3Y3 + Y4 ~ 5 4Y3 + Y4 ~6 Y3'Y4~0 last tableau Y YI C -3-8 Y4 3/5-4/5 /5 Y3 -/5 1/5 7/5 (b) Maximize -18Y3-7Y4 - Mz subject to -3Y3 - Y4 + YI = 5 4Y3 + Y4 - Y + Z = 6 Yj, z ~o last tableau Y4 Y B 5/ 9/ -7 YI -5/4-3/4 19/ Y3 1/4-1/4 3/
5 144 LINEAR PROGRAMMING 3.. The problem is minimize Xl + X subject to -Xl +X -X3 = 1 XI-X -X4 = 1 Last tableau c M -1 o -M o -1 M i.e. the problem has no feasible region (remember that its dual had an infinite objective function!) Chapter Pay-off table A should choose, or 4, in proportions 3 : 1 B should choose 1, or 4, in proportions 1: A ought to pay an entry fee of Pay-off table (1, 1) (1,) (, 1) (,) (1, 1) (1,) (, 1) (,) Two basic solutions: (0,4/7,3/7,0) and (0,3/5,/5,0)
6 ANSWERS TO PROBLEMS Chapter 5 5.l(a) Optimal distribution A Bl B B3 145 Al 3 A 6 B 4 cost 38 (b) Optimal distribution Bl B B3 Al 3 A 1 cost Optimal distributions A Bl B B3 A Bl Al 1 4 Al 1 A 4 and A Bl B B 4 cost The cut consists of SA, SB, CD, ET. Total capacity 13 B 4 B3 -I D T -4 Figure to answer of problem 5.3
7 146 LINEAR PROGRAMMING Chapter Final tableaus Xl S Xl S C C X and X X4 3-1 X 1 4 x3 3-4 X4 0 - They give the same point 6.(a) Final tableau Sl S C Y4 3-3 Y3-4/3 1 Y Yl -3 (b) Final tableau Sl S B Yl Y Y Y4-3
8 ANSWERS TO PROBLEMS 147 Chapter Last tableau X Xl Y Z B Xl Zl Equivalent constraints 3Xl + 4X"';; 19, xl +X...;; 8 Answer: Xl = 13/5, X = 14/5, B = 149/5 5 7
9 References [1] Charnes, A. (195), Optimality and degeneracy. Econometrica, 0, [] Dantzig, G. B. (1951), Maximization of a linear function of variables subject to linear inequalities, in Activity Analysis of Production and Allocation, (ed. T. C. Koopmans) J. Wiley & Sons, New York, pp [3] Easterfield, T. E. (1946), A combinatorial algorithm. J. London Math. Soc., 1, [4] Farkas, J. (190), tiber die Theorie der einfachen Ungleichungen. J.t. reine und angew. Math., 14, 1-4. [5] Ford, L. R., jun. and Fulkerson, D. R. (1957), A simple algorithm for finding maximal network flows and an application to the Hitchcock problem. Can. J. Math., 9, 1 O-lB. [6] Ford, L. R., jun. and Fulkerson, D. R. (1956), Solving the transportation problem. Man. Science, 3,4-3. [7] Gomory, R. E. (195B), Essentials of an algorithm for integer solutions to linear programs. Bull. A mer. Math. Soc., 64, 75-7B. [B] Hall, P. (1935), On representation of sub-sets. J. London Math. Soc., 10,6-30. [9] Hitchcock, F. L. (1941), The distribution of a product from several sources to numerous localities. J. Math. and Phys., 0, [ 10] Kuhn, H. W. and Tucker, A. W. (eds.) (1956), Linear Inequalities and Related Systems. Annals of Math. Study no. 38, Princeton University Press, Preface xi-xiii. [11] Land, A. H. and Doig, A. G. (1960), An automatic method of solving discrete programming problems. Econometrica, B, [1] Tucker, A. W. (1956), Dual systems of homogeneous linear relations, in Linear Inequalities and Related Systems (eds. H. W. Kuhn and A. W. Tucker), Princeton University Press, pp.3-lb. [13] Vajda, S. (197), Probabilistic Programming, Academic Press.
10 Index alternatives, 61 artificial variable, 4 assignment problem, 63, 85 basic solution, 6 basic variable, 6 breakthrough, 110 feasible point,s feasible region,s finite game, 65 fixed charge, 16 four-colour theorem, 16 graph,95 capacitated transportation problem, 88 check,, 5 complementary slackness, 54 complete tableau, 31 constraint, convexity constraint, 39 convex region, 9 cut, 99 cutting plane, 118 degenerate, 6, 9 dual feasible tableau, 51 dual network, 100 dual problem, 50 Dual Simplex Method, 5 duality, 51 duality theorem, 53 elementary matrix, 34 Inverse Matrix Method, 34 knapsack problem, 16 Lagrangean, 54 linear programming, master problem, 39 matrix game, 65 maximin, 66 minimax, 66 mixed integer programming, 117 mixed strategy, 67 multipliers, 33 necessity theorem, 53 networ}<::,95 non-basic variable, 6 Farkas' theorem, 60 objective function,
11 150 INDEX parametric programming, 43 pay-off matrix, 65 pivot, 18 planar, 100 primal problem, 50 proposal,39 pure integer programming, 117 pure strategy, 67 representative, 63 restricted tableau, 31 Revised Simplex Method, 33 saddle point, 55, 66 sensitivity analysis, 58 shadow cost, 58 shadow price, 58 Simplex Method, 16 Simplex tableau, 18 slack variable, 3 strategy line, 7 sufficiency theorem, 53 system of distinct representatives, 63 transportation problem, 78 value of a game, 66 zero-sum game, 65
Part IB Optimisation
Part IB Optimisation Theorems Based on lectures by F. A. Fischer Notes taken by Dexter Chua Easter 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after
Part 1. The Review of Linear Programming
In the name of God Part 1. The Review of Linear Programming 1.5. Spring 2010 Instructor: Dr. Masoud Yaghini Outline Introduction Formulation of the Dual Problem Primal-Dual Relationship Economic Interpretation
Worked Examples for Chapter 5
Worked Examples for Chapter 5 Example for Section 5.2 Construct the primal-dual table and the dual problem for the following linear programming model fitting our standard form. Maximize Z = 5 x 1 + 4 x
Farkas Lemma, Dual Simplex and Sensitivity Analysis
Summer 2011 Optimization I Lecture 10 Farkas Lemma, Dual Simplex and Sensitivity Analysis 1 Farkas Lemma Theorem 1. Let A R m n, b R m. Then exactly one of the following two alternatives is true: (i) x
DEPARTMENT OF STATISTICS AND OPERATIONS RESEARCH OPERATIONS RESEARCH DETERMINISTIC QUALIFYING EXAMINATION. Part I: Short Questions
DEPARTMENT OF STATISTICS AND OPERATIONS RESEARCH OPERATIONS RESEARCH DETERMINISTIC QUALIFYING EXAMINATION Part I: Short Questions August 12, 2008 9:00 am - 12 pm General Instructions This examination is
4. Duality Duality 4.1 Duality of LPs and the duality theorem. min c T x x R n, c R n. s.t. ai Tx = b i i M a i R n
2 4. Duality of LPs and the duality theorem... 22 4.2 Complementary slackness... 23 4.3 The shortest path problem and its dual... 24 4.4 Farkas' Lemma... 25 4.5 Dual information in the tableau... 26 4.6
Review Solutions, Exam 2, Operations Research
Review Solutions, Exam 2, Operations Research 1. Prove the weak duality theorem: For any x feasible for the primal and y feasible for the dual, then... HINT: Consider the quantity y T Ax. SOLUTION: To
Math 5593 Linear Programming Week 1
University of Colorado Denver, Fall 2013, Prof. Engau 1 Problem-Solving in Operations Research 2 Brief History of Linear Programming 3 Review of Basic Linear Algebra Linear Programming - The Story About
The Strong Duality Theorem 1
1/39 The Strong Duality Theorem 1 Adrian Vetta 1 This presentation is based upon the book Linear Programming by Vasek Chvatal 2/39 Part I Weak Duality 3/39 Primal and Dual Recall we have a primal linear
Summary of the simplex method
MVE165/MMG631,Linear and integer optimization with applications The simplex method: degeneracy; unbounded solutions; starting solutions; infeasibility; alternative optimal solutions Ann-Brith Strömberg
Summary of the simplex method
MVE165/MMG630, The simplex method; degeneracy; unbounded solutions; infeasibility; starting solutions; duality; interpretation Ann-Brith Strömberg 2012 03 16 Summary of the simplex method Optimality condition:
Optimization 4. GAME THEORY
Optimization GAME THEORY DPK Easter Term Saddle points of two-person zero-sum games We consider a game with two players Player I can choose one of m strategies, indexed by i =,, m and Player II can choose
Slack Variable. Max Z= 3x 1 + 4x 2 + 5X 3. Subject to: X 1 + X 2 + X x 1 + 4x 2 + X X 1 + X 2 + 4X 3 10 X 1 0, X 2 0, X 3 0
Simplex Method Slack Variable Max Z= 3x 1 + 4x 2 + 5X 3 Subject to: X 1 + X 2 + X 3 20 3x 1 + 4x 2 + X 3 15 2X 1 + X 2 + 4X 3 10 X 1 0, X 2 0, X 3 0 Standard Form Max Z= 3x 1 +4x 2 +5X 3 + 0S 1 + 0S 2
Note 3: LP Duality. If the primal problem (P) in the canonical form is min Z = n (1) then the dual problem (D) in the canonical form is max W = m (2)
Note 3: LP Duality If the primal problem (P) in the canonical form is min Z = n j=1 c j x j s.t. nj=1 a ij x j b i i = 1, 2,..., m (1) x j 0 j = 1, 2,..., n, then the dual problem (D) in the canonical
Linear Programming Duality P&S Chapter 3 Last Revised Nov 1, 2004
Linear Programming Duality P&S Chapter 3 Last Revised Nov 1, 2004 1 In this section we lean about duality, which is another way to approach linear programming. In particular, we will see: How to define
21. Solve the LP given in Exercise 19 using the big-m method discussed in Exercise 20.
Extra Problems for Chapter 3. Linear Programming Methods 20. (Big-M Method) An alternative to the two-phase method of finding an initial basic feasible solution by minimizing the sum of the artificial
Standard Form An LP is in standard form when: All variables are non-negativenegative All constraints are equalities Putting an LP formulation into sta
Chapter 4 Linear Programming: The Simplex Method An Overview of the Simplex Method Standard Form Tableau Form Setting Up the Initial Simplex Tableau Improving the Solution Calculating the Next Tableau
LINEAR AND NONLINEAR PROGRAMMING
LINEAR AND NONLINEAR PROGRAMMING Stephen G. Nash and Ariela Sofer George Mason University The McGraw-Hill Companies, Inc. New York St. Louis San Francisco Auckland Bogota Caracas Lisbon London Madrid Mexico
3. THE SIMPLEX ALGORITHM
Optimization. THE SIMPLEX ALGORITHM DPK Easter Term. Introduction We know that, if a linear programming problem has a finite optimal solution, it has an optimal solution at a basic feasible solution (b.f.s.).
Duality in LPP Every LPP called the primal is associated with another LPP called dual. Either of the problems is primal with the other one as dual. The optimal solution of either problem reveals the information
Lecture 10: Linear programming duality and sensitivity 0-0
Lecture 10: Linear programming duality and sensitivity 0-0 The canonical primal dual pair 1 A R m n, b R m, and c R n maximize z = c T x (1) subject to Ax b, x 0 n and minimize w = b T y (2) subject to
MVE165/MMG631 Linear and integer optimization with applications Lecture 5 Linear programming duality and sensitivity analysis
MVE165/MMG631 Linear and integer optimization with applications Lecture 5 Linear programming duality and sensitivity analysis Ann-Brith Strömberg 2017 03 29 Lecture 4 Linear and integer optimization with
SAMPLE QUESTIONS. b = (30, 20, 40, 10, 50) T, c = (650, 1000, 1350, 1600, 1900) T.
SAMPLE QUESTIONS. (a) We first set up some constant vectors for our constraints. Let b = (30, 0, 40, 0, 0) T, c = (60, 000, 30, 600, 900) T. Then we set up variables x ij, where i, j and i + j 6. By using
Chapter 1 Linear Programming. Paragraph 5 Duality
Chapter 1 Linear Programming Paragraph 5 Duality What we did so far We developed the 2-Phase Simplex Algorithm: Hop (reasonably) from basic solution (bs) to bs until you find a basic feasible solution
UNIT-4 Chapter6 Linear Programming
UNIT-4 Chapter6 Linear Programming Linear Programming 6.1 Introduction Operations Research is a scientific approach to problem solving for executive management. It came into existence in England during
Math Models of OR: Sensitivity Analysis
Math Models of OR: Sensitivity Analysis John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 8 USA October 8 Mitchell Sensitivity Analysis / 9 Optimal tableau and pivot matrix Outline Optimal
4.6 Linear Programming duality
4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP Different spaces and objective functions but in general same optimal
Algorithmic Game Theory and Applications. Lecture 7: The LP Duality Theorem
Algorithmic Game Theory and Applications Lecture 7: The LP Duality Theorem Kousha Etessami recall LP s in Primal Form 1 Maximize c 1 x 1 + c 2 x 2 +... + c n x n a 1,1 x 1 + a 1,2 x 2 +... + a 1,n x n
Optimization - Examples Sheet 1
Easter 0 YMS Optimization - Examples Sheet. Show how to solve the problem min n i= (a i + x i ) subject to where a i > 0, i =,..., n and b > 0. n x i = b, i= x i 0 (i =,...,n). Minimize each of the following
GYULA FARKAS WOULD ALSO FEEL PROUD
GYULA FARKAS WOULD ALSO FEEL PROUD PABLO GUERRERO-GARCÍA a, ÁNGEL SANTOS-PALOMO a a Department of Applied Mathematics, University of Málaga, 29071 Málaga, Spain (Received 26 September 2003; In final form
56:270 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
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Mathematics (2011 Admission Onwards) II SEMESTER Complementary Course
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Mathematics (2011 Admission Onwards) II SEMESTER Complementary Course MATHEMATICAL ECONOMICS QUESTION BANK 1. Which of the following is a measure
The Dual Simplex Algorithm
p. 1 The Dual Simplex Algorithm Primal optimal (dual feasible) and primal feasible (dual optimal) bases The dual simplex tableau, dual optimality and the dual pivot rules Classical applications of linear
Linear Programming: Chapter 5 Duality
Linear Programming: Chapter 5 Duality Robert J. Vanderbei September 30, 2010 Slides last edited on October 5, 2010 Operations Research and Financial Engineering Princeton University Princeton, NJ 08544
Linear and Combinatorial Optimization
Linear and Combinatorial Optimization The dual of an LP-problem. Connections between primal and dual. Duality theorems and complementary slack. Philipp Birken (Ctr. for the Math. Sc.) Lecture 3: Duality
Economic Foundations of Symmetric Programming
Economic Foundations of Symmetric Programming QUIRINO PARIS University of California, Davis B 374309 CAMBRIDGE UNIVERSITY PRESS Foreword by Michael R. Caputo Preface page xv xvii 1 Introduction 1 Duality,
Special cases of linear programming
Special cases of linear programming Infeasible solution Multiple solution (infinitely many solution) Unbounded solution Degenerated solution Notes on the Simplex tableau 1. The intersection of any basic
Section 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
Math 210 Finite Mathematics Chapter 4.2 Linear Programming Problems Minimization - The Dual Problem
Math 2 Finite Mathematics Chapter 4.2 Linear Programming Problems Minimization - The Dual Problem Richard Blecksmith Dept. of Mathematical Sciences Northern Illinois University Math 2 Website: http://math.niu.edu/courses/math2.
OPTIMISATION /09 EXAM PREPARATION GUIDELINES
General: OPTIMISATION 2 2008/09 EXAM PREPARATION GUIDELINES This points out some important directions for your revision. The exam is fully based on what was taught in class: lecture notes, handouts and
LINEAR PROGRAMMING II
LINEAR PROGRAMMING II LP duality strong duality theorem bonus proof of LP duality applications Lecture slides by Kevin Wayne Last updated on 7/25/17 11:09 AM LINEAR PROGRAMMING II LP duality Strong duality
Jørgen Tind, Department of Statistics and Operations Research, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen O, Denmark.
DUALITY THEORY Jørgen Tind, Department of Statistics and Operations Research, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen O, Denmark. Keywords: Duality, Saddle point, Complementary
OPTIMISATION 2007/8 EXAM PREPARATION GUIDELINES
General: OPTIMISATION 2007/8 EXAM PREPARATION GUIDELINES This points out some important directions for your revision. The exam is fully based on what was taught in class: lecture notes, handouts and homework.
CS711008Z Algorithm Design and Analysis
.. CS711008Z Algorithm Design and Analysis Lecture 9. Algorithm design technique: Linear programming and duality Dongbo Bu Institute of Computing Technology Chinese Academy of Sciences, Beijing, China
Introduction to Mathematical Programming IE406. Lecture 10. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 10 Dr. Ted Ralphs IE406 Lecture 10 1 Reading for This Lecture Bertsimas 4.1-4.3 IE406 Lecture 10 2 Duality Theory: Motivation Consider the following
Multicommodity 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
1 Review Session. 1.1 Lecture 2
1 Review Session Note: The following lists give an overview of the material that was covered in the lectures and sections. Your TF will go through these lists. If anything is unclear or you have questions
A Polyhedral Cone Counterexample
Division of the Humanities and Social Sciences A Polyhedral Cone Counterexample KCB 9/20/2003 Revised 12/12/2003 Abstract This is an example of a pointed generating convex cone in R 4 with 5 extreme rays,
Chap6 Duality Theory and Sensitivity Analysis
Chap6 Duality Theory and Sensitivity Analysis The rationale of duality theory Max 4x 1 + x 2 + 5x 3 + 3x 4 S.T. x 1 x 2 x 3 + 3x 4 1 5x 1 + x 2 + 3x 3 + 8x 4 55 x 1 + 2x 2 + 3x 3 5x 4 3 x 1 ~x 4 0 If we
Understanding the Simplex algorithm. Standard Optimization Problems.
Understanding the Simplex algorithm. Ma 162 Spring 2011 Ma 162 Spring 2011 February 28, 2011 Standard Optimization Problems. A standard maximization problem can be conveniently described in matrix form
Chapter 1: Linear Programming
Chapter 1: Linear Programming Math 368 c Copyright 2013 R Clark Robinson May 22, 2013 Chapter 1: Linear Programming 1 Max and Min For f : D R n R, f (D) = {f (x) : x D } is set of attainable values of
Midterm Review. Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A.
Midterm Review Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye (LY, Chapter 1-4, Appendices) 1 Separating hyperplane
1. Algebraic and geometric treatments Consider an LP problem in the standard form. x 0. Solutions to the system of linear equations
The Simplex Method Most textbooks in mathematical optimization, especially linear programming, deal with the simplex method. In this note we study the simplex method. It requires basically elementary linear
Sensitivity Analysis and Duality in LP
Sensitivity Analysis and Duality in LP Xiaoxi Li EMS & IAS, Wuhan University Oct. 13th, 2016 (week vi) Operations Research (Li, X.) Sensitivity Analysis and Duality in LP Oct. 13th, 2016 (week vi) 1 /
(P ) Minimize 4x 1 + 6x 2 + 5x 3 s.t. 2x 1 3x 3 3 3x 2 2x 3 6
The exam is three hours long and consists of 4 exercises. The exam is graded on a scale 0-25 points, and the points assigned to each question are indicated in parenthesis within the text. Problem 1 Consider
The use of shadow price is an example of sensitivity analysis. Duality theory can be applied to do other kind of sensitivity analysis:
Sensitivity analysis The use of shadow price is an example of sensitivity analysis. Duality theory can be applied to do other kind of sensitivity analysis: Changing the coefficient of a nonbasic variable
IP 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 +
FIN 550 Practice Exam Answers. A. Linear programs typically have interior solutions.
FIN 550 Practice Exam Answers Phil Dybvig. True-False 25 points A. Linear programs typically have interior solutions. False. Unless the objective is zero, all solutions are at the boundary. B. A local
Lectures 6, 7 and part of 8
Lectures 6, 7 and part of 8 Uriel Feige April 26, May 3, May 10, 2015 1 Linear programming duality 1.1 The diet problem revisited Recall the diet problem from Lecture 1. There are n foods, m nutrients,
Part IB Optimisation
Part IB Optimisation Based on lectures by F. A. Fischer Notes taken by Dexter Chua Easter 015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.
Week_4: simplex method II
Week_4: simplex method II 1 1.introduction LPs in which all the constraints are ( ) with nonnegative right-hand sides offer a convenient all-slack starting basic feasible solution. Models involving (=)
A SIMPLE ALGORITHM FOR FINDING MAXIMAL NETWORK FLOWS AND AN APPLICATION TO THE HITCHCOCK PROBLEM
A SIMPLE ALGORITHM FOR FINDING MAXIMAL NETWORK FLOWS AND AN APPLICATION TO THE HITCHCOCK PROBLEM L. R. FORD, R. AND D. R. FULKERSON Introduction. The network-flow problem, originally posed by T. Harris
(includes both Phases I & II)
Minimize z=3x 5x 4x 7x 5x 4x subject to 2x x2 x4 3x6 0 x 3x3 x4 3x5 2x6 2 4x2 2x3 3x4 x5 5 and x 0 j, 6 2 3 4 5 6 j ecause of the lack of a slack variable in each constraint, we must use Phase I to find
Resource 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
Introduction 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
min 4x 1 5x 2 + 3x 3 s.t. x 1 + 2x 2 + x 3 = 10 x 1 x 2 6 x 1 + 3x 2 + x 3 14
The exam is three hours long and consists of 4 exercises. The exam is graded on a scale 0-25 points, and the points assigned to each question are indicated in parenthesis within the text. If necessary,
Minimum cost transportation problem
Minimum cost transportation problem Complements of Operations Research Giovanni Righini Università degli Studi di Milano Definitions The minimum cost transportation problem is a special case of the minimum
The Simplex Algorithm
8.433 Combinatorial Optimization The Simplex Algorithm October 6, 8 Lecturer: Santosh Vempala We proved the following: Lemma (Farkas). Let A R m n, b R m. Exactly one of the following conditions is true:.
COMP3121/9101/3821/9801 Lecture Notes. Linear Programming
COMP3121/9101/3821/9801 Lecture Notes Linear Programming LiC: Aleks Ignjatovic THE UNIVERSITY OF NEW SOUTH WALES School of Computer Science and Engineering The University of New South Wales Sydney 2052,
Introduction to Linear and Combinatorial Optimization (ADM I)
Introduction to Linear and Combinatorial Optimization (ADM I) Rolf Möhring based on the 20011/12 course by Martin Skutella TU Berlin WS 2013/14 1 General Remarks new flavor of ADM I introduce linear and
Introduction. Very efficient solution procedure: simplex method.
LINEAR PROGRAMMING Introduction Development of linear programming was among the most important scientific advances of mid 20th cent. Most common type of applications: allocate limited resources to competing
Teaching Duality in Linear Programming - the Multiplier Approach
Teaching Duality in Linear Programming - the Multiplier Approach Jens Clausen DIKU, Department of Computer Science University of Copenhagen Universitetsparken 1 DK 2100 Copenhagen Ø June 3, 1998 Abstract
Lecture 7 Duality II
L. Vandenberghe EE236A (Fall 2013-14) Lecture 7 Duality II sensitivity analysis two-person zero-sum games circuit interpretation 7 1 Sensitivity analysis purpose: extract from the solution of an LP information
Brief summary of linear programming and duality: Consider the linear program in standard form. (P ) min z = cx. x 0. (D) max yb. z = c B x B + c N x N
Brief summary of linear programming and duality: Consider the linear program in standard form (P ) min z = cx s.t. Ax = b x 0 where A R m n, c R 1 n, x R n 1, b R m 1,and its dual (D) max yb s.t. ya c.
MATH 445/545 Homework 2: Due March 3rd, 2016
MATH 445/545 Homework 2: Due March 3rd, 216 Answer the following questions. Please include the question with the solution (write or type them out doing this will help you digest the problem). I do not
March 13, Duality 3
15.53 March 13, 27 Duality 3 There are concepts much more difficult to grasp than duality in linear programming. -- Jim Orlin The concept [of nonduality], often described in English as "nondualism," is
COT 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
CS711008Z Algorithm Design and Analysis
.. CS711008Z Algorithm Design and Analysis Lecture 9. Lagrangian duality and SVM Dongbo Bu Institute of Computing Technology Chinese Academy of Sciences, Beijing, China 1 / 136 . Outline Classification
5 The Primal-Dual Simplex Algorithm
he Primal-Dual Simplex Algorithm Again, we consider the primal program given as a minimization problem defined in standard form his algorithm is based on the cognition that both optimal solutions, i.e.,
MATH 373 Section A1. Final Exam. Dr. J. Bowman 17 December :00 17:00
MATH 373 Section A1 Final Exam Dr. J. Bowman 17 December 2018 14:00 17:00 Name (Last, First): Student ID: Email: @ualberta.ca Scrap paper is supplied. No notes or books are permitted. All electronic equipment,
Math Models of OR: Some Definitions
Math Models of OR: Some Definitions John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA September 2018 Mitchell Some Definitions 1 / 20 Active constraints Outline 1 Active constraints
Introduction to linear programming using LEGO.
Introduction to linear programming using LEGO. 1 The manufacturing problem. A manufacturer produces two pieces of furniture, tables and chairs. The production of the furniture requires the use of two different
(includes both Phases I & II)
(includes both Phases I & II) Dennis ricker Dept of Mechanical & Industrial Engineering The University of Iowa Revised Simplex Method 09/23/04 page 1 of 22 Minimize z=3x + 5x + 4x + 7x + 5x + 4x subject
COMPUTATIONAL COMPLEXITY OF PARAMETRIC LINEAR PROGRAMMING +
Mathematical Programming 19 (1980) 213-219. North-Holland Publishing Company COMPUTATIONAL COMPLEXITY OF PARAMETRIC LINEAR PROGRAMMING + Katta G. MURTY The University of Michigan, Ann Arbor, MI, U.S.A.
LINEAR PROGRAMMING 2. In many business and policy making situations the following type of problem is encountered:
LINEAR PROGRAMMING 2 In many business and policy making situations the following type of problem is encountered: Maximise an objective subject to (in)equality constraints. Mathematical programming provides
- 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
Robust Farkas Lemma for Uncertain Linear Systems with Applications
Robust Farkas Lemma for Uncertain Linear Systems with Applications V. Jeyakumar and G. Li Revised Version: July 8, 2010 Abstract We present a robust Farkas lemma, which provides a new generalization of
Algorithms and Theory of Computation. Lecture 13: Linear Programming (2)
Algorithms and Theory of Computation Lecture 13: Linear Programming (2) Xiaohui Bei MAS 714 September 25, 2018 Nanyang Technological University MAS 714 September 25, 2018 1 / 15 LP Duality Primal problem
maxz = 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
that nds a basis which is optimal for both the primal and the dual problems, given
On Finding Primal- and Dual-Optimal Bases Nimrod Megiddo (revised June 1990) Abstract. We show that if there exists a strongly polynomial time algorithm that nds a basis which is optimal for both the primal
::::: OFTECHY. .0D 0 ::: ::_ I;. :.!:: t;0i f::t l. :- - :.. :?:: : ;. :--- :-.-i. .. r : : a o er -,:I :,--:-':: : :.:
,-..., -. :', ; -:._.'...,..-.-'3.-..,....; i b... {'.'',,,.!.C.,..'":',-...,'. ''.>.. r : : a o er.;,,~~~~~~~~~~~~~~~~~~~~~~~~~.'. -...~..........".: ~ WS~ "'.; :0:_: :"_::.:.0D 0 ::: ::_ I;. :.!:: t;0i
Lecture 10: Linear programming. duality. and. The dual of the LP in standard form. maximize w = b T y (D) subject to A T y c, minimize z = c T x (P)
Lecture 10: Linear programming duality Michael Patriksson 19 February 2004 0-0 The dual of the LP in standard form minimize z = c T x (P) subject to Ax = b, x 0 n, and maximize w = b T y (D) subject to
Outline. 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
Dual Basic Solutions. Observation 5.7. Consider LP in standard form with A 2 R m n,rank(a) =m, and dual LP:
Dual Basic Solutions Consider LP in standard form with A 2 R m n,rank(a) =m, and dual LP: Observation 5.7. AbasisB yields min c T x max p T b s.t. A x = b s.t. p T A apple c T x 0 aprimalbasicsolutiongivenbyx
56:171 Operations Research Midterm Exam - October 26, 1989 Instructor: D.L. Bricker
56:171 Operations Research Midterm Exam - October 26, 1989 Instructor: D.L. Bricker Answer all of Part One and two (of the four) problems of Part Two Problem: 1 2 3 4 5 6 7 8 TOTAL Possible: 16 12 20 10
Section 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
Lecture #21. c T x Ax b. maximize subject to
COMPSCI 330: Design and Analysis of Algorithms 11/11/2014 Lecture #21 Lecturer: Debmalya Panigrahi Scribe: Samuel Haney 1 Overview In this lecture, we discuss linear programming. We first show that the
Homework Assignment 4 Solutions
MTAT.03.86: Advanced Methods in Algorithms Homework Assignment 4 Solutions University of Tartu 1 Probabilistic algorithm Let S = {x 1, x,, x n } be a set of binary variables of size n 1, x i {0, 1}. Consider
Optimisation and Operations Research
Optimisation and Operations Research Lecture 9: Duality and Complementary Slackness Matthew Roughan http://www.maths.adelaide.edu.au/matthew.roughan/ Lecture_notes/OORII/
MATH2070 Optimisation
MATH2070 Optimisation Linear Programming Semester 2, 2012 Lecturer: I.W. Guo Lecture slides courtesy of J.R. Wishart Review The standard Linear Programming (LP) Problem Graphical method of solving LP problem