ENGI 5708 Design of Civil Engineering Systems
|
|
- Solomon Stephens
- 6 years ago
- Views:
Transcription
1 ENGI 5708 Design of Civil Engineering Systems Lecture 09: Characteristics of Simplex Algorithm Solutions Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland
2 Information Update Additional Class Thursday February am EN2006 Midterm Exam Monday March am EN S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
3 Lecture 09 Objective to identify characteristics of linear programming solutions using the simplex algorithm S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
4 Is the Solution Optimal? Examine Reduced Cost Coefficients Maximize Objective Function Recall Example 8-01 Z = 10 1 s 1 s Minimize Objective Function Recall Example 8-02 Z = s + 3s Unique solution S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
5 Alternate Optima? Recall Example 6-01 Objective function Z = 140 x x Unique optimal solution at D (6,4) F E Feasible Region Basic Feasible Solutions D C A B S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
6 Alternate Optima? (cont.) Consider Modified Objective Function Z = 140 x x Solve LP Using Simplex Algorithm Alternate optima solutions at C (6,2) and D (6,4) F A E Feasible Region D Basic Feasible Solutions C B S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
7 Alternate Optima? (cont.) Basis Representations at Point C Z = 140 x x {,,, } C B = s1 x2 x1 s4 s = 4 2s + 4 5s x = 2+ s 15s x Example = 8 s 1 3 s = 4 s + 15s Z = s 32s 3 2 Modified Objective Function Z = 140 x x {,,, } C B = s1 x2 x1 s4 s = 4 2s + 4 5s x = 2+ s 15s x = 8 s 1 3 s = 4 s + 15s Z = s 32s 3 2 Alternate Optima S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
8 Infeasible Solution? Recall Example 6-01 Original constraint equations 2x + 4x 28 5x + 5x 50 x1 8 x2 6 x, x 0 F A E Feasible Region D Basic Feasible Solutions C B S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
9 Infeasible Solution? (cont.) Add Constraint Equation 2x + 4x 28 5x + 5x 50 x1 8 x2 6 x x, x x2 3 F E P2 D Feasible Region Basic Feasible Solutions C Does All Slack Basis Work? P1 B S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
10 Infeasible Solution? (cont.) All Slack Basis Representation B 0 = { s, s, s, s, s } s = 28 2x 4x 1 s = 50 5x 5x s 2 = 8 x 3 1 F E s = 6 x 4 2 s = 3 + x + x 5 No P2 D Feasible Region P1 C B S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
11 Non-Trivial Starting Point All Slack/Surplus Basis Previous slide illustrates the problem Finding an initial basic feasible solution? Artificial Variable Technique Define artificial variables Transform constraint equations Assume all slack/surplus basis Solve LP using simplex algorithm Drive artificial variable from the basis Minimize objective function Find initial feasible extreme point S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
12 Artificial Variables Recall Example 5-02 Objective function Z = 3000 x x Z = 3x + 9x Constraints 40 x + 15 x x + 35 x x x x1, x2 0 No Does All Slack Basis Work? Feasible Region S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09 B A C D
13 Artificial Variables (cont.) Define Artificial Variable, A Arbitrarily large value Transform constraint equations 40 x + 15 x 100 x1 5 x x + 35 x x + 15 x + A x + 35 x + A A 5 Find feasible solution such that A = 0 Minimize A x x 2 A S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
14 Artificial Variables (cont.) Basis Representation for Origin A to enter basis Most negative basic variable (s 2 ) to leave B 0 = { s, s, s, s } x + 15 x + A s x + 35 x + A s x1 A+ s3 5 x2 A+ s4 5 s = x + 15 x + A 1 s = x + 35 x + A 2 s = 5 x + A 3 1 s = 5 x + A 4 2 Z = 3x + 9x + A= 0+ A S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
15 Artificial Variables (cont.) Basis Representation Pivot 1 Substitute A= x 35 x + s 2 B 1 = { s, A, s, s } x 1 x 2 s = x + 15 x + A 1 s = x 20 x + s s = x + 35 x + A 2 A= x 35 x + s s = 5 x + A 3 1 s = x 35x + s s = 5 x + A 4 2 s = x 36x + s Z = x 35x + s 2 Choose x 2 to enter & s 1 to leave S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
16 Artificial Variables (cont.) Basis Representation Pivot 2 Substitute s = x 20 x + s 1 2 x = 2 + x s + s x = 2 + x s + s {,,, } 2 B = x2 A s3 s4 x1 s A= x 35 x + s 2 A= 70 x + s s s = x 35x + s 3 2 s = 75 x + s s s = x 36x + s 4 2 s = 73 x + s s Z = 70 x + s s Choose x 1 to enter & A to leave S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
17 Artificial Variables (cont.) Basis Representation Pivot 3 Substitute x = + s s A {,,, } 3 B = x2 x1 s3 s4 x = 2 + x s + s A= 70 x + s s s = 75 x + s s s = 73 x + s s x = s + s A x = + s s A s = s + s A s = + s s A Z = 0 + A Optimal solution & A = S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
18 Artificial Variables (cont.) 3 Basis Representation Pivot 3 B = { x, 2 x,, 1 s3 s4} x = s + s A x = + s s A s = s + s A B C s = + s s A Z = 0 + A Optimal solution of artificial variable problem becomes initial feasible point of actual LP problem looking to solve Feasible Region S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09 A D
19 Artificial Variables (cont.) Basis Representation Point A A = 0 & basic variables obj. function {,,, } A B = x2 x1 s3 s4 s 1 x = s + s A x = + s s A s = s + s A s = + s s A x = s + s x = + s s s = s + s s = + s s Z = 3x + 9x ( ) ( ) Z = 3 + s s + 9 s + s = s + s S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
20 Artificial Variables (cont.) Basis Representation Point D Substitute s = s 34s {,,, } D B = x2 x1 s1 s4 x = s + s x = + s s s = s + s s = + s s x = 2 + s s x = 5 s 1 3 s = 130 s 34s s = 3 s s Z = 33 + s + s Optimal solution S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
21 Reading List Pike, R.W. (2001). Optimization for Engineering Systems S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
22 References ReVelle, C.S., E.E. Whitlatch, Jr. and J.R. Wright (2004). Civil and Environmental Systems Engineering 2 nd Edition, Pearson Prentice Hall ISBN S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 09
ENGI 5708 Design of Civil Engineering Systems
ENGI 5708 Design of Civil Engineering Systems Lecture 04: Graphical Solution Methods Part 1 Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University
More informationENGI 5708 Design of Civil Engineering Systems
ENGI 5708 Design of Civil Engineering Systems Lecture 10: Sensitivity Analysis Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland
More informationThe Simplex Method. Lecture 5 Standard and Canonical Forms and Setting up the Tableau. Lecture 5 Slide 1. FOMGT 353 Introduction to Management Science
The Simplex Method Lecture 5 Standard and Canonical Forms and Setting up the Tableau Lecture 5 Slide 1 The Simplex Method Formulate Constrained Maximization or Minimization Problem Convert to Standard
More informationENGI 1313 Mechanics I
ENGI 1313 Mechanics I Lecture 25: Equilibrium of a Rigid Body Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland spkenny@engr.mun.ca
More informationENGI 1313 Mechanics I
ENGI 1313 Mechanics I Lecture 43: Course Material Review Shawn Kenny, Ph.D., P.Eng. ssistant Professor aculty of Engineering and pplied Science Memorial University of Newfoundland spkenny@engr.mun.ca inal
More informationORF 307: Lecture 2. Linear Programming: Chapter 2 Simplex Methods
ORF 307: Lecture 2 Linear Programming: Chapter 2 Simplex Methods Robert Vanderbei February 8, 2018 Slides last edited on February 8, 2018 http://www.princeton.edu/ rvdb Simplex Method for LP An Example.
More informationLecture 11 Linear programming : The Revised Simplex Method
Lecture 11 Linear programming : The Revised Simplex Method 11.1 The Revised Simplex Method While solving linear programming problem on a digital computer by regular simplex method, it requires storing
More informationStandard 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
More informationENGI 1313 Mechanics I
ENGI 1313 Mechanics I Lecture 01: Course Introduction and General Principles Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland
More informationLecture 11: Post-Optimal Analysis. September 23, 2009
Lecture : Post-Optimal Analysis September 23, 2009 Today Lecture Dual-Simplex Algorithm Post-Optimal Analysis Chapters 4.4 and 4.5. IE 30/GE 330 Lecture Dual Simplex Method The dual simplex method will
More informationIE 400: Principles of Engineering Management. Simplex Method Continued
IE 400: Principles of Engineering Management Simplex Method Continued 1 Agenda Simplex for min problems Alternative optimal solutions Unboundedness Degeneracy Big M method Two phase method 2 Simplex for
More informationSimplex Algorithm Using Canonical Tableaus
41 Simplex Algorithm Using Canonical Tableaus Consider LP in standard form: Min z = cx + α subject to Ax = b where A m n has rank m and α is a constant In tableau form we record it as below Original Tableau
More informationLinear Programming and the Simplex method
Linear Programming and the Simplex method Harald Enzinger, Michael Rath Signal Processing and Speech Communication Laboratory Jan 9, 2012 Harald Enzinger, Michael Rath Jan 9, 2012 page 1/37 Outline Introduction
More information9/23/ S. Kenny, Ph.D., P.Eng. Lecture Goals. Reading List. Students will be able to: Lecture 09 Soil Retaining Structures
Lecture 09 Soil Retaining Structures Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland spkenny@mun.ca Lecture Goals Students
More informationSlack 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
More informationFebruary 17, Simplex Method Continued
15.053 February 17, 2005 Simplex Method Continued 1 Today s Lecture Review of the simplex algorithm. Formalizing the approach Alternative Optimal Solutions Obtaining an initial bfs Is the simplex algorithm
More informationCivil Engineering Systems Analysis Lecture XII. Instructor: Prof. Naveen Eluru Department of Civil Engineering and Applied Mechanics
Civil Engineering Systems Analysis Lecture XII Instructor: Prof. Naveen Eluru Department of Civil Engineering and Applied Mechanics Today s Learning Objectives Dual Midterm 2 Let us look at a complex case
More informationSystems Analysis in Construction
Systems Analysis in Construction CB312 Construction & Building Engineering Department- AASTMT by A h m e d E l h a k e e m & M o h a m e d S a i e d 3. Linear Programming Optimization Simplex Method 135
More informationIntroduction to Mathematical Programming
Introduction to Mathematical Programming Ming Zhong Lecture 22 October 22, 2018 Ming Zhong (JHU) AMS Fall 2018 1 / 16 Table of Contents 1 The Simplex Method, Part II Ming Zhong (JHU) AMS Fall 2018 2 /
More informationSimplex Method for LP (II)
Simplex Method for LP (II) Xiaoxi Li Wuhan University Sept. 27, 2017 (week 4) Operations Research (Li, X.) Simplex Method for LP (II) Sept. 27, 2017 (week 4) 1 / 31 Organization of this lecture Contents:
More information(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
More informationChapter 3, Operations Research (OR)
Chapter 3, Operations Research (OR) Kent Andersen February 7, 2007 1 Linear Programs (continued) In the last chapter, we introduced the general form of a linear program, which we denote (P) Minimize Z
More informationMATH2070 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
More informationMATH 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
More informationIntroduction to Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras
Introduction to Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Module - 03 Simplex Algorithm Lecture 15 Infeasibility In this class, we
More informationDistributed Real-Time Control Systems. Lecture Distributed Control Linear Programming
Distributed Real-Time Control Systems Lecture 13-14 Distributed Control Linear Programming 1 Linear Programs Optimize a linear function subject to a set of linear (affine) constraints. Many problems can
More information9.1 Linear Programs in canonical form
9.1 Linear Programs in canonical form LP in standard form: max (LP) s.t. where b i R, i = 1,..., m z = j c jx j j a ijx j b i i = 1,..., m x j 0 j = 1,..., n But the Simplex method works only on systems
More informationMetode Kuantitatif Bisnis. Week 4 Linear Programming Simplex Method - Minimize
Metode Kuantitatif Bisnis Week 4 Linear Programming Simplex Method - Minimize Outlines Solve Linear Programming Model Using Graphic Solution Solve Linear Programming Model Using Simplex Method (Maximize)
More informationThe Simplex Method. Formulate Constrained Maximization or Minimization Problem. Convert to Standard Form. Convert to Canonical Form
The Simplex Method 1 The Simplex Method Formulate Constrained Maximization or Minimization Problem Convert to Standard Form Convert to Canonical Form Set Up the Tableau and the Initial Basic Feasible Solution
More informationSummary 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
More informationAlgebraic Simplex Active Learning Module 4
Algebraic Simplex Active Learning Module 4 J. René Villalobos and Gary L. Hogg Arizona State University Paul M. Griffin Georgia Institute of Technology Time required for the module: 50 Min. Reading Most
More informationLinear 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
More information56:171 Operations Research Midterm Exam--15 October 2002
Name 56:171 Operations Research Midterm Exam--15 October 2002 Possible Score 1. True/False 25 _ 2. LP sensitivity analysis 25 _ 3. Transportation problem 15 _ 4. LP tableaux 15 _ Total 80 _ Part I: True(+)
More informationMath 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
More informationPrelude to the Simplex Algorithm. The Algebraic Approach The search for extreme point solutions.
Prelude to the Simplex Algorithm The Algebraic Approach The search for extreme point solutions. 1 Linear Programming-1 x 2 12 8 (4,8) Max z = 6x 1 + 4x 2 Subj. to: x 1 + x 2
More informationNotes on Simplex Algorithm
Notes on Simplex Algorithm CS 9 Staff October 8, 7 Until now, we have represented the problems geometrically, and solved by finding a corner and moving around Now we learn an algorithm to solve this without
More informationThe Primal-Dual Algorithm P&S Chapter 5 Last Revised October 30, 2006
The Primal-Dual Algorithm P&S Chapter 5 Last Revised October 30, 2006 1 Simplex solves LP by starting at a Basic Feasible Solution (BFS) and moving from BFS to BFS, always improving the objective function,
More informationmin 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,
More informationNon-Standard Constraints. Setting up Phase 1 Phase 2
Non-Standard Constraints Setting up Phase 1 Phase 2 Maximizing with Mixed Constraints Some maximization problems contain mixed constraints, like this: maximize 3x 1 + 2x 2 subject to 2x 1 + x 2 50 (standard)
More information21. 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
More informationMATH 445/545 Test 1 Spring 2016
MATH 445/545 Test Spring 06 Note the problems are separated into two sections a set for all students and an additional set for those taking the course at the 545 level. Please read and follow all of these
More informationWeek_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 (=)
More informationFarkas 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
More informationAM 121: Intro to Optimization Models and Methods
AM 121: Intro to Optimization Models and Methods Fall 2017 Lecture 2: Intro to LP, Linear algebra review. Yiling Chen SEAS Lecture 2: Lesson Plan What is an LP? Graphical and algebraic correspondence Problems
More informationSupplementary lecture notes on linear programming. We will present an algorithm to solve linear programs of the form. maximize.
Cornell University, Fall 2016 Supplementary lecture notes on linear programming CS 6820: Algorithms 26 Sep 28 Sep 1 The Simplex Method We will present an algorithm to solve linear programs of the form
More information(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
More informationDr. Maddah ENMG 500 Engineering Management I 10/21/07
Dr. Maddah ENMG 500 Engineering Management I 10/21/07 Computational Procedure of the Simplex Method The optimal solution of a general LP problem is obtained in the following steps: Step 1. Express the
More informationSensitivity Analysis
Dr. Maddah ENMG 500 /9/07 Sensitivity Analysis Changes in the RHS (b) Consider an optimal LP solution. Suppose that the original RHS (b) is changed from b 0 to b new. In the following, we study the affect
More informationLecture 5 Simplex Method. September 2, 2009
Simplex Method September 2, 2009 Outline: Lecture 5 Re-cap blind search Simplex method in steps Simplex tableau Operations Research Methods 1 Determining an optimal solution by exhaustive search Lecture
More informationPart III: A Simplex pivot
MA 3280 Lecture 31 - More on The Simplex Method Friday, April 25, 2014. Objectives: Analyze Simplex examples. We were working on the Simplex tableau The matrix form of this system of equations is called
More informationTIM 206 Lecture 3: The Simplex Method
TIM 206 Lecture 3: The Simplex Method Kevin Ross. Scribe: Shane Brennan (2006) September 29, 2011 1 Basic Feasible Solutions Have equation Ax = b contain more columns (variables) than rows (constraints),
More informationIntroduction 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
More informationIntroduction to the Simplex Algorithm Active Learning Module 3
Introduction to the Simplex Algorithm Active Learning Module 3 J. René Villalobos and Gary L. Hogg Arizona State University Paul M. Griffin Georgia Institute of Technology Background Material Almost any
More informationWater Resources Systems: Modeling Techniques and Analysis
INDIAN INSTITUTE OF SCIENCE Water Resources Systems: Modeling Techniques and Analysis Lecture - 9 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc. Summary of the previous lecture
More information(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
More information"SYMMETRIC" PRIMAL-DUAL PAIR
"SYMMETRIC" PRIMAL-DUAL PAIR PRIMAL Minimize cx DUAL Maximize y T b st Ax b st A T y c T x y Here c 1 n, x n 1, b m 1, A m n, y m 1, WITH THE PRIMAL IN STANDARD FORM... Minimize cx Maximize y T b st Ax
More informationIntroduction to Operations Research
Introduction to Operations Research (Week 4: Linear Programming: More on Simplex and Post-Optimality) José Rui Figueira Instituto Superior Técnico Universidade de Lisboa (figueira@tecnico.ulisboa.pt) March
More informationDual 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
More informationCO350 Linear Programming Chapter 6: The Simplex Method
CO50 Linear Programming Chapter 6: The Simplex Method rd June 2005 Chapter 6: The Simplex Method 1 Recap Suppose A is an m-by-n matrix with rank m. max. c T x (P ) s.t. Ax = b x 0 On Wednesday, we learned
More informationMAT016: Optimization
MAT016: Optimization M.El Ghami e-mail: melghami@ii.uib.no URL: http://www.ii.uib.no/ melghami/ March 29, 2011 Outline for today The Simplex method in matrix notation Managing a production facility The
More informationReview 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
More information1 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
More informationLinear programs Optimization Geoff Gordon Ryan Tibshirani
Linear programs 10-725 Optimization Geoff Gordon Ryan Tibshirani Review: LPs LPs: m constraints, n vars A: R m n b: R m c: R n x: R n ineq form [min or max] c T x s.t. Ax b m n std form [min or max] c
More informationLinear 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
More informationSensitivity Analysis and Duality
Sensitivity Analysis and Duality Part II Duality Based on Chapter 6 Introduction to Mathematical Programming: Operations Research, Volume 1 4th edition, by Wayne L. Winston and Munirpallam Venkataramanan
More informationIE 5531: Engineering Optimization I
IE 5531: Engineering Optimization I Lecture 5: The Simplex method, continued Prof. John Gunnar Carlsson September 22, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I September 22, 2010
More informationLinear programming Dr. Arturo S. Leon, BSU (Spring 2010)
Linear programming (Adapted from Chapter 13 Supplement, Operations and Management, 5 th edition by Roberta Russell & Bernard W. Taylor, III., Copyright 2006 John Wiley & Sons, Inc. This presentation also
More informationLecture 5. x 1,x 2,x 3 0 (1)
Computational Intractability Revised 2011/6/6 Lecture 5 Professor: David Avis Scribe:Ma Jiangbo, Atsuki Nagao 1 Duality The purpose of this lecture is to introduce duality, which is an important concept
More informationLinear Programming, Lecture 4
Linear Programming, Lecture 4 Corbett Redden October 3, 2016 Simplex Form Conventions Examples Simplex Method To run the simplex method, we start from a Linear Program (LP) in the following standard simplex
More information56:171 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
More informationChapter 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
More informationThe augmented form of this LP is the following linear system of equations:
1 Consider the following LP given in standard form: max z = 5 x_1 + 2 x_2 Subject to 3 x_1 + 2 x_2 2400 x_2 800 2 x_1 1200 x_1, x_2 >= 0 The augmented form of this LP is the following linear system of
More informationTermination, Cycling, and Degeneracy
Chapter 4 Termination, Cycling, and Degeneracy We now deal first with the question, whether the simplex method terminates. The quick answer is no, if it is implemented in a careless way. Notice that we
More informationAM 121 Introduction to Optimization: Models and Methods Example Questions for Midterm 1
AM 121 Introduction to Optimization: Models and Methods Example Questions for Midterm 1 Prof. Yiling Chen Fall 2018 Here are some practice questions to help to prepare for the midterm. The midterm will
More informationMath 354 Summer 2004 Solutions to review problems for Midterm #1
Solutions to review problems for Midterm #1 First: Midterm #1 covers Chapter 1 and 2. In particular, this means that it does not explicitly cover linear algebra. Also, I promise there will not be any proofs.
More informationMATH 4211/6211 Optimization Linear Programming
MATH 4211/6211 Optimization Linear Programming Xiaojing Ye Department of Mathematics & Statistics Georgia State University Xiaojing Ye, Math & Stat, Georgia State University 0 The standard form of a Linear
More informationSlide 1 Math 1520, Lecture 10
Slide 1 Math 1520, Lecture 10 In this lecture, we study the simplex method which is a powerful method for solving maximization/minimization problems developed in the late 1940 s. It has been implemented
More informationLecture 9 Tuesday, 4/20/10. Linear Programming
UMass Lowell Computer Science 91.503 Analysis of Algorithms Prof. Karen Daniels Spring, 2010 Lecture 9 Tuesday, 4/20/10 Linear Programming 1 Overview Motivation & Basics Standard & Slack Forms Formulating
More informationLecture 2: The Simplex method. 1. Repetition of the geometrical simplex method. 2. Linear programming problems on standard form.
Lecture 2: The Simplex method. Repetition of the geometrical simplex method. 2. Linear programming problems on standard form. 3. The Simplex algorithm. 4. How to find an initial basic solution. Lecture
More informationMotivating examples Introduction to algorithms Simplex algorithm. On a particular example General algorithm. Duality An application to game theory
Instructor: Shengyu Zhang 1 LP Motivating examples Introduction to algorithms Simplex algorithm On a particular example General algorithm Duality An application to game theory 2 Example 1: profit maximization
More informationExample Problem. Linear Program (standard form) CSCI5654 (Linear Programming, Fall 2013) Lecture-7. Duality
CSCI5654 (Linear Programming, Fall 013) Lecture-7 Duality Lecture 7 Slide# 1 Lecture 7 Slide# Linear Program (standard form) Example Problem maximize c 1 x 1 + + c n x n s.t. a j1 x 1 + + a jn x n b j
More informationOptimisation 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/
More informationExample. 1 Rows 1,..., m of the simplex tableau remain lexicographically positive
3.4 Anticycling Lexicographic order In this section we discuss two pivoting rules that are guaranteed to avoid cycling. These are the lexicographic rule and Bland s rule. Definition A vector u R n is lexicographically
More informationAM 121: Intro to Optimization
AM 121: Intro to Optimization Models and Methods Lecture 6: Phase I, degeneracy, smallest subscript rule. Yiling Chen SEAS Lesson Plan Phase 1 (initialization) Degeneracy and cycling Smallest subscript
More informationAnn-Brith Strömberg. Lecture 4 Linear and Integer Optimization with Applications 1/10
MVE165/MMG631 Linear and Integer Optimization with Applications Lecture 4 Linear programming: degeneracy; unbounded solution; infeasibility; starting solutions Ann-Brith Strömberg 2017 03 28 Lecture 4
More informationCO350 Linear Programming Chapter 8: Degeneracy and Finite Termination
CO350 Linear Programming Chapter 8: Degeneracy and Finite Termination 27th June 2005 Chapter 8: Finite Termination 1 The perturbation method Recap max c T x (P ) s.t. Ax = b x 0 Assumption: B is a feasible
More informationToday s class. Constrained optimization Linear programming. Prof. Jinbo Bi CSE, UConn. Numerical Methods, Fall 2011 Lecture 12
Today s class Constrained optimization Linear programming 1 Midterm Exam 1 Count: 26 Average: 73.2 Median: 72.5 Maximum: 100.0 Minimum: 45.0 Standard Deviation: 17.13 Numerical Methods Fall 2011 2 Optimization
More informationSpecial 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
More informationDM545 Linear and Integer Programming. Lecture 7 Revised Simplex Method. Marco Chiarandini
DM545 Linear and Integer Programming Lecture 7 Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. 2. 2 Motivation Complexity of single pivot operation
More informationChapter 2: Linear Programming Basics. (Bertsimas & Tsitsiklis, Chapter 1)
Chapter 2: Linear Programming Basics (Bertsimas & Tsitsiklis, Chapter 1) 33 Example of a Linear Program Remarks. minimize 2x 1 x 2 + 4x 3 subject to x 1 + x 2 + x 4 2 3x 2 x 3 = 5 x 3 + x 4 3 x 1 0 x 3
More informationIntroduction to Operations Research
Introduction to Operations Research (Week 5: Linear Programming: More on Simplex) José Rui Figueira Instituto Superior Técnico Universidade de Lisboa (figueira@tecnico.ulisboa.pt) March 14-15, 2016 This
More informationENGI 1313 Mechanics I
ENGI 1313 Mechanics I Lectue 03: Foce Vectos and Paallelogam Law Shawn Kenny, Ph.D., P.Eng. Assistant Pofesso Faculty of Engineeing and Applied Science Memoial Univesity of Newfoundland spkenny@eng.mun.ca
More informationCO350 Linear Programming Chapter 6: The Simplex Method
CO350 Linear Programming Chapter 6: The Simplex Method 8th June 2005 Chapter 6: The Simplex Method 1 Minimization Problem ( 6.5) We can solve minimization problems by transforming it into a maximization
More informationCSCI5654 (Linear Programming, Fall 2013) Lecture-8. Lecture 8 Slide# 1
CSCI5654 (Linear Programming, Fall 2013) Lecture-8 Lecture 8 Slide# 1 Today s Lecture 1. Recap of dual variables and strong duality. 2. Complementary Slackness Theorem. 3. Interpretation of dual variables.
More informationx 4 = 40 +2x 5 +6x x 6 x 1 = 10 2x x 6 x 3 = 20 +x 5 x x 6 z = 540 3x 5 x 2 3x 6 x 4 x 5 x 6 x x
MATH 4 A Sensitivity Analysis Example from lectures The following examples have been sometimes given in lectures and so the fractions are rather unpleasant for testing purposes. Note that each question
More informationOperations Research I: Deterministic Models
AMS 341 (Fall, 2016) Estie Arkin Operations Research I: Deterministic Models Exam 1: Thursday, September 29, 2016 READ THESE INSTRUCTIONS CAREFULLY. Do not start the exam until told to do so. Make certain
More informationThe simplex algorithm
The simplex algorithm The simplex algorithm is the classical method for solving linear programs. Its running time is not polynomial in the worst case. It does yield insight into linear programs, however,
More informationThe Avis-Kalunzy Algorithm is designed to find a basic feasible solution (BFS) of a given set of constraints. Its input: A R m n and b R m such that
Lecture 4 Avis-Kaluzny and the Simplex Method Last time, we discussed some applications of Linear Programming, such as Max-Flow, Matching, and Vertex-Cover. The broad range of applications to Linear Programming
More informationLinear programming: algebra
: algebra CE 377K March 26, 2015 ANNOUNCEMENTS Groups and project topics due soon Announcements Groups and project topics due soon Did everyone get my test email? Announcements REVIEW geometry Review geometry
More informationCO350 Linear Programming Chapter 5: Basic Solutions
CO350 Linear Programming Chapter 5: Basic Solutions 1st June 2005 Chapter 5: Basic Solutions 1 On Monday, we learned Recap Theorem 5.3 Consider an LP in SEF with rank(a) = # rows. Then x is bfs x is extreme
More information