Transportation Simplex: Initial BFS 03/20/03 page 1 of 12
|
|
- Heather Atkinson
- 6 years ago
- Views:
Transcription
1 Dennis L. ricker Dept of Mechanical & Industrial Engineering The University of Iowa & Dept of usiness Lithuania hristian ollege Transportation Simplex: Initial FS 0/0/0 page of
2 Obtaining an Initial asic Feasible Solution ommonly-used methods: Northwest-orner Method Least-ost Method Vogel s pproximation Method (VM) Russell s Method Some of these give better quality initial solutions (which often requires fewer iterations of the simplex method in optimization) than others. Transportation Simplex: Initial FS 0/0/0 page of
3 ommon to all methods: Select a cell of the current tableau (i.e., a shipment route) ssign to that route the smaller of the available supply for the row the unsatisfied demand for the column Update the available supply & unsatisfied demand accordingly ancel row with no remaining supply or column with no remaining demand but never both! Repeat these steps until no rows & columns remain. The different methods differ only in their choice of a cell in the first step. Transportation Simplex: Initial FS 0/0/0 page of
4 Northwest-orner Method simplest to explain & use lower quality starting solution more appropriately called Upper-left orner Method Rule for selecting next cell to be assigned a shipment: hoose the upper-left corner of the remaining tableau ecause costs are ignored in the selection process, the resulting solution is generally of lower quality. Transportation Simplex: Initial FS 0/0/0 page of
5 Example (NW-orner Method) 7 6 NW- orner Min{S, D } = Min{,}= The cell is selected, and the largest shipment possible () is assigned. This satisfies the demand for destination #, so the first column will now be ignored. Transportation Simplex: Initial FS 0/0/0 page of
6 NW- orner Min{S, D } = Min{,}= The northwest corner of the remaining tableau is cell. 0 We assign it a shipment which is the smaller of (the remaining supply for ) and (the demand for dstn#). We will now ignore row, since the supply is now exhausted. etc. Transportation Simplex: Initial FS 0/0/0 page 6 of
7 Least-ost Method Rule for selecting next cell to be assigned a shipment: hoose the cell in the remaining tableau which has the lowest shipping cost Note: ignore any row or column with all costs identical, e.g., a column for a dummy destination with zero costs. Transportation Simplex: Initial FS 0/0/0 page 7 of
8 Example: The lowest cost in the tableau is in cell, namely. 7 6 Lowestcost cell Min{S, D } = Min{,}= 7 We assign the smaller of the available supply () and the unsatisfied 6 0 demand (). Since the supply of has been exhausted, that row will now be ignored. Transportation Simplex: Initial FS 0/0/0 page 8 of
9 Vogel s pproximation Method (VM) Rule for selecting next cell to be assigned a shipment: ) associate with each row a penalty for not selecting the lowest cost in that row: ) namely, the difference between the lowest and nd -lowest costs ) associate with each column a penalty for not selecting the lowest column in that row. ) Identify the row or column with the largest penalty. ) Select the lowest-cost cell in that row or column. The motivation is that, by means of this choice, the largest penalty will be avoided! Transportation Simplex: Initial FS 0/0/0 page 9 of
10 Example: The penalty for a row or column is the difference between the two smallest costs. 7 6 Row penalties = = = olumn penalties: = = 6 = = The largest penalty is in the column for destination #. The smallest cost in that column is in cell. y choosing this cell, we avoid having to use cell with a cost of 6 (an increase of )! Minimum cost in column Min{S, D } = Min{,}= Transportation Simplex: Initial FS 0/0/0 page 0 of
11 Russell s Method Motivation: approximate the dual variables Ui for each source i & V for each destination j j in order to approximate the reduced cost ( ) = U + V of each cell (i,j). ij ij i j The cell with the most negative ij is then selected. The approximations : U i = largest cost in row i (ignoring those columns whose demands are already satisfied) V = largest cost in column j (ignoring those rows whose supplies are already j exhausted) Transportation Simplex: Initial FS 0/0/0 page of
12 Example: We first compute the estimates of the dual variables, and then the estimated reduced cost. 7 6 V j 7 The smallest (i.e., most U i 7 6 = (7+)= 7 = (6+)= 9 = (7+)= 7 = (6+)= 6 =7 (7+7)= 7 7 =6 (6+7)= 6 6 = (7+)= 8 = (6+)= 8 negative ) ij is 9, in both cells and. (In this example, we = (+)= 7 = (+)= 8 = (+7)= 9 = (+)= have arbitrarily chosen 0 Minimum cost in column Min{S, D } = Min{,}= cell.) Transportation Simplex: Initial FS 0/0/0 page of
(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(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 informationUNIT 4 TRANSPORTATION PROBLEM
UNIT 4 TRANSPORTATION PROLEM Structure 4.1 Introduction Objectives 4.2 Mathematical Formulation of the Transportation Problem 4.3 Methods of Finding Initial asic Feasible Solution North-West orner Rule
More informationChapter 7 TRANSPORTATION PROBLEM
Chapter 7 TRANSPORTATION PROBLEM Chapter 7 Transportation Transportation problem is a special case of linear programming which aims to minimize the transportation cost to supply goods from various sources
More informationThe Transportation Problem
CHAPTER 12 The Transportation Problem Basic Concepts 1. Transportation Problem: BASIC CONCEPTS AND FORMULA This type of problem deals with optimization of transportation cost in a distribution scenario
More informationTRANSPORTATION PROBLEMS
Chapter 6 TRANSPORTATION PROBLEMS 61 Transportation Model Transportation models deal with the determination of a minimum-cost plan for transporting a commodity from a number of sources to a number of destinations
More informationExample: 1. In this chapter we will discuss the transportation and assignment problems which are two special kinds of linear programming.
Ch. 4 THE TRANSPORTATION AND ASSIGNMENT PROBLEMS In this chapter we will discuss the transportation and assignment problems which are two special kinds of linear programming. deals with transporting goods
More informationOPERATIONS RESEARCH. Transportation and Assignment Problems
OPERATIONS RESEARCH Chapter 2 Transportation and Assignment Problems Prof. Bibhas C. Giri Department of Mathematics Jadavpur University Kolkata, India Email: bcgiri.jumath@gmail.com 1.0 Introduction In
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 informationThe Transportation Problem
11 The Transportation Problem Question 1 The initial allocation of a transportation problem, alongwith the unit cost of transportation from each origin to destination is given below. You are required to
More informationDuality 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
More informationThe Transportation Problem. Experience the Joy! Feel the Excitement! Share in the Pleasure!
The Transportation Problem Experience the Joy! Feel the Excitement! Share in the Pleasure! The Problem A company manufactures a single product at each of m factories. i has a capacity of S i per month.
More information56:171 Operations Research Fall 1998
56:171 Operations Research Fall 1998 Quiz Solutions D.L.Bricker Dept of Mechanical & Industrial Engineering University of Iowa 56:171 Operations Research Quiz
More informationTRANSPORTATION PROBLEMS
63 TRANSPORTATION PROBLEMS 63.1 INTRODUCTION A scooter production company produces scooters at the units situated at various places (called origins) and supplies them to the places where the depot (called
More informationLecture 14 Transportation Algorithm. October 9, 2009
Transportation Algorithm October 9, 2009 Outline Lecture 14 Revisit the transportation problem Simplex algorithm for the balanced problem Basic feasible solutions Selection of the initial basic feasible
More informationOPERATIONS RESEARCH. Transportation and Assignment Problems
OPERATIONS RESEARCH Chapter 2 Transportation and Assignment Problems Prof. Bibhas C. Giri Department of Mathematics Jadavpur University Kolkata, India Email: bcgiri.jumath@gmail.com MODULE - 2: Optimality
More informationMinimum 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
More informationTransportation Problem
Transportation Problem Alireza Ghaffari-Hadigheh Azarbaijan Shahid Madani University (ASMU) hadigheha@azaruniv.edu Spring 2017 Alireza Ghaffari-Hadigheh (ASMU) Transportation Problem Spring 2017 1 / 34
More informationFundamentals of Operations Research. Prof. G. Srinivasan. Indian Institute of Technology Madras. Lecture No. # 15
Fundamentals of Operations Research Prof. G. Srinivasan Indian Institute of Technology Madras Lecture No. # 15 Transportation Problem - Other Issues Assignment Problem - Introduction In the last lecture
More informationAnswer the following questions: Q1: Choose the correct answer ( 20 Points ):
Benha University Final Exam. (ختلفات) Class: 2 rd Year Students Subject: Operations Research Faculty of Computers & Informatics Date: - / 5 / 2017 Time: 3 hours Examiner: Dr. El-Sayed Badr Answer the following
More informationM.SC. MATHEMATICS - II YEAR
MANONMANIAM SUNDARANAR UNIVERSITY DIRECTORATE OF DISTANCE & CONTINUING EDUCATION TIRUNELVELI 627012, TAMIL NADU M.SC. MATHEMATICS - II YEAR DKM24 - OPERATIONS RESEARCH (From the academic year 2016-17)
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 informationc) Place the Coefficients from all Equations into a Simplex Tableau, labeled above with variables indicating their respective columns
BUILDING A SIMPLEX TABLEAU AND PROPER PIVOT SELECTION Maximize : 15x + 25y + 18 z s. t. 2x+ 3y+ 4z 60 4x+ 4y+ 2z 100 8x+ 5y 80 x 0, y 0, z 0 a) Build Equations out of each of the constraints above by introducing
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 informationSEN301 OPERATIONS RESEARCH I LECTURE NOTES
SEN30 OPERATIONS RESEARCH I LECTURE NOTES SECTION II (208-209) Y. İlker Topcu, Ph.D. & Özgür Kabak, Ph.D. Acknowledgements: We would like to acknowledge Prof. W.L. Winston's "Operations Research: Applications
More informationTo Obtain Initial Basic Feasible Solution Physical Distribution Problems
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 9 (2017), pp. 4671-4676 Research India Publications http://www.ripublication.com To Obtain Initial Basic Feasible Solution
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 informationW P 1 30 / 10 / P 2 25 / 15 / P 3 20 / / 0 20 / 10 / 0 35 / 20 / 0
11 P 1 and W 1 with shipping cost. The column total (i.e. market requirement) corresponding to this cell is 2 while the row total (Plant capacity) is. So we allocate 2 units to this cell. Not the market
More information9.5 THE SIMPLEX METHOD: MIXED CONSTRAINTS
SECTION 9.5 THE SIMPLEX METHOD: MIXED CONSTRAINTS 557 9.5 THE SIMPLEX METHOD: MIXED CONSTRAINTS In Sections 9. and 9., you looked at linear programming problems that occurred in standard form. The constraints
More information4.4 The Simplex Method and the Standard Minimization Problem
. The Simplex Method and the Standard Minimization Problem Question : What is a standard minimization problem? Question : How is the standard minimization problem related to the dual standard maximization
More informationUNIVERSITY of LIMERICK
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Department of Mathematics & Statistics Faculty of Science and Engineering END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MS4303 SEMESTER: Spring 2018 MODULE TITLE:
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 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 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 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 informationSimplex tableau CE 377K. April 2, 2015
CE 377K April 2, 2015 Review Reduced costs Basic and nonbasic variables OUTLINE Review by example: simplex method demonstration Outline Example You own a small firm producing construction materials for
More informationMVE165/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
More informationMath Models of OR: The Revised Simplex Method
Math Models of OR: The Revised Simplex Method John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA September 2018 Mitchell The Revised Simplex Method 1 / 25 Motivation Outline 1
More informationSummary 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:
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 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 informationThe Simplex Method. Standard form (max) z c T x = 0 such that Ax = b.
The Simplex Method Standard form (max) z c T x = 0 such that Ax = b. The Simplex Method Standard form (max) z c T x = 0 such that Ax = b. Build initial tableau. z c T 0 0 A b The Simplex Method Standard
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 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 informationMath 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.
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 informationLecture Simplex Issues: Number of Pivots. ORIE 6300 Mathematical Programming I October 9, 2014
ORIE 6300 Mathematical Programming I October 9, 2014 Lecturer: David P. Williamson Lecture 14 Scribe: Calvin Wylie 1 Simplex Issues: Number of Pivots Question: How many pivots does the simplex algorithm
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 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 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 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 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 informationD1 D2 D3 - 50
CSE 8374 QM 721N Network Flows: Transportation Problem 1 Slide 1 Slide 2 The Transportation Problem The uncapacitated transportation problem is one of the simplest of the pure network models, provides
More informationMULTIPLE CHOICE QUESTIONS DECISION SCIENCE
MULTIPLE CHOICE QUESTIONS DECISION SCIENCE 1. Decision Science approach is a. Multi-disciplinary b. Scientific c. Intuitive 2. For analyzing a problem, decision-makers should study a. Its qualitative aspects
More informationIntroduction to Mathematical Programming IE406. Lecture 13. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 13 Dr. Ted Ralphs IE406 Lecture 13 1 Reading for This Lecture Bertsimas Chapter 5 IE406 Lecture 13 2 Sensitivity Analysis In many real-world problems,
More informationThe 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
More informationDeterministic Operations Research, ME 366Q and ORI 391 Chapter 2: Homework #2 Solutions
Deterministic Operations Research, ME 366Q and ORI 391 Chapter 2: Homework #2 Solutions 11. Consider the following linear program. Maximize z = 6x 1 + 3x 2 subject to x 1 + 2x 2 2x 1 + x 2 20 x 1 x 2 x
More informationF 1 F 2 Daily Requirement Cost N N N
Chapter 5 DUALITY 5. The Dual Problems Every linear programming problem has associated with it another linear programming problem and that the two problems have such a close relationship that whenever
More informationConnecting Calculus to Linear Programming
Connecting Calculus to Marcel Y., Ph.D. Worcester Polytechnic Institute Dept. of Mathematical Sciences July 27 Motivation Goal: To help students make connections between high school math and real world
More information3. Duality: What is duality? Why does it matter? Sensitivity through duality.
1 Overview of lecture (10/5/10) 1. Review Simplex Method 2. Sensitivity Analysis: How does solution change as parameters change? How much is the optimal solution effected by changing A, b, or c? How much
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 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 informationPart 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
More informationLecture 2: The Simplex method
Lecture 2 1 Linear and Combinatorial Optimization Lecture 2: The Simplex method Basic solution. The Simplex method (standardform, b>0). 1. Repetition of basic solution. 2. One step in the Simplex algorithm.
More informationLinear Programming: Simplex Method CHAPTER The Simplex Tableau; Pivoting
CHAPTER 5 Linear Programming: 5.1. The Simplex Tableau; Pivoting Simplex Method In this section we will learn how to prepare a linear programming problem in order to solve it by pivoting using a matrix
More informationCHAPTER SOLVING MULTI-OBJECTIVE TRANSPORTATION PROBLEM USING FUZZY PROGRAMMING TECHNIQUE-PARALLEL METHOD 40 3.
CHAPTER - 3 40 SOLVING MULTI-OBJECTIVE TRANSPORTATION PROBLEM USING FUZZY PROGRAMMING TECHNIQUE-PARALLEL METHOD 40 3.1 INTRODUCTION 40 3.2 MULTIOBJECTIVE TRANSPORTATION PROBLEM 41 3.3 THEOREMS ON TRANSPORTATION
More informationMath 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
More information56:171 Fall 2002 Operations Research Quizzes with Solutions
56:7 Fall Operations Research Quizzes with Solutions Instructor: D. L. Bricker University of Iowa Dept. of Mechanical & Industrial Engineering Note: In most cases, each quiz is available in several versions!
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 informationThe 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
More informationBalance An Unbalanced Transportation Problem By A Heuristic approach
International Journal of Mathematics And Its Applications Vol.1 No.1 (2013), pp.12-18(galley Proof) ISSN:(online) Balance An Unbalanced Transportation Problem By A Heuristic approach Nigus Girmay and Tripti
More informationDr. S. Bourazza Math-473 Jazan University Department of Mathematics
Dr. Said Bourazza Department of Mathematics Jazan University 1 P a g e Contents: Chapter 0: Modelization 3 Chapter1: Graphical Methods 7 Chapter2: Simplex method 13 Chapter3: Duality 36 Chapter4: Transportation
More information3 The Simplex Method. 3.1 Basic Solutions
3 The Simplex Method 3.1 Basic Solutions In the LP of Example 2.3, the optimal solution happened to lie at an extreme point of the feasible set. This was not a coincidence. Consider an LP in general form,
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 informationInteger Programming. The focus of this chapter is on solution techniques for integer programming models.
Integer Programming Introduction The general linear programming model depends on the assumption of divisibility. In other words, the decision variables are allowed to take non-negative integer as well
More informationLinear programming. Saad Mneimneh. maximize x 1 + x 2 subject to 4x 1 x 2 8 2x 1 + x x 1 2x 2 2
Linear programming Saad Mneimneh 1 Introduction Consider the following problem: x 1 + x x 1 x 8 x 1 + x 10 5x 1 x x 1, x 0 The feasible solution is a point (x 1, x ) that lies within the region defined
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 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 informationΩ R n is called the constraint set or feasible set. x 1
1 Chapter 5 Linear Programming (LP) General constrained optimization problem: minimize subject to f(x) x Ω Ω R n is called the constraint set or feasible set. any point x Ω is called a feasible point We
More informationx 1 + 4x 2 = 5, 7x 1 + 5x 2 + 2x 3 4,
LUNDS TEKNISKA HÖGSKOLA MATEMATIK LÖSNINGAR LINJÄR OCH KOMBINATORISK OPTIMERING 2018-03-16 1. a) The rst thing to do is to rewrite the problem so that the right hand side of all constraints are positive.
More informationOptimization 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
More informationThe dual simplex method with bounds
The dual simplex method with bounds Linear programming basis. Let a linear programming problem be given by min s.t. c T x Ax = b x R n, (P) where we assume A R m n to be full row rank (we will see in the
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 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 informationProfessor Alan H. Stein October 31, 2007
Mathematics 05 Professor Alan H. Stein October 3, 2007 SOLUTIONS. For most maximum problems, the contraints are in the form l(x) k, where l(x) is a linear polynomial and k is a positive constant. Explain
More information4.3 Minimizing & Mixed Constraints
Mathematics : Mattingly, Fall 6 8 4. Minimizing & Mixed Constraints So far, you have seen how to solve one type of problem: Standard Maximum. The objective function is to be maximized.. Constraints use..
More informationRevised Simplex Method
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 informationIE 5531: Engineering Optimization I
IE 5531: Engineering Optimization I Lecture 7: Duality and applications Prof. John Gunnar Carlsson September 29, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I September 29, 2010 1
More informationTransportation, Transshipment, and Assignment Problems
Transportation, Transshipment, and Assignment Problems Riset Operasi 1 6-1 Chapter Topics The Transportation Model Computer Solution of a Transportation Problem The Transshipment Model Computer Solution
More information1. 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
More informationSection Notes 9. IP: Cutting Planes. Applied Math 121. Week of April 12, 2010
Section Notes 9 IP: Cutting Planes Applied Math 121 Week of April 12, 2010 Goals for the week understand what a strong formulations is. be familiar with the cutting planes algorithm and the types of cuts
More 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 informationNonlinear Programming (Hillier, Lieberman Chapter 13) CHEM-E7155 Production Planning and Control
Nonlinear Programming (Hillier, Lieberman Chapter 13) CHEM-E7155 Production Planning and Control 19/4/2012 Lecture content Problem formulation and sample examples (ch 13.1) Theoretical background Graphical
More informationChap6 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
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 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 informationi.e., into a monomial, using the Arithmetic-Geometric Mean Inequality, the result will be a posynomial approximation!
Dennis L. Bricker Dept of Mechanical & Industrial Engineering The University of Iowa i.e., 1 1 1 Minimize X X X subject to XX 4 X 1 0.5X 1 Minimize X X X X 1X X s.t. 4 1 1 1 1 4X X 1 1 1 1 0.5X X X 1 1
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 informationChapter 5 Linear Programming (LP)
Chapter 5 Linear Programming (LP) General constrained optimization problem: minimize f(x) subject to x R n is called the constraint set or feasible set. any point x is called a feasible point We consider
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 information