Mid-term Exam of Operations Research
|
|
- Dustin Osborne
- 6 years ago
- Views:
Transcription
1 Mid-term Exam of Operations Research Economics and Management School, Wuhan University November 10, 2016 ******************************************************************************** Rules: 1. No electronic device (for either computation or communication) allowed. 2. No document or paper allowed. 3. No exit of the exam within the rst 30 mins. 4. No allowance for participation of exam after 30-min late. 5. All conclusions/deductions need appropriate justications. New variable needs introduction. ********************************************************************************* Ex. 1 [LP Modeling] The Transportation Department of Wuhan University considers reoptimizing its campus-bus schedule. An OR team is employed to carry out an analysis and to give reasonable recommendations. The study seeks a minimum number of buses that can handle the transportation needs within campus. After gathering necessary information, the OR team noticed that minimum number of buses needed uctuated with the time of the day and that the required number of buses could be approximated by constant values over successive 4-hour intervals. To carry out the required daily maintenance, each bus can operate 8 successive hours a day only and is operated only once a day. The following table summarizes the team's ndings. Question [12.5'+12.5=25']. (a)[12.5'] A non-stop duration of 8 hours is called a shift. Suppose that the Transportation Department will purchase a number of buses for the plan, and each bus costs A RMB. Write a linear programming to determine the optimal bus plan that will meet the demand. 1
2 (b)[12.5'] A non-stop duration of 4 hours is called an interval. Suppose that the Transportation Department will rent buses for the plan. The rent rates are as follows: each bus/interval costs B 1 RMB, each bus/2 succesive intervals costs B 2 RMB, where B 1 > B 2 /2. Write a linear programming to determine the optimal bus plan that willl meet the demand. Solution. (a) Consider the following 6 succesive shifts: 1) 12:00 AM-8:00 AM; 2) 4:00 AM-12:00 Noon; 3) 8:00 AM-4:00 PM; 4) 12:00 Noon-8:00 PM; 5) 4:00 PM-12:00 Midnight; 6) 8:00 PM-4:00 AM. [2'] Let x 1,..., x 6 be the number of buses planned for shift 1,..., shift 6 respectively, and let z be the purchasing cost of all these buses. [2'. Most students fail to specify the shifts and the decision variables in a fully correct way] The linear programming problem is [8.5'. "-1'" for answer without x i 0; "-2'" for answer with x 1 4 the rst ineauqlity; "-2" or more for wrongly specied decision variables x i ] Min z = A(x 1 + x 2 + x 3 + x 4 + x 5 + x 6 ) x 1 + x 6 4 s.t. x 1 + x 2 8 x 2 + x 3 10 x 3 + x 4 7 x 4 + x 5 12 x 5 + x 6 4 x 1, x 2, x 3, x 4, x 5, x 6 0 (b) Consider the following 6 succesive intervals: 1) 12:00 AM-4:00 AM; 2) 4:00 AM-8:00 AM; 3) 8:00 AM-12:00 Noon; 4) 12:00 Noon-4:00 PM; 5) 4:00 PM-8:00 PM; 6) 8:00 PM-12:00 Midnight. [2'] Let x ij be the number of buses rented at the beginning of interal i and run for j succesive intervals, for i = 1, 2,..., 6 and j = 1, 2. Let z be the rental cost for the plan. [2] The linear programming problem is [8.5'] s.t. 6 Min z = B 1 x i1 + B 2 6 i=1 i=1 x i2 x 11 + x 12 + x 62 4 x 21 + x 22 + x 12 8 x 31 + x 32 + x x 41 + x 42 + x 32 7 x 51 + x 52 + x x 61 + x 62 + x 52 4 x ij 0, i = 1,..., 6, j = 1, 2 Ex. 2 [LP: simplex method, sensitivity analysis and duality] Consider the following linear programming: Max z = x 1 + 2x 2 9x 3 + 8x 4 36x 5 2
3 s.t. 2x 2 x 3 + x 4 3x 5 40 x 1 x 2 + 2x 4 2x 5 10 x 1, x 2, x 3, x 4, x 5 0 Question [15'+15'+15'+10'=55']. (a)[15'] Solve the problem using the simplex method. Write out the simplex tableau and specify the entering/exiting variable (or the pivot number) in each step. (b)[15'] Compute the new optimal solution (z, x 1,..., x 5, s 1, s 2 ) if the constant "10" on the RHS of the second constraint is reduced to 4. (c)[15'] Write down the dual of the above problem. Solve the dual graphically and check your solution with the optimal simplex tableau in primal program. (d)[10'] Solve the primal using the complementary slackness conditions. Solution. (a) Adding s 1, s 2 as slack variables into the constaints to obtain the LP in canonical form: [1'. Sucient to introduce s 1, s 2 0] Max z = x 1 + 2x 2 9x 3 + 8x 4 36x 5 2x 2 x 3 + x 4 3x 5 + s 1 = 40 s.t. x 1 x 2 + 2x 4 2x 5 + s 2 = 10 x 1, x 2, x 3, x 4, x 5, s 1, s 2 0 The initial simplex tableau is [3'. "-1'" for each error] It is not optimal since "-1,-2,-8<0". [0.5. Most students did not write it explicitly] We observe that the entering variable is x 4 and the leaving variable is s 2. [1'. It is okey if you specify the pivot number correctly] The next simplex tableau is [3'] Eq. BV z x 1 x 2 x 3 x 4 x 5 s 1 s 2 RHS (0) z (1) s (2) x It is not optimal since "-6<0". [0.5'] We observe that the entering variable is x 2 and the leaving variable is s 1. [1'] The next simplex tableau is [3'] 3
4 Eq. BV z x 1 x 2 x 3 x 4 x 5 s 1 s 2 RHS (0) z (1) x (2) x / It is now optimal since all the coecients "1.8, 0, 6.6, 0, 23.2, 2.4, 2.4" in the objective equation are nonnegative [0.5']. The optimal solution is (x 1, x 2, x 3, x 4, x 5, s 1, s 2 ) = (0, 14, 0, 12, 0, 0, 0) and the optimal value is z = 124 [0.5']. (b) We rst compute the optimum range for the RHS constant "10" (corresponding the s 2 ). By adding to 10 on the RHS, the same operations of simplex method give us the optimal simplex tableau as above except for the last column, which is now "( , , )". To have the optimal basis unchanged in the perturbed problem, we should have and Equivalently, [ 30, 70] is the optimum range [0.5']. Now "10" is reduced to 4, we have = 4 10 = 6 which is still within the optimum range so the optimal basis and the shadow prices remain unchanged. [3'+2'=5'. Most students do not have this argument. In this case, you can obtain this 5 point only if you point out that "15.2, 9.6>0" so the perturbed nal simplex tableau remains optimal. Some students solve the new problem directly using the simplex method. Points are given if the computation is correct. ] The new optimal solution is (z, x 2, x 4) = ( ( 6), ( 6), ( 6) ) = (107.2, 15.2, 9.6) [8'. "2'" for z's value, "3'" for x 2 or x 4 's value. ] and [2'. Most fail to mention. ] (x 1, x 3, x 5) = (0, 0, 0). (c) Let y 1 and y 2 be the dual variables of the two constraints. The dual LP is written as [5'] The graphical solution to the dual program is shown in Figure 1. [4'. Important to point out each line's equation.] 4
5 We see that the optimal solution is the intersection of the lines "2y 1 y 2 = 2" and "y 1 +2y 2 = 8", which is (y1, y 2 ) = (2.4, 2.8) [3'. "-1.5" if you do not mention or we can not read from the graph 'which two lines are they']. Check with the optimal simplex tableau of the primal program: (2.4, 2.8) are (s 1, s 2 )'s coecients in the optimum tableau's objective equation [3'. It is ne if you say "at optimum v = 40y y 2 = 124" as in the primal problem]. (d) For (y1, y 2 ) = (2.4, 2.8), we see that only the two constraints "2y 1 y 2 2" and "y 1 +2y 2 8" are satised with equality. This imples that the dual variables x 1, x 3, x 5 are all equal to zero [3'. It is important to show that all other three inequalities do not bind]. Moreover, both y1 = 2.4 and y 2 = 2.8 are stictly positive, this imples that 2x 2 x 3 + x 4 3x 5 40 and "x 1 x 2 + 2x 4 2x 5 10" are satised with equality [3']. We put them together to obtain { 2x 2 + x 4 = 40 x 2 + 2x 4 = 10 and (x 2, x 4 ) = (14, 12) [2'+2'=4'. Assigning "2'" only if the solution is obtained without correct argument],which are the same as obtained in (a) by the simplex method. Ex. 3 [Optimal Transportation] Use the VAM and MODI methods to nd an optimal solution to the following transportation problem [10'+10'=20']. 5
6 Solution. Phase I: looking for an initial BF solution using the VAM method [6'. Partial point is given if you obtain a dierent initial BF solution from here, by using some other method than the VAM method.] The above transportation is not optimal since K 22 = 2 < 0.[3'] [Put 2' for specifying the loop being used for a new BF solution.] Phase II: looking for an optimal solution(transportation) using the MODI method. The modied transportation is [5'. Partial point is given for minor error in the nal solution.] We verify that it is optimal [3']. Conclusion: The optimal transportation is x 12 = 5, x 14 = 15, x 22 = 10, x 23 = 15, x 31 = 10, x 34 = 5, and all other x ij = 0; The optimal transportation cost is = 460 [1']. 6
Linear Programming. Linear programming provides methods for allocating limited resources among competing activities in an optimal way.
University of Southern California Viterbi School of Engineering Daniel J. Epstein Department of Industrial and Systems Engineering ISE 330: Introduction to Operations Research - Deterministic Models Fall
More informationSolutions for Operations Research Final Exam
Solutions for Operations Research Final Exam. (a) The buffer stock is B = i a i = a + a + a + a + a + a 6 + a 7 = + + + + + + =. And the transportation tableau corresponding to the transshipment problem
More informationAM 121: Intro to Optimization Models and Methods Fall 2017
AM 121: Intro to Optimization Models and Methods Fall 2017 Lecture 10: Dual Simplex Yiling Chen SEAS Lesson Plan Interpret primal simplex in terms of pivots on the corresponding dual tableau Dictionaries
More information5. DUAL LP, SOLUTION INTERPRETATION, AND POST-OPTIMALITY
5. DUAL LP, SOLUTION INTERPRETATION, AND POST-OPTIMALITY 5.1 DUALITY Associated with every linear programming problem (the primal) is another linear programming problem called its dual. If the primal involves
More informationAdvanced Operations Research Techniques IE316. Quiz 2 Review. Dr. Ted Ralphs
Advanced Operations Research Techniques IE316 Quiz 2 Review Dr. Ted Ralphs IE316 Quiz 2 Review 1 Reading for The Quiz Material covered in detail in lecture Bertsimas 4.1-4.5, 4.8, 5.1-5.5, 6.1-6.3 Material
More informationOutline. CS38 Introduction to Algorithms. Linear programming 5/21/2014. Linear programming. Lecture 15 May 20, 2014
5/2/24 Outline CS38 Introduction to Algorithms Lecture 5 May 2, 24 Linear programming simplex algorithm LP duality ellipsoid algorithm * slides from Kevin Wayne May 2, 24 CS38 Lecture 5 May 2, 24 CS38
More informationSection Notes 5. Review of Linear Programming. Applied Math / Engineering Sciences 121. Week of October 15, 2017
Section Notes 5 Review of Linear Programming Applied Math / Engineering Sciences 121 Week of October 15, 2017 The following list of topics is an overview of the material that was covered in the lectures
More informationBCN Decision and Risk Analysis. Syed M. Ahmed, Ph.D.
Linear Programming Module Outline Introduction The Linear Programming Model Examples of Linear Programming Problems Developing Linear Programming Models Graphical Solution to LP Problems The Simplex Method
More informationSection Notes 4. Duality, Sensitivity, and the Dual Simplex Algorithm. Applied Math / Engineering Sciences 121. Week of October 8, 2018
Section Notes 4 Duality, Sensitivity, and the Dual Simplex Algorithm Applied Math / Engineering Sciences 121 Week of October 8, 2018 Goals for the week understand the relationship between primal and dual
More informationSome Advanced Topics in Linear Programming
Some Advanced Topics in Linear Programming Matthew J. Saltzman July 2, 995 Connections with Algebra and Geometry In this section, we will explore how some of the ideas in linear programming, duality theory,
More informationMarginal and Sensitivity Analyses
8.1 Marginal and Sensitivity Analyses Katta G. Murty, IOE 510, LP, U. Of Michigan, Ann Arbor, Winter 1997. Consider LP in standard form: min z = cx, subject to Ax = b, x 0 where A m n and rank m. Theorem:
More informationIntroduction to Mathematical Programming IE496. Final Review. Dr. Ted Ralphs
Introduction to Mathematical Programming IE496 Final Review Dr. Ted Ralphs IE496 Final Review 1 Course Wrap-up: Chapter 2 In the introduction, we discussed the general framework of mathematical modeling
More informationIntroduction. Linear because it requires linear functions. Programming as synonymous of planning.
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
More informationRead: H&L chapters 1-6
Viterbi School of Engineering Daniel J. Epstein Department of Industrial and Systems Engineering ISE 330: Introduction to Operations Research Fall 2006 (Oct 16): Midterm Review http://www-scf.usc.edu/~ise330
More informationDiscuss mainly the standard inequality case: max. Maximize Profit given limited resources. each constraint associated to a resource
Sensitivity Analysis Discuss mainly the standard inequality case: ma s.t. n a i, z n b, i c i,, m s.t.,,, n ma Maimize Profit given limited resources each constraint associated to a resource Alternate
More informationSimulation. Lecture O1 Optimization: Linear Programming. Saeed Bastani April 2016
Simulation Lecture O Optimization: Linear Programming Saeed Bastani April 06 Outline of the course Linear Programming ( lecture) Integer Programming ( lecture) Heuristics and Metaheursitics (3 lectures)
More informationTribhuvan University Institute Of Science and Technology Tribhuvan University Institute of Science and Technology
Tribhuvan University Institute Of Science and Technology Tribhuvan University Institute of Science and Technology Course Title: Linear Programming Full Marks: 50 Course No. : Math 403 Pass Mark: 17.5 Level
More informationMath 414 Lecture 30. The greedy algorithm provides the initial transportation matrix.
Math Lecture The greedy algorithm provides the initial transportation matrix. matrix P P Demand W ª «2 ª2 «W ª «W ª «ª «ª «Supply The circled x ij s are the initial basic variables. Erase all other values
More informationAssignment #3 - Solutions MATH 3300A (01) Optimization Fall 2015
Assignment #3 - Solutions MATH 33A (1) Optimization Fall 15 Section 6.1 1. Typical isoprofit line is 3x 1 +c x =z. This has slope -3/c. If slope of isoprofit line is
More informationMATHEMATICS II: COLLECTION OF EXERCISES AND PROBLEMS
MATHEMATICS II: COLLECTION OF EXERCISES AND PROBLEMS GRADO EN A.D.E. GRADO EN ECONOMÍA GRADO EN F.Y.C. ACADEMIC YEAR 2011-12 INDEX UNIT 1.- AN INTRODUCCTION TO OPTIMIZATION 2 UNIT 2.- NONLINEAR PROGRAMMING
More informationLinear Programming. Linear Programming. Linear Programming. Example: Profit Maximization (1/4) Iris Hui-Ru Jiang Fall Linear programming
Linear Programming 3 describes a broad class of optimization tasks in which both the optimization criterion and the constraints are linear functions. Linear Programming consists of three parts: A set of
More informationLinear Programming Motivation: The Diet Problem
Agenda We ve done Greedy Method Divide and Conquer Dynamic Programming Network Flows & Applications NP-completeness Now Linear Programming and the Simplex Method Hung Q. Ngo (SUNY at Buffalo) CSE 531 1
More informationThe Simplex Algorithm
The Simplex Algorithm Uri Feige November 2011 1 The simplex algorithm The simplex algorithm was designed by Danzig in 1947. This write-up presents the main ideas involved. It is a slight update (mostly
More informationCivil Engineering Systems Analysis Lecture XV. Instructor: Prof. Naveen Eluru Department of Civil Engineering and Applied Mechanics
Civil Engineering Systems Analysis Lecture XV Instructor: Prof. Naveen Eluru Department of Civil Engineering and Applied Mechanics Today s Learning Objectives Sensitivity Analysis Dual Simplex Method 2
More informationMath Introduction to Operations Research
Math 300 Introduction to Operations Research Examination (50 points total) Solutions. (6 pt total) Consider the following linear programming problem: Maximize subject to and x, x, x 3 0. 3x + x + 5x 3
More informationLinear Programming. Course review MS-E2140. v. 1.1
Linear Programming MS-E2140 Course review v. 1.1 Course structure Modeling techniques Linear programming theory and the Simplex method Duality theory Dual Simplex algorithm and sensitivity analysis Integer
More informationINEN 420 Final Review
INEN 420 Final Review Office Hours: Mon, May 2 -- 2:00-3:00 p.m. Tues, May 3 -- 12:45-2:00 p.m. (Project Report/Critiques due on Thurs, May 5 by 5:00 p.m.) Tuesday, April 28, 2005 1 Final Exam: Wednesday,
More informationNew Directions in Linear Programming
New Directions in Linear Programming Robert Vanderbei November 5, 2001 INFORMS Miami Beach NOTE: This is a talk mostly on pedagogy. There will be some new results. It is not a talk on state-of-the-art
More informationLinear Programming Problems
Linear Programming Problems Two common formulations of linear programming (LP) problems are: min Subject to: 1,,, 1,2,,;, max Subject to: 1,,, 1,2,,;, Linear Programming Problems The standard LP problem
More informationLecture 9: Linear Programming
Lecture 9: Linear Programming A common optimization problem involves finding the maximum of a linear function of N variables N Z = a i x i i= 1 (the objective function ) where the x i are all non-negative
More informationRyerson Polytechnic University Department of Mathematics, Physics, and Computer Science Final Examinations, April, 2003
Ryerson Polytechnic University Department of Mathematics, Physics, and Computer Science Final Examinations, April, 2003 MTH 503 - Operations Research I Duration: 3 Hours. Aids allowed: Two sheets of notes
More informationPart 4. Decomposition Algorithms Dantzig-Wolf Decomposition Algorithm
In the name of God Part 4. 4.1. Dantzig-Wolf Decomposition Algorithm Spring 2010 Instructor: Dr. Masoud Yaghini Introduction Introduction Real world linear programs having thousands of rows and columns.
More informationEcon 172A - Slides from Lecture 8
1 Econ 172A - Slides from Lecture 8 Joel Sobel October 23, 2012 2 Announcements Important: Midterm seating assignments. Posted tonight. Corrected Answers to Quiz 1 posted. Quiz 2 on Thursday at end of
More informationMATLAB Solution of Linear Programming Problems
MATLAB Solution of Linear Programming Problems The simplex method is included in MATLAB using linprog function. All is needed is to have the problem expressed in the terms of MATLAB definitions. Appendix
More informationRecap, and outline of Lecture 18
Recap, and outline of Lecture 18 Previously Applications of duality: Farkas lemma (example of theorems of alternative) A geometric view of duality Degeneracy and multiple solutions: a duality connection
More informationCOLUMN GENERATION IN LINEAR PROGRAMMING
COLUMN GENERATION IN LINEAR PROGRAMMING EXAMPLE: THE CUTTING STOCK PROBLEM A certain material (e.g. lumber) is stocked in lengths of 9, 4, and 6 feet, with respective costs of $5, $9, and $. An order for
More informationLinear programming and duality theory
Linear programming and duality theory Complements of Operations Research Giovanni Righini Linear Programming (LP) A linear program is defined by linear constraints, a linear objective function. Its variables
More informationVARIANTS OF THE SIMPLEX METHOD
C H A P T E R 6 VARIANTS OF THE SIMPLEX METHOD By a variant of the Simplex Method (in this chapter) we mean an algorithm consisting of a sequence of pivot steps in the primal system using alternative rules
More informationEcon 172A - Slides from Lecture 9
1 Econ 172A - Slides from Lecture 9 Joel Sobel October 25, 2012 2 Announcements Important: Midterm seating assignments. Posted. Corrected Answers to Quiz 1 posted. Midterm on November 1, 2012. Problems
More informationSolving Linear Programs Using the Simplex Method (Manual)
Solving Linear Programs Using the Simplex Method (Manual) GáborRétvári E-mail: retvari@tmit.bme.hu The GNU Octave Simplex Solver Implementation As part of the course material two simple GNU Octave/MATLAB
More informationThe Simplex Algorithm with a New. Primal and Dual Pivot Rule. Hsin-Der CHEN 3, Panos M. PARDALOS 3 and Michael A. SAUNDERS y. June 14, 1993.
The Simplex Algorithm with a New rimal and Dual ivot Rule Hsin-Der CHEN 3, anos M. ARDALOS 3 and Michael A. SAUNDERS y June 14, 1993 Abstract We present a simplex-type algorithm for linear programming
More informationChapter II. Linear Programming
1 Chapter II Linear Programming 1. Introduction 2. Simplex Method 3. Duality Theory 4. Optimality Conditions 5. Applications (QP & SLP) 6. Sensitivity Analysis 7. Interior Point Methods 1 INTRODUCTION
More informationArtificial Intelligence
Artificial Intelligence Combinatorial Optimization G. Guérard Department of Nouvelles Energies Ecole Supérieur d Ingénieurs Léonard de Vinci Lecture 1 GG A.I. 1/34 Outline 1 Motivation 2 Geometric resolution
More informationPivot and Gomory Cut. A MIP Feasibility Heuristic NSERC
Pivot and Gomory Cut A MIP Feasibility Heuristic Shubhashis Ghosh Ryan Hayward shubhashis@randomknowledge.net hayward@cs.ualberta.ca NSERC CGGT 2007 Kyoto Jun 11-15 page 1 problem given a MIP, find a feasible
More informationLinear Optimization. Andongwisye John. November 17, Linkoping University. Andongwisye John (Linkoping University) November 17, / 25
Linear Optimization Andongwisye John Linkoping University November 17, 2016 Andongwisye John (Linkoping University) November 17, 2016 1 / 25 Overview 1 Egdes, One-Dimensional Faces, Adjacency of Extreme
More informationIntroduction to Operations Research
- Introduction to Operations Research Peng Zhang April, 5 School of Computer Science and Technology, Shandong University, Ji nan 5, China. Email: algzhang@sdu.edu.cn. Introduction Overview of the Operations
More informationDM545 Linear and Integer Programming. Lecture 2. The Simplex Method. Marco Chiarandini
DM545 Linear and Integer Programming Lecture 2 The Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. 2. 3. 4. Standard Form Basic Feasible Solutions
More information4.1 The original problem and the optimal tableau
Chapter 4 Sensitivity analysis The sensitivity analysis is performed after a given linear problem has been solved, with the aim of studying how changes to the problem affect the optimal solution In particular,
More informationEaster Term OPTIMIZATION
DPK OPTIMIZATION Easter Term Example Sheet It is recommended that you attempt about the first half of this sheet for your first supervision and the remainder for your second supervision An additional example
More informationCOT 6936: Topics in Algorithms! Giri Narasimhan. ECS 254A / EC 2443; Phone: x3748
COT 6936: Topics in Algorithms! Giri Narasimhan ECS 254A / EC 2443; Phone: x3748 giri@cs.fiu.edu http://www.cs.fiu.edu/~giri/teach/cot6936_s12.html https://moodle.cis.fiu.edu/v2.1/course/view.php?id=174
More informationSEN301 OPERATIONS RESEARCH I PREVIUOS EXAM QUESTIONS
SEN30 OPERATIONS RESEARCH I PREVIUOS EXAM QUESTIONS. A company is involved in the production of two items (X and Y). The resources need to produce X and Y are twofold, namely machine time for automatic
More informationGraphs that have the feasible bases of a given linear
Algorithmic Operations Research Vol.1 (2006) 46 51 Simplex Adjacency Graphs in Linear Optimization Gerard Sierksma and Gert A. Tijssen University of Groningen, Faculty of Economics, P.O. Box 800, 9700
More informationHow to Solve a Standard Maximization Problem Using the Simplex Method and the Rowops Program
How to Solve a Standard Maximization Problem Using the Simplex Method and the Rowops Program Problem: Maximize z = x + 0x subject to x + x 6 x + x 00 with x 0 y 0 I. Setting Up the Problem. Rewrite each
More informationDesign and Analysis of Algorithms (V)
Design and Analysis of Algorithms (V) An Introduction to Linear Programming Guoqiang Li School of Software, Shanghai Jiao Tong University Homework Assignment 2 is announced! (deadline Apr. 10) Linear Programming
More informationCSC 8301 Design & Analysis of Algorithms: Linear Programming
CSC 8301 Design & Analysis of Algorithms: Linear Programming Professor Henry Carter Fall 2016 Iterative Improvement Start with a feasible solution Improve some part of the solution Repeat until the solution
More informationTuesday, April 10. The Network Simplex Method for Solving the Minimum Cost Flow Problem
. Tuesday, April The Network Simplex Method for Solving the Minimum Cost Flow Problem Quotes of the day I think that I shall never see A poem lovely as a tree. -- Joyce Kilmer Knowing trees, I understand
More informationCSE 40/60236 Sam Bailey
CSE 40/60236 Sam Bailey Solution: any point in the variable space (both feasible and infeasible) Cornerpoint solution: anywhere two or more constraints intersect; could be feasible or infeasible Feasible
More informationLesson 11: Duality in linear programming
Unit 1 Lesson 11: Duality in linear programming Learning objectives: Introduction to dual programming. Formulation of Dual Problem. Introduction For every LP formulation there exists another unique linear
More informationLinear Programming. Larry Blume. Cornell University & The Santa Fe Institute & IHS
Linear Programming Larry Blume Cornell University & The Santa Fe Institute & IHS Linear Programs The general linear program is a constrained optimization problem where objectives and constraints are all
More informationOptimization of Design. Lecturer:Dung-An Wang Lecture 8
Optimization of Design Lecturer:Dung-An Wang Lecture 8 Lecture outline Reading: Ch8 of text Today s lecture 2 8.1 LINEAR FUNCTIONS Cost Function Constraints 3 8.2 The standard LP problem Only equality
More information= 5. x 3,1. x 1,4. x 3,7 =15. =10 x 6,3 = 8
Network Programming Exam CSE 8 (NTU# QN M) November, Updated December, Exam Instructions ffl This take-home" exam is open book and notes. ffl The submitted exam must be your original work, achieved with
More informationPreviously Local sensitivity analysis: having found an optimal basis to a problem in standard form,
Recap, and outline of Lecture 20 Previously Local sensitivity analysis: having found an optimal basis to a problem in standard form, if the cost vectors is changed, or if the right-hand side vector is
More informationMathematical and Algorithmic Foundations Linear Programming and Matchings
Adavnced Algorithms Lectures Mathematical and Algorithmic Foundations Linear Programming and Matchings Paul G. Spirakis Department of Computer Science University of Patras and Liverpool Paul G. Spirakis
More information56:272 Integer Programming & Network Flows Final Examination -- December 14, 1998
56:272 Integer Programming & Network Flows Final Examination -- December 14, 1998 Part A: Answer any four of the five problems. (15 points each) 1. Transportation problem 2. Integer LP Model Formulation
More informationDetecting Infeasibility in Infeasible-Interior-Point. Methods for Optimization
FOCM 02 Infeasible Interior Point Methods 1 Detecting Infeasibility in Infeasible-Interior-Point Methods for Optimization Slide 1 Michael J. Todd, School of Operations Research and Industrial Engineering,
More information16.410/413 Principles of Autonomy and Decision Making
16.410/413 Principles of Autonomy and Decision Making Lecture 17: The Simplex Method Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology November 10, 2010 Frazzoli (MIT)
More informationNATCOR Convex Optimization Linear Programming 1
NATCOR Convex Optimization Linear Programming 1 Julian Hall School of Mathematics University of Edinburgh jajhall@ed.ac.uk 5 June 2018 What is linear programming (LP)? The most important model used in
More informationLecture notes on Transportation and Assignment Problem (BBE (H) QTM paper of Delhi University)
Transportation and Assignment Problems The transportation model is a special class of linear programs. It received this name because many of its applications involve determining how to optimally transport
More informationUnconstrained Optimization Principles of Unconstrained Optimization Search Methods
1 Nonlinear Programming Types of Nonlinear Programs (NLP) Convexity and Convex Programs NLP Solutions Unconstrained Optimization Principles of Unconstrained Optimization Search Methods Constrained Optimization
More information5.3 Cutting plane methods and Gomory fractional cuts
5.3 Cutting plane methods and Gomory fractional cuts (ILP) min c T x s.t. Ax b x 0integer feasible region X Assumption: a ij, c j and b i integer. Observation: The feasible region of an ILP can be described
More informationTHE simplex algorithm [1] has been popularly used
Proceedings of the International MultiConference of Engineers and Computer Scientists 207 Vol II, IMECS 207, March 5-7, 207, Hong Kong An Improvement in the Artificial-free Technique along the Objective
More informationApproximation Algorithms
Approximation Algorithms Prof. Tapio Elomaa tapio.elomaa@tut.fi Course Basics A 4 credit unit course Part of Theoretical Computer Science courses at the Laboratory of Mathematics There will be 4 hours
More informationGraphical Analysis. Figure 1. Copyright c 1997 by Awi Federgruen. All rights reserved.
Graphical Analysis For problems with 2 variables, we can represent each solution as a point in the plane. The Shelby Shelving model (see the readings book or pp.68-69 of the text) is repeated below for
More informationA Comparative study on Algorithms for Shortest-Route Problem and Some Extensions
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: No: 0 A Comparative study on Algorithms for Shortest-Route Problem and Some Extensions Sohana Jahan, Md. Sazib Hasan Abstract-- The shortest-route
More informationMath Models of OR: The Simplex Algorithm: Practical Considerations
Math Models of OR: The Simplex Algorithm: Practical Considerations John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA September 2018 Mitchell Simplex Algorithm: Practical Considerations
More informationChap5 The Theory of the Simplex Method
College of Management, NCTU Operation Research I Fall, Chap The Theory of the Simplex Method Terminology Constraint oundary equation For any constraint (functional and nonnegativity), replace its,, sign
More informationCivil Engineering Systems Analysis Lecture XIV. Instructor: Prof. Naveen Eluru Department of Civil Engineering and Applied Mechanics
Civil Engineering Systems Analysis Lecture XIV Instructor: Prof. Naveen Eluru Department of Civil Engineering and Applied Mechanics Today s Learning Objectives Dual 2 Linear Programming Dual Problem 3
More informationTransportation Problems
Transportation Problems Transportation is considered as a special case of LP Reasons? it can be formulated using LP technique so is its solution 1 (to p2) Here, we attempt to firstly define what are them
More informationChapter 1 Linear Programming. Paragraph 4 The Simplex Algorithm
Chapter Linear Programming Paragraph 4 The Simplex Algorithm What we did so far By combining ideas of a specialized algorithm with a geometrical view on the problem, we developed an algorithm idea: Find
More informationDuality. Primal program P: Maximize n. Dual program D: Minimize m. j=1 c jx j subject to n. j=1. i=1 b iy i subject to m. i=1
Duality Primal program P: Maximize n j=1 c jx j subject to n a ij x j b i, i = 1, 2,..., m j=1 x j 0, j = 1, 2,..., n Dual program D: Minimize m i=1 b iy i subject to m a ij x j c j, j = 1, 2,..., n i=1
More informationConnexions module: m Linear Equations. Rupinder Sekhon
Connexions module: m18901 1 Linear Equations Rupinder Sekhon This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License 3.0 Abstract This chapter covers
More informationUNIT 6 MODELLING DECISION PROBLEMS (LP)
UNIT 6 MODELLING DECISION This unit: PROBLEMS (LP) Introduces the linear programming (LP) technique to solve decision problems 1 INTRODUCTION TO LINEAR PROGRAMMING A Linear Programming model seeks to maximize
More informationOptimization. A first course on mathematics for economists Problem set 6: Linear programming
Optimization. A first course on mathematics for economists Problem set 6: Linear programming Xavier Martinez-Giralt Academic Year 2015-2016 6.1 A company produces two goods x and y. The production technology
More informationUnit.9 Integer Programming
Unit.9 Integer Programming Xiaoxi Li EMS & IAS, Wuhan University Dec. 22-29, 2016 (revised) Operations Research (Li, X.) Unit.9 Integer Programming Dec. 22-29, 2016 (revised) 1 / 58 Organization of this
More informationDiscrete Optimization. Lecture Notes 2
Discrete Optimization. Lecture Notes 2 Disjunctive Constraints Defining variables and formulating linear constraints can be straightforward or more sophisticated, depending on the problem structure. The
More informationMULTIMEDIA UNIVERSITY FACULTY OF ENGINEERING PEM2046 ENGINEERING MATHEMATICS IV TUTORIAL
A. Linear Programming (LP) MULTIMEDIA UNIVERSITY FACULTY OF ENGINEERING PEM046 ENGINEERING MATHEMATICS IV TUTORIAL. Identify the optimal solution and value: (a) Maximize f = 0x + 0 x (b) Minimize f = 45x
More informationHeuristic Optimization Today: Linear Programming. Tobias Friedrich Chair for Algorithm Engineering Hasso Plattner Institute, Potsdam
Heuristic Optimization Today: Linear Programming Chair for Algorithm Engineering Hasso Plattner Institute, Potsdam Linear programming Let s first define it formally: A linear program is an optimization
More informationAMATH 383 Lecture Notes Linear Programming
AMATH 8 Lecture Notes Linear Programming Jakob Kotas (jkotas@uw.edu) University of Washington February 4, 014 Based on lecture notes for IND E 51 by Zelda Zabinsky, available from http://courses.washington.edu/inde51/notesindex.htm.
More information16.410/413 Principles of Autonomy and Decision Making
16.410/413 Principles of Autonomy and Decision Making Lecture 16: Mathematical Programming I Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology November 8, 2010 E. Frazzoli
More informationAhigh school curriculum in Algebra 2 contains both solving systems of linear equations,
The Simplex Method for Systems of Linear Inequalities Todd O. Moyer, Towson University Abstract: This article details the application of the Simplex Method for an Algebra 2 class. Students typically learn
More informationAMS : Combinatorial Optimization Homework Problems - Week V
AMS 553.766: Combinatorial Optimization Homework Problems - Week V For the following problems, A R m n will be m n matrices, and b R m. An affine subspace is the set of solutions to a a system of linear
More informationA Computer Technique for Duality Theory in Linear Programs
American Journal of Applied Mathematics 2015; 3(3): 95-99 Published online April 23, 2015 (http://www.sciencepublishinggroup.com/j/ajam) doi: 10.11648/j.ajam.20150303.13 ISSN: 2330-0043 (Print); ISSN:
More informationInteger Programming as Projection
Integer Programming as Projection H. P. Williams London School of Economics John Hooker Carnegie Mellon University INFORMS 2015, Philadelphia USA A Different Perspective on IP Projection of an IP onto
More informationCopyright 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin Introduction to the Design & Analysis of Algorithms, 2 nd ed., Ch.
Iterative Improvement Algorithm design technique for solving optimization problems Start with a feasible solution Repeat the following step until no improvement can be found: change the current feasible
More informationDiscrete Optimization 2010 Lecture 5 Min-Cost Flows & Total Unimodularity
Discrete Optimization 2010 Lecture 5 Min-Cost Flows & Total Unimodularity Marc Uetz University of Twente m.uetz@utwente.nl Lecture 5: sheet 1 / 26 Marc Uetz Discrete Optimization Outline 1 Min-Cost Flows
More information11 Linear Programming
11 Linear Programming 11.1 Definition and Importance The final topic in this course is Linear Programming. We say that a problem is an instance of linear programming when it can be effectively expressed
More informationLecture 2 Optimization with equality constraints
Lecture 2 Optimization with equality constraints Constrained optimization The idea of constrained optimisation is that the choice of one variable often affects the amount of another variable that can be
More informationLinear and Integer Programming :Algorithms in the Real World. Related Optimization Problems. How important is optimization?
Linear and Integer Programming 15-853:Algorithms in the Real World Linear and Integer Programming I Introduction Geometric Interpretation Simplex Method Linear or Integer programming maximize z = c T x
More informationFebruary 10, 2005
15.053 February 10, 2005 The Geometry of Linear Programs the geometry of LPs illustrated on DTC Announcement: please turn in homework solutions now with a cover sheet 1 Goal of this Lecture 3 mathematical
More informationLab 2: Support Vector Machines
Articial neural networks, advanced course, 2D1433 Lab 2: Support Vector Machines March 13, 2007 1 Background Support vector machines, when used for classication, nd a hyperplane w, x + b = 0 that separates
More information