C&O 370 Deterministic OR Models Winter 2011


 Lucas Hunt
 1 years ago
 Views:
Transcription
1 C&O 370 Deterministic OR Models Winter 2011 Assignment 1 Due date: Friday Jan. 21, 2011 Assignments are due at the start of class on the due date. Write your name and ID# clearly, and underline your last name. Contents 1 Problem 1: LP Review/Duality/AMPL 10 Marks 2 2 LP Formulation and AMPL; a Transportation Problem  10 Marks 5 3 LP Formulation Manufacturing 10 Marks 8 1
2 1 Problem 1: LP Review/Duality/AMPL 10 Marks Consider the primal linear programming problem, (P), given in AMPL form. (The file with the AMPL problem is also available here: examp1.mod.) var x1; var x2; var x3; var x4; var x5; var x6; var x7; var x8; minimize Expense: +(6)*x1+(9)*x2+(12)*x3+(14)*x4+(23)*x5+(5)*x6+(18)*x7 +(2)*x8; subject to T1: +(3)*x1+(5)*x2+(3)*x3+(4)*x4+(2)*x5+(2)*x6+(9)*x7 <= 6; subject to T2: +(5)*x1+(2)*x2+(15)*x3+(5)*x4+(6)*x5+(3)*x6+(6)*x7 x8= 2; subject to T3: +(3)*x1+(0)*x2+(2)*x3+(3)*x4+(10)*x5+(0)*x6+(4)*x7 <= 44; subject to T4: +(3)*x1+(6)*x2+(1)*x3+(5)*x4+(7)*x5+(5)*x6+(6)*x7 +(3)*x8 <= 9; subject to xlimit1: x1 >=0; subject to xlimit2: x2 >=0; subject to xlimit3: x3 >=0; subject to xlimit4: x4 <=0; subject to xlimit5: x5 <=0; subject to xlimit6: x6 >=0; subject to xlimit7: x7 <=0; 1. Write down the dual problem (D) of (P). (In the dual problem, the constraints should be a mixture of,, = constraints. The variables should be a mixture of,, and free.) Solution 1.1 The primal problem is: min c T x s.t. Ax ineq b b, x ineq x 0, x R 8, where ineq b, ineq x, represent the appropriate inequalities/equality for the constraints and the variables, respectively. (Note that this problem was generated randomly using the file available here: lpexamp1.m.) Then, the dual program (D) is: max b T y s.t. c ineq x A T y, y ineq b 0, y R 4. Note that the inequalities for constraints go with the inequaities for variables. (With 2
3 an equality constraint corresponding to a free variable.) We get the following: max subject to ( ) y = y y 1, y 3, y 4 0, y 2 free. 2. Write down all the complementary slackness conditions for (P),(D). Solution 1.2 We can write the primal constraint as the equality Ax s = b and the dual constraint as the equality A T y + z = c, where the slack variables s = Ax b ineq b 0, z = c A T y ineq x 0 have the appropriate signs given by their corresponding inequalities. Then the complementary slackness constraints are given by This is equivalent to x T z = s T y = 0. x z = 0 R 7, s y = 0 R 4, where corresponds to the componentwise/elementwise product (also called the Hadamard product). Therefore, all the inequalities can be expressed as x (A T y c) = 0 R 8, (Ax b) y = 0 R 4. 3
4 Explicitly, the complementary slackness conditions for the primal (P) are ( [ 3 ] ) x y ( 6) = 0 ( [5 ] ) x y ( 9) = 0 ( [3 ] ) x y 12 = 0 ( [4 ] ) x y ( 14) = 0 ( [ 2 ] ) x y ( 23) = 0 ( [2 ] ) x y 5 = 0 ( [9 ] ) x y ( 18) = 0 ( [0 ] ) x y ( 2) = 0 and the complementary slackness conditions for the dual (D) are ( [ 3 ] ) y x ( 6) ( [5 ] ) y x 2 ( [ 3 ] ) y x ( 44) ( [ 3 ] ) y x ( 9) = 0 = 0 = 0 = 0 3. Consider the following possible vector of solutions for (P). x = ( ) T First, use the given data and confirm that this vector provides a feasible solution to (P). Then, use part 2 to find a solution to (D) and also, to show (provide the details) whether or not the vector x above is optimal for (P). Solution 1.3 Substituting the given (approximate) solution x into the equality constraint T2 and solving for the unknown x 8 yields x All primal constraints except T1 are active at x. By the complementary slackness conditions for y, we conclude that the first component of the dual variable is y 1 = 0. 4
5 On the other hand, all entries of x except x 1, x 2, x 7, x 8 are approximately equal to 0. The complementary slackness conditions using x imply that the (four) 1, 2, 7, 8 constraints of the dual are active. This yields the linear system of equations y 2 y 3 = y The unique solution to this system yields the optimal dual solution 0 y = 5/ /12 (1) Since this vector has the correct signs, we conclude that we have a certificate of optimality for x. 4. Is the dual optimal solution unique? Is the primal optimal solution unique? Solution 1.4 Uniqueness holds for the dual, since the solution of the equations in (1) is unique. For the primal, the value for y above implies that the last three constraints hold as equality. Moreover, all dual constraints except those corresponding to x 3, x 4, x 5, x 6 are active at y. This implies that we have three equations with four unknowns, i.e. we only conclude that x 3 = x 4 = x 5 = x 6 = 0. We now have to check whether there are any other solutions x with the correct sign pattern. Another optimal solution (to 4 decimals) (confirmation of optimality can be done as above for x) is x = ( ) T. 2 LP Formulation and AMPL; a Transportation Problem  10 Marks Suppose that there are two canning plants (at Halifax, Winnipeg) and three markets (at Montreal, Toronto, Vancouver). Table 1 provides the data; shipping distances are in thousands of KM, shipping costs are assumed to be $90.00 per case per thousand KM, and supplies (and demands) are in numbers of cases. 5
6 Markets Montreal Toronto Vancouver Plants Shipping Distances Supplies Halifax Winnipeg Demands Table 1: shipping data for Problem 2 1. Formulate an LP problem for minimizing the transportation cost while meeting customer demand and satisfying the supply constraints. (Your solution should include a description of the sets, the main decision variables, and the constraints.) Solution 2.1 The sets are: set plants; set markets. The decision variables are: x ij  the number of cases to be shipped from plant i to market j. Then the LP becomes: min 1.244x x x x x x 23 objective s.t. 3 j=1 x 1j 350 supply constraint 3 j=1 x 2j 600 supply constraint 2 i=1 x i1 325 demand constraint 2 i=1 x i2 300 demand constraint 2 i=1 x i3 275 demand constraint x ij 0, i, j Note that the supply is 950 cases while the demand is only 900 cases. For transportation problems one often sets up a dummy market with an appropriate demand to make the supply and demand equal. In that case, one can replace the inequality constraints with equality constraints. 2. Solve the LP using the AMPL software. Submit a printed version of your LP model (including any data files), and a log of your session on AMPL that shows (i) the optimal value, and (ii) an optimal solution. Solution 2.2 The file trans.mod is: set Plant; set Market; # set of plants # set of markets param Supply {i in Plant}; param Demand {j in Market}; param ShipDist {i in Plant, j in Market}; # number of cases # number of cases # thousand KMs 6
7 var x {Plant, Market} >= 0; # cases to ship from each plant to each market minimize TransDist: sum{ i in Plant, j in Market } ShipDist[i,j] * x[i,j]; subject to PlantSupply {i in Plant}: sum{ j in Market } x[i,j] <= Supply[i]; subject to MarketDemand {j in Market}: sum{ i in Plant } x[i,j] >= Demand[j]; The file trans.dat is: set Plant := Halifax Winnipeg; set Market := Montreal, Toronto, Vancouver; param: Supply := Halifax 350 Winnipeg 600 ; param: Demand := Montreal 325 Toronto 300 Vancouver 275 ; param ShipDist: Montreal Toronto Vancouver := Halifax Winnipeg ; The output from AMPL is: ILOG AMPL , licensed to "universitywaterloo, canada". AMPL Version (SunOS 5.9) ampl: model trans.mod ampl: data trans.dat ampl: solve; ILOG CPLEX , licensed to "universitywaterloo, canada", options: e m b q CPLEX : optimal solution; objective dual simplex iterations (0 in phase I) ampl: display x; x := Halifax Montreal 325 Halifax Toronto 25 Halifax Vancouver 0 7
8 Winnipeg Montreal 0 Winnipeg Toronto 275 Winnipeg Vancouver 275 ; ampl: display PlantSupply; PlantSupply [*] := Halifax Winnipeg 0 ; ampl: display MarketDemand; MarketDemand [*] := Montreal Toronto Vancouver ; ampl: quit; 3. Repeat part 2 but replace the first number of supplies (350) and the first number of demands (325) using the two three digit numbers formed from: the first 3 digits of your student ID, and the second 3 digits of your student ID. The larger of these two numbers replaces the supply and the smaller number replaces the demand. 3 LP Formulation Manufacturing 10 Marks A liquor company produces and sells two kinds of liquor: blended whiskey and bourbon. The company purchases intermediate products in bulk, purifies them by repeated distillation, mixes them, and bottles the final product under their own brand names. In the past, the firm has always been able to sell all that it produced. The firm has been limited by its machine capacity and available cash. The bourbon requires 3 machine hours per bottle while, due to additional blending requirements, the blended whiskey requires 4 hours of machine time per bottle. There are 20,000 machine hours available in the current production period. The direct operating costs, which are principally for labor and materials, are $3.00 per bottle of bourbon and $2.00 per bottle of blended whiskey. The working capital available to finance labor and material is $4000; however, 45% of the bourbon sales revenues and 30% of the blendedwhiskey sales revenues from production in the current period will be collected during the current period and be available to finance operations. The selling price to the distributor is $6 per bottle of bourbon and $5.40 per bottle of blended whiskey. 8
9 1. Formulate a linear program that maximizes contribution subject to limitations on machine capacity and working capital. (Your solution should include a description of the sets, the main decision variables, and the constraints.) Solution 3.1 The sets are the (two) types of whiskey and the (two) types of constraints (constraints on machine hours and available cash). The variables are the number of bottles of each time of whiskey. We denote the number of bottles for the blended whiskey and the bourbon by x W and x B, respectively. max 5.4x W + 6x B objective s.t. 4x W + 3x B machine hours constraint 2x W + 3x B (.3)(5.4)x W + (.45)(6)x B available cash constraint x W, x B 0 Note that the second constraint is equivalent to.38x W +.3x B The slope of the level line for the objective function is: slope obj = 5.4/6 =.9; while the slopes for the constraints are slope hrs = 4/3 and slope cash =.1407, respectively. 2. What is the optimal production mix to schedule? Solution 3.2 From looking at the graph of the feasible set and the level lines of the objective function, we see that only the first constraint is active and that the optimal solution (production mix) is x W = 0, x B = 20000/3. 3. Can the selling prices change without changing the optimal production mix? Solution 3.3 As mentioned above, the slopes slope obj = 5.4/6 =.9 > slope hrs = 4/3. Therefore, we can change the selling price as long as the inequality of the slopes does not change, i.e. as long as slope obj = ( W )/(6 + B ) slope hrs = 4/3. The intervals for the objective coefficients are: < c W 8 and 4.05 c B < Suppose that the company could spend $400 to repair some machinery and increase its available machine hours by 2000 hours. Should the investment be made? Solution 3.4 The increase in the income would be $2000(6)/3, as the constraint for available cash is not violated. Therefore, it is worthwhile to ivest the $ What interest rate could the company afford to pay to borrow funds to finance its operations during the current period? Solution 3.5 The cash constraint is not active. Therefore, it does not pay to borrow funds. 9
4.6 Linear Programming duality
4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal
More informationIEOR 4404 Homework #2 Intro OR: Deterministic Models February 14, 2011 Prof. Jay Sethuraman Page 1 of 5. Homework #2
IEOR 4404 Homework # Intro OR: Deterministic Models February 14, 011 Prof. Jay Sethuraman Page 1 of 5 Homework #.1 (a) What is the optimal solution of this problem? Let us consider that x 1, x and x 3
More informationDefinition of a Linear Program
Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1
More information56:171. Operations Research  Sample Homework Assignments Fall 1992 Dennis Bricker Dept. of Industrial Engineering University of Iowa.
56:171 Operations Research  Sample Homework Assignments Fall 1992 Dennis Bricker Dept. of Industrial Engineering University of Iowa Homework #1 (1.) Linear Programming Model Formulation. SunCo processes
More informationLinear Programming: Chapter 11 Game Theory
Linear Programming: Chapter 11 Game Theory Robert J. Vanderbei October 17, 2007 Operations Research and Financial Engineering Princeton University Princeton, NJ 08544 http://www.princeton.edu/ rvdb RockPaperScissors
More information56:171 Operations Research Midterm Exam Solutions Fall 2001
56:171 Operations Research Midterm Exam Solutions Fall 2001 True/False: Indicate by "+" or "o" whether each statement is "true" or "false", respectively: o_ 1. If a primal LP constraint is slack at the
More informationLinear Programming. March 14, 2014
Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1
More informationLinear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.
1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that
More informationUNIT 1 LINEAR PROGRAMMING
OUTLINE Session : Session 2: Session 3: Session 4: Session 5: Session 6: Session 7: Session 8: Session 9: Session 0: Session : Session 2: UNIT LINEAR PROGRAMMING Introduction What is Linear Programming
More informationLinear Programming Notes V Problem Transformations
Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material
More information1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.
Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S
More informationLinear Programming Supplement E
Linear Programming Supplement E Linear Programming Linear programming: A technique that is useful for allocating scarce resources among competing demands. Objective function: An expression in linear programming
More informationLinear Programming. April 12, 2005
Linear Programming April 1, 005 Parts of this were adapted from Chapter 9 of i Introduction to Algorithms (Second Edition) /i by Cormen, Leiserson, Rivest and Stein. 1 What is linear programming? The first
More informationSensitivity Analysis 3.1 AN EXAMPLE FOR ANALYSIS
Sensitivity Analysis 3 We have already been introduced to sensitivity analysis in Chapter via the geometry of a simple example. We saw that the values of the decision variables and those of the slack and
More informationPractical Guide to the Simplex Method of Linear Programming
Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear
More informationLinear Programming Notes VII Sensitivity Analysis
Linear Programming Notes VII Sensitivity Analysis 1 Introduction When you use a mathematical model to describe reality you must make approximations. The world is more complicated than the kinds of optimization
More informationProximal mapping via network optimization
L. Vandenberghe EE236C (Spring 234) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:
More informationThe Graphical Simplex Method: An Example
The Graphical Simplex Method: An Example Consider the following linear program: Max 4x 1 +3x Subject to: x 1 +3x 6 (1) 3x 1 +x 3 () x 5 (3) x 1 +x 4 (4) x 1, x 0. Goal: produce a pair of x 1 and x that
More informationSolving Linear Programs using Microsoft EXCEL Solver
Solving Linear Programs using Microsoft EXCEL Solver By Andrew J. Mason, University of Auckland To illustrate how we can use Microsoft EXCEL to solve linear programming problems, consider the following
More informationLinear Programming. Solving LP Models Using MS Excel, 18
SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting
More informationLinear Programming I
Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins
More informationLinear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.
Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.
More information3.1 Solving Systems Using Tables and Graphs
Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system
More information1. Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded:
Final Study Guide MATH 111 Sample Problems on Algebra, Functions, Exponents, & Logarithms Math 111 Part 1: No calculator or study sheet. Remember to get full credit, you must show your work. 1. Determine
More informationWhat is Linear Programming?
Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to
More informationThe application of linear programming to management accounting
The application of linear programming to management accounting Solutions to Chapter 26 questions Question 26.16 (a) M F Contribution per unit 96 110 Litres of material P required 8 10 Contribution per
More informationThe Graphical Method: An Example
The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,
More informationMathematical finance and linear programming (optimization)
Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may
More information1 Solving LPs: The Simplex Algorithm of George Dantzig
Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.
More informationLECTURE 5: DUALITY AND SENSITIVITY ANALYSIS. 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method
LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method Introduction to dual linear program Given a constraint matrix A, right
More informationDuality in General Programs. Ryan Tibshirani Convex Optimization 10725/36725
Duality in General Programs Ryan Tibshirani Convex Optimization 10725/36725 1 Last time: duality in linear programs Given c R n, A R m n, b R m, G R r n, h R r : min x R n c T x max u R m, v R r b T
More informationStandard Form of a Linear Programming Problem
494 CHAPTER 9 LINEAR PROGRAMMING 9. THE SIMPLEX METHOD: MAXIMIZATION For linear programming problems involving two variables, the graphical solution method introduced in Section 9. is convenient. However,
More informationLecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization
Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a realvalued
More informationChapter 3 LINEAR PROGRAMMING GRAPHICAL SOLUTION 3.1 SOLUTION METHODS 3.2 TERMINOLOGY
Chapter 3 LINEAR PROGRAMMING GRAPHICAL SOLUTION 3.1 SOLUTION METHODS Once the problem is formulated by setting appropriate objective function and constraints, the next step is to solve it. Solving LPP
More informationLesson 22: Solution Sets to Simultaneous Equations
Student Outcomes Students identify solutions to simultaneous equations or inequalities; they solve systems of linear equations and inequalities either algebraically or graphically. Classwork Opening Exercise
More informationUsing CPLEX. =5 has objective value 150.
Using CPLEX CPLEX is optimization software developed and sold by ILOG, Inc. It can be used to solve a variety of different optimization problems in a variety of computing environments. Here we will discuss
More information3 Does the Simplex Algorithm Work?
Does the Simplex Algorithm Work? In this section we carefully examine the simplex algorithm introduced in the previous chapter. Our goal is to either prove that it works, or to determine those circumstances
More informationLecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method
Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming
More informationA Detailed Price Discrimination Example
A Detailed Price Discrimination Example Suppose that there are two different types of customers for a monopolist s product. Customers of type 1 have demand curves as follows. These demand curves include
More informationFirst Welfare Theorem
First Welfare Theorem Econ 2100 Fall 2015 Lecture 17, November 2 Outline 1 First Welfare Theorem 2 Preliminaries to Second Welfare Theorem Last Class Definitions A feasible allocation (x, y) is Pareto
More informationNonlinear Programming Methods.S2 Quadratic Programming
Nonlinear Programming Methods.S2 Quadratic Programming Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard A linearly constrained optimization problem with a quadratic objective
More information5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1
5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition
More informationLinear Equations and Inequalities
Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109  Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................
More informationLinear Programming in Matrix Form
Linear Programming in Matrix Form Appendix B We first introduce matrix concepts in linear programming by developing a variation of the simplex method called the revised simplex method. This algorithm,
More informationChapter 2 Solving Linear Programs
Chapter 2 Solving Linear Programs Companion slides of Applied Mathematical Programming by Bradley, Hax, and Magnanti (AddisonWesley, 1977) prepared by José Fernando Oliveira Maria Antónia Carravilla A
More informationChapter 5. Linear Inequalities and Linear Programming. Linear Programming in Two Dimensions: A Geometric Approach
Chapter 5 Linear Programming in Two Dimensions: A Geometric Approach Linear Inequalities and Linear Programming Section 3 Linear Programming gin Two Dimensions: A Geometric Approach In this section, we
More informationChapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints
Chapter 6 Linear Programming: The Simplex Method Introduction to the Big M Method In this section, we will present a generalized version of the simplex method that t will solve both maximization i and
More informationLinear Programming Sensitivity Analysis
Linear Programming Sensitivity Analysis Massachusetts Institute of Technology LP Sensitivity Analysis Slide 1 of 22 Sensitivity Analysis Rationale Shadow Prices Definition Use Sign Range of Validity Opportunity
More informationDuration Must be Job (weeks) Preceeded by
1. Project Scheduling. This problem deals with the creation of a project schedule; specifically, the project of building a house. The project has been divided into a set of jobs. The problem is to schedule
More informationIntroduction to AMPL A Tutorial
Introduction to AMPL A Tutorial September 13, 2000 AMPL is a powerful language designed specifically for mathematical programming. AMPL has many features and options; however this tutorial covers a small
More informationWeek 5 Integral Polyhedra
Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory
More informationDuality in Linear Programming
Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis, we saw that the shadowprice interpretation of the optimal simplex multipliers is a very useful concept. First, these shadow
More informationApplied Algorithm Design Lecture 5
Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design
More informationThe Transportation Problem: LP Formulations
The Transportation Problem: LP Formulations An LP Formulation Suppose a company has m warehouses and n retail outlets A single product is to be shipped from the warehouses to the outlets Each warehouse
More informationconstraint. Let us penalize ourselves for making the constraint too big. We end up with a
Chapter 4 Constrained Optimization 4.1 Equality Constraints (Lagrangians) Suppose we have a problem: Maximize 5, (x 1, 2) 2, 2(x 2, 1) 2 subject to x 1 +4x 2 =3 If we ignore the constraint, we get the
More informationStudy Guide 2 Solutions MATH 111
Study Guide 2 Solutions MATH 111 Having read through the sample test, I wanted to warn everyone, that I might consider asking questions involving inequalities, the absolute value function (as in the suggested
More informationQuestion 2: How do you solve a linear programming problem with a graph?
Question 2: How do you solve a linear programming problem with a graph? Now that we have several linear programming problems, let s look at how we can solve them using the graph of the system of inequalities.
More informationMinimizing costs for transport buyers using integer programming and column generation. Eser Esirgen
MASTER STHESIS Minimizing costs for transport buyers using integer programming and column generation Eser Esirgen DepartmentofMathematicalSciences CHALMERS UNIVERSITY OF TECHNOLOGY UNIVERSITY OF GOTHENBURG
More informationAbsolute Value Equations and Inequalities
Key Concepts: Compound Inequalities Absolute Value Equations and Inequalities Intersections and unions Suppose that A and B are two sets of numbers. The intersection of A and B is the set of all numbers
More information9.4 THE SIMPLEX METHOD: MINIMIZATION
SECTION 9 THE SIMPLEX METHOD: MINIMIZATION 59 The accounting firm in Exercise raises its charge for an audit to $5 What number of audits and tax returns will bring in a maximum revenue? In the simplex
More informationLECTURE: INTRO TO LINEAR PROGRAMMING AND THE SIMPLEX METHOD, KEVIN ROSS MARCH 31, 2005
LECTURE: INTRO TO LINEAR PROGRAMMING AND THE SIMPLEX METHOD, KEVIN ROSS MARCH 31, 2005 DAVID L. BERNICK dbernick@soe.ucsc.edu 1. Overview Typical Linear Programming problems Standard form and converting
More informationSupport Vector Machine (SVM)
Support Vector Machine (SVM) CE725: Statistical Pattern Recognition Sharif University of Technology Spring 2013 Soleymani Outline Margin concept HardMargin SVM SoftMargin SVM Dual Problems of HardMargin
More informationLargest FixedAspect, AxisAligned Rectangle
Largest FixedAspect, AxisAligned Rectangle David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 19982016. All Rights Reserved. Created: February 21, 2004 Last Modified: February
More informationEXCEL SOLVER TUTORIAL
ENGR62/MS&E111 Autumn 2003 2004 Prof. Ben Van Roy October 1, 2003 EXCEL SOLVER TUTORIAL This tutorial will introduce you to some essential features of Excel and its plugin, Solver, that we will be using
More informationInternational Doctoral School Algorithmic Decision Theory: MCDA and MOO
International Doctoral School Algorithmic Decision Theory: MCDA and MOO Lecture 2: Multiobjective Linear Programming Department of Engineering Science, The University of Auckland, New Zealand Laboratoire
More informationBasic Components of an LP:
1 Linear Programming Optimization is an important and fascinating area of management science and operations research. It helps to do less work, but gain more. Linear programming (LP) is a central topic
More informationSpecial Situations in the Simplex Algorithm
Special Situations in the Simplex Algorithm Degeneracy Consider the linear program: Maximize 2x 1 +x 2 Subject to: 4x 1 +3x 2 12 (1) 4x 1 +x 2 8 (2) 4x 1 +2x 2 8 (3) x 1, x 2 0. We will first apply the
More informationDiscrete (and Continuous) Optimization Solutions of Exercises 1 WI4 131
Discrete (and Continuous) Optimization Solutions of Exercises 1 WI4 131 Kees Roos Technische Universiteit Delft Faculteit Informatietechnologie en Systemen Afdeling Informatie, Systemen en Algoritmiek
More informationMath 120 Final Exam Practice Problems, Form: A
Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,
More informationCORPORATE FINANCE # 2: INTERNAL RATE OF RETURN
CORPORATE FINANCE # 2: INTERNAL RATE OF RETURN Professor Ethel Silverstein Mathematics by Dr. Sharon Petrushka Introduction How do you compare investments with different initial costs ( such as $50,000
More informationGrade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 %
Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the
More informationAn Expressive Auction Design for Online Display Advertising. AUTHORS: Sébastien Lahaie, David C. Parkes, David M. Pennock
An Expressive Auction Design for Online Display Advertising AUTHORS: Sébastien Lahaie, David C. Parkes, David M. Pennock Li PU & Tong ZHANG Motivation Online advertisement allow advertisers to specify
More informationSample Midterm Solutions
Sample Midterm Solutions Instructions: Please answer both questions. You should show your working and calculations for each applicable problem. Correct answers without working will get you relatively few
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. #approximation algorithm.
Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of three
More informationWeek 2 Quiz: Equations and Graphs, Functions, and Systems of Equations
Week Quiz: Equations and Graphs, Functions, and Systems of Equations SGPE Summer School 014 June 4, 014 Lines: Slopes and Intercepts Question 1: Find the slope, yintercept, and xintercept of the following
More informationWeek 1: Functions and Equations
Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.12.2, and Chapter
More information7.1 Modelling the transportation problem
Chapter Transportation Problems.1 Modelling the transportation problem The transportation problem is concerned with finding the minimum cost of transporting a single commodity from a given number of sources
More informationHomework # 3 Solutions
Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8
More information(a) Let x and y be the number of pounds of seed and corn that the chicken rancher must buy. Give the inequalities that x and y must satisfy.
MA 44 Practice Exam Justify your answers and show all relevant work. The exam paper will not be graded, put all your work in the blue book provided. Problem A chicken rancher concludes that his flock
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationJianlin Cheng, PhD Computer Science Department University of Missouri, Columbia Fall, 2013
Jianlin Cheng, PhD Computer Science Department University of Missouri, Columbia Fall, 2013 Princeton s class notes on linear programming MIT s class notes on linear programming Xian Jiaotong University
More informationOPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2
More informationAlgebra 2: Q1 & Q2 Review
Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short
More informationMAT12X Intermediate Algebra
MAT12X Intermediate Algebra Workshop I  Exponential Functions LEARNING CENTER Overview Workshop I Exponential Functions of the form y = ab x Properties of the increasing and decreasing exponential functions
More informationNonlinear Optimization: Algorithms 3: Interiorpoint methods
Nonlinear Optimization: Algorithms 3: Interiorpoint methods INSEAD, Spring 2006 JeanPhilippe Vert Ecole des Mines de Paris JeanPhilippe.Vert@mines.org Nonlinear optimization c 2006 JeanPhilippe Vert,
More informationLecture 3: Linear Programming Relaxations and Rounding
Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationIntroduction to Linear Programming (LP) Mathematical Programming (MP) Concept
Introduction to Linear Programming (LP) Mathematical Programming Concept LP Concept Standard Form Assumptions Consequences of Assumptions Solution Approach Solution Methods Typical Formulations Massachusetts
More information1 Calculus of Several Variables
1 Calculus of Several Variables Reading: [Simon], Chapter 14, p. 30031. 1.1 Partial Derivatives Let f : R n R. Then for each x i at each point x 0 = (x 0 1,..., x 0 n) the ith partial derivative is defined
More informationOperation Research. Module 1. Module 2. Unit 1. Unit 2. Unit 3. Unit 1
Operation Research Module 1 Unit 1 1.1 Origin of Operations Research 1.2 Concept and Definition of OR 1.3 Characteristics of OR 1.4 Applications of OR 1.5 Phases of OR Unit 2 2.1 Introduction to Linear
More informationSolving Linear Programs
Solving Linear Programs 2 In this chapter, we present a systematic procedure for solving linear programs. This procedure, called the simplex method, proceeds by moving from one feasible solution to another,
More informationAirport Planning and Design. Excel Solver
Airport Planning and Design Excel Solver Dr. Antonio A. Trani Professor of Civil and Environmental Engineering Virginia Polytechnic Institute and State University Blacksburg, Virginia Spring 2012 1 of
More informationReadings. D Chapter 1. Lecture 2: Constrained Optimization. Cecilia Fieler. Example: Input Demand Functions. Consumer Problem
Economics 245 January 17, 2012 : Example Readings D Chapter 1 : Example The FOCs are max p ( x 1 + x 2 ) w 1 x 1 w 2 x 2. x 1,x 2 0 p 2 x i w i = 0 for i = 1, 2. These are two equations in two unknowns,
More informationChapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling
Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NPhard problem. What should I do? A. Theory says you're unlikely to find a polytime algorithm. Must sacrifice one
More information1 Portfolio mean and variance
Copyright c 2005 by Karl Sigman Portfolio mean and variance Here we study the performance of a oneperiod investment X 0 > 0 (dollars) shared among several different assets. Our criterion for measuring
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationThe CobbDouglas Production Function
171 10 The CobbDouglas Production Function This chapter describes in detail the most famous of all production functions used to represent production processes both in and out of agriculture. First used
More informationSeveral Views of Support Vector Machines
Several Views of Support Vector Machines Ryan M. Rifkin Honda Research Institute USA, Inc. Human Intention Understanding Group 2007 Tikhonov Regularization We are considering algorithms of the form min
More informationLabor Demand. Labor Economics VSE Praha March 2009
Labor Demand Labor Economics VSE Praha March 2009 Labor Economics: Outline Labor Supply Labor Demand Equilibrium in Labor Market et cetera Labor Demand Model: Firms Firm s role in: Labor Market consumes
More informationRegression Using Support Vector Machines: Basic Foundations
Regression Using Support Vector Machines: Basic Foundations Technical Report December 2004 Aly Farag and Refaat M Mohamed Computer Vision and Image Processing Laboratory Electrical and Computer Engineering
More information