# Linear Programming: Introduction

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Linear Programming: Introduction Frédéric Giroire F. Giroire LP - Introduction 1/28

2 Course Schedule Session 1: Introduction to optimization. Modelling and Solving simple problems. Modelling combinatorial problems. Session 2: Duality or Assessing the quality of a solution. Session 3: Solving problems in practice or using solvers (Glpk or Cplex). F. Giroire LP - Introduction 2/28

3 Motivation Why linear programming is a very important topic? A lot of problems can be formulated as linear programmes, and There exist efficient methods to solve them or at least give good approximations. Solve difficult problems: e.g. original example given by the inventor of the theory, Dantzig. Best assignment of 70 people to 70 tasks. Magic algorithmic box. F. Giroire LP - Introduction 3/28

4 What is a linear programme? Optimization problem consisting in maximizing (or minimizing) a linear objective function of n decision variables subject to a set of constraints expressed by linear equations or inequalities. Originally, military context: "programme"="resource planning". Now "programme"="problem" Terminology due to George B. Dantzig, inventor of the Simplex Algorithm (1947) F. Giroire LP - Introduction 4/28

5 Terminology x 1,x 2 : Decision variables max 350x x 2 subject to x 1 + x x 1 + 6x x x x 1,x 2 0 Objective function Constraints F. Giroire LP - Introduction 5/28

6 Terminology x 1,x 2 : Decision variables max 350x x 2 subject to x 1 + x x 1 + 6x x x x 1,x 2 0 Objective function Constraints In linear programme: objective function + constraints are all linear Typically (not always): variables are non-negative If variables are integer: system called Integer Programme (IP) F. Giroire LP - Introduction 6/28

7 Terminology Linear programmes can be written under the standard form: Maximize n j=1 c jx j Subject to: n j=1 a ijx j b i for all 1 i m x j 0 for all 1 j n. (1) the problem is a maximization; all constraints are inequalities (and not equations); all variables are non-negative. F. Giroire LP - Introduction 7/28

8 Example 1: a resource allocation problem A company produces copper cable of 5 and 10 mm of diameter on a single production line with the following constraints: The available copper allows to produces meters of cable of 5 mm diameter per week. A meter of 10 mm diameter copper consumes 4 times more copper than a meter of 5 mm diameter copper. Due to demand, the weekly production of 5 mm cable is limited to meters and the production of 10 mm cable should not exceed 40% of the total production. Cable are respectively sold 50 and 200 euros the meter. What should the company produce in order to maximize its weekly revenue? F. Giroire LP - Introduction 8/28

9 Example 1: a resource allocation problem A company produces copper cable of 5 and 10 mm of diameter on a single production line with the following constraints: The available copper allows to produces meters of cable of 5 mm diameter per week. A meter of 10 mm diameter copper consumes 4 times more copper than a meter of 5 mm diameter copper. Due to demand, the weekly production of 5 mm cable is limited to meters and the production of 10 mm cable should not exceed 40% of the total production. Cable are respectively sold 50 and 200 euros the meter. What should the company produce in order to maximize its weekly revenue? F. Giroire LP - Introduction 8/28

10 Example 1: a resource allocation problem Define two decision variables: x 1 : the number of thousands of meters of 5 mm cables produced every week x 2 : the number of thousands meters of 10 mm cables produced every week The revenue associated to a production (x 1,x 2 ) is z = 50x x 2. The capacity of production cannot be exceeded x 1 + 4x F. Giroire LP - Introduction 9/28

11 Example 1: a resource allocation problem The demand constraints have to be satisfied x (x 1 + x 2 ) x 1 15 Negative quantities cannot be produced x1 0,x2 0. F. Giroire LP - Introduction 10/28

12 Example 1: a resource allocation problem The model: To maximize the sell revenue, determine the solutions of the following linear programme x 1 and x 2 : max z = 50x x 2 subject to x 1 + 4x x 1 + 6x 2 0 x 1 15 x 1,x 2 0 F. Giroire LP - Introduction 11/28

13 Example 2: Scheduling m = 3 machines n = 8 tasks Each task lasts x units of time Objective: affect the tasks to the machines in order to minimize the duration Here, the 8 tasks are finished after 7 units of times on 3 machines. F. Giroire LP - Introduction 12/28

14 Example 2: Scheduling m = 3 machines n = 8 tasks Each task lasts x units of time Objective: affect the tasks to the machines in order to minimize the duration Now, the 8 tasks are accomplished after 6.5 units of time: OPT? m n possibilities! (Here 3 8 = 6561) F. Giroire LP - Introduction 12/28

15 Example 2: Scheduling m = 3 machines n = 8 tasks Each task lasts x units of time Solution: LP model. min subject to t 1 i n t i x j i t ( j,1 j m) 1 j m x j i = 1 ( i,1 i n) with x j i = 1 if task i is affected to machine j. F. Giroire LP - Introduction 12/28

16 Solving Difficult Problems Difficulty: Large number of solutions. Choose the best solution among 2 n or n! possibilities: all solutions cannot be enumerated. Complexity of studied problems: often NP-complete. Solving methods: Optimal solutions: Graphical method (2 variables only). Simplex method. Approximations: Theory of duality (assert the quality of a solution). Approximation algorithms. F. Giroire LP - Introduction 13/28

17 Graphical Method The constraints of a linear programme define a zone of solutions. The best point of the zone corresponds to the optimal solution. For problem with 2 variables, easy to draw the zone of solutions and to find the optimal solution graphically. F. Giroire LP - Introduction 14/28

18 Graphical Method Example: max 350x x 2 subject to x 1 + x x 1 + 6x x x x 1,x 2 0 F. Giroire LP - Introduction 15/28

19 Graphical Method F. Giroire LP - Introduction 16/28

20 Graphical Method F. Giroire LP - Introduction 17/28

21 Graphical Method F. Giroire LP - Introduction 18/28

22 Graphical Method F. Giroire LP - Introduction 19/28

23 Graphical Method F. Giroire LP - Introduction 20/28

24 Graphical Method F. Giroire LP - Introduction 21/28

25 Computation of the optimal solution The optimal solution is at the intersection of the constraints: x 1 + x 2 = 200 (2) 9x 1 + 6x 2 = 1566 (3) We get: x 1 = 122 x 2 = 78 Objective = F. Giroire LP - Introduction 22/28

26 Optimal Solutions: Different Cases F. Giroire LP - Introduction 23/28

27 Optimal Solutions: Different Cases Three different possible cases: a single optimal solution, an infinite number of optimal solutions, or no optimal solutions. F. Giroire LP - Introduction 23/28

28 Optimal Solutions: Different Cases Three different possible cases: a single optimal solution, an infinite number of optimal solutions, or no optimal solutions. If an optimal solution exists, there is always a corner point optimal solution! F. Giroire LP - Introduction 23/28

29 Solving Linear Programmes F. Giroire LP - Introduction 24/28

30 Solving Linear Programmes The constraints of an LP give rise to a geometrical shape: a polyhedron. If we can determine all the corner points of the polyhedron, then we calculate the objective function at these points and take the best one as our optimal solution. The Simplex Method intelligently moves from corner to corner until it can prove that it has found the optimal solution. F. Giroire LP - Introduction 25/28

31 Solving Linear Programmes Geometric method impossible in higher dimensions Algebraical methods: Simplex method (George B. Dantzig 1949): skim through the feasible solution polytope. Similar to a "Gaussian elimination". Very good in practice, but can take an exponential time. Polynomial methods exist: Leonid Khachiyan 1979: ellipsoid method. But more theoretical than practical. Narendra Karmarkar 1984: a new interior method. Can be used in practice. F. Giroire LP - Introduction 26/28

32 But Integer Programming (IP) is different! Feasible region: a set of discrete points. Corner point solution not assured. No "efficient" way to solve an IP. Solving it as an LP provides a relaxation and a bound on the solution. F. Giroire LP - Introduction 27/28

33 Summary: To be remembered What is a linear programme. The graphical method of resolution. Linear programs can be solved efficiently (polynomial). Integer programs are a lot harder (in general no polynomial algorithms). In this case, we look for approximate solutions. F. Giroire LP - Introduction 28/28

### Lecture 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

### Lecture 2: August 29. Linear Programming (part I)

10-725: Convex Optimization Fall 2013 Lecture 2: August 29 Lecturer: Barnabás Póczos Scribes: Samrachana Adhikari, Mattia Ciollaro, Fabrizio Lecci Note: LaTeX template courtesy of UC Berkeley EECS dept.

### Linear 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.

### 1 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.

### Can linear programs solve NP-hard problems?

Can linear programs solve NP-hard problems? p. 1/9 Can linear programs solve NP-hard problems? Ronald de Wolf Linear programs Can linear programs solve NP-hard problems? p. 2/9 Can linear programs solve

### Optimization Modeling for Mining Engineers

Optimization Modeling for Mining Engineers Alexandra M. Newman Division of Economics and Business Slide 1 Colorado School of Mines Seminar Outline Linear Programming Integer Linear Programming Slide 2

### Lecture 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

### 3.1 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

### Linear 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

### Linear 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

### What 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

### Module1. x 1000. y 800.

Module1 1 Welcome to the first module of the course. It is indeed an exciting event to share with you the subject that has lot to offer both from theoretical side and practical aspects. To begin with,

### 3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max

SOLUTION OF LINEAR PROGRAMMING PROBLEMS THEOREM 1 If a linear programming problem has a solution, then it must occur at a vertex, or corner point, of the feasible set, S, associated with the problem. Furthermore,

### Chapter 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

### Linear 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

### Chapter 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 NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one

### Linear 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

### Standard 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,

### Discrete Optimization

Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using

### Definition 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

### Special 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

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.

Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of

### 7.4 Linear Programming: The Simplex Method

7.4 Linear Programming: The Simplex Method For linear programming problems with more than two variables, the graphical method is usually impossible, so the simplex method is used. Because the simplex method

### Basic 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

### 5.1 Bipartite Matching

CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson

### Converting a Linear Program to Standard Form

Converting a Linear Program to Standard Form Hi, welcome to a tutorial on converting an LP to Standard Form. We hope that you enjoy it and find it useful. Amit, an MIT Beaver Mita, an MIT Beaver 2 Linear

### Practical 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

### Mathematical 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

### Applied 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

### Linear 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

### LECTURE 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

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.

Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three

### The 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,

### CHAPTER 11: BASIC LINEAR PROGRAMMING CONCEPTS

Linear programming is a mathematical technique for finding optimal solutions to problems that can be expressed using linear equations and inequalities. If a real-world problem can be represented accurately

### Linear 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

### Unit 1. Today I am going to discuss about Transportation problem. First question that comes in our mind is what is a transportation problem?

Unit 1 Lesson 14: Transportation Models Learning Objective : What is a Transportation Problem? How can we convert a transportation problem into a linear programming problem? How to form a Transportation

### 24. The Branch and Bound Method

24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no

### Chapter 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

### Optimal Scheduling for Dependent Details Processing Using MS Excel Solver

BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 8, No 2 Sofia 2008 Optimal Scheduling for Dependent Details Processing Using MS Excel Solver Daniela Borissova Institute of

### Chapter 13: Binary and Mixed-Integer Programming

Chapter 3: Binary and Mixed-Integer Programming The general branch and bound approach described in the previous chapter can be customized for special situations. This chapter addresses two special situations:

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

### Operation 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

### Modeling and solving linear programming with R. Jose M Sallan Oriol Lordan Vicenc Fernandez

Modeling and solving linear programming with R Jose M Sallan Oriol Lordan Vicenc Fernandez Open Access Support If you find this book interesting, we would appreciate that you supported its authors and

### Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood

PERFORMANCE EXCELLENCE IN THE WOOD PRODUCTS INDUSTRY EM 8720-E October 1998 \$3.00 Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood A key problem faced

### . P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2

4. Basic feasible solutions and vertices of polyhedra Due to the fundamental theorem of Linear Programming, to solve any LP it suffices to consider the vertices (finitely many) of the polyhedron P of the

### Jeffrey Kantor. August 28, 2006

Mosel August 28, 2006 Announcements Mosel Enrollment and Registration - Last Day for Class Change is August 30th. Homework set 1 was handed out today. A copy is available on the course web page http://jkantor.blogspot.com

### Interior Point Methods and Linear Programming

Interior Point Methods and Linear Programming Robert Robere University of Toronto December 13, 2012 Abstract The linear programming problem is usually solved through the use of one of two algorithms: either

### 3 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

### Jianlin 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

### 1 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

### Largest Fixed-Aspect, Axis-Aligned Rectangle

Largest Fixed-Aspect, Axis-Aligned Rectangle David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 1998-2016. All Rights Reserved. Created: February 21, 2004 Last Modified: February

### Linear 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

### LU Factorization Method to Solve Linear Programming Problem

Website: wwwijetaecom (ISSN 2250-2459 ISO 9001:2008 Certified Journal Volume 4 Issue 4 April 2014) LU Factorization Method to Solve Linear Programming Problem S M Chinchole 1 A P Bhadane 2 12 Assistant

### 5 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

### USING EXCEL SOLVER IN OPTIMIZATION PROBLEMS

USING EXCEL SOLVER IN OPTIMIZATION PROBLEMS Leslie Chandrakantha John Jay College of Criminal Justice of CUNY Mathematics and Computer Science Department 445 West 59 th Street, New York, NY 10019 lchandra@jjay.cuny.edu

### Linear 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

### Solving Quadratic Equations

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

### Question 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.

### 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

### INTEGER PROGRAMMING. Integer Programming. Prototype example. BIP model. BIP models

Integer Programming INTEGER PROGRAMMING In many problems the decision variables must have integer values. Example: assign people, machines, and vehicles to activities in integer quantities. If this is

### Study 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

### EdExcel Decision Mathematics 1

EdExcel Decision Mathematics 1 Linear Programming Section 1: Formulating and solving graphically Notes and Examples These notes contain subsections on: Formulating LP problems Solving LP problems Minimisation

### UNIT 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

### Math 1314 Lesson 8: Business Applications: Break Even Analysis, Equilibrium Quantity/Price

Math 1314 Lesson 8: Business Applications: Break Even Analysis, Equilibrium Quantity/Price Cost functions model the cost of producing goods or providing services. Examples: rent, utilities, insurance,

### CHAPTER 9. Integer Programming

CHAPTER 9 Integer Programming An integer linear program (ILP) is, by definition, a linear program with the additional constraint that all variables take integer values: (9.1) max c T x s t Ax b and x integral

### Solving NP Hard problems in practice lessons from Computer Vision and Computational Biology

Solving NP Hard problems in practice lessons from Computer Vision and Computational Biology Yair Weiss School of Computer Science and Engineering The Hebrew University of Jerusalem www.cs.huji.ac.il/ yweiss

### 26 Linear Programming

The greatest flood has the soonest ebb; the sorest tempest the most sudden calm; the hottest love the coldest end; and from the deepest desire oftentimes ensues the deadliest hate. Th extremes of glory

### Linear Programming. brewer s problem simplex algorithm implementation linear programming

Linear Programming brewer s problem simplex algorithm implementation linear programming References: The Allocation of Resources by Linear Programming, Scientific American, by Bob Bland Algs in Java, Part

### 11. APPROXIMATION ALGORITHMS

11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Lecture slides by Kevin Wayne Copyright 2005

### Linear Programming Problems

Linear Programming Problems Linear programming problems come up in many applications. In a linear programming problem, we have a function, called the objective function, which depends linearly on a number

### 0.1 Linear Programming

0.1 Linear Programming 0.1.1 Objectives By the end of this unit you will be able to: formulate simple linear programming problems in terms of an objective function to be maximized or minimized subject

### Eleonóra STETTNER, Kaposvár Using Microsoft Excel to solve and illustrate mathematical problems

Eleonóra STETTNER, Kaposvár Using Microsoft Excel to solve and illustrate mathematical problems Abstract At the University of Kaposvár in BSc and BA education we introduced a new course for the first year

### Solution of a Large-Scale Traveling-Salesman Problem

Chapter 1 Solution of a Large-Scale Traveling-Salesman Problem George B. Dantzig, Delbert R. Fulkerson, and Selmer M. Johnson Introduction by Vašek Chvátal and William Cook The birth of the cutting-plane

### Transportation Polytopes: a Twenty year Update

Transportation Polytopes: a Twenty year Update Jesús Antonio De Loera University of California, Davis Based on various papers joint with R. Hemmecke, E.Kim, F. Liu, U. Rothblum, F. Santos, S. Onn, R. Yoshida,

### Solving Integer Programming with Branch-and-Bound Technique

Solving Integer Programming with Branch-and-Bound Technique This is the divide and conquer method. We divide a large problem into a few smaller ones. (This is the branch part.) The conquering part is done

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### How to speed-up hard problem resolution using GLPK?

How to speed-up hard problem resolution using GLPK? Onfroy B. & Cohen N. September 27, 2010 Contents 1 Introduction 2 1.1 What is GLPK?.......................................... 2 1.2 GLPK, the best one?.......................................

### Minimizing 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

### Linear Programming: Theory and Applications

Linear Programming: Theory and Applications Catherine Lewis May 11, 2008 1 Contents 1 Introduction to Linear Programming 3 1.1 What is a linear program?...................... 3 1.2 Assumptions.............................

### Chapter 11 Monte Carlo Simulation

Chapter 11 Monte Carlo Simulation 11.1 Introduction The basic idea of simulation is to build an experimental device, or simulator, that will act like (simulate) the system of interest in certain important

### OPRE 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

### Simplex method summary

Simplex method summary Problem: optimize a linear objective, subject to linear constraints 1. Step 1: Convert to standard form: variables on right-hand side, positive constant on left slack variables for

### 1. Graphing Linear Inequalities

Notation. CHAPTER 4 Linear Programming 1. Graphing Linear Inequalities x apple y means x is less than or equal to y. x y means x is greater than or equal to y. x < y means x is less than y. x > y means

### Operations Research. Inside Class Credit Hours: 51 Prerequisite:Linear Algebra Number of students : 55 Semester: 2 Credit: 3

Operations Research Title of the Course:Operations Research Course Teacher:Xiaojin Zheng No. of Course: GOS10011 Language:English Students: postgraduate Inside Class Credit Hours: 51 Prerequisite:Linear

### Algorithm Design and Analysis

Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;

### 5 Systems of Equations

Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate

### Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

### 56:171. 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

### PhysicsAndMathsTutor.com

D Linear programming - Simplex algorithm. The tableau below is the initial tableau for a maximising linear programming problem in x, y and z. Basic variable x y z r s t Value r 4 7 5 0 0 64 s 0 0 0 6 t

### 9.4 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

### Chapter 2: Introduction to Linear Programming

Chapter 2: Introduction to Linear Programming You may recall unconstrained optimization from your high school years: the idea is to find the highest point (or perhaps the lowest point) on an objective

### Where Do We Meet? Students will represent and analyze algebraically a wide variety of problem solving situations.

Beth Yancey MAED 591 Where Do We Meet? Introduction: This lesson covers objectives in the algebra and geometry strands of the New York State standards for Algebra I. The students will use the graphs of

### 1 Linear Programming. 1.1 Introduction. Problem description: motivate by min-cost flow. bit of history. everything is LP. NP and conp. P breakthrough.

1 Linear Programming 1.1 Introduction Problem description: motivate by min-cost flow bit of history everything is LP NP and conp. P breakthrough. general form: variables constraints: linear equalities

### Math 1314 Lesson 8 Business Applications: Break Even Analysis, Equilibrium Quantity/Price

Math 1314 Lesson 8 Business Applications: Break Even Analysis, Equilibrium Quantity/Price Three functions of importance in business are cost functions, revenue functions and profit functions. Cost functions

### International 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

### Discuss the size of the instance for the minimum spanning tree problem.

3.1 Algorithm complexity The algorithms A, B are given. The former has complexity O(n 2 ), the latter O(2 n ), where n is the size of the instance. Let n A 0 be the size of the largest instance that can

### SOLVING EQUATIONS WITH EXCEL

SOLVING EQUATIONS WITH EXCEL Excel and Lotus software are equipped with functions that allow the user to identify the root of an equation. By root, we mean the values of x such that a given equation cancels

### How the European day-ahead electricity market works

How the European day-ahead electricity market works ELEC0018-1 - Marché de l'énergie - Pr. D. Ernst! Bertrand Cornélusse, Ph.D.! bertrand.cornelusse@ulg.ac.be! October 2014! 1 Starting question How is