# Basic Components of an LP:

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 1

2 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 in optimization. It provides a powerful tool in modeling many applications. LP has attracted most of its attention in optimization during the last six decades for two main reasons: Applicability: There are many realworld applications that can be modeled as linear programming; Solvability: There are theoretically and practically efficient techniques for solving large-scale problems. Hi! My name is Cathy. I will guide you in tutorials during the semester. In this tutorial, we introduce the basic elements of an LP and present some examples that can be modeled as an LP. In the next tutorials, we will discuss solution techniques. 2

3 Basic Components of an LP: Each optimization problem consists of three elements: decision variables: describe our choices that are under our control; objective function: describes a criterion that we wish to minimize (e.g., cost) or maximize (e.g., profit); constraints: describe the limitations that restrict our choices for decision variables. Formally, we use the term linear programming (LP) to refer to an optimization problem in which the objective function is linear and each constraint is a linear inequality or equality. I ll discuss these features soon. 3

4 An Introductory Example I am a bit confused about the LP elements. Can you give me more details. Oh! I forgot to introduce myself. I am Tom; a new member of the class. I am interested in learning linear programming. I will be with you during the tutorials. Let s start with an example. I ll describe it first in words, and then we ll translate it into a linear program. 4

5 An Introductory Example Problem Statement: A company makes two products (say, P and Q) using two machines (say, A and B). Each unit of P that is produced requires 50 minutes processing time on machine A and 30 minutes processing time on machine B. Each unit of Q that is produced requires 24 minutes processing time on machine A and 33 minutes processing time on machine B. Machine A is going to be available for 40 hours and machine B is available for 35 hours. The profit per unit of P is \$25 and the profit per unit of Q is \$30. Company policy is to determine the production quantity of each product in such a way as to maximize the total profit given that the available resources should not be exceeded Task: The aim is to formulate the problem of deciding how much of each product to make in the current week as an LP. 5

6 Step 1: Defining the Decision Variables We often start with identifying decision variables (i.e., what we want to determine among those things which are under our control). Tom! Can you identify the decision variables for our example? The company wants to determine the optimal product to make in the current week. So there are two decision variables: x: the number of units of P y: the number of units of Q Good job! Let s move on to the second step. 6

7 Step 2: Choosing an Objective Function We usually seek a criterion (or a measure) to compare alternative solutions. This yields the objective function. Tom! It is now your turn to identify the objective function. We want to maximize the total profit. The profit per each unit of product P is \$25 and profit per each unit of product Q is \$30. Therefore, the total profit is 25x+30y if we produce x units of P and y units of Q. This leads to the following objective function: max 40x+35y 7 Note that: 1: The objective function is linear in terms of decision variables x and y (i.e., it is of the form ax + by, where a and b are constant). 2: We typically use the variable z to denote the value of the objective. So the objective function can be stated as: max z=25x+30y

8 Step 3: Identifying the Constraints In many practical problems, there are limitations (such as resource / physical / strategic / economical) that restrict our decisions. We describe these limitations using mathematical constraints. Tom! What are the constraints in our example? The amount of time that machine A is available restricts the quantities to be manufactured. If we produce x units of P and y units of Q, machine A should be used for 50x+24y minutes since each unit of P requires 50 minutes processing time on machine A and each unit of Q requires 24 minutes processing time on machine A. On the other hand, machine A is available for 40 hours or equivalently for 2400 minutes. This imposes the following constraint: 50x + 24y Similarly, the amount of time that machine B is available imposes the following constraint: 30x + 33y These constrains are linear inequalities since in each constraint the left-hand side of the inequality sign is a linear function in terms of the decision variables x and y and the right hand side is constant.

9 Step 3: Identifying the Constraints Note: In most problems, the decision variables are required to be nonnegative, and this should be explicitly included in the formulation. This is the case here. So you need to include the following two nonnegativity constraints as well: x 0 and y 0 I see your point. So the constraints we are subject to (s.t.) are : 50x + 24y 2400, (machine A time) 30x + 33y 2100, (machine B time) x 0, y 0. 9

10 LP for the Example Well done Tom! You did a good job. Now, write the LP by putting all the LP elements together. Here is the LP: max z= 25x + 30y s.t. 50x + 24y 2400, 30x + 33y 2100, x 0, y 0. Note: To be realistic, we would require integrality for the decision variables x and y. It will lead to an integer program if we include integrality. Integer programs are harder to solve and we will consider them in later lectures. For the moment, we leave out integrality and consider this LP in this tutorial. 10 Now let s consider a generalization of this example in order to test whether you have understood the concepts.

11 A Manufacturing Example Problem Statement: An operations manager is trying to determine a production plan for the next week. There are three products (say, P, Q, and Q) to produce using four machines (say, A and B, C, and D). Each of the four machines performs a unique process. There is one machine of each type, and each machine is available for 2400 minutes per week. The unit processing times for each machine is given in Table 1. Table 1: Machine Data Unit Processing Time (min) Machine Product P Product Q Product R Availability (min) A B C D Total processing time

12 A Manufacturing Example Problem Statement (cont.): The unit revenues and maximum sales for the week are indicated in Table 2. Storage from one week to the next is not permitted. The operating expenses associated with the plant are \$6000 per week, regardless of how many components and products are made. The \$6000 includes all expenses except for material costs. Table 2: Product Data Item Product P Product Q Product R Revenue per unit \$90 \$100 \$70 Material cost per unit \$45 \$40 \$20 Profit per unit \$45 \$60 \$50 Maximum sales Task: Here we seek the optimal product mix-- that is, the amount of each product that should be manufactured during the present week in order to maximize profits. Formulate this as an LP.

13 Step 1: Defining the Decision Variables Tom! You are supposed to do this example by yourself. Remember that the first step is to define the decision variables. Try to come up with the correct definition. If you need help or want to check your solution, click here to see the answer. Otherwise, you can continue by identifying the objective function. Good idea! Let me try. It should not be a difficult task. We are trying to select the optimal product mix, so we define three decision variables as follows: p: number of units of product P to produce, q: number of units of product Q to produce, r: number of units of product R to produce. 13

14 Step 2: Choosing an Objective Function Click here to if you want to see the objective function. Otherwise, continue to describe the constraints. Let me review the problem statement to write the constraints Our objective is to maximize profit: Profit = (90-45)p + (100-40)q + (70-20r 6000 = 45p + 60q + 50r 6000 Note: The operating costs are not a function of the variables in the problem. If we were to drop the \$6000 term from the profit function, we would still obtain the same optimal mix of products. Thus, the objective function is z = 45p + 60q + 50r 14

15 Step 3: Identifying the Constraints Click here to if you want to see the constraints. It is good to see the complete model now to see my answer. The amount of time a machine is available and the maximum sales potential for each product restrict the quantities to be manufactured. Since we know the unit processing times for each machine, the constraints can be written as linear inequalities as follows: 20p+10q +10r 2400 (Machine A) 12p+28q+16r 2400 (Machine B) 15p+6q+16r 2400 (Machine C ) 10p+15q+0r 2400 (Machine D) Observe that the unit for these constraints is minutes per week. Both sides of an inequality must be in the same unit. The market limitations are written as simple upper bounds. Market constraints: P 100, Q 40, R 60. Logic indicates that we should also include nonnegativity restrictions on the variables. Nonnegativity constraints: P 0, Q 0, R 0. 15

16 A Manufacturing Example Click here to if you want to see complete model. Let me review the problem statement to write the constraints By combining the objective function and the constraints, we obtain the LP model as follows: max z=45p+60q+50r s.t. 20p+10q+10r p+28q+16r p+6q+16r p+15q+0r p q 40 0 r 60 16

17 A Manufacturing Example: Optimal Solution The optimal solution to this LP is P=81.82,Q=16.36,R=60 with the corresponding objective value z=\$7664. To compute the profit for the week, we reduce this value by \$6000 for operating expenses and get \$1664. By setting the production quantities to the amount specified by the solution, we find machine usage as shown in the following table. Machine Available usage(min) Actual usage (min) A B C D Product Maximum sales Units produced P Q R

18 Post Office Problem: Let s consider one more example. Problem Statement: A post office requires different numbers of fulltime employees on different days of the week. Each full-time employee must work five consecutive days and then receive two days off. In the following table, the number of employees required on each day of the week is specified. 18 Day Number of full-time Employees Required 1=Monday 17 2=Tuesday 13 3=Wednesday 15 4=Thursday 19 5=Friday 14 6=Saturday 16 7=Sunday 11 Task: Try to formulate an LP that the post office can use to minimize the number of full-time employees who are needed to satisfy these constraints.

19 Step 1: Defining the Decision Variables Remember that we often start with identifying decision variables. There might be several ways for defining decision variables, and you should be careful to pick the right one. In our example, we want to decide home many employees we must hire. One may think of defining decision variables as follows: y i : number of employees who work on day i. You can check this is not the right choice. To see why, notice that we cannot enforce the constraint that each employee works five consecutive days. In this case, there is a choice of decision variables that ensures we satisfy the constraint that each employee works five consecutive days. Tom! Can you come up with decision variables that can be used to formulate this problem? 19

20 Step 1: Defining the Decision Variables Cathy, I think I see what you mean. I think we need to define seven decision variables as follows: x i : number of employees whose five consecutive days of work begin on day i. (They work on days i, i+1, i+2, i+3, and i+4). We define it for i=1,2,,7. For example, x 1 is the number of people beginning work on Monday. Great choice! That s exactly what I was looking for. Now, try to formulate the objective function and the constraints. 20

21 Step 2 & 3: Choosing an Objective and Identifying the Constraints Our aim is to minimize the number of hired employees. The objective function is min z= x 1 +x 2 +x 3 +x 4 +x 5 +x 6 +x 7 The post office must ensure that enough employees are working on each day of the week. For example, at least 17 employees must be working on Monday. To ensure that at least 17 employees are working on Monday,we require that the constraint x 1 +x 4 +x 5 +x 6 +x 7 17 be satisfied. We have similar constraints for other days. Well done, Tom. You can now put together all the constraints and the objective function to have an LP for the post office problem. 21

22 The LP Model for the Post Office Problem min z= x 1 +x 2 +x 3 +x 4 +x 5 +x 6 +x 7 s.t. x 1 +x 4 +x 5 +x 6 +x 7 17 (Monday constraint) x 1 +x 2 +x 5 +x 6 +x 7 13 (Tuesday constraint) x 1 +x 2 +x 3 +x 6 +x 7 15 (Wednesday constraint) x 1 +x 2 +x 3 +x 4 +x 7 19 (Thursday constraint) x 1 +x 2 +x 3 +x 4 +x 5 14 (Friday constraint) x 2 +x 3 +x 4 +x 5 +x 6 16 (Saturday constraint) x 3 +x 4 +x 5 +x 6 +x 7 11 (Sunday constraint) x i 0 for i=1,2,,7 (Nonnegativity constraints) Having formulated the LP as above, I am curious what the optimal solution is. 22

23 Optimal Solution for the Post Office Problem The optimal solution to this LP is z=67/3, x 1 =4/3, x 2 =10/3, x 3 =2, x 4 =22/3, x 5 =0, x 6 =10/3, x 7 =5. Because we are only allowing full-time employees, however, the variables must be integers (to be realistic). If we include integrality, we get an integer program. Integer programming techniques can be used to show that an optimal solution to the post office problem is z=23, x 1 =4, x 2 =4, x 3 =2, x 4 =6, x 5 =0, x 6 =6, x 7 =3. Note: Bartholdi, Orlin and Ratliff (1980) have developed an efficient technique to determine the minimum number of employees required when each worker receives two consecutive days off. 23

24 Creating a Fair Schedule for Employees The optimal solution we found requires 4 workers to start on Monday, 4 on Tuesday, and so on. The workers who starts on Saturday will be unhappy because they never receive a weekend day off. This is not fair. Good point Tom! By rotating the schedules of the employees over a 23- week period, a fairer schedule can be obtained. To see how this is done, consider the following schedule: Weeks 1-4: start on Monday Weeks 5-8: start on Tuesday Weeks 9-10: start on Wednesday Weeks 11-16: start on Thursday Weeks 17-20: start on Saturday Weeks 21-23: start on Sunday 24

25 Creating a Fair Schedule for Employees (cont.) Employee 1 follows this schedule for a 23-week period. Employee 2 starts with week 2 of the schedule (starting on Monday for 3 weeks, then on Tuesday for 4 weeks, and closing with 3 weeks starting on Sunday and 1 week on Monday).We continue in this fashion to generate a 23-week schedule for each employee. For example, employee 13 will have the following schedule : Weeks 1-4 : start on Thursday Weeks 5-8: start on Saturday Weeks 9-11: start on Sunday Weeks 12-15: start on Monday Weeks :start on Tuesday Weeks 20-21:start on Wednesday Weeks :start on Thursday This method of scheduling treats each employee equally. 25

26 So, is there anything more to say? We will continue our discussion on algebraic formulation of linear programming in the second tutorial. We can say goodbye to the students and wish them well. 26

### 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

### 1st January 2014 Wednesday

1st January 2014 Wednesday January 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 February _ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

### Math 407A: Linear Optimization

Math 407A: Linear Optimization Lecture 4: LP Standard Form 1 1 Author: James Burke, University of Washington LPs in Standard Form Minimization maximization Linear equations to linear inequalities Lower

### 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

### Lecture 4 Linear Programming Models: Standard Form. August 31, 2009

Linear Programming Models: Standard Form August 31, 2009 Outline: Lecture 4 Standard form LP Transforming the LP problem to standard form Basic solutions of standard LP problem Operations Research Methods

### Unique Visits. Returning Visits Total 1,538 1, Average Unique Visits

Side 1 af 9 30-10- 15:11:04 30 October Summary Log: No Limit Total 1,538 1,186 771 415 Average 5 4 3 1 Day Date Sunday 30th October 7 3 3 0 Saturday 29th October 2 1 0 1 Friday 28th October 2 2 1 1 Thursday

### Introduction to Linear Programming and Graphical Method

Course: COMMERCE (CBCS) Subject: Business Mathematics Lesson: Introduction to Linear Programming and Graphical Method Authors Name: Dr. Gurmeet Kaur, Associate Professor, Daulat Ram College,University

### 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

### AM PM AM PM AM PM AM PM AM PM AM PM AM PM AM PM

RAILROAD BRIDGE WINGS NECK R.R. BRIDGE SANDWICH CURRENT TURNS JANUARY ---- 1230 0612 1848 Thursday 1 0442 1708 ---- 1200* 0554 1813 0033 1312* 0747 2023 0123 1406* ---- ---- ---- 2346 0042 1324 0706 1948

### DRIVING TEST ON MONDAY

DRIVING TEST ON MONDAY - 17.10.2016 1 HCK1605DRV000004 35 HCK1605DRV000109 69 HCK1605DRV000197 2 HCK1605DRV000005 36 HCK1605DRV000110 70 HCK1605DRV000200 3 HCK1605DRV000008 37 HCK1605DRV000115 71 HCK1605DRV000204

### Iso profit or Iso cost method for solving LPP graphically

Unit Lesson 4: Graphical solution to a LPP Learning Outcomes How to get an optimal solution to a linear programming model using Iso profit (or Iso cost method) Iso profit or Iso cost method for solving

### Monday 26 th Tuesday 27 th Wednesday 28 th Thursday 29 th Friday 30 th Saturday 31 st Sunday 1 st

Monday 26 th Tuesday 27 th Wednesday 28 th Thursday 29 th Friday 30 th Saturday 31 st Sunday 1 st BOXING DAY BOXING DAY PUBLIC HOLIDAY \$12 with hire skates 12.15 12.45 \$16 ¾ Public ¼ Figure Skating New

### Chapter 1, Operations Research (OR)

Chapter 1, Operations Research (OR) Kent Andersen February 7, 2007 The term Operations Research refers to research on operations. In other words, the study of how to operate something in the best possible

### Fixture List 2018 FIFA World Cup Preliminary Competition

Fixture List 2018 FIFA World Cup Preliminary Competition MATCHDAY 1 4-6 September 2016 4 September Sunday 18:00 Group C 4 September Sunday 20:45 Group C 4 September Sunday 20:45 Group C 4 September Sunday

### 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

### Linear Programming: Introduction

Linear Programming: Introduction Frédéric Giroire F. Giroire LP - Introduction 1/28 Course Schedule Session 1: Introduction to optimization. Modelling and Solving simple problems. Modelling combinatorial

### SUPPLEMENT TO CHAPTER

SUPPLEMENT TO CHAPTER 6 Linear Programming SUPPLEMENT OUTLINE Introduction and Linear Programming Model, 2 Graphical Solution Method, 5 Computer Solutions, 14 Sensitivity Analysis, 17 Key Terms, 22 Solved

### 15.053/8 February 26, 2013

15.053/8 February 26, 2013 Sensitivity analysis and shadow prices special thanks to Ella, Cathy, McGraph, Nooz, Stan and Tom 1 Quotes of the Day If the facts don't fit the theory, change the facts. --

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5 Modular Arithmetic One way to think of modular arithmetic is that it limits numbers to a predefined range {0,1,...,N

### 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

### LINEAR PROGRAMMING PROBLEM: A GEOMETRIC APPROACH

59 LINEAR PRGRAMMING PRBLEM: A GEMETRIC APPRACH 59.1 INTRDUCTIN Let us consider a simple problem in two variables x and y. Find x and y which satisfy the following equations x + y = 4 3x + 4y = 14 Solving

### Cruise Line Agencies of Alaska. Cruise Ship Calendar for 2016 FOR PORT(S) = KTN AND SHIP(S) = ALL AND VOYAGES = ALL

Cruise Line Agencies of Alaska Cruise Ship Calendar for 2016 FOR PORT(S) = KTN AND SHIP(S) = ALL AND VOYAGES = ALL Page 1 of 5 Sunday, May 1 07:0-18:0 Monday, May 2 Tuesday, May 3 Wednesday, May 4 Thursday,

### Introductory Notes on Demand Theory

Introductory Notes on Demand Theory (The Theory of Consumer Behavior, or Consumer Choice) This brief introduction to demand theory is a preview of the first part of Econ 501A, but it also serves as a prototype

### Office of Human Resources Holiday Calendar

2013 Independence Day Thursday, July 4, 2013 Thursday, July 4, 2013 Labor Day Monday, September 2, 2013 Monday, September 2, 2013 Veterans Day Monday, November 11, 2013 Monday, November 11, 2013 Thanksgiving

### 3y 1 + 5y 2. y 1 + y 2 20 y 1 0, y 2 0.

1 Linear Programming A linear programming problem is the problem of maximizing (or minimizing) a linear function subject to linear constraints. The constraints may be equalities or inequalities. 1.1 Example

### Cruise Line Agencies of Alaska. Cruise Ship Calendar for 2016 FOR PORT(S) = KTN AND SHIP(S) = ALL AND VOYAGES = ALL

Cruise Line Agencies of Alaska Cruise Ship Calendar for 06 FOR PORT(S) = KTN AND SHIP(S) = ALL AND VOYAGES = ALL 6: Friday, April 5, 06 Cruise Line Agencies of Alaska, Cruise Ship Calendar for 06 Page

### Linear Programming: Using the Excel Solver

Outline: Linear Programming: Using the Excel Solver We will use Microsoft Excel Solver to solve the four LP examples discussed in last class. 1. The Product Mix Example The Outdoor Furniture Corporation

### Discrete (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

### Linear Programming Models: Graphical and Computer Methods

Linear Programming Models: Graphical and Computer Methods Learning Objectives Students will be able to: 1. Understand the basic assumptions and properties of linear programming (LP). 2. Graphically solve

### MATHEMATICAL NOTES #1 - DEMAND AND SUPPLY CURVES -

MATHEMATICAL NOTES #1 - DEMAND AND SUPPLY CURVES - Linear Equations & Graphs Remember that we defined demand as the quantity of a good consumers are willing and able to buy at a particular price. Notice

### max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x

Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study

### Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

### Using Excel to Find Numerical Solutions to LP's

Using Excel to Find Numerical Solutions to LP's by Anil Arya and Richard A. Young This handout sketches how to solve a linear programming problem using Excel. Suppose you wish to solve the following LP:

### 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

### 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

### 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

### Introduction to Linear Programming.

Chapter 1 Introduction to Linear Programming. This chapter introduces notations, terminologies and formulations of linear programming. Examples will be given to show how real-life problems can be modeled

### Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs

Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs These questions are from previous years and should you give you some idea of what to expect on Quiz 1. 1. Consider the

### Sensitivity 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

### Lesson 3: Graphical method for solving LPP. Graphical Method of solving Linear Programming Problems

Unit Lesson 3: Graphical method for solving LPP. Learning outcome.finding the graphical solution to the linear programming model Graphical Method of solving Linear Programming Problems Introduction Dear

### Section Notes 4. Duality, Sensitivity, Dual Simplex, Complementary Slackness. Applied Math 121. Week of February 28, 2011

Section Notes 4 Duality, Sensitivity, Dual Simplex, Complementary Slackness Applied Math 121 Week of February 28, 2011 Goals for the week understand the relationship between primal and dual programs. know

### Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

### 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

### Degeneracy in Linear Programming

Degeneracy in Linear Programming I heard that today s tutorial is all about Ellen DeGeneres Sorry, Stan. But the topic is just as interesting. It s about degeneracy in Linear Programming. Degeneracy? Students

### Stanford University CS261: Optimization Handout 6 Luca Trevisan January 20, In which we introduce the theory of duality in linear programming.

Stanford University CS261: Optimization Handout 6 Luca Trevisan January 20, 2011 Lecture 6 In which we introduce the theory of duality in linear programming 1 The Dual of Linear Program Suppose that we

### Components of LP Problem. Chapter 13. Linear Programming (LP): Model Formulation & Graphical Solution. Introduction. Components of LP Problem (Cont.

Chapter 13 Linear Programming (LP): Model Formulation & Graphical Solution Components of LP Problem Decision Variables Denoted by mathematical symbols that does not have a specific value Examples How much

### Lecture notes 1: Introduction to linear and (mixed) integer programs

Lecture notes 1: Introduction to linear and (mixed) integer programs Vincent Conitzer 1 An example We will start with a simple example. Suppose we are in the business of selling reproductions of two different

### 2014 Juvenile Justice Center Holidays

2014 Juvenile Justice Center Holidays Wednesday January 1 New Year s Day Monday January 20 Martin Luther King s Birthday Monday February 17 Presidents Day Monday May 26 Memorial Day Friday July 4 Independence

### 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

### APPLICATION OF CONDITIONS OF SERVICE PUBLIC HOLIDAY WORKING

APPLICATION OF CONDITIONS OF SERVICE PUBLIC HOLIDAY WORKING CHRISTMAS AND NEW YEAR PUBLIC HOLIDAYS 1. General Principles 1.1 It is recognised that because of varying working patterns and arrangements,

### 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

### 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,

### January. Success Principles SUNDAY MONDAY TUESDAY WEDNESDAY. How to Get From Where You Are to Where You Want to Be

g he Success Principles How to Get From Where You Are to Where You Want to Be January You must take personal responsibility. You cannot change the circumstances, the seasons, or the wind, but you can change

### 8-week Early Registration

FALL 2016 16-week Accelerated I April 25 -August 8 (Payment Due Aug 8) August 9 27 August 9 27 August 9 - August 9 - October September 24 22 Senior Registration August 25-27 August 25-27 September 22-24

### Establishing a mathematical model (Ch. 2): 1. Define the problem, gathering data;

Establishing a mathematical model (Ch. ): 1. Define the problem, gathering data;. Formulate the model; 3. Derive solutions;. Test the solution; 5. Apply. Linear Programming: Introduction (Ch. 3). An example

### is redundant. If you satisfy the other four constraints, then you automatically satisfy x 1 0. (In fact, if the first two constraints hold, then x 1 m

Linear Programming Notes II: Graphical Solutions 1 Graphing Linear Inequalities in the Plane You can solve linear programming problems involving just two variables by drawing a picture. The method works

### Notes for Recitation 5

6.042/18.062J Mathematics for Computer Science September 24, 2010 Tom Leighton and Marten van Dijk Notes for Recitation 5 1 Exponentiation and Modular Arithmetic Recall that RSA encryption and decryption

### 1. 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

### 1. Setting up a small sample model to be solved by LINDO. 5. How do solve LP with Unrestricted variables (i.e. non-standard form)?

Contents: 1. Setting up a small sample model to be solved by LINDO 2. How to prepare the model to be solved by LINDO? 3. How to Enter the (input) Model to LINDO [Windows]? 4. How do I Print my work or

MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose

### Chapter 3: Elementary Number Theory and Methods of Proof. January 31, 2010

Chapter 3: Elementary Number Theory and Methods of Proof January 31, 2010 3.4 - Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem Quotient-Remainder Theorem Given

### 2.1 Systems of Linear Equations

. Systems of Linear Equations Question : What is a system of linear equations? Question : Where do systems of equations come from? In Chapter, we looked at several applications of linear functions. One

### Functions: Piecewise, Even and Odd.

Functions: Piecewise, Even and Odd. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway September 21-22, 2015 Tutorials, Online Homework.

### 3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes

Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

### Product Mix as a Framing Exercise: The Role of Cost Allocation. Anil Arya The Ohio State University. Jonathan Glover Carnegie Mellon University

Product Mix as a Framing Exercise: The Role of Cost Allocation Anil Arya The Ohio State University Jonathan Glover Carnegie Mellon University Richard Young The Ohio State University December 1999 Product

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### Cruise Line Agencies of Alaska. Cruise Ship Calendar for 2012 FOR PORT(S) = KTN AND SHIP(S) = ALL AND VOYAGES = ALL

Cruise Line Agencies of Alaska Cruise Ship Calendar for 0 FOR PORT(S) = KTN AND SHIP(S) = ALL AND VOYAGES = ALL :8 Tuesday, January 0, 0 Cruise Line Agencies of Alaska, Cruise Ship Calendar for 0 Page

### Cruise Line Agencies of Alaska. Cruise Ship Calendar for 2016 FOR PORT(S) = JNU AND SHIP(S) = ALL AND VOYAGES = ALL

Cruise Line Agencies of Alaska Cruise Ship Calendar for 2016 FOR PORT(S) = JNU AND SHIP(S) = ALL AND VOYAGES = ALL Page 1 of 5 Sunday, April 24 Monday, April 25 Tuesday, April 26 Wednesday, April 27 Thursday,

### Introduction to Nonlinear Programming

Week 1 to Nonlinear Programming OPR 992 Applied Mathematical Programming OPR 992 - Applied Mathematical Programming - p. 1/14 What is Nonlinear Programming (NLP)? Beaver Creek Pottery The LP Model for

### Mathematics for Computer Science

mcs 01/1/19 1:3 page i #1 Mathematics for Computer Science revised Thursday 19 th January, 01, 1:3 Eric Lehman Google Inc. F Thomson Leighton Department of Mathematics and the Computer Science and AI Laboratory,

### MATH 22. THE FUNDAMENTAL THEOREM of ARITHMETIC. Lecture R: 10/30/2003

MATH 22 Lecture R: 10/30/2003 THE FUNDAMENTAL THEOREM of ARITHMETIC You must remember this, A kiss is still a kiss, A sigh is just a sigh; The fundamental things apply, As time goes by. Herman Hupfeld

### Cruise Line Agencies of Alaska. Cruise Ship Calendar for 2013 FOR PORT(S) = KTN AND SHIP(S) = ALL AND VOYAGES = ALL

Cruise Line Agencies of Alaska Cruise Ship Calendar for 0 FOR PORT(S) = KTN AND SHIP(S) = ALL AND VOYAGES = ALL 9:5 Friday, January 0, 0 Cruise Line Agencies of Alaska, Cruise Ship Calendar for 0 Page

### Chapter 2 Introduction to Linear Programming

Chapter 2 Introduction to Linear Programming The key takeaways for the reader from this chapter are listed below: A good understanding of linear programming (LP) problems Formulation of the two-variable

### Plans for Monday, May 19, 2014 By: Cheryl Casey

Plans for Monday, May 19, 2014 Students will continue graphing and writing inequalities HW: Graphing and Writing Inequalities, due Tuesday Students will graph and write inequalities HW: Graphing and Writing

### By W.E. Diewert. July, Linear programming problems are important for a number of reasons:

APPLIED ECONOMICS By W.E. Diewert. July, 3. Chapter : Linear Programming. Introduction The theory of linear programming provides a good introduction to the study of constrained maximization (and minimization)

### Linear Programming with Post-Optimality Analyses

Linear Programming with Post-Optimality Analyses Wilson Problem: Wilson Manufacturing produces both baseballs and softballs, which it wholesales to vendors around the country. Its facilities permit the

### Notes 10: An Equation Based Model of the Macroeconomy

Notes 10: An Equation Based Model of the Macroeconomy In this note, I am going to provide a simple mathematical framework for 8 of the 9 major curves in our class (excluding only the labor supply curve).

### COMP 250 Fall Mathematical induction Sept. 26, (n 1) + n = n + (n 1)

COMP 50 Fall 016 9 - Mathematical induction Sept 6, 016 You will see many examples in this course and upcoming courses of algorithms for solving various problems It many cases, it will be obvious that

### Comparison of Optimization Techniques in Large Scale Transportation Problems

Journal of Undergraduate Research at Minnesota State University, Mankato Volume 4 Article 10 2004 Comparison of Optimization Techniques in Large Scale Transportation Problems Tapojit Kumar Minnesota State

### EXCEL 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 plug-in, Solver, that we will be using

### CHAPTER 17. Linear Programming: Simplex Method

CHAPTER 17 Linear Programming: Simplex Method CONTENTS 17.1 AN ALGEBRAIC OVERVIEW OF THE SIMPLEX METHOD Algebraic Properties of the Simplex Method Determining a Basic Solution Basic Feasible Solution 17.2

### Linear Programming: Basic Concepts

Linear Programming: Basic Concepts Table of Contents Three Classic Applications of LP The Wyndor Glass Company Product Mix Problem Formulating the Wyndor Problem on a Spreadsheet The Algebraic Model for

### Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

### Lecture 3: Summations and Analyzing Programs with Loops

high school algebra If c is a constant (does not depend on the summation index i) then ca i = c a i and (a i + b i )= a i + b i There are some particularly important summations, which you should probably

### Beginner Health Workout

Note: This 12-week program is designed for the beginner walker who wants to improve overall health and increase energy. Walks start at 10 minutes or less and gradually work up to 30+ minutes. Health experts

### 4.1 Solving a System of Linear Inequalities

4.1 Solving a System of Linear Inequalities Question 1: How do you graph a linear inequality? Question : How do you graph a system of linear inequalities? In Chapter, we were concerned with systems of

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### Quiz Sensitivity Analysis

1 Quiz Sensitivity Analysis 1. The difference between the right-hand side (RHS) values of the constraints and the final (optimal) value assumed by the left-hand side (LHS) formula for each constraint is

### LINEAR PROGRAMMING: MODEL FORMULATION AND 2-1

LINEAR PROGRAMMING: MODEL FORMULATION AND GRAPHICAL SOLUTION 2-1 Chapter Topics Model Formulation A Maximization Model Example Graphical Solutions of Linear Programming Models A Minimization Model Example

### Linear Programming Exercises. Week 1. Exercise 1 Consider the case of the Betta Machine Products Company described in the lecture notes.

Linear Programming Exercises Week 1 Exercise 1 Consider the case of the Betta Machine Products Company described in the lecture notes. (a) Use a graphical method to obtain the new optimal solution when

### Integer Programming Formulation

Integer Programming Formulation 1 Integer Programming Introduction When we introduced linear programs in Chapter 1, we mentioned divisibility as one of the LP assumptions. Divisibility allowed us to consider

### 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,

### The Simplex Method. What happens when we need more decision variables and more problem constraints?

The Simplex Method The geometric method of solving linear programming problems presented before. The graphical method is useful only for problems involving two decision variables and relatively few problem

### The Simplex Method. yyye

Yinyu Ye, MS&E, Stanford MS&E310 Lecture Note #05 1 The Simplex Method Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/

### Outline. Linear Programming (LP): Simplex Search. Simplex: An Extreme-Point Search Algorithm. Basic Solutions

Outline Linear Programming (LP): Simplex Search Benoît Chachuat McMaster University Department of Chemical Engineering ChE 4G03: Optimization in Chemical Engineering 1 Basic Solutions

### Monday Tuesday Wednesday Thursday Friday Saturday Sunday AUGUST 29 30

Library Research Workshops Fall 2016 Learn how to develop a focused research topic, search reliable reference sources for overviews, and narrow & broaden research topics. 75min Evaluating Websites Learn

### 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