1. Introduction. Scheduling Theory

Save this PDF as:
Size: px
Start display at page:

Download "1. Introduction. Scheduling Theory"

Transcription

1 . Itroductio. Itroductio As a idepedet brach of Operatioal Research, Schedulig Theory appeared i the begiig of the 50s. I additio to computer systems ad maufacturig, schedulig theory ca be applied to may areas icludig agriculture, health care ad trasport. Operatioal Research Mathematical Programmig Combiatorial Optimisatio Schedulig Theory Schedulig deals with the problems of optimal arragemet, sequecig ad timetablig. Schedulig is a decisiomakig process of allocatig limited resources to activities over time. Resources (machies): machies at a workshop, ruways at a airport, crews at a costructio site, processig uits i a computig eviromet. Tasks (obs): operatios i a workshop, takeoffs ad ladigs, stages at a costructio proect, computer programs. A schedule is a ob sequece determied for every machie of the processig system. Stadard schedulig requiremets say: a ob caot be processed by two or more machies at a time, a machie caot process two or more obs at the same time. Depedig o the type of schedulig system, specific costraits should be satisfied (obs may be released at differet times, there may be allowed preemptio of obs by other obs, etc.).

2 . Itroductio Gatt chart is a horizotal bar chart that graphically displays the time relatioships betwee the differet tasks i a proect: the Xaxis represets the time, the Yaxis represets machies, a colour ad/or patter code may be used to idicate operatios of the same ob. M M M 3 M time Schedulig theory covers more tha 0,000 differet models. The models are specified accordig to threefield classificatio α β γ where α specifies the machie eviromet, β specifies the ob characteristics, ad γ determies the optimality criterio. Machie eviromet Sigle stage systems Multistage systems If there is a sigle machie (m=), each ob should be processed by that machie exactly oce. If there are several parallel machies {M, M,, M m }, each ob ca be processed by ay machie. α = P sigle (dedicated) machie p p i processig time of detical parallelmachies, = p o machie i ob processig time of ob Each ob should be processed o each machie from the set {M, M,, M m }. All machies are differet. F α = J O flow shop, ob is processed first o machie, the o machie,..., ad fially o machiem ob shop, eachob has its ow route to follow ope shop, each ob ca be processed by the machiesi a arbitrary order

3 . Itroductio Job eviromet There are obs N={,,}. Processig time of ob o machie i is p i. If there is a sigle machie, the processig time of ob does ot deped o machie umber ad it is deoted by p. For ob there may be give also r release time (the time the ob arrives at the system), d due date (the time the ob is promised to the customer), w weight (the importace of ob ). Preemptio (pmt) implies the processig of ay ob ca be iterrupted ad resumed later. Optimality criterio The schedule ca be characterised by startig or completio times of all operatios of the obs. The obective is to costruct a schedule, that miimizes a give obective fuctio F. Usually fuctio F depeds o ob completio times C, =,,, where C is the completio time of the last operatio of ob. The most commo obective fuctios are Makespa Total completio time Total weighted completio time C max = max {C =,,} = = w C = C C = w C Other obective fuctios deped o due dates d. We defie for each ob L = C d E = max{0, d C } T = max{0, C d } 0 if C d U = otherwise lateess of ob earliess, tardiess, uit pealty., The correspodig obective fuctios ca be defied as follows: Maximum lateess Total (weighted) tardiess Total (weighted) umber of late obs L max = max {L =,,} = T ( = w T ) = U ( = w U ) 3

4 . Itroductio Examples: ) r, pmt L max is the problem of fidig a preemptive schedule o oe machie for a set of obs with give release times such that maximum lateess is miimised. ) P p = C max is the problem of schedulig obs with uit processig times o m idetical parallel machies such that the makespa is miimised. 3) J3 p i = C max is the problem of miimisig maximum completio time i a threemachie ob shop with uit processig times. Iclass exercise : Cosider a schedulig problem with readers ad two books. Classify the followig schedulig models: Machies Jobs Obective α β γ Two volumes of oe book Fiish readig as soo as possible F C max Two volumes of oe book Miimise the cost of late book retur Two differet (idepedet) books Fiish readig as soo as possible Two differet (idepedet) books (each reader has its ow readig sequece ) Fiish readig as soo as possible Iclass exercise : Classify the examples of schedulig problems: Publishig idustry: typesettig, actual pritig, bidig, packagig. Differet items have differet processig times depedig o the book size, the umber of copies, etc. The obective is to produce all items as soo as possible. Clothig idustry: cuttig, sewig, pressig, packig. The obective is to produce all items as soo as possible. Steel mills: differet rods or girders pass through the set of rollers i their ow orders with their ow temperatures ad pressure settigs. The obective is to produce all items as soo as possible. Repair of cars i a garage: replace tires, repair gear box, check brakes, repair headlights, etc. The obective is to repair all cars i a garage as soo as possible. Completig several pieces of CW so that the maximum lateess is miimised. CW is released at time r, requires p days for completio ad has a due date d. Revisio schedule: startig o 3//004, revise the material of modules by their exam dates. Revisio time for module is p. F4 C max? Literature review for FYP should be based o library books. Book ca be read i p days ad it should be retured by its due date d. The library charges 30p per day o each overdue book. The obective is to miimise the total fie. 4

5 . Itroductio Complexity Hierarchy There is a certai complexity hierarchy amog schedulig systems ad obective fuctios as show i the diagrams below. The arrows go to harder problems. Σw T Σw U J Σw C ΣT ΣU P F O ΣC L max C max C max reduces to L max by settig d =0 for all. ΣC reduces to Σw C by settig w = for all. ΣC reduces to ΣΤ by settig d =0 for all. Σw C reduces to Σw Τ by settig d =0 for all. I this course we demostrate most schedulig methods ad techiques. Most algorithms will be illustrated usig schedulig software systems LEKIN ad LiSA. LiSA Library of Schedulig Algorithms, OttovoGuericke Uiversität, Magdeburg, Germay, Proect Leader: Professor Heidemarie Bräsel, LEKIN Educatioal Schedulig System, Ster School of Busiess, New York Uiversity, Proect Leader: Professor Michael Piedo, The followig books are cosidered to be stadard texts o the topic:. J.Blazewicz, K.H. Ecker, E.Pesch, G.Schmidt, J. Weglarz Schedulig Computer ad Maufacturig Processes, Spriger, 996, 00.. P. Brucker Schedulig Algorithms, Spriger, 995, 998, S. Frech Sequecig ad schedulig, Ellis Horwood, M. Piedo Schedulig: Theory, Algorithms, ad Systems, Pretice Hall, 995, M. Piedo Operatios Schedulig with Applicatios i Maufacturig ad Services, Irwi/McGraw Hill,

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

More information

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature. Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.

More information

Stochastic Online Scheduling with Precedence Constraints

Stochastic Online Scheduling with Precedence Constraints Stochastic Olie Schedulig with Precedece Costraits Nicole Megow Tark Vredeveld July 15, 2008 Abstract We cosider the preemptive ad o-preemptive problems of schedulig obs with precedece costraits o parallel

More information

Incremental calculation of weighted mean and variance

Incremental calculation of weighted mean and variance Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

More information

Chatpun Khamyat Department of Industrial Engineering, Kasetsart University, Bangkok, Thailand ocpky@hotmail.com

Chatpun Khamyat Department of Industrial Engineering, Kasetsart University, Bangkok, Thailand ocpky@hotmail.com SOLVING THE OIL DELIVERY TRUCKS ROUTING PROBLEM WITH MODIFY MULTI-TRAVELING SALESMAN PROBLEM APPROACH CASE STUDY: THE SME'S OIL LOGISTIC COMPANY IN BANGKOK THAILAND Chatpu Khamyat Departmet of Idustrial

More information

Modified Line Search Method for Global Optimization

Modified Line Search Method for Global Optimization Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

More information

W. Sandmann, O. Bober University of Bamberg, Germany

W. Sandmann, O. Bober University of Bamberg, Germany STOCHASTIC MODELS FOR INTERMITTENT DEMANDS FORECASTING AND STOCK CONTROL W. Sadma, O. Bober Uiversity of Bamberg, Germay Correspodig author: W. Sadma Uiversity of Bamberg, Dep. Iformatio Systems ad Applied

More information

Foundations of Operations Research

Foundations of Operations Research Foudatios of Operatios Research Master of Sciece i Computer Egieerig Roberto Cordoe roberto.cordoe@uimi.it Tuesday 13.15-15.15 Thursday 10.15-13.15 http://homes.di.uimi.it/~cordoe/courses/2014-for/2014-for.html

More information

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics

More information

INVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology

INVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology

More information

ODBC. Getting Started With Sage Timberline Office ODBC

ODBC. Getting Started With Sage Timberline Office ODBC ODBC Gettig Started With Sage Timberlie Office ODBC NOTICE This documet ad the Sage Timberlie Office software may be used oly i accordace with the accompayig Sage Timberlie Office Ed User Licese Agreemet.

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

Baan Service Master Data Management

Baan Service Master Data Management Baa Service Master Data Maagemet Module Procedure UP069A US Documetiformatio Documet Documet code : UP069A US Documet group : User Documetatio Documet title : Master Data Maagemet Applicatio/Package :

More information

Engineering Data Management

Engineering Data Management BaaERP 5.0c Maufacturig Egieerig Data Maagemet Module Procedure UP128A US Documetiformatio Documet Documet code : UP128A US Documet group : User Documetatio Documet title : Egieerig Data Maagemet Applicatio/Package

More information

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig

More information

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

More information

CS100: Introduction to Computer Science

CS100: Introduction to Computer Science Review: History of Computers CS100: Itroductio to Computer Sciece Maiframes Miicomputers Lecture 2: Data Storage -- Bits, their storage ad mai memory Persoal Computers & Workstatios Review: The Role of

More information

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Cosider a legth- sequece x[ with a -poit DFT X[ where Represet the idices ad as +, +, Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Usig these

More information

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.

More information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

More information

Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT

Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee

More information

1 Computing the Standard Deviation of Sample Means

1 Computing the Standard Deviation of Sample Means Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

TruStore: The storage. system that grows with you. Machine Tools / Power Tools Laser Technology / Electronics Medical Technology

TruStore: The storage. system that grows with you. Machine Tools / Power Tools Laser Technology / Electronics Medical Technology TruStore: The storage system that grows with you Machie Tools / Power Tools Laser Techology / Electroics Medical Techology Everythig from a sigle source. Cotets Everythig from a sigle source. 2 TruStore

More information

MTO-MTS Production Systems in Supply Chains

MTO-MTS Production Systems in Supply Chains NSF GRANT #0092854 NSF PROGRAM NAME: MES/OR MTO-MTS Productio Systems i Supply Chais Philip M. Kamisky Uiversity of Califoria, Berkeley Our Kaya Uiversity of Califoria, Berkeley Abstract: Icreasig cost

More information

THE CARDINALITY CONSTRAINED MULTIPLE KNAPSACK PROBLEM

THE CARDINALITY CONSTRAINED MULTIPLE KNAPSACK PROBLEM THE CARDINALITY CONSTRAINED MULTIPLE KNAPSACK PROBLEM A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY MURAT ASLAN IN PARTIAL FULFILLMENT

More information

Multiplexers and Demultiplexers

Multiplexers and Demultiplexers I this lesso, you will lear about: Multiplexers ad Demultiplexers 1. Multiplexers 2. Combiatioal circuit implemetatio with multiplexers 3. Demultiplexers 4. Some examples Multiplexer A Multiplexer (see

More information

Institute of Actuaries of India Subject CT1 Financial Mathematics

Institute of Actuaries of India Subject CT1 Financial Mathematics Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i

More information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS

More information

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,

More information

Convention Paper 6764

Convention Paper 6764 Audio Egieerig Society Covetio Paper 6764 Preseted at the 10th Covetio 006 May 0 3 Paris, Frace This covetio paper has bee reproduced from the author's advace mauscript, without editig, correctios, or

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say

More information

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

More information

NATIONAL SENIOR CERTIFICATE GRADE 11

NATIONAL SENIOR CERTIFICATE GRADE 11 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 007 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad a -page formula sheet. Please tur over Mathematics/P DoE/November

More information

Asymptotic Growth of Functions

Asymptotic Growth of Functions CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

More information

Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:

Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows: Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network

More information

Subject CT5 Contingencies Core Technical Syllabus

Subject CT5 Contingencies Core Technical Syllabus Subject CT5 Cotigecies Core Techical Syllabus for the 2015 exams 1 Jue 2014 Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which ca be used to model ad value

More information

Domain 1: Designing a SQL Server Instance and a Database Solution

Domain 1: Designing a SQL Server Instance and a Database Solution Maual SQL Server 2008 Desig, Optimize ad Maitai (70-450) 1-800-418-6789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a

More information

Department of Computer Science, University of Otago

Department of Computer Science, University of Otago Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

More information

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics

More information

Effective Project Scheduling Under Workspace Congestion and Workflow Disturbance Factors

Effective Project Scheduling Under Workspace Congestion and Workflow Disturbance Factors Abstract Effective Project Schedulig Uder Workspace Cogestio ad Workflow Disturbace Factors Vitaly Semeov, Ato Aichki, Sergey Morozov, Oleg Tarlapa, Vladislav Zolotov (Istitute for System Programmig RAS,

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios

More information

Statistical inference: example 1. Inferential Statistics

Statistical inference: example 1. Inferential Statistics Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either

More information

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL. Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory

More information

Irreducible polynomials with consecutive zero coefficients

Irreducible polynomials with consecutive zero coefficients Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem

More information

Queuing Systems: Lecture 1. Amedeo R. Odoni October 10, 2001

Queuing Systems: Lecture 1. Amedeo R. Odoni October 10, 2001 Queuig Systems: Lecture Amedeo R. Odoi October, 2 Topics i Queuig Theory 9. Itroductio to Queues; Little s Law; M/M/. Markovia Birth-ad-Death Queues. The M/G/ Queue ad Extesios 2. riority Queues; State

More information

Lecture 3. denote the orthogonal complement of S k. Then. 1 x S k. n. 2 x T Ax = ( ) λ x. with x = 1, we have. i = λ k x 2 = λ k.

Lecture 3. denote the orthogonal complement of S k. Then. 1 x S k. n. 2 x T Ax = ( ) λ x. with x = 1, we have. i = λ k x 2 = λ k. 18.409 A Algorithmist s Toolkit September 17, 009 Lecture 3 Lecturer: Joatha Keler Scribe: Adre Wibisoo 1 Outlie Today s lecture covers three mai parts: Courat-Fischer formula ad Rayleigh quotiets The

More information

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy

More information

Research Method (I) --Knowledge on Sampling (Simple Random Sampling)

Research Method (I) --Knowledge on Sampling (Simple Random Sampling) Research Method (I) --Kowledge o Samplig (Simple Radom Samplig) 1. Itroductio to samplig 1.1 Defiitio of samplig Samplig ca be defied as selectig part of the elemets i a populatio. It results i the fact

More information

AP Calculus AB 2006 Scoring Guidelines Form B

AP Calculus AB 2006 Scoring Guidelines Form B AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success

More information

I. Why is there a time value to money (TVM)?

I. Why is there a time value to money (TVM)? Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios

More information

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is 0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

More information

CCH Accountants Starter Pack

CCH Accountants Starter Pack CCH Accoutats Starter Pack We may be a bit smaller, but fudametally we re o differet to ay other accoutig practice. Util ow, smaller firms have faced a stark choice: Buy cheaply, kowig that the practice

More information

Theorems About Power Series

Theorems About Power Series Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

More information

CONTROL CHART BASED ON A MULTIPLICATIVE-BINOMIAL DISTRIBUTION

CONTROL CHART BASED ON A MULTIPLICATIVE-BINOMIAL DISTRIBUTION www.arpapress.com/volumes/vol8issue2/ijrras_8_2_04.pdf CONTROL CHART BASED ON A MULTIPLICATIVE-BINOMIAL DISTRIBUTION Elsayed A. E. Habib Departmet of Statistics ad Mathematics, Faculty of Commerce, Beha

More information

Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2

Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2 74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is

More information

Maximum Likelihood Estimators.

Maximum Likelihood Estimators. Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio

More information

Running Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis

Running Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.

More information

5 Boolean Decision Trees (February 11)

5 Boolean Decision Trees (February 11) 5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected

More information

A probabilistic proof of a binomial identity

A probabilistic proof of a binomial identity A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

More information

APPENDIX V TWO-YEAR COLLEGE SURVEY

APPENDIX V TWO-YEAR COLLEGE SURVEY APPENDIX V TWO-YEAR COLLEGE SURVEY GENERAL INSTRUCTIONS Coferece Board of the Mathematical Scieces SURVEY OF PROGRAMS i MATHEMATICS AND COMPUTER SCIENCE i TWO-YEAR COLLEGES 1990 This questioaire should

More information

Output Analysis (2, Chapters 10 &11 Law)

Output Analysis (2, Chapters 10 &11 Law) B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

More information

Diploma in Secretarial Administration

Diploma in Secretarial Administration Istitute of Fiace Diploma i Secretarial Admiistratio Awarded by the Lodo Chamber of Commerce ad Idustry (LCCI) Startig October 2007 ope for erollmet from July 2007 Be smart start right eroll ow! Eglish

More information

The Forgotten Middle. research readiness results. Executive Summary

The Forgotten Middle. research readiness results. Executive Summary The Forgotte Middle Esurig that All Studets Are o Target for College ad Career Readiess before High School Executive Summary Today, college readiess also meas career readiess. While ot every high school

More information

Desktop Management. Desktop Management Tools

Desktop Management. Desktop Management Tools Desktop Maagemet 9 Desktop Maagemet Tools Mac OS X icludes three desktop maagemet tools that you might fid helpful to work more efficietly ad productively: u Stacks puts expadable folders i the Dock. Clickig

More information

Prof. Dr. Liggesmeyer, 2. Fault Tree Analysis (DIN 25424, IEC 61025) Reliability Block Diagrams (IEC 61078)

Prof. Dr. Liggesmeyer, 2. Fault Tree Analysis (DIN 25424, IEC 61025) Reliability Block Diagrams (IEC 61078) (icherheit ud Zuverlässigkeit eigebetteter ysteme) afety ad Reliability Aalysis Models: Overview Classificatio of afety / Reliability Aalysis Techiques Focused Property afety, Reliability, Availability...

More information

(VCP-310) 1-800-418-6789

(VCP-310) 1-800-418-6789 Maual VMware Lesso 1: Uderstadig the VMware Product Lie I this lesso, you will first lear what virtualizatio is. Next, you ll explore the products offered by VMware that provide virtualizatio services.

More information

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

More information

FI A CIAL MATHEMATICS

FI A CIAL MATHEMATICS CHAPTER 7 FI A CIAL MATHEMATICS Page Cotets 7.1 Compoud Value 117 7.2 Compoud Value of a Auity 118 7.3 Sikig Fuds 119 7.4 Preset Value 122 7.5 Preset Value of a Auity 122 7.6 Term Loas ad Amortizatio 123

More information

Estimating Probability Distributions by Observing Betting Practices

Estimating Probability Distributions by Observing Betting Practices 5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,

More information

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The

More information

Infinite Sequences and Series

Infinite Sequences and Series CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

More information

A GUIDE TO LEVEL 3 VALUE ADDED IN 2013 SCHOOL AND COLLEGE PERFORMANCE TABLES

A GUIDE TO LEVEL 3 VALUE ADDED IN 2013 SCHOOL AND COLLEGE PERFORMANCE TABLES A GUIDE TO LEVEL 3 VALUE ADDED IN 2013 SCHOOL AND COLLEGE PERFORMANCE TABLES Cotets Page No. Summary Iterpretig School ad College Value Added Scores 2 What is Value Added? 3 The Learer Achievemet Tracker

More information

Reliability Analysis in HPC clusters

Reliability Analysis in HPC clusters Reliability Aalysis i HPC clusters Narasimha Raju, Gottumukkala, Yuda Liu, Chokchai Box Leagsuksu 1, Raja Nassar, Stephe Scott 2 College of Egieerig & Sciece, Louisiaa ech Uiversity Oak Ridge Natioal Lab

More information

On Formula to Compute Primes. and the n th Prime

On Formula to Compute Primes. and the n th Prime Applied Mathematical cieces, Vol., 0, o., 35-35 O Formula to Compute Primes ad the th Prime Issam Kaddoura Lebaese Iteratioal Uiversity Faculty of Arts ad cieces, Lebao issam.kaddoura@liu.edu.lb amih Abdul-Nabi

More information

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig

More information

MADE BY VOLVO. IN INDIA.

MADE BY VOLVO. IN INDIA. MADE BY VOLVO. IN INDIA. The quality of Volvo trucks ad our maufacturig systems is well exhibited amog the scores of Volvo trucks i operatio i Idia, amog the toughest of coditios & applicatios. The Volvo

More information

Time Value of Money. First some technical stuff. HP10B II users

Time Value of Money. First some technical stuff. HP10B II users Time Value of Moey Basis for the course Power of compoud iterest $3,600 each year ito a 401(k) pla yields $2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle

More information

On the Capacity of Hybrid Wireless Networks

On the Capacity of Hybrid Wireless Networks O the Capacity of Hybrid ireless Networks Beyua Liu,ZheLiu +,DoTowsley Departmet of Computer Sciece Uiversity of Massachusetts Amherst, MA 0002 + IBM T.J. atso Research Ceter P.O. Box 704 Yorktow Heights,

More information

Chapter 5: Inner Product Spaces

Chapter 5: Inner Product Spaces Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

More information

INDEPENDENT BUSINESS PLAN EVENT 2016

INDEPENDENT BUSINESS PLAN EVENT 2016 INDEPENDENT BUSINESS PLAN EVENT 2016 The Idepedet Busiess Pla Evet ivolves the developmet of a comprehesive proposal to start a ew busiess. Ay type of busiess may be used. The Idepedet Busiess Pla Evet

More information

Rainbow options. A rainbow is an option on a basket that pays in its most common form, a nonequally

Rainbow options. A rainbow is an option on a basket that pays in its most common form, a nonequally Raibow optios INRODUCION A raibow is a optio o a basket that pays i its most commo form, a oequally weighted average of the assets of the basket accordig to their performace. he umber of assets is called

More information

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13 EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

More information

3. Greatest Common Divisor - Least Common Multiple

3. Greatest Common Divisor - Least Common Multiple 3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

More information

FVS SERIES. Vertical Laboratory Autoclaves. Simplest solution for routine laboratory sterilization

FVS SERIES. Vertical Laboratory Autoclaves. Simplest solution for routine laboratory sterilization FVS SERIES Vertical Laboratory Autoclaves Simplest solutio for routie laboratory sterilizatio The FVS is compact, portable, vertical Laboratory Autoclave, with very small overall dimesios for easy hadlig

More information

Trading the randomness - Designing an optimal trading strategy under a drifted random walk price model

Trading the randomness - Designing an optimal trading strategy under a drifted random walk price model Tradig the radomess - Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore

More information

LECTURE 13: Cross-validation

LECTURE 13: Cross-validation LECTURE 3: Cross-validatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Three-way data partitioi Itroductio to Patter Aalysis Ricardo Gutierrez-Osua Texas A&M

More information

Domain 1 - Describe Cisco VoIP Implementations

Domain 1 - Describe Cisco VoIP Implementations Maual ONT (642-8) 1-800-418-6789 Domai 1 - Describe Cisco VoIP Implemetatios Advatages of VoIP Over Traditioal Switches Voice over IP etworks have may advatages over traditioal circuit switched voice etworks.

More information

1 Correlation and Regression Analysis

1 Correlation and Regression Analysis 1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

More information

Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread and Boxplots Discrete Math, Section 9.4 Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

More information

BaanERP 5.0c. EDI User Guide

BaanERP 5.0c. EDI User Guide BaaERP 5.0c A publicatio of: Baa Developmet B.V. P.O.Box 143 3770 AC Bareveld The Netherlads Prited i the Netherlads Baa Developmet B.V. 1999. All rights reserved. The iformatio i this documet is subject

More information

Computing First-Order Logic Programs by Fibring Artificial Neural Networks

Computing First-Order Logic Programs by Fibring Artificial Neural Networks Computig First-Order Logic Programs by Fibrig Artificial Neural Netorks Sebastia Bader Departmet of Computer Sciece Techische Uiversität Dresde Germay Artur S. d Avila Garcez Departmet of Computig City

More information

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here). BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

More information

Math C067 Sampling Distributions

Math C067 Sampling Distributions Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters

More information

Perfect Packing Theorems and the Average-Case Behavior of Optimal and Online Bin Packing

Perfect Packing Theorems and the Average-Case Behavior of Optimal and Online Bin Packing SIAM REVIEW Vol. 44, No. 1, pp. 95 108 c 2002 Society for Idustrial ad Applied Mathematics Perfect Packig Theorems ad the Average-Case Behavior of Optimal ad Olie Bi Packig E. G. Coffma, Jr. C. Courcoubetis

More information

Research Article Sign Data Derivative Recovery

Research Article Sign Data Derivative Recovery Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov

More information

GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.

GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number. GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea - add up all

More information

Capacity Management for Contract Manufacturing

Capacity Management for Contract Manufacturing OPERATIONS RESEARCH Vol. 55, No. 2, March April 2007, pp. 367 377 iss 0030-364X eiss 526-5463 07 5502 0367 iforms doi 0.287/opre.060.0359 2007 INFORMS Capacity Maagemet for Cotract Maufacturig Diwakar

More information

CS100: Introduction to Computer Science

CS100: Introduction to Computer Science Course Iformatio CS100: Itroductio to Computer Sciece Lecture 1: Itroductio (Survey, Pictures) Istructor: Xiaoya Li Lecture: Mo. & Wed. 11:00am 12:15pm Room: Kedade Hall 305 Labs: Wed or Thu 1:00pm 2:50pm

More information

Savings and Retirement Benefits

Savings and Retirement Benefits 60 Baltimore Couty Public Schools offers you several ways to begi savig moey through payroll deductios. Defied Beefit Pesio Pla Tax Sheltered Auities ad Custodial Accouts Defied Beefit Pesio Pla Did you

More information

CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

More information