The EOQ Inventory Formula


 Jacob Norton
 2 years ago
 Views:
Transcription
1 Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of a given item to order. A great deal of literature as dealt wit tis problem (unfortunately many of te best books on te subject are out of print). Many formulas and algoritms ave been created. Of tese te simplest formula is te most used: Te EOQ (economic order quantity) or Lot Size formula. Te EOQ formula as been independently discovered many times in te last eigty years. We will see tat te EOQ formula is simplistic and uses several unrealistic assumptions. Tis raises te question, wic we will address: given tat it is so unrealistic, wy does te formula work so well? Indeed, despite te many more sopisticated formulas and algoritms available, even large corporations use te EOQ formula. In general, large corporations tat use te EOQ formula do not want te public or competitors to know tey use someting so unsopisticated. Hence you migt wonder ow I can state tat large corporations do use te EOQ formula. Let s just say tat I ave good sources of information tat I feel can be relied upon. Te Variables of te EOQ Problem Let us assume tat we are interested in optimal inventory policies for widgets. Te EOQ formula uses four variables. Tey are: D: Te demand for widgets in quantity per unit time. Demand can be tougt of as a rate. Q: Te order quantity. Tis is te variable we want to optimize. All te oter variables are fixed quantities. C: Te order cost. Tis is te flat fee carged for making any order and is independent of Q. : Holding costs per widget per unit time. If we store x widgets for one unit of time, it costs us Te EOQ problem can be summarized as determining te order quantity Q, tat balances te order cost C and te olding costs to minimize total costs. Te greater Q is, te less we will spend on orders, since we order less often. On te oter and, te greater Q is te more we spend on inventory. Note tat te price of widgets is a variables tat does not interest us. Tis is because we plan to meet te demand for widgets. Hence te value of Q as noting to do wit tis quantity. If we put te price of widgets into our problem formulation, wen we finally ave finally solved te optimal value for Q, it will not involve tis term.
2 Te Assumptions of te EOQ Model Te underlying assumptions of te EOQ problem can be represented by Figure 1. Te idea is tat orders for widgets arrive instantly and all at once. Secondly, te demand for widgets is perfectly steady. Note tat it is relatively easy to modify tese assumptions; Hadley and Witin [1963] cover many suc cases. Despite te fact tat many more elaborate models ave been constructed for inventory problem te EOQ model is by far te most used. Figure 1 Te EOQ Process An Incorrect Solution Solving for te EOQ, tat is te quantity tat minimizes total costs, requires tat we formulate wat te costs are. Te order period is te block of time between two consecutive orders. Te lengt of te order period, wic we will denote by P, is Q/D. For example, if te order quantity is 0 widgets and te rate of demand is five widgets per day, ten te order period is 0/5, or four days. Let T p be te total costs per order period. By definition, te order cost per order period will be C. During te order period te inventory will go steadily from Q, te order amount, to zero. Hence te average inventory is Q/ and te inventory costs per period is te average cost, Q/, times te lengt of te period, Q/D. Hence te total cost per period is: T P C Q Q C Q D D If we take te derivative of T p wit respect to Q and set it to zero, we get Q = 0. Te problem is solved by te device of not ordering anyting. Tis indeed minimizes te inventory costs but at te small inconvenience of not meeting demand and terefore going out of business. Tis is wat many people, peraps most people do, wen trying to solve for te EOQ te first time. Te Classic EOQ Derivation Te first step to solving te EOQ problem is to correctly state te inventory costs formula. Tis can be done by taking te cost per period T p and dividing by te lengt of te period, Q/D, to get te total cost per unit time, T u : T u CD Q Q
3 In tis formula te order cost per unit time is CD/Q and Q/ is te average inventory cost per unit time. If we take te derivative of T u wit respect to Q and set tat equal to 0, we can solve for te economic order quantity (were te exponent * implies tat tis is te optimal order quantity): Q * T u * CD If we plug Q * into te formula for T u, we get te optimal cost, per unit time: CD Te Algebraic Solution It never urts to solve a problem in two different ways. Usually eac solution tecnique will yield its own insigts. In any case, getting te same answer by two different metods, is a great way to verify te result. If we multiply te formula for T u on bot sides by Q, we get, after a little rearranging, a quadratic equation in Q: Q QT u CD Next we divide troug by / to cange our leading coefficient to 1. Completing te square, we get: 0 Q T u CD Tu 0 Note tat tis equation as two variables. T u is a function of Q (and could well be written as T u (Q)). Hence we ave a curve in te Cartesian plane wit axes labeled Q and T u. We want te value of Q tat minimizes T u. Notice tat te term rearranging te equation, we get: T u CD Q T u is a constant. Writing tat constant as k, and k 3
4 If we set te quadratic term to zero, ten T u k increases te size of T u. Hence te optimal size of T u is 3050 Q * Any cange in te quadratic term from zero k wic just appens to be CD Q T u wic is te value we found earlier. Te quadratic term is zero if and only if. Q * * T u Tis gives us te identity. An Example It is useful at tis point to consider a numerical example. Te demand for klabitz s is 50 per week. Te order cost is $30 (regardless of te size of te order), and te olding cost is $6 per klabitz per week. Plugging tese figures into te EOQ formula we get: Tis brings up a little mentioned drawback of te EOQ formula. Te EOQ formula is not an integer formula. It would be more appropriate if we ordered klabitz s by te gallon. Most of te time, te nearest integer will be te optimal integer amount. In tis case, te total inventory cost T u is $ per week wen we order klabitz s. If instead, we order 3 klabitz s te cost is $134.. Total Cost Min Order Quantity Figure Total Cost as a Function of Order Quantity 4
5 Order Cost Min Order Quantity Holding Cost Figure 3 Order and Holding Costs A grap of tis problem is illuminating: Figure. Because te grap is so flat at te optimal point, tere is very little penalty if we order a sligtly suboptimal quantity. We can better understand te grap if we view te combined graps of te order costs and te olding costs given in Figure 3. Te basic sapes of all tree graps (total costs, order costs, olding costs) are always te same. Te grap of order costs is a yperbola; te grap of olding costs is linear; and as a dtu result te grap of te total costs (T u ) is convex. Tis can also be seen in tat te function is dq increasing and te function dt dq CD Q 3 is positive every were. If we plug te optimal quantity, Q *, into tis last formula we get: dt dq * 5 CD 4CD
6 Ordinarily tis last quantity is very small, wic indicates tat te total cost of inventory T u canges very slowly wit Q (in te optimal region). Hence te assumptions of te EOQ model do not ave to be accurate because te problem usually is tolerant of errors. If you study closely te graps in Figure 3, it may seem clear to you tat teir sum, T u, reaces a minimum precisely were te two graps intersect; tat is at te point were order costs and olding costs are equal. Te gives us te equality way to derive te EOQ formula. Q CD Q. Solving tat equality is te easiest Wy Use te EOQ Formula At All? A problem tat occurs in applied matematics more tan pure is tat we ang onto formulas and tecniques tat ave been made (mostly) obsolete by tecnology. Try to tink wat it was like to solve a problem like te one ere forty years ago. Te simple operation of division was eiter done by and, or by use of logaritms out of a table, or less exactly by using a slide rule. It was not practical to simply calculate te total cost of inventory for a large set of order quantities and to compare answers. Te EOQ formula simplified te problem to a minimal number of calculations. However, now it is quite simple to calculate total costs of inventory for undreds of order quantities, and tis can be done from scratc in less time tan it use to take to employ te EOQ formula. We can do it wit a spreadseet. Te spreadseet in Figure 4, takes te problem from te previous example, and computes for a large variety of quantities te order costs, olding costs, and total costs. It took about 15 minutes to set tis spreadseet up, and most of tat time was spent on formatting (for example putting in lines). Te formulas for order costs and olding costs were inserted to calculate te respective entries for te order quantity of one. Te total cost entry is defined to be te sum of te first two entries (altoug its column comes first). Ten te formulas are copied down te lengt of te order quantity column. Te only trick is tat you must remember to use absolute addresses for te fixed parameters. Once te spreadseet is set up, te values for C, D, and can be canged and te entire spreadseet will recalculate in less tan a second. Note also, tat te EOQ formula itself is calculated at te top of te spreadseet: its result can be compared wit te columns. Te spreadseet also provided te graps in Figure and Figure 3. Spreadseets are superb for many problems in discrete matematics. Knowing tis, software companies ave endowed current spreadseets wit undreds of scientific functions. Just like formulas, te spreadseet can be made to incorporate assumptions more realistic tan tose in te EOQ model. In many cases it is easier to do tis wit spreadseets and more illuminating. 6
7 Demand/unit time order cost olding cost item /unit time EOQ = Order Quantity Total Cost Order Cost Holding Cost Figure 4 A Spreadseet Analysis of te Inventory Problem 7
8 Hadley, G. and T. M. Witin. Analysis of Inventory Systems Englewood Cliffs, New Jersey: PrenticeHall. 8
Instantaneous Rate of Change:
Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over
More informationTangent Lines and Rates of Change
Tangent Lines and Rates of Cange 922005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims
More informationLecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function
Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between
More information7.6 Complex Fractions
Section 7.6 Comple Fractions 695 7.6 Comple Fractions In tis section we learn ow to simplify wat are called comple fractions, an eample of wic follows. 2 + 3 Note tat bot te numerator and denominator are
More informationDerivatives Math 120 Calculus I D Joyce, Fall 2013
Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te
More information2 Limits and Derivatives
2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line
More informationACT Math Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as
More information2.0 5Minute Review: Polynomial Functions
mat 3 day 3: intro to limits 5Minute Review: Polynomial Functions You sould be familiar wit polynomials Tey are among te simplest of functions DEFINITION A polynomial is a function of te form y = p(x)
More informationVerifying Numerical Convergence Rates
1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and
More informationTrapezoid Rule. y 2. y L
Trapezoid Rule and Simpson s Rule c 2002, 2008, 200 Donald Kreider and Dwigt Lar Trapezoid Rule Many applications of calculus involve definite integrals. If we can find an antiderivative for te integrand,
More informationMath 113 HW #5 Solutions
Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten
More information11.2 Instantaneous Rate of Change
11. Instantaneous Rate of Cange Question 1: How do you estimate te instantaneous rate of cange? Question : How do you compute te instantaneous rate of cange using a limit? Te average rate of cange is useful
More informationOptimal Pricing Strategy for Second Degree Price Discrimination
Optimal Pricing Strategy for Second Degree Price Discrimination Alex O Brien May 5, 2005 Abstract Second Degree price discrimination is a coupon strategy tat allows all consumers access to te coupon. Purcases
More informationExam 2 Review. . You need to be able to interpret what you get to answer various questions.
Exam Review Exam covers 1.6,.1.3, 1.5, 4.14., and 5.15.3. You sould know ow to do all te omework problems from tese sections and you sould practice your understanding on several old exams in te exam
More informationMATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1  BASIC DIFFERENTIATION
MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1  BASIC DIFFERENTIATION Tis tutorial is essential prerequisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of
More informationSAT Subject Math Level 1 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses
More informationThis supplement is meant to be read after Venema s Section 9.2. Throughout this section, we assume all nine axioms of Euclidean geometry.
Mat 444/445 Geometry for Teacers Summer 2008 Supplement : Similar Triangles Tis supplement is meant to be read after Venema s Section 9.2. Trougout tis section, we assume all nine axioms of uclidean geometry.
More information2.28 EDGE Program. Introduction
Introduction Te Economic Diversification and Growt Enterprises Act became effective on 1 January 1995. Te creation of tis Act was to encourage new businesses to start or expand in Newfoundland and Labrador.
More informationME422 Mechanical Control Systems Modeling Fluid Systems
Cal Poly San Luis Obispo Mecanical Engineering ME422 Mecanical Control Systems Modeling Fluid Systems Owen/Ridgely, last update Mar 2003 Te dynamic euations for fluid flow are very similar to te dynamic
More informationAn inquiry into the multiplier process in ISLM model
An inquiry into te multiplier process in ISLM model Autor: Li ziran Address: Li ziran, Room 409, Building 38#, Peing University, Beijing 00.87,PRC. Pone: (86) 0062763074 Internet Address: jefferson@water.pu.edu.cn
More informationA.4. Rational Expressions. Domain of an Algebraic Expression. What you should learn. Why you should learn it
A6 Appendi A Review of Fundamental Concepts of Algebra A.4 Rational Epressions Wat you sould learn Find domains of algebraic epressions. Simplify rational epressions. Add, subtract, multiply, and divide
More informationSection 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of
More informationFinite Difference Approximations
Capter Finite Difference Approximations Our goal is to approximate solutions to differential equations, i.e., to find a function (or some discrete approximation to tis function) tat satisfies a given relationsip
More informationP.4 Rational Expressions
7_0P04.qp /7/06 9:4 AM Page 7 Section P.4 Rational Epressions 7 P.4 Rational Epressions Domain of an Algebraic Epression Te set of real numbers for wic an algebraic epression is defined is te domain of
More informationSolution Derivations for Capa #7
Solution Derivations for Capa #7 1) Consider te beavior of te circuit, wen various values increase or decrease. (Select Iincreases, Ddecreases, If te first is I and te rest D, enter IDDDD). A) If R1
More informationGeometric Stratification of Accounting Data
Stratification of Accounting Data Patricia Gunning * Jane Mary Horgan ** William Yancey *** Abstract: We suggest a new procedure for defining te boundaries of te strata in igly skewed populations, usual
More informationUnderstanding the Derivative Backward and Forward by Dave Slomer
Understanding te Derivative Backward and Forward by Dave Slomer Slopes of lines are important, giving average rates of cange. Slopes of curves are even more important, giving instantaneous rates of cange.
More informationACTIVITY: Deriving the Area Formula of a Trapezoid
4.3 Areas of Trapezoids a trapezoid? How can you derive a formula for te area of ACTIVITY: Deriving te Area Formula of a Trapezoid Work wit a partner. Use a piece of centimeter grid paper. a. Draw any
More informationCan a LumpSum Transfer Make Everyone Enjoy the Gains. from Free Trade?
Can a LumpSum Transfer Make Everyone Enjoy te Gains from Free Trade? Yasukazu Icino Department of Economics, Konan University June 30, 2010 Abstract I examine lumpsum transfer rules to redistribute te
More informationResearch on the Antiperspective Correction Algorithm of QR Barcode
Researc on te Antiperspective Correction Algoritm of QR Barcode Jianua Li, YiWen Wang, YiJun Wang,Yi Cen, Guoceng Wang Key Laboratory of Electronic Tin Films and Integrated Devices University of Electronic
More informationComputer Science and Engineering, UCSD October 7, 1999 GoldreicLevin Teorem Autor: Bellare Te GoldreicLevin Teorem 1 Te problem We æx a an integer n for te lengt of te strings involved. If a is an nbit
More informationModule 2. The Science of Surface and Ground Water. Version 2 CE IIT, Kharagpur
Module Te Science of Surface and Ground Water Version CE IIT, Karagpur Lesson 6 Principles of Ground Water Flow Version CE IIT, Karagpur Instructional Objectives On completion of te lesson, te student
More informationSAT Math MustKnow Facts & Formulas
SAT Mat MustKnow Facts & Formuas Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationas: fractions, tat is, anyting expressabe as a ratio of integers Reas: integers pus rationas
More information1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution
1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis
More informationSurface Areas of Prisms and Cylinders
12.2 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10.B G.11.C Surface Areas of Prisms and Cylinders Essential Question How can you find te surface area of a prism or a cylinder? Recall tat te surface area of
More informationA strong credit score can help you score a lower rate on a mortgage
NET GAIN Scoring points for your financial future AS SEEN IN USA TODAY S MONEY SECTION, JULY 3, 2007 A strong credit score can elp you score a lower rate on a mortgage By Sandra Block Sales of existing
More informationFinite Volume Discretization of the Heat Equation
Lecture Notes 3 Finite Volume Discretization of te Heat Equation We consider finite volume discretizations of te onedimensional variable coefficient eat equation, wit Neumann boundary conditions u t x
More informationCollege Planning Using Cash Value Life Insurance
College Planning Using Cas Value Life Insurance CAUTION: Te advisor is urged to be extremely cautious of anoter college funding veicle wic provides a guaranteed return of premium immediately if funded
More information2.23 Gambling Rehabilitation Services. Introduction
2.23 Gambling Reabilitation Services Introduction Figure 1 Since 1995 provincial revenues from gambling activities ave increased over 56% from $69.2 million in 1995 to $108 million in 2004. Te majority
More informationThe Derivative. Not for Sale
3 Te Te Derivative 3. Limits 3. Continuity 3.3 Rates of Cange 3. Definition of te Derivative 3.5 Grapical Differentiation Capter 3 Review Etended Application: A Model for Drugs Administered Intravenously
More informationf(x) f(a) x a Our intuition tells us that the slope of the tangent line to the curve at the point P is m P Q =
Lecture 6 : Derivatives and Rates of Cange In tis section we return to te problem of finding te equation of a tangent line to a curve, y f(x) If P (a, f(a)) is a point on te curve y f(x) and Q(x, f(x))
More informationReinforced Concrete Beam
Mecanics of Materials Reinforced Concrete Beam Concrete Beam Concrete Beam We will examine a concrete eam in ending P P A concrete eam is wat we call a composite eam It is made of two materials: concrete
More informationCompute the derivative by definition: The four step procedure
Compute te derivative by definition: Te four step procedure Given a function f(x), te definition of f (x), te derivative of f(x), is lim 0 f(x + ) f(x), provided te limit exists Te derivative function
More informationWhat is Advanced Corporate Finance? What is finance? What is Corporate Finance? Deciding how to optimally manage a firm s assets and liabilities.
Wat is? Spring 2008 Note: Slides are on te web Wat is finance? Deciding ow to optimally manage a firm s assets and liabilities. Managing te costs and benefits associated wit te timing of cas in and outflows
More informationPipe Flow Analysis. Pipes in Series. Pipes in Parallel
Pipe Flow Analysis Pipeline system used in water distribution, industrial application and in many engineering systems may range from simple arrangement to extremely complex one. Problems regarding pipelines
More informationSections 3.1/3.2: Introducing the Derivative/Rules of Differentiation
Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here
More informationProof of the Power Rule for Positive Integer Powers
Te Power Rule A function of te form f (x) = x r, were r is any real number, is a power function. From our previous work we know tat x x 2 x x x x 3 3 x x In te first two cases, te power r is a positive
More informationSAT Math Facts & Formulas
Numbers, Sequences, Factors SAT Mat Facts & Formuas Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reas: integers pus fractions, decimas, and irrationas ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences:
More informationConstrained Probabilistic Economic Order Quantity Model under Varying Order Cost and Zero Lead Time Via Geometric Programming
Journal of Matematics and Statistics 7 (): 7, 0 ISSN 596 0 Science Publications Constrained Probabilistic Economic Order Quantity Model under Varying Order Cost and Zero Lead Time Via Geometric Programming
More informationThe modelling of business rules for dashboard reporting using mutual information
8 t World IMACS / MODSIM Congress, Cairns, Australia 37 July 2009 ttp://mssanz.org.au/modsim09 Te modelling of business rules for dasboard reporting using mutual information Gregory Calbert Command, Control,
More informationImproved dynamic programs for some batcing problems involving te maximum lateness criterion A P M Wagelmans Econometric Institute Erasmus University Rotterdam PO Box 1738, 3000 DR Rotterdam Te Neterlands
More informationAverage and Instantaneous Rates of Change: The Derivative
9.3 verage and Instantaneous Rates of Cange: Te Derivative 609 OBJECTIVES 9.3 To define and find average rates of cange To define te derivative as a rate of cange To use te definition of derivative to
More informationArea of Trapezoids. Find the area of the trapezoid. 7 m. 11 m. 2 Use the Area of a Trapezoid. Find the value of b 2
Page 1 of. Area of Trapezoids Goal Find te area of trapezoids. Recall tat te parallel sides of a trapezoid are called te bases of te trapezoid, wit lengts denoted by and. base, eigt Key Words trapezoid
More informationProjective Geometry. Projective Geometry
Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts,
More informationTAKEHOME EXP. # 2 A Calculation of the Circumference and Radius of the Earth
TAKEHOME EXP. # 2 A Calculation of te Circumference and Radius of te Eart On two dates during te year, te geometric relationsip of Eart to te Sun produces "equinox", a word literally meaning, "equal nigt"
More informationA STUDY ON NUMERICAL COMPUTATION OF POTENTIAL DISTRIBUTION AROUND A GROUNDING ROD DRIVEN IN SOIL
A STUDY ON NUMERICAL COMPUTATION OF POTENTIAL DISTRIBUTION AROUND A GROUNDING ROD DRIVEN IN SOIL Ersan Şentürk 1 Nurettin Umurkan Özcan Kalenderli 3 email: ersan.senturk@turkcell.com.tr email: umurkan@yildiz.edu.tr
More informationDetermine the perimeter of a triangle using algebra Find the area of a triangle using the formula
Student Name: Date: Contact Person Name: Pone Number: Lesson 0 Perimeter, Area, and Similarity of Triangles Objectives Determine te perimeter of a triangle using algebra Find te area of a triangle using
More informationChapter 7 Numerical Differentiation and Integration
45 We ave a abit in writing articles publised in scientiþc journals to make te work as Þnised as possible, to cover up all te tracks, to not worry about te blind alleys or describe ow you ad te wrong idea
More informationDistances in random graphs with infinite mean degrees
Distances in random graps wit infinite mean degrees Henri van den Esker, Remco van der Hofstad, Gerard Hoogiemstra and Dmitri Znamenski April 26, 2005 Abstract We study random graps wit an i.i.d. degree
More information1. [5 points] Find an equation of the line that passes through the points ( 3, 2) and (1, 10) = 2, b = 2 2 ( 3) = 8. Equation: y = 2x + 8.
Mat 12 Final Eam Sample Answer Keys 1. [ points] Find an equation of te line tat passes troug te points 3, 2) and 1, 1) m = 1 2 1 + 3 = 2, b = 2 2 3) =. Equation: y = 2 +. 2. Evaluate te it a) [ points]
More informationFINITE DIFFERENCE METHODS
FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to
More informationHardness Measurement of Metals Static Methods
Hardness Testing Principles and Applications Copyrigt 2011 ASM International Konrad Herrmann, editor All rigts reserved. www.asminternational.org Capter 2 Hardness Measurement of Metals Static Metods T.
More informationA LowTemperature Creep Experiment Using Common Solder
A LowTemperature Creep Experiment Using Common Solder L. Roy Bunnell, Materials Science Teacer Soutridge Hig Scool Kennewick, WA 99338 roy.bunnell@ksd.org Copyrigt: Edmonds Community College 2009 Abstract:
More informationVOL. 6, NO. 9, SEPTEMBER 2011 ISSN ARPN Journal of Engineering and Applied Sciences
VOL. 6, NO. 9, SEPTEMBER 0 ISSN 896608 0060 Asian Researc Publising Network (ARPN). All rigts reserved. BIT ERROR RATE, PERFORMANCE ANALYSIS AND COMPARISION OF M x N EQUALIZER BASED MAXIMUM LIKELIHOOD
More informationMath WarmUp for Exam 1 Name: Solutions
Disclaimer: Tese review problems do not represent te exact questions tat will appear te exam. Tis is just a warmup to elp you begin studying. It is your responsibility to review te omework problems, webwork
More informationLecture 10. Limits (cont d) Onesided limits. (Relevant section from Stewart, Seventh Edition: Section 2.4, pp. 113.)
Lecture 10 Limits (cont d) Onesided its (Relevant section from Stewart, Sevent Edition: Section 2.4, pp. 113.) As you may recall from your earlier course in Calculus, we may define onesided its, were
More informationNew Vocabulary volume
. Plan Objectives To find te volume of a prism To find te volume of a cylinder Examples Finding Volume of a Rectangular Prism Finding Volume of a Triangular Prism 3 Finding Volume of a Cylinder Finding
More information1 Density functions, cummulative density functions, measures of central tendency, and measures of dispersion
Density functions, cummulative density functions, measures of central tendency, and measures of dispersion densityfunctionsintro.tex October, 9 Note tat tis section of notes is limitied to te consideration
More information 1  Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz
CS106B Spring 01 Handout # May 3, 01 Huffman Encoding and Data Compression Handout by Julie Zelenski wit minor edits by Keit Scwarz In te early 1980s, personal computers ad ard disks tat were no larger
More informationIn other words the graph of the polynomial should pass through the points
Capter 3 Interpolation Interpolation is te problem of fitting a smoot curve troug a given set of points, generally as te grap of a function. It is useful at least in data analysis (interpolation is a form
More informationSchedulability Analysis under Graph Routing in WirelessHART Networks
Scedulability Analysis under Grap Routing in WirelessHART Networks Abusayeed Saifulla, Dolvara Gunatilaka, Paras Tiwari, Mo Sa, Cenyang Lu, Bo Li Cengjie Wu, and Yixin Cen Department of Computer Science,
More informationSimilar interpretations can be made for total revenue and total profit functions.
EXERCISE 37 Tings to remember: 1. MARGINAL COST, REVENUE, AND PROFIT If is te number of units of a product produced in some time interval, ten: Total Cost C() Marginal Cost C'() Total Revenue R() Marginal
More informationOptimized Data Indexing Algorithms for OLAP Systems
Database Systems Journal vol. I, no. 2/200 7 Optimized Data Indexing Algoritms for OLAP Systems Lucian BORNAZ Faculty of Cybernetics, Statistics and Economic Informatics Academy of Economic Studies, Bucarest
More informationDate: Section P.2: Exponents and Radicals. Properties of Exponents: Example #1: Simplify. a.) 3 4. b.) 2. c.) 3 4. d.) Example #2: Simplify. b.) a.
Properties of Exponents: Section P.2: Exponents and Radicals Date: Example #1: Simplify. a.) 3 4 b.) 2 c.) 34 d.) Example #2: Simplify. a.) b.) c.) d.) 1 Square Root: Principal n th Root: Example #3: Simplify.
More informationElectromagnetism Physics 15b
Electromagnetism Pysics 15b Lecture #7 Capacitance Purcell 3.5 3.8 Wat We Did Last Time Studied electric field near and around conductors Just outside a conductor surface, E = 4πσ ˆn E = 0 in an empty
More informationTRADING AWAY WIDE BRANDS FOR CHEAP BRANDS. Swati Dhingra London School of Economics and CEP. Online Appendix
TRADING AWAY WIDE BRANDS FOR CHEAP BRANDS Swati Dingra London Scool of Economics and CEP Online Appendix APPENDIX A. THEORETICAL & EMPIRICAL RESULTS A.1. CES and Logit Preferences: Invariance of Innovation
More informationDiscovering Area Formulas of Quadrilaterals by Using Composite Figures
Activity: Format: Ojectives: Related 009 SOL(s): Materials: Time Required: Directions: Discovering Area Formulas of Quadrilaterals y Using Composite Figures Small group or Large Group Participants will
More informationModule 1: Introduction to Finite Element Analysis Lecture 1: Introduction
Module : Introduction to Finite Element Analysis Lecture : Introduction.. Introduction Te Finite Element Metod (FEM) is a numerical tecnique to find approximate solutions of partial differential equations.
More informationAnalysis of Algorithms  Recurrences (cont.) 
Analysis of Algoritms  Recurrences (cont.)  Andreas Ermedal MRTC (Mälardalens Real Time Researc Center) andreas.ermedal@md.se Autumn 004 Recurrences: Reersal Recurrences appear frequently in running
More informationComparison between two approaches to overload control in a Real Server: local or hybrid solutions?
Comparison between two approaces to overload control in a Real Server: local or ybrid solutions? S. Montagna and M. Pignolo Researc and Development Italtel S.p.A. Settimo Milanese, ITALY Abstract Tis wor
More information6. Differentiating the exponential and logarithm functions
1 6. Differentiating te exponential and logaritm functions We wis to find and use derivatives for functions of te form f(x) = a x, were a is a constant. By far te most convenient suc function for tis purpose
More informationGEOMETRY. Grades 6 8 Part 1 of 2. Lessons & Worksheets to Build Skills in Measuring Perimeter, Area, Surface Area, and Volume. Poster/Teaching Guide
Grades 6 8 Part 1 of 2 Poster/Teacing Guide GEOMETRY Lessons & Workseets to Build Skills in Measuring Perimeter, Area, Surface Area, and Volume Aligned wit NCTM Standards Mat Grants Available Lesson 1
More informationExercises for numerical integration. Øyvind Ryan
Exercises for numerical integration Øyvind Ryan February 25, 21 1. Vi ar r(t) = (t cos t, t sin t, t) Solution: Koden blir % Oppgave.1.11 b) quad(@(x)sqrt(2+t.^2),,2*pi) a. Finn astigeten, farten og akselerasjonen.
More informationM(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)
Insertion and Deletion in VL Trees Submitted in Partial Fulfillment of te Requirements for Dr. Eric Kaltofen s 66621: nalysis of lgoritms by Robert McCloskey December 14, 1984 1 ackground ccording to Knut
More informationStrategic trading in a dynamic noisy market. Dimitri Vayanos
LSE Researc Online Article (refereed) Strategic trading in a dynamic noisy market Dimitri Vayanos LSE as developed LSE Researc Online so tat users may access researc output of te Scool. Copyrigt and Moral
More informationIs Gravity an Entropic Force?
Is Gravity an Entropic Force? San Gao Unit for HPS & Centre for Time, SOPHI, University of Sydney, Sydney, NSW 006, Australia; EMail: sgao7319@uni.sydney.edu.au Abstract: Te remarkable connections between
More informationReliability and Route Diversity in Wireless Networks
Reliability and Route Diversity in Wireless Networks Amir E. Kandani, Jinane Abounadi, Eytan Modiano, and Lizong Zeng Abstract We study te problem of communication reliability of wireless networks in a
More informationSAMPLE DESIGN FOR THE TERRORISM RISK INSURANCE PROGRAM SURVEY
ASA Section on Survey Researc Metods SAMPLE DESIG FOR TE TERRORISM RISK ISURACE PROGRAM SURVEY G. ussain Coudry, Westat; Mats yfjäll, Statisticon; and Marianne Winglee, Westat G. ussain Coudry, Westat,
More informationWe consider the problem of determining (for a short lifecycle) retail product initial and
Optimizing Inventory Replenisment of Retail Fasion Products Marsall Fiser Kumar Rajaram Anant Raman Te Warton Scool, University of Pennsylvania, 3620 Locust Walk, 3207 SHDH, Piladelpia, Pennsylvania 191046366
More informationA New Hybrid Metaheuristic Approach for Stratified Sampling
A New Hybrid Metaeuristic Approac for Stratified Sampling Timur KESKİNTÜRK 1, Sultan KUZU 2, Baadır F. YILDIRIM 3 1 Scool of Business, İstanbul University, İstanbul, Turkey 2 Scool of Business, İstanbul
More informationChapter 11. Limits and an Introduction to Calculus. Selected Applications
Capter Limits and an Introduction to Calculus. Introduction to Limits. Tecniques for Evaluating Limits. Te Tangent Line Problem. Limits at Infinit and Limits of Sequences.5 Te Area Problem Selected Applications
More informationWriting Mathematics Papers
Writing Matematics Papers Tis essay is intended to elp your senior conference paper. It is a somewat astily produced amalgam of advice I ave given to students in my PDCs (Mat 4 and Mat 9), so it s not
More informationFor Sale By Owner Program. We can help with our for sale by owner kit that includes:
Dawn Coen Broker/Owner For Sale By Owner Program If you want to sell your ome By Owner wy not:: For Sale Dawn Coen Broker/Owner YOUR NAME YOUR PHONE # Look as professional as possible Be totally prepared
More informationRecall from last time: Events are recorded by local observers with synchronized clocks. Event 1 (firecracker explodes) occurs at x=x =0 and t=t =0
1/27 Day 5: Questions? Time Dilation engt Contraction PH3 Modern Pysics P11 I sometimes ask myself ow it came about tat I was te one to deelop te teory of relatiity. Te reason, I tink, is tat a normal
More informationCyber Epidemic Models with Dependences
Cyber Epidemic Models wit Dependences Maocao Xu 1, Gaofeng Da 2 and Souuai Xu 3 1 Department of Matematics, Illinois State University mxu2@ilstu.edu 2 Institute for Cyber Security, University of Texas
More informationCHAPTER 8: DIFFERENTIAL CALCULUS
CHAPTER 8: DIFFERENTIAL CALCULUS 1. Rules of Differentiation As we ave seen, calculating erivatives from first principles can be laborious an ifficult even for some relatively simple functions. It is clearly
More informationWeek #15  Word Problems & Differential Equations Section 8.2
Week #1  Word Problems & Differential Equations Section 8. From Calculus, Single Variable by HugesHallett, Gleason, McCallum et. al. Copyrigt 00 by Jon Wiley & Sons, Inc. Tis material is used by permission
More information4.1 Rightangled Triangles 2. 4.2 Trigonometric Functions 19. 4.3 Trigonometric Identities 36. 4.4 Applications of Trigonometry to Triangles 53
ontents 4 Trigonometry 4.1 Rigtangled Triangles 4. Trigonometric Functions 19 4.3 Trigonometric Identities 36 4.4 pplications of Trigonometry to Triangles 53 4.5 pplications of Trigonometry to Waves 65
More informationAn Interest Rate Model
An Interest Rate Model Concepts and Buzzwords Building Price Tree from Rate Tree Lognormal Interest Rate Model Nonnegativity Volatility and te Level Effect Readings Tuckman, capters 11 and 12. Lognormal
More information