Financing Terms in the EOQ Model

Size: px
Start display at page:

Download "Financing Terms in the EOQ Model"

Transcription

1 Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 August 6, Intoduction This note discusses two tems that ae often omitted fom the standad economic-ode-quantity (EOQ) model when the holding cost is based on financing costs. Both of these tems wee fist identified in Poteus (1985). One was identified explicitly, and the othe was implicit in the calculation of an appoximation bound. This note was witten fo anyone teaching the EOQ model to gaduate business students. The motivation fo this note is the need to econcile the EOQ model with what students may have leaned about woking capital in an accounting couse. To eview, total cost in the EOQ model is assumed to be the sum of puchase costs, odeing costs, and holding costs, namely cλ + λ q k + q h, whee λ is a constant demand pe unit of time, c is a constant, pe-unit puchase cost, k is a fixed cost pe ode, and h is a constant, pe-unit holding cost. The decision vaiable, q, is the ode size. (Fo an ealy histoy of the model, see Elenkotte (1990). Fo a moden teatment, see, fo example, Poteus (00) o Zipkin (000).) When the holding cost is based on financing costs, the usual appoach is to set h = c, whee is the cost of money, typically an inteest ate. This gives a total cost of C(q) =cλ + λ q k + q c and an optimal ode quantity of q = λk λk h = c. Fo students who undestand woking capital, it appeas that the total cost is undestated. (The analysis that follows is called the conventional appoach in Appendix C of Poteus (00).) Since the cost of an ode is, thecostpeunitis k q + c. This implies that the aveage woking capital equied by the cycle-stock inventoy is q k q + c = 1 (), 1

2 and so the cost of this woking capital is This suggests a total cost of (). C 1 (q) = cλ + λ q k + q c + k = C(q)+ k. The exta tem can be intepeted as the cost of the woking capital needed fo the fixed cost potion of the cycle stock inventoy. The exta tem is not elevant to the ode size decision, so the usual appoach of setting h = c yields the optimal decision. Since the EOQ model was designed to answe only the ode size question, using C(q) instead of C 1 (q) does no ham. But if the EOQ model is used to evaluate diffeent vendos, the exta tem in C 1 (q) can be elevant. Conside the following example. Example Demand λ,000 Cost of Money 0% Vendo A chages a fixed cost of $4,000 and a pe-unit cost of $0. Vendo B chages a fixed cost of $1,000 and a pe-unit cost of $0.50. With Vendo A, q =8, 000, soletq EOQ =8, 000. With Vendo B, q =, 951, soletq EOQ =4, 000. The best choice of vendo depends upon whethe o not the k/ tem is included in the analysis: Vendo A Vendo B C(q EOQ ) $67, 000 $67, 00 k/ $400 $100 C 1 (q EOQ ) $67, 400 $67, 00 Pesent Value Analysis At the close of the pevious section, we indulged in a (often employed) slip of logic: we took a model designed fo detemining an optimal ode size and used it fo a diffeent pupose, namely evaluating diffeent vendos. To see if this slip can be justified, we conside the tue economic cost associated with a paticula choice of vendo. In this simple (and classic) vesion of the EOQ model, it suffices to conside the pesent value of the payments. (See, fo example, Poteus (00, Appendix C.1) o Zipkin (000, Section.7).) The pesent value of the payments is PV(q) = ()+ e q λ = 1 e q/λ, + e q λ +... whee q/λ is the elevant inteest ate, and the facto e q/λ epesents continuous discounting. The question now is whethe we can find a elationship between PV(q) and C 1 (q). We can. Conside the

3 following expansion of PV(q): PV(q) = 1 e q/λ = q λ q + q... = q q λ 6λ λ (1 λ + q q ±...) 6λ 4λ λ() = 1+ q q λ + q 1λ 4 q 4 70λ = 1 cλ + λ q k + q c + k q() + q () 1λ 70λ +... (1) Fom equation (1), it follows that lim PV(q) C 1(q) =0, () 0 showing that the k/ tem should be included in the EOQ model. Befoe poceeding, it is useful to be specific about ou pefomance citeion. Let k A and c A be the cost paametes associated with Vendo A. Define PV A (q) =PV(q; k A,c A,). Define PV B similaly. We want to find a function C(q; k, c, ) such that PV A (q) PV B (q) C A (q) C B (q). (*) Ideally, the ode size q would be chosen to optimize the elevant function, but fo pactical easons, we will use ode sizes that ae easy to compute and close to the optimal q. Given ou pefomance citeion, it tuns out that equation () is misleading. (The autho thanks Paul Zipkin fo this obsevation.) Since the optimal q depends upon, we need to conside equation (1) with q = q. PV(q ) = C 1(q ) = C 1(q ) = C 1(q ) + q ( )... 1λ + k λk 1λ c + c 1λ k + 7λc + k 6... λk... c Thus, lim PV(q ) C 1(q ) = k 0 6. This suggests that if q is chosen optimally, the total cost in the EOQ model will be bette epesented by C (q) =C 1 (q)+ k 6. The additional k/6 facto can be intepeted as incemental inteest due to compounding. The fist ode appoximation of this incemental inteest is epesented by the q()/1λ tem in equation (1). With optimal ode sizes, the incemental inteest on the k/ pat of the cycle stock cost, namely q k/1λ, goestozeoas goes to zeo. But the incemental inteest on the cq/ pat of the cycle stock cost does not; c (q ) /1λ equals k/6 fo all.

4 Since lim PV(q ) C (q ) =0, 0 the function C satisfies ou pefomance citeion (*) in the limit. Retuning to the Example, the table below lists the costs associated with the diffeent models. In the pesent value model, the optimal quantities fo Vendo A and Vendo B ae 7, 94 and, 95, espectively. Vendo A Vendo B C(q EOQ ) $67, 000 $67, 00 C 1 (q EOQ ) $67, 400 $67, 00 C (q EOQ ) $67, 5 $67, P V (q EOQ ) $67, 57 $67, 5 P V (q opt ) $67, 56 $67, Discounted Aveage Value Poteus (1985) povides the fist fomal teatment of the analysis in the peceding section by using the discounted aveage value appoach. Poteus (00) descibes the discounted aveage value... ove a specific time inteval... [as] the cost ate that, if incued continuously ove that inteval, yields the same NPV as the actual steam. The goal is to find a function DAV (q) such that PV(q) = DAV (q)+ = DAV (q) 1 e. DAV (q) e + DAV (q) e +... Theoem 1 of Poteus (1985), fo the paametes given in the fist section above, states that λ() DAV (q) = 1+ q q λ + q 1λ + O( ) = C 1 (q)+ q() + O( ). 1λ Fo q = q, DAV (q ) = C 1 (q )+ k λk 1λ c + c 1λ λk + O( ) c k = C (q ) λc + O( ). Theoem 1 implies that lim (DAV (q) C 1(q)) = 0 0 and lim (DAV 0 (q ) C (q )) = 0. () As mentioned in the vey beginning, the k/6 tem in C (q ) is implicit in Poteus (1985), but it is not specifically identified. And it is discussed vebally as inteest on... inteest... chaged. Finally, note that since 1/ (1 e ) does not depend on q, DAV (q) satisfies the pefomance citeion (*) in the limit. 4

5 4 Compound Inteest Effect We have just shown how adding two tems to the taditional EOQ model leads to a bette appoximation of the pesent value model. The intuitive explanation fo the fist of these tems, namely k/, is staightfowad. In fact, it was the motivation fo this note. In this section, we povide some intuition fo the second tem, namely k/6. In the two analyses above, we have descibed it as inteest on inteest. In this section, we will ty to isolate the souce of this tem. We stat by asking whee, in the pesent value model, is the cycle stock? In the taditional EOQ model, the cycle stock cost can actually be seen. Each ode cycle, units aive and ae dawn down. Consequently, we see the inventoy value cycle in the same way. But in the pesent value model, whee is the cycling? We have an inceasing step function, in which the total costs incued jump with each ode. Futhe, the evenue steam does not depend on the ode policy, so it can t be used to explain the missing cycle stock. To answe this question, conside Figue 1 below. It depicts the balance ove time. (λ is denoted by l.) kl/q + lc 0 q/l q/l... 1 Figue 1 To find the missing cycle stock, conside the aveage balance in Figue 1: 1 k ()+1 λq + cλ. The fist pat of the expession is the missing cycle stock. It is the amount by which the balance exceeds the dotted, diagonal line. As the ode fequency inceases, this aea deceases. Loosely speaking, one can think of this as money being spent on units befoe the money is needed to be spent. This paallels nicely with thinking of cycle stock as units on-hand befoe they ae needed. To demonstate that k/6 is due to a compound inteest effect, fist conside what the simple inteest would be on the balances in Figue 1. This would be just the inteest ate times the aveage balance, namely k ()+ λq + cλ. (4) To intoduce a compound inteest effect, we next detemine how the simple inteest in equation (4) accumulates ove time. This is depicted in Figue below. (λ is denoted by l, is denoted by a, and n = λ/q.) Note that the value at t =1educes to the value in equation (4). 5

6 (aq/l) ()* (1/)(n )* (n+1) 6(aq/l)() (aq/l)() (aq/l)() 0 q/l q/l... 1 Figue To appoximate the effects of compound inteest, we compute inteest on the inteest. Let i denote the ode inteval in the above gaph, i.e. i anges fom 1 to n = λ/q. The aveage simple inteest in inteval i is ³ q i(i 1) SI i (q) = () + λ i ³ q i = (). λ The inteest on the simple inteest in inteval i is then ³ q II i (q) = SI i (q) λ ³ = q i (). λ Summing ove the intevals yields nx ³ II(q) = II i (q) = q X n i () λ i=1 ³ i=1 = q n () λ 6 + n 4 + n 1 λ = () 6q q. 1λ At q = q, II(q ) lim = c(q ) = k 0 1λ 6. Thus, we see why c (q ) /1λ can be thought of as a compound inteest effect on the cq pat of the cycle stock. 6

7 Refeences [1] Elenkotte, Donald, Fod Whitman Hais and the Economic Ode Quantity Model, Opeations Reseach, 1990, 8:6, [] Poteus, Evan L., Undiscounted Appoximations of Discounted Regeneative Models, Opeations Reseach Lettes, 1985,, [] Poteus, Evan L., Foundations of Stochastic Inventoy Theoy. Stanfod: Stanfod Univesity Pess, 00. [4] Zipkin, Paul H., Foundations of Inventoy Management. New Yok: McGaw-Hill,

The LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero.

The LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero. Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the

More information

Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapter 3 Savings, Present Value and Ricardian Equivalence Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

More information

Continuous Compounding and Annualization

Continuous Compounding and Annualization Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

More information

ON THE (Q, R) POLICY IN PRODUCTION-INVENTORY SYSTEMS

ON THE (Q, R) POLICY IN PRODUCTION-INVENTORY SYSTEMS ON THE R POLICY IN PRODUCTION-INVENTORY SYSTEMS Saifallah Benjaafa and Joon-Seok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poduction-inventoy

More information

BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR. John R. Graham Adapted from S. Viswanathan FUQUA SCHOOL OF BUSINESS

BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR. John R. Graham Adapted from S. Viswanathan FUQUA SCHOOL OF BUSINESS BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR John R. Gaham Adapted fom S. Viswanathan FUQUA SCHOOL OF BUSINESS DUKE UNIVERSITY 1 In this lectue we conside the effect of govenment

More information

Valuation of Floating Rate Bonds 1

Valuation of Floating Rate Bonds 1 Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned

More information

est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.

est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years. 9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,

More information

Trigonometric Functions of Any Angle

Trigonometric Functions of Any Angle Tigonomet Module T2 Tigonometic Functions of An Angle Copight This publication The Nothen Albeta Institute of Technolog 2002. All Rights Reseved. LAST REVISED Decembe, 2008 Tigonometic Functions of An

More information

The Binomial Distribution

The Binomial Distribution The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between

More information

AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM

AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,

More information

International Monetary Economics Note 1

International Monetary Economics Note 1 36-632 Intenational Monetay Economics Note Let me biefly ecap on the dynamics of cuent accounts in small open economies. Conside the poblem of a epesentative consume in a county that is pefectly integated

More information

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360! 1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

More information

Episode 401: Newton s law of universal gravitation

Episode 401: Newton s law of universal gravitation Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce

More information

An Introduction to Omega

An Introduction to Omega An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei isk-ewad chaacteistics? The Finance Development Cente 2002 1 Fom

More information

Promised Lead-Time Contracts Under Asymmetric Information

Promised Lead-Time Contracts Under Asymmetric Information OPERATIONS RESEARCH Vol. 56, No. 4, July August 28, pp. 898 915 issn 3-364X eissn 1526-5463 8 564 898 infoms doi 1.1287/ope.18.514 28 INFORMS Pomised Lead-Time Contacts Unde Asymmetic Infomation Holly

More information

1.4 Phase Line and Bifurcation Diag

1.4 Phase Line and Bifurcation Diag Dynamical Systems: Pat 2 2 Bifucation Theoy In pactical applications that involve diffeential equations it vey often happens that the diffeential equation contains paametes and the value of these paametes

More information

CHAPTER 10 Aggregate Demand I

CHAPTER 10 Aggregate Demand I CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income

More information

Compound Interest Doubling Time Rule: Extensions and Examples from Antiquities

Compound Interest Doubling Time Rule: Extensions and Examples from Antiquities Communications in Mathematical Finance, vol. 5, no. 2, 2016, 1-11 ISSN: 2241-1968 (pint), 2241 195X (online) Scienpess Ltd, 2016 Compound Inteest Doubling Time Rule: Extensions and Examples fom Antiquities

More information

Problem Set # 9 Solutions

Problem Set # 9 Solutions Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new high-speed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease

More information

Economics 326: Input Demands. Ethan Kaplan

Economics 326: Input Demands. Ethan Kaplan Economics 326: Input Demands Ethan Kaplan Octobe 24, 202 Outline. Tems 2. Input Demands Tems Labo Poductivity: Output pe unit of labo. Y (K; L) L What is the labo poductivity of the US? Output is ouhgly

More information

STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts

More information

1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2

1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2 Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the

More information

Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing

Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow

More information

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

More information

Wasting the Percentage of On-hand Inventory in an Instantaneous Economic Order Quantity Model Due to Deterioration

Wasting the Percentage of On-hand Inventory in an Instantaneous Economic Order Quantity Model Due to Deterioration Jounal of mathematics and compute Science 7 (2013) 154-159 Wasting the Pecentage of On-hand Inventoy in an Instantaneous Economic Ode Quantity Model Due to Deteioation Aticle histoy: Received Mach 2013

More information

The Supply of Loanable Funds: A Comment on the Misconception and Its Implications

The Supply of Loanable Funds: A Comment on the Misconception and Its Implications JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently Fields-Hat

More information

875 Grocery Products Purchase Order

875 Grocery Products Purchase Order 875 Gocey Poducts Puchase Ode Functional Goup ID=OG Intoduction: This X12 Tansaction Set contains the fomat establishes the data contents of the Gocey Poducts Puchase Ode Tansaction Set (875) fo use within

More information

Semipartial (Part) and Partial Correlation

Semipartial (Part) and Partial Correlation Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated

More information

(3) Bipolar Transistor Current Sources

(3) Bipolar Transistor Current Sources B73 lectonics Analysis & Design (3) Bipola Tansisto Cuent Souces Leaning utcome Able to descibe and: Analyze and design a simple twotansisto BJT cuent-souce cicuit to poduce a given bias cuent. Analyze

More information

Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project

Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.

More information

Carter-Penrose diagrams and black holes

Carter-Penrose diagrams and black holes Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

More information

Lecture 8 Topic 5: Multiple Comparisons (means separation)

Lecture 8 Topic 5: Multiple Comparisons (means separation) Lectue 8 Topic 5: Multiple Compaisons (means sepaation) ANOVA: H 0 : µ 1 = µ =... = µ t H 1 : The mean of at least one teatment goup is diffeent If thee ae moe than two teatments in the expeiment, futhe

More information

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it. Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing

More information

Experiment 6: Centripetal Force

Experiment 6: Centripetal Force Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee

More information

Channel selection in e-commerce age: A strategic analysis of co-op advertising models

Channel selection in e-commerce age: A strategic analysis of co-op advertising models Jounal of Industial Engineeing and Management JIEM, 013 6(1):89-103 Online ISSN: 013-0953 Pint ISSN: 013-843 http://dx.doi.og/10.396/jiem.664 Channel selection in e-commece age: A stategic analysis of

More information

USB True Emulation for KVM Switches. Transparent and reliable USB KVM switching technology.

USB True Emulation for KVM Switches. Transparent and reliable USB KVM switching technology. Tanspaent and eliable KVM switching technology. 0118 724-746-5500 965 6000 www.blackbox.co.uk blackbox.com Table of Contents Intoduction... 3 Enumeated Switching... 4 Emulated Switching... 5 Tue Emulation...

More information

Section 5-3 Angles and Their Measure

Section 5-3 Angles and Their Measure 5 5 TRIGONOMETRIC FUNCTIONS Section 5- Angles and Thei Measue Angles Degees and Radian Measue Fom Degees to Radians and Vice Vesa In this section, we intoduce the idea of angle and two measues of angles,

More information

Ilona V. Tregub, ScD., Professor

Ilona V. Tregub, ScD., Professor Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation

More information

So we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1)

So we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1) Lectue 17 Cicula Motion (Chapte 7) Angula Measue Angula Speed and Velocity Angula Acceleation We ve aleady dealt with cicula motion somewhat. Recall we leaned about centipetal acceleation: when you swing

More information

Displacement, Velocity And Acceleration

Displacement, Velocity And Acceleration Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,

More information

Introduction to Electric Potential

Introduction to Electric Potential Univesiti Teknologi MARA Fakulti Sains Gunaan Intoduction to Electic Potential : A Physical Science Activity Name: HP: Lab # 3: The goal of today s activity is fo you to exploe and descibe the electic

More information

Power and Sample Size Calculations for the 2-Sample Z-Statistic

Power and Sample Size Calculations for the 2-Sample Z-Statistic Powe and Sample Size Calculations fo the -Sample Z-Statistic James H. Steige ovembe 4, 004 Topics fo this Module. Reviewing Results fo the -Sample Z (a) Powe and Sample Size in Tems of a oncentality Paamete.

More information

The Time Value of Money

The Time Value of Money he ime Value of Money Inteest Rates and Futue Value Inteest ates ae a facto in the valuation of vitually all financial instuments. While all money maket ates () ae quoted on an annual basis (PR nnual Pecentage

More information

Theory and practise of the g-index

Theory and practise of the g-index Theoy and pactise of the g-index by L. Egghe (*), Univesiteit Hasselt (UHasselt), Campus Diepenbeek, Agoalaan, B-3590 Diepenbeek, Belgium Univesiteit Antwepen (UA), Campus Die Eiken, Univesiteitsplein,

More information

The Role of Gravity in Orbital Motion

The Role of Gravity in Orbital Motion ! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State

More information

Nontrivial lower bounds for the least common multiple of some finite sequences of integers

Nontrivial lower bounds for the least common multiple of some finite sequences of integers J. Numbe Theoy, 15 (007), p. 393-411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to

More information

Software Engineering and Development

Software Engineering and Development I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining

More information

11.5 Graphs of Polar Equations

11.5 Graphs of Polar Equations 9 Applications of Tigonomet.5 Gaphs of Pola Equations In this section, we discuss how to gaph equations in pola coodinates on the ectangula coodinate plane. Since an given point in the plane has infinitel

More information

Infinite-dimensional Bäcklund transformations between isotropic and anisotropic plasma equilibria.

Infinite-dimensional Bäcklund transformations between isotropic and anisotropic plasma equilibria. Infinite-dimensional äcklund tansfomations between isotopic and anisotopic plasma equilibia. Infinite symmeties of anisotopic plasma equilibia. Alexei F. Cheviakov Queen s Univesity at Kingston, 00. Reseach

More information

Database Management Systems

Database Management Systems Contents Database Management Systems (COP 5725) D. Makus Schneide Depatment of Compute & Infomation Science & Engineeing (CISE) Database Systems Reseach & Development Cente Couse Syllabus 1 Sping 2012

More information

VISCOSITY OF BIO-DIESEL FUELS

VISCOSITY OF BIO-DIESEL FUELS VISCOSITY OF BIO-DIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use

More information

An Analysis of Manufacturer Benefits under Vendor Managed Systems

An Analysis of Manufacturer Benefits under Vendor Managed Systems An Analysis of Manufactue Benefits unde Vendo Managed Systems Seçil Savaşaneil Depatment of Industial Engineeing, Middle East Technical Univesity, 06531, Ankaa, TURKEY secil@ie.metu.edu.t Nesim Ekip 1

More information

Things to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.

Things to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request. Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to

More information

Coordinate Systems L. M. Kalnins, March 2009

Coordinate Systems L. M. Kalnins, March 2009 Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

More information

Universal Cycles. Yu She. Wirral Grammar School for Girls. Department of Mathematical Sciences. University of Liverpool

Universal Cycles. Yu She. Wirral Grammar School for Girls. Department of Mathematical Sciences. University of Liverpool Univesal Cycles 2011 Yu She Wial Gamma School fo Gils Depatment of Mathematical Sciences Univesity of Livepool Supeviso: Pofesso P. J. Giblin Contents 1 Intoduction 2 2 De Buijn sequences and Euleian Gaphs

More information

Multiband Microstrip Patch Antenna for Microwave Applications

Multiband Microstrip Patch Antenna for Microwave Applications IOSR Jounal of Electonics and Communication Engineeing (IOSR-JECE) ISSN: 2278-2834, ISBN: 2278-8735. Volume 3, Issue 5 (Sep. - Oct. 2012), PP 43-48 Multiband Micostip Patch Antenna fo Micowave Applications

More information

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses, 3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

More information

Economics 212 Microeconomic Theory I Final Exam. June Faculty of Arts and Sciences Queen s University Answer Key

Economics 212 Microeconomic Theory I Final Exam. June Faculty of Arts and Sciences Queen s University Answer Key Instuctions Economics 1 Micoeconomic Theoy I Final Exam June 008 Faculty of Ats and Sciences ueen s Univesity Anse Key The exam is thee hous in length. The exam consists of to sections: Section A has five

More information

Equity compensation plans New Income Statement impact on guidance Earnings Per Share Questions and answers

Equity compensation plans New Income Statement impact on guidance Earnings Per Share Questions and answers Investos/Analysts Confeence: Accounting Wokshop Agenda Equity compensation plans New Income Statement impact on guidance Eanings Pe Shae Questions and answes IAC03 / a / 1 1 Equity compensation plans The

More information

CIRCUITS LABORATORY EXPERIMENT 7

CIRCUITS LABORATORY EXPERIMENT 7 CIRCUITS LABORATORY EXPERIMENT 7 Design of a Single Tansisto Amplifie 7. OBJECTIVES The objectives of this laboatoy ae to: (a) Gain expeience in the analysis and design of an elementay, single tansisto

More information

Introduction to Stock Valuation. Background

Introduction to Stock Valuation. Background Intoduction to Stock Valuation (Text efeence: Chapte 5 (Sections 5.4-5.9)) Topics backgound dividend discount models paamete estimation gowth oppotunities pice-eanings atios some final points AFM 271 -

More information

ON NEW CHALLENGES FOR CFD SIMULATION IN FILTRATION

ON NEW CHALLENGES FOR CFD SIMULATION IN FILTRATION ON NEW CHALLENGES FOR CFD SIMULATION IN FILTRATION Michael Dedeing, Wolfgang Stausbeg, IBS Filtan, Industiestasse 19 D-51597 Mosbach-Lichtenbeg, Gemany. Oleg Iliev(*), Zaha Lakdawala, Faunhofe Institut

More information

New proofs for the perimeter and area of a circle

New proofs for the perimeter and area of a circle New poofs fo the peimete and aea of a cicle K. Raghul Kuma Reseach Schola, Depatment of Physics, Nallamuthu Gounde Mahalingam College, Pollachi, Tamil Nadu 64001, India 1 aghul_physics@yahoo.com aghulkumak5@gmail.com

More information

Monte Carlo Simulation-Based Supply Chain Disruption Management for Wargames

Monte Carlo Simulation-Based Supply Chain Disruption Management for Wargames CREATE Reseach Achive Non-published Reseach Repots 21 Monte Calo Simulation-Based Supply Chain Disuption Management fo Wagames Jun Zhuang Univesity of Buffalo, The State Univesity of New Yok, jzhuang@buffalo.edu

More information

9:6.4 Sample Questions/Requests for Managing Underwriter Candidates

9:6.4 Sample Questions/Requests for Managing Underwriter Candidates 9:6.4 INITIAL PUBLIC OFFERINGS 9:6.4 Sample Questions/Requests fo Managing Undewite Candidates Recent IPO Expeience Please povide a list of all completed o withdawn IPOs in which you fim has paticipated

More information

Lab #7: Energy Conservation

Lab #7: Energy Conservation Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 1-4 Intoduction: Pehaps one of the most unusual

More information

UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

More information

On Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables.

On Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables. C.Candan EE3/53-METU On Coelation Coefficient The coelation coefficient indicates the degee of linea dependence of two andom vaiables. It is defined as ( )( )} σ σ Popeties: 1. 1. (See appendi fo the poof

More information

Exam #1 Review Answers

Exam #1 Review Answers xam #1 Review Answes 1. Given the following pobability distibution, calculate the expected etun, vaiance and standad deviation fo Secuity J. State Pob (R) 1 0.2 10% 2 0.6 15 3 0.2 20 xpected etun = 0.2*10%

More information

8-1 Newton s Law of Universal Gravitation

8-1 Newton s Law of Universal Gravitation 8-1 Newton s Law of Univesal Gavitation One of the most famous stoies of all time is the stoy of Isaac Newton sitting unde an apple tee and being hit on the head by a falling apple. It was this event,

More information

4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to . Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

More information

Concept and Experiences on using a Wiki-based System for Software-related Seminar Papers

Concept and Experiences on using a Wiki-based System for Software-related Seminar Papers Concept and Expeiences on using a Wiki-based System fo Softwae-elated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wth-aachen.de,

More information

Skills Needed for Success in Calculus 1

Skills Needed for Success in Calculus 1 Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

More information

EAS Groundwater Hydrology Lecture 13: Well Hydraulics 2 Dr. Pengfei Zhang

EAS Groundwater Hydrology Lecture 13: Well Hydraulics 2 Dr. Pengfei Zhang EAS 44600 Goundwate Hydology Lectue 3: Well Hydaulics D. Pengfei Zhang Detemining Aquife Paametes fom Time-Dawdown Data In the past lectue we discussed how to calculate dawdown if we know the hydologic

More information

30 H. N. CHIU 1. INTRODUCTION. Recherche opérationnelle/operations Research

30 H. N. CHIU 1. INTRODUCTION. Recherche opérationnelle/operations Research RAIRO Rech. Opé. (vol. 33, n 1, 1999, pp. 29-45) A GOOD APPROXIMATION OF THE INVENTORY LEVEL IN A(Q ) PERISHABLE INVENTORY SYSTEM (*) by Huan Neng CHIU ( 1 ) Communicated by Shunji OSAKI Abstact. This

More information

Review of Vectors. Appendix A A.1 DESCRIBING THE 3D WORLD: VECTORS. 3D Coordinates. Basic Properties of Vectors: Magnitude and Direction.

Review of Vectors. Appendix A A.1 DESCRIBING THE 3D WORLD: VECTORS. 3D Coordinates. Basic Properties of Vectors: Magnitude and Direction. Appendi A Review of Vectos This appendi is a summa of the mathematical aspects of vectos used in electicit and magnetism. Fo a moe detailed intoduction to vectos, see Chapte 1. A.1 DESCRIBING THE 3D WORLD:

More information

Exponentially Weighted Moving Average Charts for Monitoring the Process Generalized Variance

Exponentially Weighted Moving Average Charts for Monitoring the Process Generalized Variance Geogia Southen Univesity Digital Commons@Geogia Southen Electonic Theses & Dissetations Jack N Aveitt College of Gaduate Studies COGS Summe 014 Exponentially Weighted Moving Aveage Chats fo Monitoing the

More information

L19 Geomagnetic Field Part I

L19 Geomagnetic Field Part I Intoduction to Geophysics L19-1 L19 Geomagnetic Field Pat I 1. Intoduction We now stat the last majo topic o this class which is magnetic ields and measuing the magnetic popeties o mateials. As a way o

More information

Chapter 6. Gradually-Varied Flow in Open Channels

Chapter 6. Gradually-Varied Flow in Open Channels Chapte 6 Gadually-Vaied Flow in Open Channels 6.. Intoduction A stea non-unifom flow in a pismatic channel with gadual changes in its watesuface elevation is named as gadually-vaied flow (GVF). The backwate

More information

Risk Sensitive Portfolio Management With Cox-Ingersoll-Ross Interest Rates: the HJB Equation

Risk Sensitive Portfolio Management With Cox-Ingersoll-Ross Interest Rates: the HJB Equation Risk Sensitive Potfolio Management With Cox-Ingesoll-Ross Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,

More information

NUCLEAR MAGNETIC RESONANCE

NUCLEAR MAGNETIC RESONANCE 19 Jul 04 NMR.1 NUCLEAR MAGNETIC RESONANCE In this expeiment the phenomenon of nuclea magnetic esonance will be used as the basis fo a method to accuately measue magnetic field stength, and to study magnetic

More information

An Immunological Approach to Change Detection: Algorithms, Analysis and Implications

An Immunological Approach to Change Detection: Algorithms, Analysis and Implications An Immunological Appoach to Change Detection: Algoithms, Analysis and Implications Patik D haeselee Dept. of Compute Science Univesity of New Mexico Albuqueque, NM, 87131 patik@cs.unm.edu Stephanie Foest

More information

A framework for the selection of enterprise resource planning (ERP) system based on fuzzy decision making methods

A framework for the selection of enterprise resource planning (ERP) system based on fuzzy decision making methods A famewok fo the selection of entepise esouce planning (ERP) system based on fuzzy decision making methods Omid Golshan Tafti M.s student in Industial Management, Univesity of Yazd Omidgolshan87@yahoo.com

More information

Capital Asset Pricing Model (CAPM) at Work

Capital Asset Pricing Model (CAPM) at Work Capital Asset Picing Model (CAPM) at Wok Some o the Intuition behind CAPM Applications o CAPM Estimation and Testing o CAPM Intuition behind Equilibium Suppose you obseved the ollowing chaacteistics o

More information

Voltage ( = Electric Potential )

Voltage ( = Electric Potential ) V-1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is

More information

Deteriorated Economic Order Quantity (EOQ) Model with Variable Ordering Cost

Deteriorated Economic Order Quantity (EOQ) Model with Variable Ordering Cost Thailand Statistician Januay 2014; 12(1): 83-95 http://statassoc.o.th Contibuted pape Deteioated Economic Ode Quantity (EOQ) Model with Vaiable Odeing Cost Monalisha Pattnaik Depatment of Business Administation,

More information

INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS

INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in

More information

The Doppler Effect in Warning Signals: Perceptual Investigations into Aversive Reactions Related to Pitch Changes

The Doppler Effect in Warning Signals: Perceptual Investigations into Aversive Reactions Related to Pitch Changes The Dopple Effect in Waning Signals: Peceptual Investigations into Avesive Reactions Related to Pitch Changes Clemens Kuhn-Rahloff & Matin Neukom Institute fo Compute Music and Sound Technology (ICST,

More information

Personal Saving Rate (S Households /Y) SAVING AND INVESTMENT. Federal Surplus or Deficit (-) Total Private Saving Rate (S Private /Y) 12/18/2009

Personal Saving Rate (S Households /Y) SAVING AND INVESTMENT. Federal Surplus or Deficit (-) Total Private Saving Rate (S Private /Y) 12/18/2009 1 Pesonal Saving Rate (S Households /Y) 2 SAVING AND INVESTMENT 16.0 14.0 12.0 10.0 80 8.0 6.0 4.0 2.0 0.0-2.0-4.0 1959 1961 1967 1969 1975 1977 1983 1985 1991 1993 1999 2001 2007 2009 Pivate Saving Rate

More information

Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E

Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known

More information

YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH

YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH nd INTERNATIONAL TEXTILE, CLOTHING & ESIGN CONFERENCE Magic Wold of Textiles Octobe 03 d to 06 th 004, UBROVNIK, CROATIA YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH Jana VOBOROVA; Ashish GARG; Bohuslav

More information

Lesson C3 2. Exploring Genetics. Performance Standard: 2. Discuss the implications of genetic variation.

Lesson C3 2. Exploring Genetics. Performance Standard: 2. Discuss the implications of genetic variation. Lesson C3 2 Exploing Genetics Unit C. Basic Pinciples of Agicultual/Hoticultual Science Poblem Aea 3. Undestanding Cells, Genetics, and Repoduction Lesson 2. Exploing Genetics New Mexico Content Standad:

More information

Single Particle State A A

Single Particle State A A LECTURE 3 Maxwell Boltzmann, Femi, and Bose Statistics Suppose we have a gas of N identical point paticles in a box of volume V. When we say gas, we mean that the paticles ae not inteacting with one anothe.

More information

Deflection of Electrons by Electric and Magnetic Fields

Deflection of Electrons by Electric and Magnetic Fields Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An

More information

Psychology 282 Lecture #2 Outline. Review of Pearson correlation coefficient:

Psychology 282 Lecture #2 Outline. Review of Pearson correlation coefficient: Psychology 282 Lectue #2 Outline Review of Peason coelation coefficient: z z ( n 1) Measue of linea elationship. Magnitude Stength Sign Diection Bounded by +1.0 and -1.0. Independent of scales of measuement.

More information

Chapte 3 Is Gavitation A Results Of Asymmetic Coulomb Chage Inteactions? Jounal of Undegaduate Reseach èjurè Univesity of Utah è1992è, Vol. 3, No. 1, pp. 56í61. Jeæey F. Gold Depatment of Physics, Depatment

More information

LAL Update. Letter From the President. Dear LAL User:

LAL Update. Letter From the President. Dear LAL User: LAL Update ASSOCIATES OF CAPE COD, INCORPORATED OCTOBER 00 VOLUME 0, NO. Lette Fom the Pesident Dea LAL Use: This Update will claify some of the statistics used with tubidimetic and chomogenic LAL tests.

More information

Firstmark Credit Union Commercial Loan Department

Firstmark Credit Union Commercial Loan Department Fistmak Cedit Union Commecial Loan Depatment Thank you fo consideing Fistmak Cedit Union as a tusted souce to meet the needs of you business. Fistmak Cedit Union offes a wide aay of business loans and

More information

Explicit, analytical solution of scaling quantum graphs. Abstract

Explicit, analytical solution of scaling quantum graphs. Abstract Explicit, analytical solution of scaling quantum gaphs Yu. Dabaghian and R. Blümel Depatment of Physics, Wesleyan Univesity, Middletown, CT 06459-0155, USA E-mail: ydabaghian@wesleyan.edu (Januay 6, 2003)

More information

Top K Nearest Keyword Search on Large Graphs

Top K Nearest Keyword Search on Large Graphs Top K Neaest Keywod Seach on Lage Gaphs Miao Qiao, Lu Qin, Hong Cheng, Jeffey Xu Yu, Wentao Tian The Chinese Univesity of Hong Kong, Hong Kong, China {mqiao,lqin,hcheng,yu,wttian}@se.cuhk.edu.hk ABSTRACT

More information