Handbook of Industrial Automation edited by Richard L Shell Ernest L Hall University of Cincinnati Cincinnati, Ohio Marcel Dekker, Inc TM Copyright © 2000 by Marcel Dekker, Inc All Rights Reserved Copyright © 2000 Marcel Dekker, Inc New York • Basel ISBN: 0-8247-0373-1 This book is printed on acid-free paper Headquarters Marcel Dekker, Inc 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities For more information, write to Special Sales/ Professional Marketing at the headquarters address above Copyright # 2000 by Marcel Dekker, Inc All Rights Reserved Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, micro®lming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA Copyright © 2000 Marcel Dekker, Inc Preface This handbook is designed as a comprehensive reference for the industrial automation engineer Whether in a small or large manufacturing plant, the industrial or manufacturing engineer is usually responsible for using the latest and best technology in the safest, most economic manner to build products This responsibility requires an enormous knowledge base that, because of changing technology, can never be considered complete The handbook will provide a handy starting reference covering technical, economic, certain legal standards, and guidelines that should be the ®rst source for solutions to many problems The book will also be useful to students in the ®eld as it provides a single source for information on industrial automation The handbook is also designed to present a related and connected survey of engineering methods useful in a variety of industrial and factory automation applications Each chapter is arranged to permit review of an entire subject, with illustrations to provide guideposts for the more complex topics Numerous references are provided to other material for more detailed study The mathematical de®nitions, concepts, equations, principles, and application notes for the practicing industrial automation engineer have been carefully selected to provide broad coverage Selected subjects from both undergraduate- and graduate-level topics from industrial, electrical, computer, and mechanical engineering as well as material science are included to provide continuity and depth on a variety of topics found useful in our work in teaching thousands of engineers who work in the factory environment The topics are presented in a tutorial style, without detailed proofs, in order to incorporate a large number of topics in a single volume The handbook is organized into ten parts Each part contains several chapters on important selected topics Part 1 is devoted to the foundations of mathematical and numerical analysis The rational thought process developed in the study of mathematics is vital in developing the ability to satisfy every concern in a manufacturing process Chapters include: an introduction to probability theory, sets and relations, linear algebra, calculus, differential equations, Boolean algebra and algebraic structures and applications Part 2 provides background information on measurements and control engineering Unless we measure we cannot control any process The chapter topics include: an introduction to measurements and control instrumentation, digital motion control, and in-process measurement Part 3 provides background on automatic control Using feedback control in which a desired output is compared to a measured output is essential in automated manufacturing Chapter topics include distributed control systems, stability, digital signal processing and sampled-data systems Part 4 introduces modeling and operations research Given a criterion or goal such as maximizing pro®t, using an overall model to determine the optimal solution subject to a variety of constraints is the essence of operations research If an optimal goal cannot be obtained, then continually improving the process is necessary Chapter topics include: regression, simulation and analysis of manufacturing systems, Petri nets, and decision analysis iii Copyright © 2000 Marcel Dekker, Inc iv Preface Part 5 deals with sensor systems Sensors are used to provide the basic measurements necessary to control a manufacturing operation Human senses are often used but modern systems include important physical sensors Chapter topics include: sensors for touch, force, and torque, fundamentals of machine vision, low-cost machine vision and three-dimensional vision Part 6 introduces the topic of manufacturing Advanced manufacturing processes are continually improved in a search for faster and cheaper ways to produce parts Chapter topics include: the future of manufacturing, manufacturing systems, intelligent manufacturing systems in industrial automation, measurements, intelligent industrial robots, industrial materials science, forming and shaping processes, and molding processes Part 7 deals with material handling and storage systems Material handling is often considered a necessary evil in manufacturing but an ef®cient material handling system may also be the key to success Topics include an introduction to material handling and storage systems, automated storage and retrieval systems, containerization, and robotic palletizing of ®xed- and variable-size parcels Part 8 deals with safety and risk assessment Safety is vitally important, and government programs monitor the manufacturing process to ensure the safety of the public Chapter topics include: investigative programs, government regulation and OSHA, and standards Part 9 introduces ergonomics Even with advanced automation, humans are a vital part of the manufacturing process Reducing risks to their safety and health is especially important Topics include: human interface with automation, workstation design, and physical-strength assessment in ergonomics Part 10 deals with economic analysis Returns on investment are a driver to manufacturing systems Chapter topics include: engineering economy and manufacturing cost recovery and estimating systems We believe that this handbook will give the reader an opportunity to quickly and thoroughly scan the ®eld of industrial automation in suf®cient depth to provide both specialized knowledge and a broad background of speci®c information required for industrial automation Great care was taken to ensure the completeness and topical importance of each chapter We are grateful to the many authors, reviewers, readers, and support staff who helped to improve the manuscript We earnestly solicit comments and suggestions for future improvements Richard L Shell Ernest L Hall Copyright © 2000 Marcel Dekker, Inc Contents Preface iii Contributors Part 1 ix Mathematics and Numerical Analysis 1.1 Some Probability Concepts for Engineers Enrique Castillo and Ali S Hadi 1.2 Introduction to Sets and Relations Diego A Murio 1.3 Linear Algebra William C Brown 1.4 A Review of Calculus Angelo B Mingarelli 1.5 Ordinary Differential Equations Jane Cronin 1.6 Boolean Algebra Ki Hang Kim 1.7 Algebraic Structures and Applications J B Srivastava Part 2 1 Measurements and Computer Control 2.1 Measurement and Control Instrumentation Error-Modeled Performance Patrick H Garrett 2.2 Fundamentals of Digital Motion Control Ernest L Hall, Krishnamohan Kola, and Ming Cao v Copyright © 2000 Marcel Dekker, Inc vi 2.3 Contents In-Process Measurement William E Barkman Part 3 Automatic Control 3.1 Distributed Control Systems Dobrivoje Popovic 3.2 Stability Allen R Stubberud and Stephen C Stubberud 3.3 Digital Signal Processing Fred J Taylor 3.4 Sampled-Data Systems Fred J Taylor Part 4 Modeling and Operations Research 4.1 Regression Richard Brook and Denny Meyer 4.2 A Brief Introduction to Linear and Dynamic Programming Richard B Darst 4.3 Simulation and Analysis of Manufacturing Systems Benita M Beamon 4.4 Petri Nets Frank S Cheng 4.5 Decision Analysis Hiroyuki Tamura Part 5 Sensor Systems 5.1 Sensors: Touch, Force, and Torque Richard M Crowder 5.2 Machine Vision Fundamentals Prasanthi Guda, Jin Cao, Jeannine Gailey, and Ernest L Hall 5.3 Three-Dimensional Vision Joseph H Nurre 5.4 Industrial Machine Vision Steve Dickerson Part 6 6.1 Manufacturing The Future of Manufacturing M Eugene Merchant Copyright © 2000 Marcel Dekker, Inc Contents vii 6.2 Manufacturing Systems Jon Marvel and Ken Bloemer 6.3 Intelligent Manufacturing in Industrial Automation George N Saridis 6.4 Measurements John Mandel 6.5 Intelligent Industrial Robots Wanek Golnazarian and Ernest L Hall 6.6 Industrial Materials Science and Engineering Lawrence E Murr 6.7 Forming and Shaping Processes Shivakumar Raman 6.8 Molding Processes Avraam I Isayev Part 7 Material Handling and Storage 7.1 Material Handling and Storage Systems William Wrennall and Herbert R Tuttle 7.2 Automated Storage and Retrieval Systems Stephen L Parsley 7.3 Containerization A Kader Mazouz and C P Han 7.4 Robotic Palletizing of Fixed- and Variable-Size/Content Parcels Hyder Nihal Agha, William H DeCamp, Richard L Shell, and Ernest L Hall Part 8 Safety, Risk Assessment, and Standards 8.1 Investigation Programs Ludwig Benner, Jr 8.2 Government Regulation and the Occupational Safety and Health Administration C Ray Asfahl 8.3 Standards Verna Fitzsimmons and Ron Collier Part 9 Ergonomics 9.1 Perspectives on Designing Human Interfaces for Automated Systems Anil Mital and Arunkumar Pennathur 9.2 Workstation Design Christin Shoaf and Ashraf M Genaidy Copyright © 2000 Marcel Dekker, Inc viii 9.3 Contents Physical Strength Assessment in Ergonomics Sean Gallagher, J Steven Moore, Terrence J Stobbe, James D McGlothlin, and Amit Bhattacharya Part 10 Economic Analysis 10.1 Engineering Economy Thomas R Huston 10.2 Manufacturing-Cost Recovery and Estimating Systems Eric M Malstrom and Terry R Collins Index 863 Copyright © 2000 Marcel Dekker, Inc x Contributors William H DeCamp Steve Dickerson Motoman, Inc., West Carrollton, Ohio Department of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia Verna Fitzsimmons Cincinnati, Ohio Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati, Jeannine Gailey Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio Sean Gallagher Pittsburgh Research Laboratory, National Institute for Occupational Safety and Health, Pittsburgh, Pennsylvania Patrick H Garrett Department of Electrical and Computer Engineering and Computer Science, University of Cincinnati, Cincinnati, Ohio Ashraf M Genaidy Cincinnati, Ohio Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati, Wanek Golnazarian General Dynamics Armament Systems, Burlington, Vermont Prasanthi Guda Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio Ali S Hadi Department of Statistical Sciences, Cornell University, Ithaca, New York Ernest L Hall Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio C P Han Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, Florida Thomas R Huston Cincinnati, Ohio Avraam I Isayev Ki Hang Kim Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati, Department of Polymer Engineering, The University of Akron, Akron, Ohio Mathematics Research Group, Alabama State University, Montgomery, Alabama Krishnamohan Kola Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio Eric M Malstromy Department of Industrial Engineering, University of Arkansas, Fayetteville, Arkansas John Mandelà Jon Marvel National Institute of Standards and Technology, Gaithersburg, Maryland Padnos School of Engineering, Grand Valley State University, Grand Rapids, Michigan A Kader Mazouz Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, Florida James D McGlothlin Purdue University, West Lafayette, Indiana M Eugene Merchant Institute of Advanced Manufacturing Sciences, Cincinnati, Ohio Denny Meyer Institute of Information and Mathematical Sciences, Massey University±Albany, Palmerston North, New Zealand Angelo B Mingarelli Anil Mital School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada Department of Industrial Engineering, University of Cincinnati, Cincinnati, Ohio J Steven Moore Department of Occupational and Environmental Medicine, The University of Texas Health Center, Tyler, Texas * Retired y Deceased Copyright © 2000 Marcel Dekker, Inc Contributors Diego A Murio xi Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio Lawrence E Murr Paso, Texas Department of Metallurgical and Materials Engineering, The University of Texas at El Paso, El Joseph H Nurre School of Electrical Engineering and Computer Science, Ohio University, Athens, Ohio Stephen L Parsley ESKAY Corporation, Salt Lake City, Utah Arunkumar Pennathur Dobrivoje Popovic University of Texas at El Paso, El Paso, Texas Institute of Automation Technology, University of Bremen, Bremen, Germany Shivakumar Raman Department of Industrial Engineering, University of Oklahoma, Norman, Oklahoma George N Saridis Professor Emeritus, Electrical, Computer, and Systems Engineering Department, Rensselaer Polytechnic Institute, Troy, New York Richard L Shell Cincinnati, Ohio Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati, Christin Shoaf Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio J B Srivastava Department of Mathematics, Indian Institute of Technology, Delhi, New Delhi, India Terrence J Stobbe Industrial Engineering Department, West Virginia University, Morgantown, West Virginia Allen R Stubberud California Department of Electrical and Computer Engineering, University of California Irvine, Irvine, Stephen C Stubberud Hiroyuki Tamura ORINCON Corporation, San Diego, California Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, Japan Fred J Taylor Department of Electrical and Computer Engineering and Department of Computer and Information Science Engineering, University of Florida, Gainesville, Florida Herbert R Tuttle Graduate Engineering Management, University of Kansas, Lawrence, Kansas William Wrennall The Leawood Group Ltd., Leawood, Kansas Copyright © 2000 Marcel Dekker, Inc 22 Castillo and Hadi is Example 11 The central moments of the Bernoulli random variable are all equal to p In effect, its characteristic function is X …t† ˆ …a 0 e itx  1 1 eitx a eita À 1  ˆ dx ˆ a a it 0 iat Some important properties of the characteristic function are: ˆ 1: 1: X …t†j " …Àt† ˆ X …t†, where z is the conjugate of z X If Z ˆ aX ‡ b, where X is a random variable, and a and b are real constants, we have itb Z …t† ˆ e X …at†, where X …t† and Z …t† are the characteristic functions of Z and X, respectively The characteristic function of the sum of two independent random variables is the product of their characteristic functions, that is, X‡Y …t† ˆ X …t† Y …t† The characteristic function of a linear convex combination of random variables is the linear convex combination of their characteristic functions with the same coef®cients: aFx ‡bFy …t† ˆ a X …t† ‡ b Y …t† The characteristic function of the sum of a random number N iid random variables fX1 ; F F F ; Xn g is given by   log X …t† …t† ˆ N S i j X where X …t†, N …t†, and S …t† are the character€ istic functions of Xi , N and S ˆ N Xi , respeciˆ1 tively One of the main applications of the characteristic function is to obtain the central moments of the corresponding random variable In fact, in we differentiate the characteristic function k times with respect to t, we get …I …k† ik xk eitx f …x† dx X …t† ˆ X …t† ˆ peit ‡ q and, according to Eq (20), we get mk ˆ …k† X …0† ik   it Àp X …t† ˆ 1 À a The moments with respect to the origin, according to Eq (20), are mk ˆ …k† X …0† k i from which we have mk ˆ …k† X …0† k i where mk is the kth central moment of X Copyright © 2000 Marcel Dekker, Inc …20† ˆ p…p À 1† F F F …p À k ‡ 1† ak Tables 5 and 6 give the cf of several discrete and continuous random variables Next, we can extend the characteristic function to multivariate distributions as follows Let X ˆ …X1 ; F F F ; Xn † be an n-dimensional random variable The characteristic function of X is de®ned as …I …I FFF ei…t1 x1 ‡t2 x2 ‡ÁÁÁ‡tn xn † X …t† ˆ ÀI ÀI dFX …x1 ; F F F ; xn † where the integral always exists The moments with respect to the origin can be obtained by mr1 ;FFF;rk ˆ Example 13 ÀI pik ˆp ik Example 12 The characteristic function of the gamma random variable Gamma…p; a† is ÀI which for t ˆ 0 gives …I …k† …0† ˆ ik xk dF…x† ˆ ik mk X ˆ …r1 ‡ÁÁÁ‡rk † …0; F F F ; 0† X …r1 ‡ÁÁÁ‡rk † i Consider the random variable with pdf f …x1 ; F F F ; xn † V‰ n b ` …i eÀi xi † ˆ iˆ1 b X 0 if 0 xi < IY i ˆ 1; F F F ; n otherwise Its characteristic function is Some Probability Concepts for Engineers X …t† …I 23 …I ei…t1 x1 ‡ÁÁÁ‡tn xn † dFX …x1 ; F F F ; xn † ÀI ÀI 2 3 …I …I n n ˆ ‰ ˆ FFF exp i ti xi …i eÀi xi † ˆ FFF 0 0 iˆ1 iˆ1 dx1 F F F dxn …I …I ‰ n ˆ FFF …i exi …iti Ài † † dx1 F F F dxn 0 ˆ ˆ n ‰ iˆ1 n ‰ 0 i …I 0 iˆ1 exi …iti Ài † dxi  n ˆ X1 ˆ h1 …Y1 ; F F F ; Yn † IQ Example 15 The characteristic function of the multinominal random variable, M…nY p1 ; F F F ; pk †, can be written as '…t1 ; F F F ; tk † ˆ ˆ exp k ˆ jˆ1 3 itj xj p…x1 ; x2 ; F F F ; xk † ˆ n3 x1 3x2 3 F F F xk 3 …p1 e † TRANSFORMATIONS OF RANDOM VARIABLES One-to-One Transformations Theorem 7 Transformations of Continuous Random Variables: Let …X1 ; F F F ; Xn † be an n-dimensional random variable with pdf f …x1 ; F F F ; xn † de®ned on the set A and let Copyright © 2000 Marcel Dekker, Inc Xn ˆ hn …Y1 ; F F F ; Yn † is the inverse transformation of Eq (21) and j det…J†j the absolute value of the determinant of the Jacobian matrix J of the transformation The ijth element of J is given by @Xi =@Yj Example 16 Let X and Y be two independent normal N…0; 1† random variables Then the joint pdf is 2 2 1 1 fX;Y …x; y† ˆ p eÀx =2 p eÀy =2 2 2 ÀI < x; y < I U ˆX ‡Y it1 x1 jˆ1 1.8.1 X2 ˆ h2 …Y1 ; F F F ; Yn † F F F Consider the transformation …p2 eit2 †x2 F F F …pk eitk †xk 2 3n k ˆ it j ˆ pj e 1.8 be a one-to-one continuous transformation from the set A to the set B Then, the pdf of the random variable …Y1 ; F F F ; Yn † on the set B, is where kj tk tj gU T fˆ gU Tf n k;jˆ1 gU '…t1 ; F F F ; tn † ˆ expTif t k k À gU Tf 2 eS R d kˆ1 2 …21† Yn ˆ gn …X1 ; F F F ; Xn † f …h1 …y1 ; F F F ; yn †; h2 …y1 ; F F F ; yn †; F F F ; Example 14 The characteristic function of the multinormal random variable is ˆ Y2 ˆ g2 …X1 ; F F F ; Xn † F F F hn …y1 ; y2 ; F F F ; yn ††j det…J†j  n ‰ i it À1 ˆ 1À i  À iti iˆ1 i iˆ1 i PH Y1 ˆ g1 …X1 ; F F F ; Xn † V ˆX ÀY which implies that X ˆ …U ‡ V†=2 Y ˆ …U À V†=2 Then the Jacobian matrix is    @X=@U @X=@V 1=2 Jˆ ˆ @Y=@U @Y=@V 1=2 1=2 À1=2  with j det…J†j ˆ 1=2 Thus, the joint density of U and V becomes & !' 1 1 u ‡ v2 u À v2 g…u; v† ˆ exp À ‡ 4 2 2 2 1 Àu2 =4 Àv2 =4 e e À I < u; v < I ˆ 4 24 Castillo and Hadi which is the product of a function of u and a function of v de®ned in a rectangle Thus, U and V are independent N…0; 2† random variables 1.8.2 Other Transformations If the transformation Eq (21) is not one-to-one, the above method is not applicable Assume that for each point …x1 ; F F F ; xn † in A there is one point in B, but each point in B, has more than one point in A Assume further that there exists a ®nite partition …A1 ; F F F ; An †, of A, such that the restriction of the given transformation to each Ai , is a one-to-one transformation Then, there exist transformations of B in Ai de®ned by X1 ˆ h1i …Y1 ; F F F ; Yn † X2 ˆ h2i …Y1 ; F F F ; Yn † F F F Xn ˆ hni …Y1 ; F F F ; Yn † with jacobians Ji i ˆ 1; F F F ; m Then, taking into account that the probability of the union of disjoint sets is the sum of the probabilities of the individual sets, we obtain the pdf of the random variable …Y1 ; F F F ; Yn †: g…y1 ; F F F ; yn † ˆ 1.9 m ˆ iˆ1 f …h1i ; F F F ; hni †j det…Ji †j x† ˆ ÀI f …x† dx We then generate a sequence of random numbers fu1 ; F F F ; un g from U…0; 1† and obtain the corresponding values fx1 ; F F F ; xn g by solving F…xi † ˆ ui ; i ˆ 1; F F F n; which gives xi ˆ H À1 …ui †, where H À1 …ui † is the inverse of the cdf evaluated at ui For example, Fig 16 shows the cdf F…x† and two values x1 and x2 corresponding to the uniform U…0; 1† numbers u1 and u2 Theorem 9 Simulating Normal Random Variables: Let X and Y be independent standard uniform random variables U…0; 1† Then, the random variables U and V de®ned by U ˆ …À2 log X†1=2 sin…2Y† V ˆ …À2 log X†1=2 cos…2Y† are independent N…0; 1† random variables 1.9.2 The Multivariate Case In the multivariate case …X1 ; F F F ; Xn †, we can simulate using the conditional cdfs: F…x1 †; F…x2 jx1 †; F F F ; F…xn jx1 ; F F F ; xnÀ1 † as follows First we simulate X1 with F…x1 † obtaining x1 Once we have simulated XkÀ1 obtaining xkÀ1 , we simulate Xk using F…xk jx1 ; F F F ; xkÀ1 †, and we continue the process until we have simulated all X's We repeat the whole process as many times as desired SIMULATION OF RANDOM VARIABLES A very useful application of the change-of-variables technique discussed in the previous section is that it provides a justi®cation of an important method for simulating any random variable using the standard uniform variable U…0; 1† 1.9.1 F…x† ˆ p…X …x 1.10 ORDER STATISTICS AND EXTREMES Let …X1 ; F F F ; Xn † be a random sample coming from a pdf f …x† and cdf F…x† Arrange …X1 ; F F F ; Xn † in an The Univariate Case Theorem 8 Let X be a univariate random variable with cdf F…x† Then, the random variable U ˆ F…x† is distributed as a standard uniform variable U…0; 1† Example 17 Simulating from a Probability Distribution: To generate a sample from a probability distribution f …x†, we ®rst compute the cdf, Copyright © 2000 Marcel Dekker, Inc Figure 16 Sampling from a probability distribution f …x† using the corresponding cdf F…x†: Some Probability Concepts for Engineers 25 increasing order of magnitude and let X1Xn Á Á Á XnXn be the ordered values Then, the rth element of this new sequence, XrXn , is called the rth order statistic of the sample Order statistics are very important in practice, especially so for the minimum, X1Xn and the maximum, XnXn because they are the critical values which are used in engineering, physics, medicine, etc (see, e.g., Castillo and Hadi [9±11]) In this section we study the distributions of order statistics 1.10.1 Distributions of Order Statistics The cdf of the rth order statistic XrXn is [12, 13] FrXn …x† ˆ P‰XrXn xŠ ˆ 1 À Fm…x† …r À 1† 2 3 n ˆ n ˆ F k …x†‰1 À F…x†ŠnÀk k kˆr 2 3… F…x† n ˆr urÀ1 …1 À u†nÀr du 0 r ˆ IF…x† …r; n À r ‡ 1† …22† where m…x† is the number of elements in the sample with value Xj x and Ip …a; b† is the incomplete beta function, which is implicitly de®ned in Eq (22) If the population is absolutely continuous, then the pdf of Xrn is given by the derivative of Eq (22) with respect to x:   n fXrXn …x† ˆ r F rÀ1 …x†‰1 À F…x†ŠnÀr f …x† r …23† FrÀ1 …x†‰1 À F…x†ŠnÀr f …x† ˆ …r; n À r ‡ 1† where …a; b† is the beta function Example 19 Distribution of the maximum order statistic Letting r ˆ n in Eqs (22) and (23) we obtain the cdf and the pdf of the maximum order statistic: FXnXn …x† ˆ F…x†n and fXnXn …x† ˆ nF nÀ1 …x†f …x† 1.10.2 Distributions of Subsets of Order Statistics Let Xr1 Xn ; F F F ; Xrk Xn , be the subset of k order statistics of orders r1 < F F F < rk , of a random sample of size n coming from a population with pdf f …x† and cdf F…x† With the aim of obtaining the joint distribution of this set, consider the event xj Xrj Xn < xj ‡ Áxj ; 1 j k for small values of Áxj , 1 j k (see Fig 17) That is, k values in the sample belong to the intervals …xj ; xj ‡ Áxj † for 1 j k and the rest are distributed in such a way that exactly …rj À rjÀ1 À 1† belong to the interval …xjÀ1 ‡ ÁxjÀ1 ; xj † for 1 j k, where Áx0 ˆ 0, r0 ˆ 0, rk‡1 ˆ n ‡ 1, x0 ˆ ÀI and xk‡1 ˆ I Consider the following multinomial experiment with the 2k ‡ 1 possible outcomes associated with the 2k ‡ 1 intervals illustrated in Fig 17 We obtain a sample of size n from the population and determine to which of the intervals they belong Since we assume independence and replacement, the numbers of elements in each interval is a multinomial random variable with parameters fnY f …x1 †Áx1 ; F F F ; f …xk †Áxk ; ‰F…x1 † À F…x0 †Š; ‰F…x2 † À F…x1 †Š; F F F ; ‰F…xk‡1 † À F…xk †Šg where the parameters are n (the sample size) and the probabilities associated with the 2k ‡ 1 intervals Consequently, we can use the results for multinomial random variables to obtain the joint pdf of the k order statistics and obtain Example 18 Distribution of the minimum order statistic Letting r ˆ 1 in Eqs (22) and (23) we obtain the cdf and pdf of the minimum order statistic: n ˆ n  k F …x†‰1 À F…x†ŠnÀk FX1Xn …x† ˆ k kˆ1 ˆ 1 À ‰1 À F…x†Šn and fX1Xn …x† ˆ n‰1 À F…x†ŠnÀ1 f …x† Copyright © 2000 Marcel Dekker, Inc Figure 17 An illustration of the multinomial experiment used to determine the joint pdf of a subset of k order statistics 26 Castillo and Hadi fr1 ;FFF;rk Xn …x1 ; F F F ; xk † ˆ n3 k ‰ iˆ1 f …xi † k‡1 ‰ 1.10.4 jˆ1 We have seen that the cdf of the maximum Zn and minimum Wn of a sample of size n coming from a population with cdf F…x† are Hn …x† ˆ P‰Zn xŠ ˆ and Ln …x† ˆ P‰Wn xŠ ˆ 1 À ‰1 À F…x†Šn F n …x† When n tends to in®nity we have & 1 if F…x† ˆ 1 n lim Hn …x† ˆ lim F …x† ˆ n3I n3I 0 if F…x† < 1 ‰F…xj † À F…xjÀ1 †Šrj ÀrjÀ1 À1 …rj À rjÀ1 À 1†3 …24† 1.10.3 Distributions of Particular Order Statistics 1.10.3.1 and Joint Distribution of Maximum and Minimum n lim Ln …x† ˆ lim 1 À ‰1 À F…x†Š ˆ Setting k ˆ 2, r1 ˆ 1 and r2 ˆ n in Eq (24), we obtain the joint distribution of the maximum and the minimum of a sample of size n, which becomes f1;nXn …x1 ; x2 † ˆ n…n À 1† f …x1 † f …x2 †‰F…x2 † À F…x1 †ŠnÀ2 x1 x2 1.10.3.2 n3I n3f …x1 † f …x2 †F …x1 †‰1 À F…x2 †Š …i À 1†3…n À i À 1†3 x1 x2 0 if F…x† ˆ 0 1 if F…x† > 0 lim Hn …an ‡ bn x† ˆ lim F n …an ‡ bn x† ˆ H…x† n3I n3I Vx …25† Setting k ˆ 2, r1 ˆ i and r2 ˆ i ‡ 1 in Eq (24), we get the joint density of the statistics of orders i and i ‡ 1: fi;i‡1Xn …x1 ; x2 † ˆ n3I & which means that the limit distributions are degenerate With the aim of avoiding degeneracy, we look for linear transformations Y ˆ an ‡ bn x, where an and bn are constants, depending on n, such that the limit distributions Joint Distribution of Two Consecutive Order Statistics iÀ1 Limiting Distributions of Order Statistics and lim Ln …cn ‡ dn x† ˆ lim 1 À ‰1 À F…cn ‡ dn x†Šn nÀiÀ1 n3I n3I ˆ L…x† Vx …26† are not degenerate 1.10.3.3 Joint Distribution of Any Two Order Statistics The joint distribution of the statistics of orders r and s …r < s† is f…r;sXn …xr ; xs † ˆ n3f …xr † f …xs †F rÀ1 …xr †‰F…xs † À F…xr †ŠsÀrÀ1 ‰1 À F…xs †ŠnÀs …r À 1†3…s À r À 1†3…n À s†3 xr xs 1.10.3.4 iˆ1 Copyright © 2000 Marcel Dekker, Inc f …xi † x1 The problem of limit distribution can then be stated 2 The joint density of all order statistics can be obtained from Eq (24) setting k ˆ n and obtain n ‰ as: 1 Joint Distribution of all Order Statistics f1;FFF;nXn …x1 ; F F F ; xn † ˆ n3 De®nition 13 Domain of Attraction of a Given Distribution: A given distribution, F…x†, is said to belong to the domain of attraction for maxima of H…x†, if Eq (25) holds for at least one pair of sequences fan g and fbn > 0g Similarly, when F…x† satis®es (26) we say that it belongs to the domain of attraction for minima of L…x† ÁÁÁ xn 3 Find conditions under which Eqs (25) and (26) are satis®ed Give rules for building the sequences fan g; fbn g; fcn g, and fdn g Find what distributions can occur as H…x† and L…x† The answer of the third problem is given by the following theorem [14±16] Some Probability Concepts for Engineers 27 Theorem 10 Feasible Limit Distribution for Maxima: The only nondegenerate distributions H…x† satisfying Eq (25) are & Frechet: H1;g …x† ˆ Weibull: H2;g …x† ˆ exp…ÀxÀg † 0 & 1 exp‰À…Àx†g Š if x > 0 otherwise if 0 otherwise and Gumbel: H3;0 …x† ˆ exp‰À exp…Àx†Š ÀI < x < I Theorem 11 Feasible Limit Distribution for Minima: The only nondegenerate distributions L…x† satisfying Eq (26) are & Frechet: L1;g …x† ˆ Weibull: L2;g …x† ˆ 1 À exp‰À…Àx†Àg Š if x < 0 1 otherwise & 1 À exp…Àxg † if x > 0 0 otherwise and Gumbel: L3;0 …x† ˆ 1 À exp…À exp x† ÀI < x < I To know the domains of attraction of a given distribution and the associated sequences, the reader is referred to Galambos [16] Some important implications of his theorems are: 1 2 3 4 5 Only three distributions (Frechet, Weibull, and Gumbel) can occur as limit distributions for maxima and minima Rules for determining if a given distribution F…x† belongs to the domain of attraction of these three distributions can be obtained Rules for obtaining the corresponding sequences fan g and fbn g or fcn g and fdn g …i ˆ 1; F F F† can be obtained A distribution with no ®nite end in the associated tail cannot belong to the Weibull domain of attraction A distribution with ®nite end in the associated tail cannot belong to the Frechet domain of attraction Next we give another more ef®cient alternative to solve the same problem We give two theorems [13, 17] that allow this problem to be solved The main advantage is that we use a single rule for the three cases Copyright © 2000 Marcel Dekker, Inc Theorem 12 Domain of Attraction for Maxima of a Given Distribution: A necessary and suf®cient condition for the continuous cdf F…x† to belong to the domain of attraction for maxima of Hc …x† is that F À1 …1 À "† À F À1 …1 À 2"† ˆ 2c "30 F À1 …1 À 2"† À F À1 …1 À 4"† lim where c is a constant This implies that: If c < 0, F…x† belongs to the Weibull domain of attraction for maxima If c ˆ 0, F…x† belongs to the Gumbel domain of attraction for maxima If c > 0, F…x† belongs to the Frechet domain of attraction for maxima Theorem 13 Domain of Attraction for Minima of a Given Distribution: A necessary and suf®cient condition for the continuous cdf F…x† to belong to the domain of attraction for minima of Lc …x† is that F À1 …"† À F À1 …2"† ˆ 2c "30 F À1 …2"† À F À1 …4"† lim This implies that: If c < 0, F…x† belongs to the Weibull domain of attraction for minima If c ˆ 0, F…x† belongs to the Gumbel domain of attraction for minima If c > 0, F…x† belongs to the Frechet domain of attraction for minima Table 7 shows the domains of attraction for maxima and minima of some common distributions 1.11 PROBABILITY PLOTS One of the graphical methods commonly used by engineers is the probability plot The basic idea of probability plots, of a biparametric family of distributions, consists of modifying the random variable and the probability drawing scales in such a manner that the cdfs become a family of straight lines In this way, when the cdf is drawn a linear trend is an indication of the sample coming from the corresponding family In addition, probability plots can be used to estimate the parameters of the family, once we have checked that the cdf belongs to the family However, in practice we do not usually know the exact cdf We, therefore, use the empirical cdf as an approximation to the true cdf Due to the random character of samples, even in the case of the sample Some Probability Concepts for Engineers 29  x À  F…xY ; † ˆ È  …29† where  and  are the mean and the standard deviation, respectively, and ȅx† is the cdf of the standard normal variable N…0; 1† Then, according to Eqs (27) and (28), Eq (29) gives  ˆ g…x† ˆ x  ˆ h…y† ˆ ÈÀ1 …y† aˆ 1  bˆ À  and the family of straight lines becomes  ˆ a ‡ b ˆ À  …30† Once the normality assumption has been checked, estimation of the parameters  and  is straightforward In fact, setting  ˆ 0 and  ˆ 1 in Eq (30), we obtain ˆ0 ˆ1 A A 0 ˆ … À †= 1 ˆ … À †= A A ˆ  ˆ  ‡  …31† Figure 18 shows a normal probability plot, where the ordinate axis has been transformed by  ˆ ÈÀ1 …y†, whereas the abscissa axis remains untransformed Note that we show the probability scale Y and the reduced scale  1.11.4 The Log-Normal Probability Plot The case of the log-normal probability plot can be reduced to the case of the normal plot if we take into account that X is log-normal iff Y ˆ log…X† is normal Consequently, we transform X into log…x† and obtain a normal plot Thus, the only change consists of transforming the X scale to a logarithmic scale (see Fig 19) The mean à and the standard deviation  à of the lognormal distribution can then be estimated by  2  2 2  Ã2 ˆ e2 e2 À e à ˆ e‡ =2 where  and  are the values obtained according to Eq (31) 1.11.5 The Gumbel Probability Plot The Gumbel cdf for maxima is  ! xÀ F…xY ; † ˆ exp À exp À  …32† Let p ˆ F…xY ; † Then taking logarithms of 1=p twice we get Figure 18 An example of a normal probability plot Copyright © 2000 Marcel Dekker, Inc ÀI < x < I Some Probability Concepts for Engineers 31 Figure 21 An example of a Weibull probability plot for minima maxima in which the ordinate axis has been transformed according to Eq (33) and the abscissa axis remains unchanged 1.11.6 The Weibull Probability Plot The Weibull cdf for maxima is 4   5 Àx y ˆ F…xY ; ; † ˆ exp À  ÀI < x  ˆ 0 ˆ … ‡ log † …34† A  ˆ 1 ˆ … ‡ log † A  ˆ À log  1  ˆ À log  Figure 21 shows a Weibull probability plot for minima  Letting p ˆ F…xY ; ... Marcel Dekker AG Hutgasse 4, Postfach 812 , CH-40 01 Basel, Switzerland tel: 4 1- 6 1- 2 6 1- 8 482; fax: 4 1- 6 1- 2 6 1- 8 896 World Wide Web http://www.dekker.com The publisher offers discounts on this book when... ‰F…xj † À F…xj? ?1 †Šrj Àrj? ?1 ? ?1 …rj À rj? ?1 À 1? ?3 …24† 1. 10.3 Distributions of Particular Order Statistics 1. 10.3 .1 and Joint Distribution of Maximum and Minimum n lim Ln …x† ˆ lim À ? ?1 À F…x†Š ˆ... Robotic Palletizing of Fixed- and Variable-Size/Content Parcels Hyder Nihal Agha, William H DeCamp, Richard L Shell, and Ernest L Hall Part Safety, Risk Assessment, and Standards 8 .1 Investigation