Grey Information: Theory and Practical Applications

511 Pages·2006·2.722 MB·English

by  Sifeng Liu PhD, Yi Lin PhD (auth.)

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Sifeng Liu and Yi Lin
Grey
Information
Theory and Practical Applications
With 60 Figures
Sifeng Liu, PhD Yi Lin, PhD
College of Economics and Management Department of Mathematics
Nanjang University of Aeronautics and Slippery Rock University
Astronautics Slippery Rock, PA 16057
Nanjing, 210016 USA
CHINA
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Control Number: 2005924711
Advanced Information and Knowledge Processing ISSN 1610-3947
ISBN-10: 1-85233-995-0
ISBN-13: 978-185233-995-1
Printed on acid-free paper
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Preface
Rapid formation and development of new theories of systems science have
become an important part of modern science and technology. For ex-
ample, since the 1940s, there have appeared systems theory, information
theory, fuzzy mathematics, cybernetics, dissipative structures, synergetics,
catastrophetheory,chaostheory,bifurcations,ultracirculations,dynamics,
and many other systems theories. Grey systems theory is also one of such
systems theories that appeared initially in the 1980s.
When the research of systems science and the method and technology
of systems engineering are applied in various traditional disciplines, such
as management science, decision science, and various scientific disciplines,
a whole new group of new results and breakthroughs are obtained. Such
a historical background has provided the environment and soil for grey
systems theory to form and to develop rapidly in the past 20-plus years.
More specifically, in 1982, Professor Deng Ju-Long published the first
research paper in the area of grey systems in the international journal
entitled Systems and Control Letters, published by North-Holland Co. His
paper was titled “Control Problems of Grey Systems.” The publication
of this paper signalled the birth of grey systems theory after many years
of e(cid:30)ective research of the founding father. This new theory soon caught
the attention of the international academic community and practitioners
of science. Many well-known scholars, such as Chinese academicians Qian
Xueshen, Song Jian, and Zhang Zhongjun. Professor Roger W. Brockett
of Harvard University, a former editor-in-chief of the journal Systems and
Control Letters, and several formal Soviet academicians, all provided very
positive comments on this new theory and o(cid:30)ered their support.
vi Preface
Intheshorttimeperiodofabouttwodecades,thetheoryofgreysystems
has been developed and is maturing rapidly. It has been widely applied to
analyses, modeling, predictions, decision making, and control, with signif-
icant consequences, of various systems, including, but not limited to, so-
cial, economic, scientific and technological, agricultural, industrial, trans-
portation, mechanical, petrological, meteorological, ecological, hydrologi-
cal, geological, financial, medical, legal, military, etc., systems. Research
papers on grey systems have been cited by many scholars around the
globe and been reviewed by internationally authoritative review period-
icals. Currently, eighty-some universities worldwide, located in countries
such as Australia, China, Japan, Taiwan, and the United States of Amer-
ica, have o(cid:30)ered courses or workshops on grey systems, and hundreds of
graduate students are applying the methodology of grey systems in their
research and their writing of dissertations. There have been many inter-
national conferences listing grey systems as a special topic. All of these
represent the fact that grey systems theory with its strong vitality has al-
readystoodintheforestofscientifictheories,andthefactthatitsposition
as a transfield scientific theory has been well established.
Startingin1982,wehavegraduallyrecognizedthemeaningandvalueof
the theory of grey systems, and started to learn and to study this theory.
Itisnodoubtthattrudginginanyscientificdisciplineisnoteasy,andthat
it is more di(cid:33)cult to explore and to pioneer in a new theory. To this end,
we have devoted the best years of our lives.
ThisresearchhasbeenfundedinsuccessionbytheChinaNaturalScience
Foundation,HenanProvinceNaturalScienceFoundation(China),SoftSci-
ence Foundation, Science Foundation for Prominent Young Scientists, Na-
tional Science Foundation for Cross-Century Academic Leaders, etc. And,
ourworkhasbroughtforwardnewprogressandbreakthroughsintheareas
ofgreysequenceoperators(includingweakeningoperatorsandstrengthen-
ingoperators),generalizeddegreesofgreyincidence(includingtheabsolute
degree of grey incidence, relative degree of grey incidence, and synthetic
degree of grey incidence), finding positioned solutions of linear and non-
linear programming models with grey parameters, G—E combined models,
fixed weight grey clusterings, grey incidence clusterings, measurement of
grey information, etc. All these results have obtained wide acceptance in
theacademiccommunity.Thisbookissurelythecrystallizationofourwork
of many years in the past.
During the entire period of creating this book, we have always put our
emphasis on the scientificability, readability, and practical applicability,
tried to present the material in a logical, systematic, and simple structure,
andfollowedtheprincipleofeliminatingallmistakesinourreasoning.This
bookcontainsatotaloftwelvechapters,coveringthetheoreticalfoundation
of grey systems theory, fundamental methods, and the main topics in grey
systems theory, including grey sequence generation, grey systems analysis,
modeling, predictions, decision making, optimization, control, etc. In the
Preface vii
final chapter, we briefly describe some main topics on numerical computa-
tions of some of the major models presented in the book.
This book can be and most parts of this book have been, in the past
fifteen years, used as a textbook for upper-level undergraduate and grad-
uate students majoring in systems science, economics, and administration,
and as a self-study book for students and scholars in areas such as geo-
science, engineering, agriculture, medicine, meteorology, natural sciences,
bioscience, etc. Best of all, this book can be and most parts of this book
have been used as a reference by state employees, politicians, administra-
tors,planners,andpolicy-makersinthepastyearsandmanyyearstocome.
Here in this book, we have absorbed the research work by Professor Deng
Ju-Longandmanyothers.Withitscurrentpresentationofthismanuscript,
thereadercanexpecttolearngreysystemstheoryinasystematicfashion.
And, at the finish of this book, he or she can expect to be at the cutting
edge of this new and exciting theory and applications.
Over the years, many people have been involved in the research, discus-
sion,andwritingofvariouspartsofthisbook,including,butnotlimitedto,
ZhuYongda,YangLing,LiXiuli,GuoTianbang,DongYaoguo,GuoHong,
HouYunxian,ZhaoLi,JiaYong,DonaldMcNeil,LinWen,ZengGuoqing,
RomanDeNu,NarendraPatel,LiuQuanfeng,XuXian,AdnanMahmood,
Hector Sabelli, Sun Suan, Cao Dianli, Liu Hongbin, Shi Benguang, Kim-
berly Forrest, Achim Sydow, Yang Wanzai, Wang Ziliang, Tan Xuerui,
Zhao Deying, Wang Lianghua, Genti Zaimi, Ye Rongjun, Li Bingjun, Li
Beiyou, Xu Chaozhi, Han Jianjun, Zhang Tao, Rebecca Martin, and Wan
Yagang. Our parents, wives, and children have been patient and sacrificial
insupportingourresearchandrelatedwriting.Agreatdealofsupportand
encouragementhasbeengiventousovertheyearsfromourcolleaguesand
theadministratorsatHenanAgricultureUniversity,InternationalInstitute
forGeneralSystemsStudies,andSlipperyRockUniversity.Finally,butnot
least, the editors and sta(cid:30) members at our publisher and the referees have
done a great deal for the final publication of this book. We would like to
use this opportunity to express our sincere appreciation to all the people,
both listed and not listed above, for their teaching, role models, guidance,
support, and encouragement. Without these people, this book would have
been impossible.
Sifeng Liu, Ph. D.
Nanjing University of Aeronautics and Astronautics, China
and
Yi Lin, Ph. D.
International Institute for General Systems Studies, USA
July 30, 2004
Contents
1 Introduction 1
1.1 Scientific Background for the Appearance of Grey Systems
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fundamental Concepts and Principles of Grey Systems . . . 3
1.2.1 Fundational Concepts of Grey Systems. . . . . . . . 3
1.2.2 Fundamental Principles of Grey Systems . . . . . . . 5
1.3 Comparison Between Several Nondeterministic Methods . . 7
1.4 Main Contents in Grey Systems Theory . . . . . . . . . . . 8
1.5 Role of Grey Systems Theory in the Development of Science 10
1.6 Positions of Grey Systems Theory in the Spectrum
of Interdisciplinary Sciences . . . . . . . . . . . . . . . . . . 11
1.7 Grey Systems in the Content of Uncertain Information . . . 13
1.7.1 Grey Uncertainties . . . . . . . . . . . . . . . . . . . 15
1.7.2 Stochastic Uncertainty . . . . . . . . . . . . . . . . . 15
1.7.3 Unascertainty . . . . . . . . . . . . . . . . . . . . . . 16
1.7.4 Fuzzy Uncertainty . . . . . . . . . . . . . . . . . . . 17
1.7.5 Rough Uncertainty . . . . . . . . . . . . . . . . . . . 17
1.7.6 Soros Reflexive Uncertainty . . . . . . . . . . . . . . 20
2 Grey Numbers and Their Operations 23
2.1 Grey Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Whitenization of Grey Numbers and Degree of Greyness . . 26
2.3 Operations of Interval Grey Numbers. . . . . . . . . . . . . 30
2.4 Measures of Grey Numbers . . . . . . . . . . . . . . . . . . 33
x Contents
2.5 Information Content of Grey Numbers . . . . . . . . . . . . 38
3 Grey Equations and Grey Matrices 45
3.1 Grey Algebraic Equations and Grey Di(cid:30)erential Equations . 45
3.2 Grey Matrices and Their Operations . . . . . . . . . . . . . 46
3.3 Several Special Grey Matrices . . . . . . . . . . . . . . . . . 50
3.4 Singularities of Grey Matrices . . . . . . . . . . . . . . . . . 52
3.5 Grey Characteristic Values and Vectors . . . . . . . . . . . 54
4 Generation of Grey Sequences 57
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Generation Based on Average . . . . . . . . . . . . . . . . . 59
4.3 Operators of Sequences . . . . . . . . . . . . . . . . . . . . 61
4.4 Smooth Sequences . . . . . . . . . . . . . . . . . . . . . . . 70
4.5 Stepwise and Smooth Ratios. . . . . . . . . . . . . . . . . . 73
4.6 Accumulating and Inverse Accumulating Generation
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.7 Randomness of Sequences of Accumulating Generations . . 79
4.8 Grey Exponentiality of Accumulating Generations . . . . . 81
5 Grey Incidence Analysis 85
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Grey Incidence Factors and Set of Grey Incidence Operators 87
5.3 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4 Degrees of Grey Incidences . . . . . . . . . . . . . . . . . . 93
5.5 Absolute Degree of Grey Incidence . . . . . . . . . . . . . . 101
5.6 Relative Degree of Grey Incidence . . . . . . . . . . . . . . 113
5.7 Synthetic Degree of Grey Incidence . . . . . . . . . . . . . . 117
5.8 Order of Grey Incidences . . . . . . . . . . . . . . . . . . . 118
5.9 Preference Analysis . . . . . . . . . . . . . . . . . . . . . . . 120
5.10 Practical Applications . . . . . . . . . . . . . . . . . . . . . 132
6 Grey Clusters and Grey Statistical Evaluations 139
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.2 Clusters of Grey Incidences . . . . . . . . . . . . . . . . . . 140
6.3 Clusters with Variable Weights . . . . . . . . . . . . . . . . 144
6.4 Clusters with Fixed Weights . . . . . . . . . . . . . . . . . . 153
6.5 GreyEvaluationBasedonTriangularWhitenizationFunctions 162
6.6 Grey Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.7 Entropy of Coe(cid:33)cient Vector of Grey Evaluations. . . . . . 169
6.8 Practical Examples . . . . . . . . . . . . . . . . . . . . . . . 174
7 Grey Systems Modeling 191
7.1 The Thought of Five-Step-Modeling . . . . . . . . . . . . . 191
7.2 Grey Di(cid:30)erential Equations . . . . . . . . . . . . . . . . . . 194
Contents xi
7.3 Model: GM(1,1). . . . . . . . . . . . . . . . . . . . . . . . . 197
7.4 Model: Remnant GM(1,1) . . . . . . . . . . . . . . . . . . . 217
7.5 Model Group of GM(1,1) Type . . . . . . . . . . . . . . . . 222
7.6 GM(1,N) and GM(0,N) . . . . . . . . . . . . . . . . . . . . 228
7.7 GM(2,1) and Verhulst Model . . . . . . . . . . . . . . . . . 235
8 Grey Combined Models 245
8.1 Econometric Models . . . . . . . . . . . . . . . . . . . . . . 246
8.1.1 Choice of Variables to Be Used in Modeling . . . . . 246
8.1.2 Econometric Models . . . . . . . . . . . . . . . . . . 247
8.2 Cobb-Douglas Model . . . . . . . . . . . . . . . . . . . . . . 254
8.3 Markov Model . . . . . . . . . . . . . . . . . . . . . . . . . 258
8.3.1 Grey Moving Probability Markov Model . . . . . . . 258
8.3.2 Grey State Markov Model . . . . . . . . . . . . . . . 260
8.4 Combined Time Series Model . . . . . . . . . . . . . . . . . 262
8.5 Combined Predictions . . . . . . . . . . . . . . . . . . . . . 265
8.5.1 The Combined Prediction Model . . . . . . . . . . . 266
8.5.2 Combined Predictions with Changing Structure . . . 267
8.5.3 Combined Predictions with Fixed Structure . . . . . 268
9 Grey Prediction 275
9.1 Test of Grey Prediction Models . . . . . . . . . . . . . . . . 275
9.2 Predictions of Sequences . . . . . . . . . . . . . . . . . . . . 277
9.3 Interval Predictions. . . . . . . . . . . . . . . . . . . . . . . 281
9.4 Disaster Predictions . . . . . . . . . . . . . . . . . . . . . . 289
9.5 Seasonal Disaster Predictions . . . . . . . . . . . . . . . . . 293
9.6 Stock-Market-Like Predictions . . . . . . . . . . . . . . . . 299
9.7 Systems Predictions . . . . . . . . . . . . . . . . . . . . . . 305
9.8 Practical Applications . . . . . . . . . . . . . . . . . . . . . 311
10 Grey Decisions 315
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
10.2 Grey Target Decisions . . . . . . . . . . . . . . . . . . . . . 318
10.3 Grey Incidence Decisions. . . . . . . . . . . . . . . . . . . . 325
10.4 Grey Development Decisions . . . . . . . . . . . . . . . . . 336
10.5 Grey Statistical Decisions . . . . . . . . . . . . . . . . . . . 341
10.6 Grey Cluster Decisions . . . . . . . . . . . . . . . . . . . . . 347
10.7 Multiple-Target-SituationDecisionswithaSynthesizedTarget 351
10.8 Grey Stratified Decisions. . . . . . . . . . . . . . . . . . . . 358
11 Grey Programming 367
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
11.2 Linear Programming Models with Grey Parameters . . . . . 369
11.3 Grey Linear Programming of Prediction Type . . . . . . . . 373
11.4 Several Theorems on Positioned Solutions of LPGP . . . . . 377