Quasi-Fuzzy Computation Model

Quasi-Fuzzy Computation Model

) Quasi-fuzzy Computation Model by Chien-Chung Wang Thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in the School of Computer Science and Engineering of the University of New South Wales Sydney, NSW 2052, Australia January 1997 PLEASE TYPE UNIVERSITY OF NEW SOUTH WALES Thesis/Project Report Sheet SumameorFamilyname: ..... af.lW g .............................................................................................................................................................................. Firstname·. ........ C:hien.::Chung .............................................. Othername/s: ..................................................................................................... Abbreviation for degreeas given in the Universitycalendar: .... J?.hP. ................................................................................................................................... School: ..... Ccrnpute.r... Scie.nc.e... aod.. Engineex:in�acu1cy:...... Engineex:ing ..................................................................... nae:... Quas;i.-:-:f:uzzy .Cornoutation .. Model .................................................................................................................................. Abstract 350 words maximum: (PLEASE TYPE) The thesis presents a new model of computation on fuzzy numbers. It permits for linear complexity of the arithmetic canputations, while retaining a good approximation to their membership functions. For func tion evaluation each argument is approximated by the rrembership curve, whose geometric shape is a family of trapezoids, placed one above the other. The heights and the number of trapezoids are the same for all arguments. The computation is perforrred by the pointwise evaluation separately for the corresponding vertices of the multitrapezoidal curves� The concluding step is to convert resulting multitrapezoidal shape to a fuzzy number of the type rratching that of the arguments. This method is universal, in that it can be applied to any fuzzy num­ bers. In particular, it serves well in arithmetic on mixed type arguments - a continuous and discrete ones. This method is based on a new concept of fuzzy multisets. They permit any given element of the basic da:nain to occur with several, different membership grades. This is more specific than the older notion of fuzzy bags which constitutes crisp multisets on fuzzy domains. It is used to formalize several constructs of fuzzy theory, both from quasi­ fuzzy computations and from other applications. Two applications are discussed, showing the usefulness of these con­ cepts. One is the economic inventory problem on fuzzy quantities, the other deals with multiple possibility assignments. The thesis concludes by considering a few possible lines of future investigations. Quasi-fuzzy arithmetic on nonnumerical domains is con­ sidered; a very general form of multisets is outlined; uncertainty measures for multipossibility assignments are suggested. Declaration relating to disposition of project report/thesis I am fullyaware of the polic y of theUniversity rel atingto theretention and use ofhigherdegreeprojectreportsand theses,namelythatthe University retainsthe copies submittedfor examinatio n andis freeto allow themto beconsulted or borrowed.Subject to thepr ovisionsof theCop yrightAct 1968,the University may issueaprojectreportorthesisinwholeorinpart,inphotostatormic rofilmorothercopyingmedium. (applicable to doctoratesonly) . Ialaoau thorisethe pub Hcationby University Mic rofilmsof a350 wo �bstr actin DissertationAbstracts International .. ....1"'1n ..� ... Lt1-...... r.. t.. .. t?..t........ Signatur6 (/ Witness {! Date TheUniversityrecognisesthattheremaybeexceptionalcircumstancesrequiringrestrictionsoncopyingorconditionsonuse.Requestsforrestrictionforaperiodof upto2yearsmustbemadeinwritingtotheRegistrar.Requestsforalongerperiodofrestrictionmaybeconsideredin exceptionalclrcumstancesifaccompaniedby a letterof supportfrom the Supervisor or Headof School.Such requestsm ust besubmitted with the thesis/project report. FOR OFFICE USE ONLY THISSHEET IS TO BEOLUEDTOTHEINSIDE FRONT COVER OFTHESIS THE Abstract The thesis presents a new model of computations on fuzzy numbers. It per­ mits for linear complexity of the arithmetic computations, while retaining a good approximation to their membership functions. For function evaluation each ar­ gument is approximated by the membership curve, whose geometric shape is a family of trapezoids, placed one above the other. The heights and the number of trapezoids are the same for all arguments. The computation is performed by the pointwise evaluation separately for the corresponding vertices of the multitrape­ zoidal curves. The concluding step is to convert resulting multitrapezoidal shape to a fuzzy number of the type matching that of the arguments. This method is universal, in that it can be applied to any fuzzy numbers. In particular, it serves well in arithmetic on mixed type arguments - a continuous and discrete ones. This method is based on a new concept of fuzzy multisets. They permit any given element of the basic domain to occur with several, different membership grades. This is more specific than the older notion of fuzzy bags which constitutes crisp multisets on fuzzy domains. It is used to formalize several constructs of fuzzy theory, both from quasi-fuzzy computations and from other applications. Two applications are discussed, showing the usefulness of these concepts. One is the economic inventory problem on fuzzy quantities, the other deals with mul­ tiple possibility assignments. The thesis concludes by considering a few possible lines of future investi­ gations. Quasi-fuzzy arithmetic on nonnumerical domains is considered; a very general form of multisets is outlined; uncertainty measures for multipossibility assignments are suggested. i Acknowledgements I am much indebted to my PhD Advisor, Professor Arthur Ramer, for his guid­ ance and support throught the four years of my doctorate research. I express special thanks to my MBA advisor, Professor Shan-Huo Chen, for his valuable discussions and insights. I would like to thank my colleagues and friends, Banchong Harangsri, Hairong Yu, Qi Chen, Jiaming Li, Yuling Chen, Werasak Kurutach, Jinsong Ouyang, Bing Ngu, Ivan Lou, Martin Li, Raymond Liu, and Shinn Sun, for their assistance in these four years. I would like to acknowledge the support of the Ministry of Defense, Republic of China, for offering me the scholarship during my studies 1993-96. I thank the University of New South Wales for providing travel fees for the international conferences in USA, Brazil and Taiwan. Lastly, I would like to express my gratitude to my family for their constant support and encouragement. 11 Contents 1 Overview 1 2 Definitions and notations 4 2.1 Basic terminology . 4 2.2 Fuzzy sets ..... 8 2.3 Trapezoidal and multitrapezoidal numbers 13 2.4 Multisets and fuzzy multisets . ...... 17 3 Arithmetic of fuzzy sets 20 3.1 Earlier computational models . 23 3.2 Quasi-fuzzy computation . 29 3.3 Multitrapezoidal approximations 30 3.4 Approximating discrete fuzzy numbers . 36 3.5 Conversion of the results ...... 41 3.6 Properties of quasi-fuzzy arithmetic 43 4 Fuzzy Multisets 47 4.1 Fuzzy multisets ........ 48 4.2 Arithmetic on fuzzy multisets . 50 4.3 Coordinate arithmetic on fuzzy multisets . 53 111 5 Two applications 56 5.1 Economic order quantity 57 5.2 Multi-possibility 66 6 Concluding remarks 70 Bibliography 72 iv Chapter 1 Overview Fuzzy quantities are more informative than the crisp values. When a numeri­ cal concept, like price, temperature, distance, or many others, is described by a crisp real number there is no room for expressing any imprecission of the values. When the same concept is described by a fuzzy number, several possible val­ ues are indicated, together with the likehoods that these values are in fact precise [10, 21, 26, 42]. Arithmetic operations on crisp numbers are one step procedures. In the analy­ sis of computer algorithms each of the four arithmetic operations is considered as a constant time computation. Arithmetic operations on the fuzzy numbers require processing sets of values [9, 11, 17]. Traditional method of extension principle [43, 47] has quadratic complexity. In addition, this method, whether implemented directly or through a-cuts [10, 19], may lead to unexpected, sometimes paradoxi­ cal results. If the fuzzy numbers are restricted to the simplest possible shape - trapezoids, an alternative method of function principle [6, 7] can be applied. It can be ex­ pected that combining certain features of these two methods would be beneficial. Our investigations were motivated by combining the advantages of the shapes 1 based on trapezoids with the relative simplicity of the a-cuts. It led to proposing a new model for fuzzy arithmetic, termed quasi-fuzzy computations. In this model all the fuzzy numbers, whether continuous or discrete, are ap­ proximated by certain piecewise-linear membership functions. These functions have a geometric outline of a family of trapezoids, placed one above the other. The resulting fuzzy numbers and their membership functions are termed multi­ trapezoidal. Computations on such multitrapezoidal structures are conducted by operations on pairs of the matching vertices. In case of single trapezoids with four vertices, this corresponds to function principle. For multi trapezoids the number of vertices can be any multiple 4n, where the number n corresponds

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