Buckley j j fuzzy probabilities new approach and applications (,2005)(t)(179s)

James J Buckley Fuzzy Probabilities Studies in Fuzziness and Soft Computing Editor-in-chief Prof Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul Newelska 01-447 Warsaw, Poland E-mail: kacprzyk@ibspan.waw.pl http://www.springer.de/cgi-bin/search_book.pl?series = 2941 Further volumes of this series can be found at our homepage Vol 94 M MacCrimmon and P Tillers (Eds.) The Dynamics of Judicial Proof, 2002 ISBN 3-7908-1459-8 Vol 95 T Y Lin, Y Y Yao and L A Zadeh (Eds.) Data Mining, Rough Sets and Granular Computing, 2002 ISBN 3-7908-1461-X Vol 96 M Schmitt, H.-N Teodorescu, A Jain, A Jain, S Jain and L C Jain (Eds.) Computational Intelligence Processing in Medical Diagnosis, 2002 ISBN 3-7908-1463-6 Vol 97 T Calvo, G Mayor and R Mesiar (Eds.) Aggregation Operators, 2002 ISBN 3-7908-1468-7 Vol 98 L C Jain, Z Chen and N Ichalkaranje (Eds.) 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Beyond Two: Theory and Applications of Multiple Valued Logic, 2003 ISBN 3-7908-1541-1 James J Buckley Fuzzy Probabilities New Approach and Applications 13 Professor James J Buckley University of Alabama at Birmingham Mathematics Department Birmingham, AL 35294 ISSN 1434-9922 ISBN 3-540-25033-6 Springer Berlin Heidelberg New York Library of Congress Control Number: 2005921518 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitations, broadcasting, reproduction on microfi lm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Violations are liable to prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: data delivered by authors Cover design: E Kirchner, Springer-Verlag, Heidelberg Printed on acid free paper 89/3141/M - To Julianne and Helen Contents Introduction 1.1 Introduction 1.2 References 1 Fuzzy Sets 2.1 Introduction 2.2 Fuzzy Sets 2.2.1 Fuzzy Numbers 2.2.2 Alpha-Cuts 2.2.3 Inequalities 2.2.4 Discrete Fuzzy Sets 2.3 Fuzzy Arithmetic 2.3.1 Extension Principle 2.3.2 Interval Arithmetic 2.3.3 Fuzzy Arithmetic 2.4 Fuzzy Functions 2.4.1 Extension Principle 2.4.2 Alpha-Cuts and Interval Arithmetic 2.4.3 Differences 2.5 Finding the Minimum of a Fuzzy Number 2.6 Ordering Fuzzy Numbers 2.7 Fuzzy Probabilities 2.8 Fuzzy Numbers from Confidence Intervals 2.9 Computing Fuzzy Probabilities 2.9.1 First Problem 2.9.2 Second Problem 2.10 Figures 2.11 References 7 11 12 12 13 14 14 16 16 17 19 21 21 23 24 26 28 28 Fuzzy Probability Theory 3.1 Introduction 3.2 Fuzzy Probability 3.3 Fuzzy Conditional Probability 31 31 32 36 11 11 CONTENTS viii 3.4 Fuzzy Independence 3.5 Fuzzy Bayes' Formula 3.6 Applications 3.6.1 Blood Types 3.6.2 Resistance to Surveys 3.6.3 Testing for HIV 3.6.4 Color Blindness 3.6.5 Fuzzy Bayes 3.7 References Discrete Fuzzy Random Variables 4.1 Introduction 4.2 Fuzzy Binomial 4.3 Fuzzy Poisson 4.4 Applications 4.4.1 Fuzzy Poisson Approximating Fuzzy Binomial 4.4.2 Overbooking 4.4.3 Rapid Response Team 4.5 References Fuzzy Queuing Theory 5.1 Introduction 5.2 Regular, Finite Markov Chains 5.3 Fuzzy Queuing Theory 5.4 Applications 5.4.1 Machine Servicing Problem 5.4.2 Fuzzy Queuing Decision Problem 5.5 References Fuzzy Markov Chains 6.1 6.2 6.3 6.4 6.5 Introduction Regular Markov Chains Absorbing Markov Chains Application: Decision Model References Fuzzy Decisions Under Risk 7.1 Introduction 7.2 Without Data 7.3 WithData 7.4 References CONTENTS ix Continuous Fuzzy Random Variables 8.1 Introduction 8.2 Fuzzy Uniform 8.3 Fuzzy Normal 8.4 Fuzzy Negative Exponential 8.5 Applications 8.5.1 Fuzzy Uniform 8.5.2 Fuzzy Normal Approximation to Fuzzy Binomial 8.5.3 Fuzzy Normal Approximation to Fuzzy Poisson 8.5.4 Fuzzy Normal 8.5.5 Fuzzy Negative Exponential 8.6 References Fuzzy Inventory Control 9.1 Introduction 9.2 Single Period Model 9.3 Multiple Periods 9.4 References 10 Joint Fuzzy Probability Distributions 10.1 Introduction 10.2 Continuous Case 10.2.1 Fuzzy Marginals 10.2.2 Fuzzy Conditionals 10.2.3 Fuzzy Correlation 10.2.4 Fuzzy Bivariate Normal 10.3 References 11 Applications of Joint Distributions 11.1 Introduction 11.2 Political Polls 11.2.1 Fuzzy Marginals 11.2.2 Fuzzy Conditionals 11.2.3 Fuzzy Correlation 11.3 Fuzzy Reliability Theory 11.4 References 12 Functions of a Fuzzy Random Variable 12.1 Introduction 12.2 Discrete Fuzzy Random Variables 12.3 Continuous Fuzzy Random Variables 152 16.2.3 CHAPTER 16 CONCLUSIONS AND FUTURE RESEARCH Chapter We modeled our fuzzy queuing theory after regular, finite, Markov chains We showed that we obtain steady state fuzzy probabilities when we use restricted fuzzy arithmetic We applied these results to the following queuing systems: c parallel, and identical, servers; finite system capacity; and finite, or infinite, calling source 16.2.4 Chapter We showed that the basic properties of regular, and absorbing, finite Markov chains carry over to fuzzy Markov chains when you use restricted fuzzy arithmetic 16.2.5 Chapter Here we looked at the classical decision making problem under risk The probabilities of the states of nature are usually "personal", or subjective, probabilities Sometimes these probabilities are estimated by "experts" Hence these probabilities are good candidates for fuzzy probabilities We looked at two cases: (1) fuzzy decisions under risk without data; and (2) fuzzy decisions under risk with data In the second case we used fuzzy Bayes' formula to update the prior fuzzy probabilities to the posterior fuzzy probabilities 16.2.6 Chapter In this chapter we studied continuous fuzzy random variables We looked at the following types of continuous fuzzy random variables: the fuzzy uniform, the fuzzy normal; and the fuzzy negative exponential In each case we discussed how to compute fuzzy probabilities and how to determine their fuzzy mean and their fuzzy variance 16.2.7 Chapter Here we are interested in the "probabilistic" inventory control problems, in particular those with probabilistic demand Suppose demand is modeled as a normal probability density The mean and variance must be estimated from past data and are good candidates for fuzzy number values We looked at two cases using the fuzzy normal density function for demand The first case was a simple single item, one period, inventory model with fuzzy demand using the decision criterion of minimizing expect costs The second case we expanded the model to multiple periods and now we wish to maximize expected profit 16.2 SUMMARY 16.2.8 153 Chapter 10 This chapter extends the results of Chapter and t o multivariable fuzzy probability (mass) density functions We studied fuzzy marginals, fuzzy conditionals and fuzzy correlation Also, we discussed the fuzzy bivariate normal density 16.2.9 Chapter 11 Here we had an application of a discrete fuzzy trinomial probability mass function, with its fuzzy marginals and its fuzzy conditional probability mass functions The other application was using a joint discrete fuzzy probability distribution, a fuzzy Poisson and a fuzzy binomial, in reliability theory 16.2.10 Chapter 12 Let X be a fuzzy random variable and Y = f (X) In this chapter we show, through five examples, how to find the fuzzy probability (mass) density function for Y 16.2.11 Chapter 13 This chapter generalizes Chapter 12 If X1 and X2 are fuzzy random variables and YI = fi ( X I , Xz), Yz = f2(X1, X2), find the joint fuzzy probability (mass) density function for (Yl,Y2) We first look at how to solve the problem, through three examples, when the transformation is one-to-one Then we see how to solve the problem, for two of the order statistics, when the transformation is not one-to-one 16.2.12 Chapter 14 Limit laws are important in probability theory We present only one in this chapter, the law of large numbers, using the fuzzy normal 16.2.13 Chapter 15 We define the fuzzy moment generating function We use this, just like the crisp moment generating function is used in crisp probability theory, t o find the fuzzy probability (mass) density function for the sum of independent, identically distributed, fuzzy random variables 154 CHAPTER 16 CONCLUSIONS AND FUTURE RESEARCH 16.3 Research Agenda 16.3.1 Chapter What is needed is a numerical optimization method for computing the max/min of a non-linear function subject to both linear and inequality constraints in order to find the a-cuts of fuzzy probabilities It would be nice if this numerical method could be coupled with a graphical procedure so we can graph these fuzzy probabilities and have the ability to export these graphs to LaTeX2, More work can be done on the basic properties of our fuzzy probability including fuzzy conditional probability and fuzzy independence 16.3.2 Chapter There are other discrete fuzzy random variables to be considered including: fuzzy uniform; fuzzy geometric; fuzzy negative binomial, 16.3.3 Chapter There are lots of other queuing systems to investigate 16.3.4 Chapter There are many other results on Markov chains to study using fuzzy probabilities and restricted fuzzy arithmetic 16.3.5 Chapter There are other decision making under risk models that can be investigated using fuzzy probabilities and restricted fuzzy arithmetic 16.3.6 Chapter There are other continuous fuzzy random variables to study including: the fuzzy beta; the fuzzy Chi-square; the fuzzy gamma, 16.3.7 Chapter There are lots of other probabilistic inventory control problems to consider using fuzzy probabilities 16.3.8 Chapter 10 We only discussed the joint fuzzy probability density for two fuzzy random variables Extend to n fuzzy random variables Also, related to correlation is linear regression Is fuzzy regression related to fuzzy correlation? , 16.4 CONCLUSIONS 16.3.9 Chapter 11 There are other applications of fuzzy probabilities to reliability theory 16.3.10 Chapter 12 Develop a general theory of finding the fuzzy probability (mass) density of Y = f (X) What to in Example 12.3.1 where we could not solve the problem? 16.3.11 Chapter 13 Generalize to n fuzzy random variables distribution and the fuzzy F-distribution 16.3.12 Also derive the fuzzy t- Chapter 14 Develop more general limit laws for fuzzy probability Is there a fuzzy central limit theorem? 16.3.13 Chapter 15 Extend the fuzzy moment generating function to more than one fuzzy random varia.ble Develop the theory of limiting fuzzy moment generating functions 16.4 Conclusions Classical probability theory is the foundation of classical statistics We propose fuzzy probability theory to be the foundation of a new fuzzy statistics Index algorithm evolutionary, 36, 64, 73, 97, 112 genetic, 36, 64, 73, 97 alpha-cut, 1, 3, 9, 13, 14, 16, 22, 23, 27, 32, 34, 35, 41, 42, 46, 47, 52, 54, 56, 59, 60, 64, 66, 68, 73, 75, 76, 86, 88, 91, 96-98, 101, 103, 106, 110, 112, 116, 119, 122, 126, 127, 133, 137, 141, 148 approximations fuzzy binomial by fuzzy normal, 101 fuzzy binomial by fuzzy Poisson, 57 fuzzy Poisson by fuzzy normal, 104 Bayes' formula, 2, 40 fuzzy, 3, 31, 40, 41, 46, 47, 151, 152 blood types, 41 Chebyshev's inequality, 145 cockpit ejection seat design, 105 color blindness, 45 confidence interval, 2, 21, 22, 58, 106, 115 constraints linear, 2, 23 sum one, 1, 31 zero final inventory, 112 covariance, 120 crisp function, matrix, number, 3, 8, 31 probability, 3, 39, 40, 58, 81 set, 3, solution, subset, decisions under risk, 85 solution aspiration level, 85, 87 max expected payoff, 85, 86, 88 with data, 88 without data, 86 defuzzify, 112 Dempster-Shafer, 71 distribution function, 134 dynamic programming, 71, 81, 111,112 fuzzy, 71 EOQ, 109 estimator fuzzy, interval, 22 point, 2, 22, 115 expert opinion, 21, 51, 52, 101 extension, 14, 16 extension principle, 12-14, 16, 71 figures LaTeX, 28 Maple, 28 fuzzify, 109 fuzzy arithmetic, 11, 13, 36 INDEX addition, 11 division, 11 multiplication, 11, 39 subtraction, 11 fuzzy constraints, 71 fuzzy correlation, 4, 120, 128, 153, 154 example, 121, 128 fuzzy covariance, 120 fuzzy decision theory, fuzzy decisions under risk, 85, 154 solution aspiration level, 85, 87 max expected payoff, 85, 86, 90 with data, 90, 152 without data, 86, 152 fuzzy distribution function, 134136, 143 fuzzy dynamic programming, 81, 112 fuzzy F-distribution, 155 fuzzy function, 3, 14, 16 application, 15 fuzzy goals, 71 fuzzy independence, 3, 4, 31, 38, 151 properties, 39 strong, 38, 39 weak, 38, 40 fuzzy inventory control, 4, 87, 109, 154 multiple periods, 152 single period, 152 fuzzy law of large numbers, 4, 145, 153 fuzzy limit laws, 155 moment generating functions, 155 fuzzy Markov chains, 3, 71, 152 absorbing, 77, 152 decision model, 79 decision problem, 71 regular, 75, 152 fuzzy moment generating function, 147, 148, 153 uniqueness, 149 fuzzy number, 1, 2, 8, 11, 12, 14, 15, 21, 26, 27, 31, 32, 36, 38, 41, 43, 44, 51, 52, 58, 63, 65, 68, 69, 72, 73, 85, 87,90,106,110-112,115, 116, 122, 126 confidence interval, 3, 21 core, 10, 18, 20, 38, 59, 110 maximum, 3, 17 minimum, 3, 17, 65, 110 support, 10, 126 trapezoidal, 8, 60 trapezoidal shaped, 9, 60 triangular, 8, 22, 37, 39, 40, 42, 43, 45, 47, 48, 75, 88, 126, 128 triangular shaped, 8, 22, 39, 64, 76, 88 fuzzy numbers, 35 height of intersection, 20 ordering, 3, 19, 65, 82, 86, 88 partitioning, 20 fuzzy order statistics, 142 fuzzy probability, 1-3, 9, 21, 27, 32, 39-43, 52, 54, 55, 57, 58, 60, 68, 79, 81, 86, 90, 95-99,101-103,106,116, 122, 127, 133, 145, 151, 152 computing, 3, 23 calculus, 24 feasible, 24, 34, 37, 42, 43, 45, 46, 75, 76, 79, 88, 91, 126, 134 first problem, 24 graphical, 24, 27 second problem, 26 conditional, 3, 4, 31, 36, 38, 41, 44-46, 107, 132, 151 properties, 37 determing, INDEX figures, 3, 28 generalized addition law, 33 new approach, posterior, 2, 152 prior, 2, 152 properties, 32 steady state, 61, 63, 64, 152 transition matrix, 63, 65, 66, 68, 69, 71, 72, 79 fuzzy probability density, 2, 134136, 139, 148 beta, 154 Chi-square, 137, 154 gamma, 150, 154 negative exponential, 4, 27, 99, 107, 136, 142, 143, 149, 150, 152 forgetfullness, 107 fuzzy mean, 100 fuzzy variance, 100 normal, 2, 4, 27, 97, 103, 105, 106, 109, 110, 112, 136, 141, 145, 149, 150, 152, 153 fuzzy mean, 22, 98 fuzzy variance, 98 uniform, 4, 95, 101, 135, 152 fuzzy mean, 96 fuzzy variance, 96 fuzzy probability distribution bivariate normal, 4, 121, 153 conditional, 122 marginal, 122 conditional, 4, 118 example, 119 fuzzy mean, 119, 127 fuzzy variance, 119, 127 discrete, 21, 31, 65, 69, 81 conditional, 127 fuzzy mean, 34, 35 fuzzy variance, 34, 35 marginal, 126 joint, 4, 139-143, 153 continuous, 115, 125 discrete, 125, 132, 153 marginal, 4, 116, 141, 142, 153 example, 117 fuzzy mean, 116 fuzzy variance, 116 posterior, 41, 46, 48, 90 prior, 46, 48, 90 fuzzy probability mass function, binomial, 2, 3, 51, 57, 58, 68, 101, 103, 126, 127, 132, 148, 149, 151, 153 fuzzy mean, 52, 103 fuzzy variance, 52, 103, 126, 149 geometric, 154 negative binomial, 154 Poisson, 3, 4, 51, 54, 55, 57, 105, 107, 131, 134, 141, 149-151,153 fuzzy mean, 56 fuzzy variance, 56 trinomial, 126, 153 uniform, 154 fuzzy queuing theory, 3,61,63,64, 152 decision problem, 68 fuzzy random variable, 2, 69, 111 continuous, 2, function of, 134 discrete, 2, function of, 133 functions of, 4, 133, 153 fuzzy random variables functions of, 139 max, 143 min, 143 one-to-one, 140, 153 other, 142, 153 sums, 147, 153 independent, 139, 141 fuzzy regression, 154 fuzzy relation, 72 fuzzy reliability theory, 4, 125, INDEX 129, 153, 155 fuzzy set, 3, discrete, 11 fuzzy statistics, 155 fuzzy subset, 3, 7, 11 fuzzy t-distribution, 155 greatest lower bound, 12 inequalities, 11 inf, 12 interval arithmetic, 2, 12-14, 16, 32, 34, 74 addition, 12 division, 12 multiplication, 12 subtraction, 12 inventory control, 87, 109 max expected profit, 111 expected cost, 110 multiple periods, 111 single period, 109 Jacobian, 140-142 law of large numbers, 145 least upper bound, 12 less than or equal, 3, 11, 19, 46, 145 linear programming, 24 machine servicing problem, 65 Maple, 58, 59, 98 impliciplot, 28 simplex, 42, 88 Markov chains, 63, 72, absorbing, 71, 77 regular, 61, 71, 75 maximum, 2, 12 membership function, 7, 15, 110 minimum, moment generating function, 147 uniqueness, 147 multiobjective problem, 17, 110 numerical optimization method, 154 optimization problem, 23 non-linear, 35 order statistics, 142 overbooking, 58 political polls, 125 possibility distribution, 71 possibility theory, 79 probability conditional, 40, 44, 47, 130 crisp, fuzzy, imprecise, interval, 1, transition matrix, 61, 62, 64, 66, 72-74, 76, 77 uncertain, probability density function, 21, 134, 139 Chi-square, 137 negative exponential, 27, 99, 107 forgetfullness, 107 normal, 22, 27, 97, 103, 104, 109, 137, 141, 145 support, 134, 140 uncertain parameter, 22, 26 uniform, 95, 101, 142 probability distribution, 85 bivariate normal, 121 conditional, 122 correlation, 122 marginal, 122 conditional, 89, 118 mean, 118 variance, 118 discrete, 1, 31, 32, 71 joint, 139 continuous, 115 discrete, 89, 130 marginal, 89 posterior, 40, 89 INDEX prior, 40, 89 probability mass function, 21 binomial, 51, 52, 57, 58, 66, 101, 130 Poisson, 54, 57, 60, 104, 107, 130 trinomial, 125 uncertain parameter, 22, 26 project scheduling, 21 queuing system, 152, 154 calling source, 61, 65, 68 servers, 61, 65, 68 system capacity, 61, 65, 68 random sample, 2, 21, 22, 41, 42, 44, 51, 115 random variables independent, 139 rapid response team, 59 resistance to surveys, 42 restricted fuzzy arithmetic, 2, 32, 52,63, 71-73,75, 77, 151, 152, 154 single objective problem, 18, 110, 112 states of nature, 40, 46, 85 statistic, 22 subjective probability, 21 sup, 12 testing HIV, 44 transitive, 20 uncertainty, 1, 9, 31, 32, 35, 44, 45, 59, 63, 64, 66, 68, 72, 77, 95, 106,109, 115,125 undominated, 18, 19 utility theory, 85 List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Triangular Fuzzy Number Trapezoidal Fuzzy Number M Triangular Shaped Fuzzy-Number P he Fuzzy ~ u m b e r c = A ~ Computations for the Min of a Fuzzy Number Determining v(N 2) Fuzzy Mean in Example 2.7.1 Fuzzy Variance in Example 3.2.2 Fuzzy Probability in the Blood Type Application Fuzzy Probability in the Survey Application Fuzzy Probability of HIV Given Test Positive Fuzzy Probability of Male Given Color Blind Fuzzy Probability of Female Given Color Blind P(A1) Using the Fuzzy Prior P(A1)Using the Fuzzy Posterior < 4.1 4.2 4.3 4.4 4.5 Fuzzy Fuzzy Fuzzy Fuzzy Fuzzy Variance in Example 4.2.2 Probability in Example 4.3.1 Probability in Example 4.3.2 Probability of Overbooking Probability of Multiple Attacks 6.1 6.2 6.3 6.4 Fuzzy Fuzzy Fuzzy Fuzzy Number 5;T1 in Example Number FT, in Example Number FGl in Example Number Fz2 in Example 6.3.2 6.3.2 6.3.2 6.3.2 7.1 Fuzzy Expected Payoff ?rll Z = zl in Example 7.3.1 7.2 Fuzzy Expected Payoff ?rZl.Z = zl in Example 7.3.1 7.3 Fuzzy Expected payoff Z31.Z = 21.in Example 7.3.1 8.1 Fuzzy Probability in Example 8.2.1 8.2 Fuzzy Probability in Example 8.3.1 LIST OF FIGURES 164 8.3 Fuzzy Probability for the Fuzzy Exponential 100 8.4 Fuzzy Probability 714, 91 for the Fuzzy Uniform 102 8.5 Fuzzy Probability in the Ejection Seat Example 107 11.1 11.2 11.3 11.4 Fuzzy Fuzzy Fuzzy Fuzzy Variance in Example 11.2.1.1 Probability in Example 11.2.2.1 Conditional Variance in Example 11.2.2.1 Correlation in Example 11.2.3.1 127 128 129 130 List of Tables 3.1 Alpha-Cuts of P(0') 42 4.1 Fuzzy Poisson Approximation to Fuzzy Binomial 58 5.3.1 65 67 67 67 6.1 Alpha-cuts of the Fuzzy Numbers ifi in Example 6.2.2 77 5.1 5.2 5.3 5.4 Alpha-cuts of the Fuzzy Probabilities in Example The Transition Matrix PI in Example 5.4.1 The Transition Matrix Pz in Example 5.4.1 The Transition Matrix P3 in Example 5.4.1 7.1 7.2 7.3 7.4 7.5 7.6 Decision Problem in Example 7.2.1 Crisp Solution in Example 7.2.2 Fuzzy Expected Values in Example 7.2.2 Conditional Probabilities in Example 7.3.1 Posterior Probabilities in Example 7.3.1 Final Expected Payoff in Example 7.3.1 8.1 8.2 8.3 8.4 Alpha-cuts of the Fuzzy Probability in Example 8.3.1 Fuzzy Normal Approximation to Fuzzy Binomial Fuzzy Normal Approximation t o Fuzzy Poisson Alpha-cuts of the P[140, 2001 86 87 89 89 90 90 98 104 105 106 Printing: Binding: Strauss GmbH, Mörlenbach Schäffer, Grünstadt ... 1-19 J. J .Buckley: Ranking Alternatives Using Fuzzy Numbers, Fuzzy Sets and Systems, 15(1985), pp.21-31 J. J .Buckley: Fuzzy Hierarchical Analysis, Fuzzy Sets and Systems, 17(1985), pp 233-247 J. J .Buckley. .. References J. J .Buckley and E.Eslami: Uncertain Probabilities I: The Discrete Case, Soft Computing To appear J. J .Buckley and E.Eslami: Uncertain Probabilities 11: The Continuous Case, under review J. J .Buckley. .. for Fuzzy Functions? Fuzzy Sets and Systems, 101 (1999), pp 323-330 J. J Buckley and Y Qu: On Using a-cuts t o Evaluate Fuzzy Equations, Fuzzy Sets and Systems, 38 (1990), pp 309-312 J J .Buckley,