Fuzzy Mathematics in Economics and Engineering 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-binlsearch_book.pl?series=2941 Further volumes of this series can be found at our homepage Vol 81 V Dimitrov and V Korotkich (Eds.) Fuzzy Logic, 2002 ISBN 3-7908-1425-3 Vol 71 K Leiviska (Ed.) Industrial Applications of Soft Computing, 2001 ISBN 3-7908-1388-5 Vol 82 Ch Carlsson and R Fuller Fuzzy Reasoning in Decision Making and Optimization, 2002 ISBN 3-7908-1428-8 Vol 72 M Mares Fuzzy Cooperative Games, 2001 ISBN 3-7908-1392-3 Vol 73 Y Yoshida (Ed.) Dynamical Aspects in Fuzzy Decision, 2001 ISBN 3-7908-1397-4 Vol 74 H.-N Teodorescu, L.C Jain and A Kandel (Eds.) Hardware Implementation of Intelligent Systems, 2001 ISBN 3-7908-1399-0 Vol 75 V Loia and S Sessa (Eds.) Soft Computing Agents, 2001 ISBN 3-7908-1404-0 Vol 76 D Ruan, J Kacprzyk and M Fedrizzi (Eds.) Soft Computingfor Risk Evaluation and Management, 2001 ISBN 3-7908-1406-7 Vol 77 W Liu Propositional, Probabilistic and Evidential Reasoning, 2001 ISBN 3-7908- I 414-8 Vol 78 U Seiffert and L C Jain (Eds.) Self-Organizing Neural Networks, 2002 ISBN 3-7908-1417-2 Vol 83 S Barro and R Marin (Eds.) Fuzzy Logic in Medicine, 2002 ISBN 3-7908-1429-6 Vol 84 L.C Jain and J Kacprzyk (Eds.) New Learning Paradigms in Soft Computing, 2002 ISBN 3-7908-1436-9 Vol 85 D Rutkowska Neuro-Fuzzy Architectures and Hybrid Learning, 2002 ISBN 3-7908-1438-5 Vol 86 M.B Gorzalczany Computational Intelligence Systems and Applications, 2002 ISBN 3-7908-1439-3 Vol 87 C Bertoluzza, M.A Gil and D.A Ralescu (Eds.) Statistical Modeling, Analysis and Management of Fuzzy Data, 2002 ISBN 3-7908-1440-7 Vol 88 R.P Srivastava and T.J Mock (Eds.) Belief Functions in Business Decisions, 2002 ISBN 3-7908-1451-2 Vol 79 A Osyczka Evolutionary Algorithms for Single and Multicriteria Design Optimization, 2002 ISBN 3-7908-1418-0 Vol 89 B Bouchon-Meunier, J Gutierrez-Rios, L Magdalena and R.R Yager (Eds.) Technologies for Constructing Intelligent Systems 1, 2002 ISBN 3-7908-1454-7 Vol 80 P Wong, F Aminzadeh and M Nikravesh (Eds.) Soft Computing for Reservoir Characterization and Modeling, 2002 ISBN 3-7908-1421-0 Vol 90 B Bouchon-Meunier, J Gutierrez-Rios, L Magdalena and R.R Yager (Eds.) Technologies for Constructing Intelligent Systems 2, 2002 ISBN 3-7908-1455-5 James J Buckley Esfandiar Eslami Thomas Feuring Fuzzy Mathematics in Economics and Engineering With 69 Figures and 27 Tables Springer-Verlag Berlin Heidelberg GmbH Professor James J Buckley University of Alabama at Birmingham Mathematics Department Birmingham, AL 35294 USA buckley@math.uab.edu Professor Esfandiar Eslami Shahid Bahonar University Department of Mathematics Kerman Iran eslami@arg3.uk.ac.ir eslami @math.uab.edu Dr Thomas Feuring University of Siegen Electrical Engineering and Computer Science HolderlinstraBe 57068 Siegen Germany I Thanks to the University of Shahid Bahonar Kerman, Iran, for financial support during my sabbatical leave at UAB Thanks to UAB for producing a good atmosphere to research and teaching Special thanks to Prof James J BuckJey for his kind cooperation that made all possible ISSN 1434-9922 ISBN 978-3-7908-2505-3 ISBN 978-3-7908-1795-9 (eBook) DOI 10.1007/978-3-7908-1795-9 Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Buckley, James J.: Fuzzy mathematics in economics and engineering: with 27 tables / James J Buckley; Esfandiar Eslami; Thomas Feuring - Heidelberg; New York: Physica-VerI., 2002 (Studies in fuzziness and soft computing; Vol 91) 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, recitation broadcasting, reproduction on microfilm 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 Physica-Verlag Violations are liable for prosecution under the German Copyright Law © Springer-Verlag Berlin Heidelberg 2002 Originally published by Physica-Verlag Heidelberg in 2002 Softcover reprint of the hardcover I st edition 2002 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 Hardcover Design: Erich Kirchner, Heidelberg To Julianne, Birgit and Mehra Helen, Lioba, Jason, Pooya, Peyman and Payam Contents Introduction Bibliography Fuzzy Sets 2.1 Fuzzy Sets 2.1.1 Fuzzy Numbers 2.1.2 Alpha-Cuts 2.1.3 Inequalities 2.1.4 Discrete Fuzzy Sets 2.2 Fuzzy Arithmetic 2.2.1 Extension Principle 2.2.2 Interval Arithmetic 2.2.3 Fuzzy Arithmetic 2.3 Fuzzy Functions 2.3.1 Extension Principle 2.3.2 Alpha-Cuts and Interval Arithmetic 2.3.3 Differences 2.4 Possibility Theory Bibliography 5 9 9 10 Solving Fuzzy Equations 3.1 AX +B = C 3.2 New Solutions 3.3 Systems of Fuzzy Linear Equations 3.4 Applications 3.4.1 Fuzzy Linear Equation 3.4.2 Fuzzy Quadratic Equation 3.4.3 System of Linear Equations 3.5 Fuzzy Input-Output Analysis 3.5.1 The Open Model 3.5.2 Fuzzy Model 3.6 Summary and Conclusions Bibliography 19 19 11 12 13 13 14 15 17 22 24 33 33 34 35 39 39 41 43 45 CONTENTS viii Fuzzy Mathematics in Finance 4.1 Future Value 4.2 Present Value 4.3 Annuities 4.3.1 Future Value 4.3.2 Present Value 4.4 Portfolio Analysis 4.4.1 NPV Method 4.4.2 IRR Method 4.5 Summary and Conclusions Bibliography 47 48 Fuzzy Non-Linear Regression 5.1 Univariate Non-Linear Fuzzy Regression 5.1.1 Testing the EA 5.1.2 Application 5.2 Multivariate Non-Linear Fuzzy Regression 5.2.1 Testing 5.2.2 Application 5.3 Conclusions and Results Bibliography 69 70 Operations Research 6.1 Fuzzy Linear Programming 6.1.1 Maximize Z 6.1.2 Fuzzy Inequality 6.1.3 Evolutionary Algorithm 6.1.4 Applications 6.1.5 Summary and Conclusions 6.2 Fuzzy PERT 6.2.1 Job Times Fuzzy Numbers 6.2.2 Job Times Discrete Fuzzy Sets 6.2.3 Summary 6.3 Fuzzy Inventory Control 6.3.1 Demand Not Fuzzy 6.3.2 Fuzzy Demand 6.3.3 Backorders 6.3.4 Evolutionary Algorithm 6.4 Fuzzy Queuing Theory 6.4.1 Service 6.4.2 Arrivals 6.4.3 Finite or Infinite System Capacity 6.4.4 Machine Servicing Problem 6.4.5 Fuzzy Queuing Decision Problem 6.4.6 Summary and Conclusions 81 81 82 85 88 89 97 98 102 104 104 105 109 111 117 50 53 53 54 55 55 59 62 65 71 72 74 75 75 76 77 118 118 120 121 122 124 126 129 CONTENTS ix 6.5 Fuzzy Network Analysis 6.5.1 Fuzzy Shortest Route 6.5.2 Fuzzy Min-Cost Capacitated Network 6.5.3 Evolutionary Algorithm 6.5.4 Summary and Conclusions 6.6 Summary and Conclusions Bibliography 129 130 132 136 137 137 139 Fuzzy Differential Equations 7.1 Fuzzy Initial Conditions 7.1.1 Electrical Circuit 7.1.2 Vibrating Mass 7.1.3 Dynamic Supply and Demand 7.2 Other Fuzzy Parameters 7.3 Summary and Conclusions Bibliography 145 146 150 153 155 158 161 163 Fuzzy Difference Equations 8.1 Difference Equations 8.2 Fuzzy Initial Conditions 8.2.1 Classical Solution 8.2.2 Extension Principle Solution 8.2.3 Interval Arithmetic Solution 8.2.4 Summary 8.3 Recursive Solutions 8.4 Applications 8.4.1 National Income 8.4.2 Transmission of Information 8.4.3 Fuzzy Fibonacci Numbers 8.5 Summary and Conclusions Bibliography 165 166 167 167 169 172 174 175 176 176 178 179 180 183 Fuzzy Partial Differential Equations 9.1 Elementary Partial Differential Equations 9.2 Classical Solution 9.3 Extension Principle Solution 9.4 Summary and Conclusions Bibliography 185 185 187 190 194 197 10 Fuzzy Eigenvalues 10.1 Fuzzy Eigenvalue Problem 10.1.1 Algorithm 10.2 Fuzzy Input-Output Analysis 10.3 Fuzzy Hierarchical Analysis 10.3.1 The Amax-Method 199 199 203 206 209 210 CONTENTS x 10.3.2 Fuzzy Amax-Method 10.3.3 Fuzzy Geometric Row Mean Method 10.4 Summary and Conclusions Bibliography 212 222 224 227 11 Fuzzy Integral Equations 11.1 Resolvent Kernel Method 11.1.1 Classical Solution 11.1.2 Second Solution Method 11.2 Symmetric Kernel Method 11.2.1 Classical Solution 11.2.2 Second Solution Method 11.3 Summary and Conclusions Bibliography 229 230 231 235 237 237 239 240 241 12 Summary and Conclusions 12.1 Summary 12.1.1 Chapter 3: Solving Fuzzy Equations 12.1.2 Chapter 4: The Fuzzy Mathematics in Finance 12.1.3 Chapter 5: Fuzzy Non-Linear Regression 12.1.4 Chapter 6: Operations Research 12.1.5 Chapter 7: Fuzzy Differential Equations 12.1.6 Chapter 8: The Fuzzy Difference Equations 12.1 Chapter 9: Fuzzy Partial Differential Equations 12.1.8 Chapter 10: Fuzzy Eigenvalues 12.1.9 Chapter 11: Fuzzy Integral Equations 12.2 Research Agenda , , 12.2.1 Chapter 3: Solving Fuzzy Equations 12.2.2 Chapter 4: The Fuzzy Mathematics of Finance 12.2.3 Chapter 5: Fuzzy Non-Linear Regression 12.2.4 Chapter 6: Operations Research , 12.2.5 Chapter 7: Fuzzy Differential Equations 12.2.6 Chapter 8: Fuzzy Difference Equations 12.2.7 Chapter 9: Fuzzy Partial Differential Equations 12.2.8 Chapter 10: Fuzzy Eigenvalues 12.2,9 Chapter 11: Fuzzy Integral Equations 12.3 Conclusions , , , 243 243 243 244 244 245 247 248 248 249 249 250 250 250 250 250 251 252 252 252 252 252 13 Evolutionary Algorithms 13.1 Introduction , 13.2 General Purpose Algorithm Bibliography , ' 253 253 253 257 Index 259 CONTENTS Xl List of Figures 267 List of Tables 271 13.2 GENERAL PURPOSE ALGORITHM 255 Randomly choose two members of the Q-population Assume they are qa and qb, where (13.2) and (13.3) We delete the (J term since it is not effected by crossover Now randomly choose an integer k in {I, 2, ,n} For example, suppose k = Then form q~emp and q~emp, two possible members of the temporary population, as follows (13.4) q~emp = (X a 1, , Xa6, Xb7,···, Xbn), and temp _ qb - ( ) Xb1,···, Xb6, Xa7, ,X an (13.5) This is the one point crossover operation We interchange (X a7,···, xan) and (Xb7,···, Xbn) However, we are not finished We test q~emp and q~emp to see if they satisfy the constraints If q~emp satisfies the constraints, then we take it into the temporary population and discard q~emp If q~emp fails to satisfy the constraints but q~emp satisfies them, then we discard q~emp and take q~emp into the temporary population When both fail to satisfy the constraints, both are discarded and we randomly choose two more individuals from the Q-population and perform the crossover operation again Continue picking two from the Q-population until we have P members in the temporary population The reader may now generalize to a multipoint crossover operator In our multipoint crossover operator, we first randomly choose the number of crossover points The crossover operator may need to be modified in certain fuzzy optimization problems Next, the mutation operator transforms the temporary population into the next generation There are many types of mutation operators A simple mutation operation is to randomly change a position Xi, :::; i :::; n, for some individuals in the temporary population First randomly choose s members from the temporary population , put them in a set S and rename them T1,·· , T s The size of S could be 10% to 50% of the total temporary population For each Ti = (Til,···, Tin) in S (omit the (J) randomly generate an integer k E {I,···, n} Suppose k = Then randomly generate a real number T in some interval and replace Ti4 with T giving (13.6) If ri satisfies the constraints, then place it into the next generation Otherwise, discard it and randomly generate positions k and replacements T until it satisfies the constraints Of course, more than one Tij may be replaced this way This simple mutation operator ignored the mutation rate (J Mutation is the driving force of evolutionary algorithms It is to randomly spread the population out over the whole search space in search of the optimum solution Now let us consider a more complicated mutation operation 256 CHAPTER 13 EVOLUTIONARY ALGORITHMS that uses the mutation rate 0- First choose s members from the temporary population and put them in set S as before and rename them rl, , r s with ri = (ril,···, rin), 1::; i ::; s Calculate, for each ::; i ::; s rij = rij + o-N(O, 1), (13.7) for ::; j ::; n, where N(O,l) stands for a normally distributed random variable having zero mean and standard deviation one We then check to see if a ri satisfies the constraints If it does, then take it into the next generation Continue the mutation operation until we get s new ri, ::; i ::; s, individuals to place in the next generation of size P We also mutate the 0- for each of these new members For each i, ::; i ::; s, replace 0- with a slightly altered o-i and place this new mutation rate o-i in the last position in ri Then a new ri is (13.8) ::; i ::; s We place all of these ri, ::; i ::; s, into the next generation The mutation operator may have to be changed to meet the fuzzy optimization problem For example, in the fuzzy min-cost capacitated network problem in Section 6.5, mutation was to simply interchange two adjacent elements in the vector v = (Xl,· , X n , 0-), not including 0- Now we have our next generation: (1) P - s individuals created by crossover from the previous generation; and (2) s individuals first altered by crossover and then mutation The selection operation now selects the fittest Q individuals for crossover and mutation We continue this way from generation to generation until we see the results stabilizing (converging) and we stop the algorithm and record the results Run the evolutionary algorithm, with new random initial populations, a number of times to check and see if we always end up in the same place By repeating the experiment again and again we gain confidence in the results as being good approximate answers to the fuzzy optimization problem There are, of course, many variations of the above description If you download a genetic, or evolutionary, algorithm it will probably differ from that discussed above However, the whole procedure appears quite robust and many different algorithms will give good approximate solutions to fuzzy optimization problems Bibliography [1) J.J Buckley and T Feuring: Fuzzy and Neural: Interactions and Applications, Physica-Verlag, Heidelburg, 1999 Index algorithm, 204 alpha-cut, 7, 12-14,26 alpha-cut and interval arithmetic, 47, 49, 52, 60, 70, 73, 108, 111, 125, 207, 243 are, 130, 133 arrival rate, 119 arrivals, 120 arrivals distribution, 118 backlogging, 108 backorders, 117 best, 131, 244 boundary conditions, 185, 187, 190, 194 fuzzy, 187, 190 boundary value problem, 194 calling capacity finite, 126 calling source, 118, 121, 246 finite, 126 infinite, 122 capacity constraints, 133 cash flow, 55, 59 central value, 124 centroid, 124, 129 Chen's method, 87,96 classical linear programming, 81 closed form, 186, 248 closed input-output analysis, 206 Leontief, 206 compound interest, 48 compound interest formula, 48 consistency, 220 constraints, 109, 253-255 fuzzy, 253 convex, 31 cost function, 124 Cramer's rule, 26 crashing, 98, 251 crisp linear program, 91 crisp matrix, 207 irreducible, 207 reducible, 207 crisp solution, 91 crisp subset, crossover, 116, 131, 134,254-256 data, 69, 71, 72, 74-76 decision variables, 253 defuzzify, 110 delivery lag, 246 demand, 105, 246 departure rate, 119 departures, 120 diet problem, 92 difference equation, 165, 171, 172, 176,180,248 general solution, 166 homogeneous, 166 linear, 165, 176, 248 second order, 165,176,248 differential equation, 247 homogeneous, 148, 158, 248 linear, 145, 247 ordinary, 145 second order, 145, 247 discounted present value method, 55 distance measure, 70 dominance, 95 260 dynamic demand, 155, 157 dynamic programming, 108 fuzzy, 108 dynamic supply, 155, 157 EA, 2, 216, 221, 244 earliest start time, 100 economic order quantity, 105 eigenfunction, 237, 238 normalized, 237, 238 eigenvalue, 237, 238 electrical circuit, 150 equations linear fuzzy, 244 equilibrium price, 155 equilibrium supply, 155 error, 75 error function, 69, 70, 75 evolutionary algorithm, 2, 27, 69, 71,73,75,76,85,88,91, 96, 109, 110, 112, 129, 131, 159, 204, 205, 244, 246, 247, 250, 251, 253 extension, 12, 14 extension principle, 9, 11, 12, 14, 25, 48, 52, 70, 73, 125, 186 feasibility, 116 feasible, 134 feasible set, 129 feasible vector, 82 FHA, 209, 211, 212, 219, 224 Fibonacci numbers, 179 fuzzy, 179 Fibonacci sequence, 180 fuzzy, 180 figures, LaTeX2f,2 Maple, final demands, 40 finance, 244, 250 fitness function, 131, 134, 253 fittest individuals, 253, 254 INDEX floats, 100 flow constraints, 133 forcing function, 162 Fredholm integral equation, 249 future amount, 50 future value, 47, 53, 54, 244, 250 fuzzification, 87, 130, 133, 134, 190, 224 fuzzified, 187 fuzzify, 22, 26, 33, 47, 48, 55, 132, 145, 165, 172, 185-187, 195, 209, 230, 235, 239, 243, 252 fuzzifying, 162, 181, 202, 203 fuzziness, 27, 147, 177, 222 fuzzy amplitude, 154 fuzzy annuity, 54 fuzzy arithmetic, 9, 11, 108 addition, division, multiplication, subtraction, fuzzy assignment problem, 132 fuzzy beats, 154 fuzzy cash flow, 56, 57 fuzzy channel capacity, 179 fuzzy complex numbers, 252 fuzzy data, 73 fuzzy demand, 111, 112, 117 fuzzy distance, 130 fuzzy eigenvalue, 199, 201, 202, 224, 249, 252 fuzzy eigenvalue problem, 199 fuzzy eigenvector, 199, 202, 203, 205, 206, 209, 211, 224, 249, 252 fuzzy equation, 19, 33 fuzzy difference equation, 19, 252 fuzzy differential equation, 19, 251 fuzzy integral equation, 19 fuzzy linear equation, 19, 33 INDEX fuzzy quadratic, 33, 34 solution strategy, 23, 27 systems, 24, 33, 35 joint solution, 25, 27, 36 marginals, 26, 31, 36 fuzzy flow, 133 fuzzy Fourier series, 195, 252 fuzzy function, 12, 14, 229 best, 69 exponential, 71, 244 linear, 71, 244 logarithmic, 244 polynomial, 71, 244 fuzzy goal, 84, 245 fuzzy hierarchical analysis, 199, 209, 249 fuzzy inequality, 86 fuzzy initial value problem, 149, 176 fuzzy input-output analysis, 199, 249 fuzzy input-output model, 208 fuzzy integral equations, 252 fuzzy internal rate of return, 60 fuzzy inventory control, 105, 109, 251 fuzzy inventory model, 108 fuzzy linear programming, 81, 83 fully fuzzified, 81, 90 multiobjective, 94 fuzzy logic, 81 fuzzy mathematics, 51, 250 fuzzy matrix, 199, 203 symmetric, 199 fuzzy max, 87 fuzzy max-flow problem, 132 fuzzy min, 87 fuzzy cost, 247 fuzzy min-cost capacitated flow problem, 129 fuzzy model, 207 consistent, 207, 208 fuzzy net present value, 56 fuzzy network, 247 261 fuzzy network analysis, 129, 251 fuzzy non-linear regression, 69, 70 fuzzy number, 5, 9, 10, 13, 70 addition, 34 bounded, 83 core, multiplication, 34 normalized, 112 ranking, 58, 86, 87, 220 trapezoidal, 6, 100, 207, 245 trapezoidal shaped, 6, 111, 229, 245 triangular, 5, 19, 70, 82, 145, 166, 180, 185, 186, 191, 244, 245, 247, 248, 254 triangular shaped, 6, 19, 70, 82, 145, 166, 187, 201, 229, 245, 249 fuzzy optimization, 253, 255 fuzzy optimization problem, 129, 133, 134 fuzzy parameters, 158 fuzzy partial differential equations, 252 fuzzy penalty cost, Ill, 112 fuzzy PERT, 251 fuzzy portfolio analysis, 250 fuzzy profit, 246 fuzzy queuing decision problem, 118 fuzzy queuing theory, 129, 251 fuzzy reciprocal matrix, 219 fuzzy set, 5, 81 discrete, 100, 246 support, fuzzy shortest route, 129, 130, 132, 247, 251 fuzzy subset, normalized, 15 fuzzy total cost, 247 fuzzy transportation problem, 132 fuzzy trigonometry, 250 fuzzy variable, 15, 120, 245 discrete, 127 INDEX 262 non-interactive, 16, 121 fuzzy vector, 208 fuzzy weight vector, 216, 220, 221 generation, 253 genetic algorithm, 69 geometric means, 209 geometric row mean, 222 greatest lower bound, 10 HA, 209,211 Hamming distance, 87 heat conduction, 186 heat equation, 195 height of the intersection, 130 hierarchical analysis, 209, 224 consistency, 211 Saaty's method, 209 Amax-method, 210, 224 fuzzy Amax-method, 212 hierarchical structure, 210 holding cost, 105, 111, 246 ideal point, 83, 95 incoming inventory, 105, 111 independent variable, 12 industry economy, 39, 40 inf, 10 infinite series, 186 inflation, 155 initial cash outlay, 59 initial conditions, 145, 165, 169, 172, 176, 247 crisp, 145 fuzz~ 145, 147, 156, 161, 165, 167, 180 uncertain, 154 initial inventory, 110 initial population, 254 input-output model, 39 closed model, 40 fuzzy model, 41-43 open model, 39 integral equation, 229 Fredholm, 229 fuzzy, 238 interest rate, 48, 244 fuzzy, 246 internal rate of return, 55, 59 interrelated activities, 98 interval arithmetic, 10-14, 20, 22, 26, 35, 52, 232, 235 addition, 10 division, 10 multiplication, 10 subtraction, 10 interval condition, 154, 156, 171 inventory control, 81, 105, 245 multi-stage, 108 inventory control problem, 246 inventory problem, 108 IRR method, 59 job time, 98 optimistic, 102 pessimistic, 102 joint distribution, 121, 122 kernel, 240, 252 non-negative, 240 separable, 240 symmetric, 240 Kerre's method, 87, 96 latest start times, 100 least upper bound, 10 library, 70, 74, 75 basic library, 71 linear programming fully fuzzified, 250 fuzzy, 250, 254 multiobjective, 245 single objective, 245 machine servicing problem, 124 marginal propensity to consume, 176 Markov chains, 129 mathematics of finance, 47 max-flow problem, 129, 132 263 INDEX max-min powers, 123 max-min product, 118, 246 maximization problem, 95 maximize, 82, 107, 108 maximum, 10 membership function, message signals, 178 method of undetermined coefficients, 167 min-cost capacitated flow problem, 129, 132 minimization, 95, 124, 131 minimization problem, 94 minimize, 107-109, 126, 133 multivariate analysis, 74 mutation, 116, 131, 134, 253-256 mutation rate, 255 national income, 176 net present value, 55 network analysis, 81, 245 next generation, 255 node, 130, 132 sink, 132, 134 source, 132, 134 noise, 71 non-interactive, 100, 121, 122 NPV method,55 objective function, 83, 85, 96, 127 single objective, 125 operations research, 81, 245, 250, 251 optimal, 91, 96, 127, 129, 131, 133, 134 optimization problem, 81, 85, 124, 133 multiobjective, 82, 88, 107 single objective, 107 vector, 125 ordering cost, 105, 106, 246 outgoing inventory, 105, 111 parallel servers, 118 parameters, 81 partial differential equation, 190, 248 crisp, 185 elementary, 186 fuzzy, 185, 187 elemantary, 194 non-homogeneous, 185, 194, 248 penalty cost, 105, 108 permutation matrix, 207 PERT, 81, 98,129,245 possibilistic, 104 probabilistic, 100, 104 portfolio analysis, 244 positive reciprocal matrix, 210, 212 consistent, 221 fuzzy, 212 reasonably consistent, 221 possibility distribution, 15, 102, 104, 127, 128, 245 joint, 102 possibility theory, 15, 129, 251 present value, 47, 50, 54, 55, 244, 250 present worth method, 55 probability density, 15 probability theory, 15 product mix problem, 90 profit, 117 profit function, 117 project evaluation and review technique, 98 proposed investment project, 55 purchase cost, 106 queuing queuing queuing queuing decision models, 251 problem, 126 system, 118,251 theory, 81, 245 random variables, 245 ranking investment alternatives, 59 reciprocal matrix, 209 264 recombination, 253 references, regression, 244 multivariate, 244 non-linear, 244 fuzzy, 244, 250 univariate, 244 regular annuity, 47, 53, 54, 244, 250 resolvent kernel, 233, 236-238, 249 resolvent kernel method, 230, 233, 240 search space, 216, 255 selection, 253, 254, 256 separable kernel, 237 service discipline, 118 service time distribution, 118 shortage cost, 106, III shortest route problem, 129 solution, 19 approximate, 253 closed form, 109 crisp, 47, 148, 243 joint solution, 244 partial, 252 recursive, 175 undominated, 125 solution concept, 131 solution method, 20 alpha-cut and interval arithmetic, 22, 27, 56, 145, 156, 162, 165, 167, 172, 185, 199, 203, 230, 240, 243, 244, 248, 249 classical, 20, 21, 26, 27, 33, 35, 47, 51, 145, 146, 148, 156, 159, 161, 165, 167, 174, 185, 187, 194, 199, 229, 231, 234, 237, 240, 243, 244, 247-249 crisp, 22 evolutionary algorithm, 69 extension principle, 22, 26, 27, 47, 57, 145, 149, 156, INDEX 161, 165, 167, 169, 174, 185, 190, 194, 199, 202, 230, 240, 243, 244, 247249 new solutions, 22 solution sequences, 84 solution set, 85 solution strategy, 52, 243 spectrum, 237 steady-state possibility distribution, 119 steady-state distribution, 121, 123 sup, 10 supply, 155 symmetric and separable kernel procedure, 230 symmetric kernel, 237 symmetric kernel method, 237 system capacity, 118, 246 finite, 122 infinite, 122 technological matrix, 40 fuzzy, 41 temporary population, 254, 255 termination date, 53 testing, 71, 75 total cost, 126 total flow, 133 total inventory cost, 106 total outputs, 40 transition matrix, 118, 119, 121, 246 transmission of information, 178 transportation problem, 132 transshipment problem, 132 uncertainty, 7, 56, 145, 186 undominated, 82, 83 undominated solutions, 85, 88 universal approximator, 72-74 vagueness, 40 vibrating beams, 186 vibrating mass, 153 INDEX wave equation, 195 weights, 209, 210 fuzzy weights, 211 worst, 244 Wronskian, 149 265 List of Figures 2.1 2.2 2.3 2.4 Triangular Fuzzy Number N Trapezoidal Fuzzy Number M Triangular Shaped Fuzzy Number P The Fuzzy Number C = A B 12 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 Solution to Example 3.1.2 Solution to Example 3.2.1 Solution to Example 3.2.2 Xcl and X JI in Example 3.3.1 X c2 and X J2 in Example 3.3.1 Support of the Joint Solution in Example 3.3.2 XJl,Xel,X n in Example 3.3.2 X J2 ,X e2 ,X/2 in Example 3.3.2 Solution P e in the First Application Fuzzy Interest Rate in the Second Application Support for the Joint Solution in the Third Application XJl for the Third Application X J2 for the Third Application Fuzzy Total Output for Industry I in Example 3.5.1 Fuzzy Total Output for Industry II in Example 3.5.1 21 24 25 29 30 31 32 32 34 36 37 38 38 42 43 4.1 4.2 4.3 4.4 The Value Se = A(l + r)n in Example 4.1.1 Future Value of a Fuzzy Annuity in Example 4.3.1.1 Fuzzy Net Present Values in Examples 4.4.1.1 and 4.4.1.2 Fuzzy Internal Rates of Return in Example 4.4.2.2 50 54 59 62 6.1 6.2 Graphical Description of Z2, Fuzzy Goal G I for Z2 • • 6.3 The Shape of Ei 6.4 6.5 n A2 and Al = 2:: Aij X j 83 84 86 j=1 Ranking Fuzzy Numbers Based on Chen's Method X I ,X2,X3 Obtained with Kerre's Inequality (CI = 100,c2 = 200, C3 = 400), Product Mix Problem 88 92 LIST OF FIGURES 268 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25 6.26 6.27 6.28 6.29 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 X ,X 2,X Obtained with Chen's Inequality (Cl = 150,c2 = 100, C3 = 300), Product Mix Problem 93 Z Obtained with Kerre's Inequality (Cl = 100,c2 = 200,C3 = 400), Product Mix Problem 94 Z Obtained with Chen's Inequality (Cl = 150, C2 = 100, C3 = 300), Product Mix Problem 95 95 Fuzzy Goal G for the Diet Problem Fuzzy Goal G for the Diet Problem 96 Fuzzy Goal G for the Diet Problem 96 X ,X 2,X Obtained with Kerre's Inequality (Cl = 400,C2 = 400, C3 = 900), Diet Problem 98 Xr,X2,X3 Obtained with Chen's Inequality (Cl = 450,C2 = 300,C3 = 1050), Diet Problem 99 Z Obtained with Kerre's Inequality (Cl = 400,C2 = 400,C3 = 900), Diet Problem 100 Z Obtained with Chen's Inequality (Cl = 450,C2 = 300,C3 = 1050), Diet Problem 101 Project Network 102 Possibility Distribution for Project Duration in Example 6.2.1.1104 Inventory Problem 105 The Fuzzy Set Xi + Zi - Di 112 113 Storage from Figure 6.19 Penalty from Figure 6.19 113 The Fuzzy Set Xi + Zi - Di 114 Storage from Figure 6.22 114 Penalty from Figure 6.22 115 XN+r, One Case of Final Inventory 115 XN+1, Another Case of Final Inventory 116 The Fuzzy Set B w in Example 6.4.4.1 127 Fuzzy Min-Cost Capacitated Flow Problem in Example 6.5.2.1 135 Optimal Fuzzy Cost in Example 6.5.2.1 135 Electrical Circuit in Application 7.1.1 Extension Principle Solution for the Fuzzy Part of Qe (t) in the Electrical Circuit Vibrating Mass in Application 7.1.2 Extension Principle Solution Ye(t) in the Vibrating Mass Application Fuzzy Price in Application 7.1.3 Fuzzy Supply in Application 7.1.3 Extension Principle Solution in Example 7.2.1 Y e (x) Solution in Example 7.2.2 10.1 Hierarchical Structure 10.2 Fuzzy Numbers in FHA: Trapezoidal 151 152 153 154 156 157 159 161 211 213 LIST OF FIGURES 10.3 10.4 10.5 10.6 10.7 10.8 10.9 Fuzzy Numbers in FHA: Triangle Fuzzy Numbers in FHA: More than a to Fuzzy Numbers in FHA: Less than to Fuzzy Numbers in FHA: Between a/I and ,,(/1 Fuzzy Numbers in FHA: At Least a/I Fuzzy Numbers in FHA: At Most 0/1 The Final Fuzzy Weights in the Application 269 213 214 214 215 215 216 222 List of Tables 3.1 3.2 Input-Output Table Data for Example 3.5.1 39 42 4.1 Fuzzy Net Cash Flows for Examples 404.1.1 and 4.4.1.2 57 6.1 6.2 90 6.5 6.6 6.7 6.8 6.9 6.10 6.11 Approximate Times Product Pi is in Department D j Results for the Product Mix Problem Using Different Values for Ci (i=1,2,3) Approximate Units of Food Fj in Product Pi Results for the Diet Problem Using Different Values for Ci (i=1,2,3) Possibility Distributions for Job Times in Example 6.2.1 Data for Example 6.3.1.1 Data for Example 6.3.2.1 The Transition Matrices in Example 6.4.4.1 Fuzzy Distances Between the Nodes in Example 6.5.1.1 Undominated Fuzzy Shortest Routes in Example 6.5.1.1 Optimal Fuzzy Flows in Example 6.5.2.1 97 103 110 116 126 132 132 136 7.1 Finding Qe(t) in Application 7.1.1 151 8.1 Extension Principle Y e Solution in the National Income Application when ~ = 0.9, () = 2.0 Extension Principle Y e Solution in the National Income Application for ~ = 0.9 and () = 0.5 Extension Principle Y e Solution in the Transmission of Information Application Fuzzy Fibonacci Sequence 6.3 604 8.2 8.3 804 10.1 Alpha-cuts of the Fuzzy Eigenvalue Xe in Example 10.1.3 10.2 Alpha-cuts of the Fuzzy Eigenvector X in Example 10.1.3 10.3 Alpha-cuts of Fuzzy Eigenvalue and Eigenvector for Example 10.1.4 lOA Fuzzy Eigenvalue and Eigenvector in Example 10.2.1 91 92 177 178 179 180 205 205 206 209 LIST OF TABLES 272 10.5 10.6 10.7 10.8 Testing the Evolutionary Algorithm: WI Testing the Evolutionary Algorithm: W2 Testing the Evolutionary Algorithm: W3 Alpha-cuts of the Final Weights for Both Methods 218 218 218 224 ... Buckley et al., Fuzzy Mathematics in Economics and Engineering © Springer-Verlag Berlin Heidelberg 2002 20 CHAPTER SOLVING FUZZY EQUATIONS This shows a major problem in solving fuzzy equations:... Constructing Intelligent Systems 2, 2002 ISBN 3-7908-1455-5 James J Buckley Esfandiar Eslami Thomas Feuring Fuzzy Mathematics in Economics and Engineering With 69 Figures and 27 Tables Springer-Verlag... et al., Fuzzy Mathematics in Economics and Engineering © Springer-Verlag Berlin Heidelberg 2002 CHAPTER INTRODUCTION differential equation in another chapter Our recommendation reading for applications