Multistage Fuzzy Control a model-based approach to fuzzy control and decision making Fuzzy techniques are used to cope with imprecision in the control process. This authoritative book explains the essential principles of fuzzy logic and describes both the theoretical and practical advantages of the new model-based, prescriptive approach. Professor Kacprzyk offers a comprehensive and in depth examination of the issues underlying multistage control and decision analysis, addressing in particular fuzzy dynamic systems, fuzzy events, fuzzy probabilities and fuzzy quantifiers. The text also comprises an introduction to the basic concepts of fuzzy sets, fuzzy logic and fuzzy systems, complemented by real-world examples of the use of the model-based prescriptive approach to improve the efficiency of fuzzy control systems. Highly experienced in fuzzy control research, the author identifies new trends in the development of fuzzy sets and their direct application to decision-making processes. Fuzzy control engineers, researchers and postgraduate students will find this an ideal reference, offering a wealth of ideas for enhancing the performance of fuzzy control systems and equipping them with the tools to resolve genuine problems. Multistage Fuzzy Control is an essential handbook for those wishing to resolve real-world problems in control and decision analysis through the use of fuzzy-logic-based methods.

Janusz Kacprzyk is a Polish engineer and mathematician, notable for his multiple contributions to the field of computational and artificial intelligence tools like fuzzy sets, mathematical optimization.

Foreword

ix

1 Introduction

1

(18)

1.1 FUZZY LOGIC CONTROL--A DESCRIPTIVE APPROACH

4

(6)

1.2 MULTISTAGE FUZZY CONTROL--A PERSCRIPTIVE APPROACH

10

(5)

1.3 A BRIEF SURVEY OF THE CONTENTS

15

(2)

1.4 ACKNOWLEDGEMENTS

17

(2)

2 Basic Elements of Fuzzy Sets and Fuzzy Systems

19

(48)

2.1 BASIC ELEMENTS OF FUZZY SETS THEORY

20

(26)

2.1.1 The idea of a fuzzy set

20

(3)

2.1.2 Basic definition and properties related to fuzzy sets

23

(6)

2.1.3 Basic operations on fuzzy sets

29

(6)

2.1.4 Fuzzy relations

35

(2)

2.1.5 Linguistic variable, fuzzy conditional statement, and compositional rule of inference

37

(3)

2.1.6 The extension principle

40

(1)

2.1.7 Fuzzy numbers

41

(2)

2.1.8 Fuzzy events and their probabilities

43

(2)

2.1.9 Defuzzification of fuzzy sets

45

(1)

2.2 FUZZY LOGIC--BASIC ISSUES

46

(2)

2.3 FUZZY-LOGIC-BASED CALCULI OF LINGUISTICALLY QUANTIFIED STATEMENTS

48

(10)

2.3.1 Algebraic or consensory calculus

49

(2)

2.3.2 Substitution or competitive calculus

51

(2)

2.3.3 Fuzzy linguistic quantifiers and the ordered weighted averaging (OWA) operators

53

(5)

2.4 DETERMINSTIC, STOCHASTIC AND FUZZY SYSTEMS UNDER CONTROL

58

(9)

2.4.1 Deterministic system under control

58

(1)

2.4.2 Stochastic system under control

59

(1)

2.4.3 Fuzzy system under control

60

(7)

3 A General Setting for Multistage Control under Fuzziness

67

(18)

3.1 DECISION MAKING IN A FUZZY ENVIRONMENT--BELLMAN AND ZADEH'S APPROACH

67

(11)

3.1.1 The concept of a fuzzy goal, fuzzy constraint, and fuzzy decision

67

(7)

3.1.2 Multiple fuzzy goals and/or fuzzy constraints

74

(2)

3.1.3 Fuzzy goals and fuzzy constraints defined in different spaces

76

(2)

3.2 MULTISTAGE DECISION MAKING (CONTROL) IN A FUZZY ENVIRONMENT

78

(7)

4 Control Processes with a Fixed and Specified Termination Time

85

(92)

4.1 CONTROL OF A DETERMINISTIC SYSTEM

85

(52)

4.1.1 Solution by dynamic programming

86

(4)

4.1.2 Solution by branch-and-bound

90

(6)

4.1.3 Solution by a generic algorithm

96

(9)

4.1.4 Solution by a neural network

105

(8)

4.1.5 Using fuzzy linguistic quantifiers in multistage fuzzy control

113

(24)

4.2 CONTROL OF A STOCHASTIC SYSTEM

137

(15)

4.2.1 Problem formulation by Bellman and Zadeh

139

(9)

4.2.2 Problem formulation by Kacprzyk and Staniewski

148

(4)

4.3 CONTROL OF A FUZZY SYSTEM

152

(25)

4.3.1 Solution by dynamic programming

154

(8)

4.3.2 Solution by branch-and-bound

162

(4)

4.3.3 Solution by interpolative reasoning

166

(2)

4.3.4 Solution by a genetic algorithm

168

(9)

5 Control Processes with an Implicity Specified Termination Time

177

(12)

5.1 AN ITERATIVE APPROACH

178

(4)

5.2 A GRAPH-THEORETIC APPROACH

182

(4)

5.3 A BRANCH-AND-BOUND APPROACH

186

(3)

6 Control Processes with a Fuzzy Termination Time

189

(26)

6.1 CONTROL OF A DETERMINISTIC SYSTEM

191

(9)

6.1.1 Solution by dynamic programming

192

(5)

6.1.2 Solution by branch-and-bound

197

(3)

6.2 CONTROL OF A STOCHASTIC SYSTEM

200

(10)

6.2.1 Kacprzyk's approach

202

(4)

6.2.2 Stein's approach

206

(2)

6.2.3 Using fuzzy probability of a fuzzy event

208

(2)

6.3 CONTROL OF A FUZZY SYSTEM

210

(5)

6.3.1 Solution by dynamic programming

210

(2)

6.3.2 Solution by branch-and-bound

212

(3)

7 Control Processes with an Infinite Termination Time

215

(32)

7.1 CONTROL OF A DETERMINISTIC SYSTEM

218

(5)

7.2 CONTROL OF A STOCHASTIC SYSTEM

223

(12)

7.2.1 Determination of EuD(a / x0)

224

(2)

7.2.2 Determination of an optimal stationary strategy

226

(9)

7.3 CONTROL OF A FUZZY SYSTEM

235

(12)

7.3.1 Approximation by reference fuzzy sets

237

(1)

7.3.2 Approximation formulation and solution of the problem

238

(9)

8 Examples of Applications

247

(62)

8.1 SOCIOECONOMIC REGIONAL DEVELOPMENT

249

(24)

8.1.1 A fuzzy model of socioeconomic regional development

249

(4)

8.1.2 Evaluation of development

253

(7)

8.1.3 Stability of development

260

(8)

8.1.4 A multistage fuzzy control model of regional development planning

268

(1)

8.1.5 Example of application to the development planning of a rural region

269

(4)

8.2 OPTIMAL FLOOD CONTROL IN A WATER RESOURCES SYSTEM

273

(9)

8.2.1 A fuzzy formulation of the flood control problem

273

(1)

8.2.2 A multistage fuzzy control model for flood control

274

(4)

8.2.3 Solution using a branch-and-bound algorithm

278

(2)

8.2.4 A numerical example

280

(2)

8.3 RESEARCH AND DEVELOPMENT (R&D) PLANNING

282

(4)

8.3.1 Optimal selection of tasks

283

(1)

8.3.2 Optimal funding of technologies

284

(1)

8.3.3 Optimal system funding

285

(1)

8.4 SCHEDULING GENERATION UNIT COMMITMENT IN A POWER SYSTEM

286

(5)

8.4.1 A general problem formulation

286

(2)

8.4.2 Determination of the membership function of the fuzzy objective function

288

(1)

8.4.3 Determination of the membership function of the fuzzy load demand

288

(1)

8.4.4 Determination of the membership function for the spinning reserve requirement

289

(1)

8.4.5 A fuzzy dynamic programming algorithm

290

(1)

8.4.6 A numerical example

290

(1)

8.5 INTRA-OPERATIVE ANESTHESIA ADMINISTRATION

291

(5)

8.6 RESOURCE ALLOCATION

296

(5)

8.7 INVENTORY CONTROL

301

(4)

8.8 A BRIEF REVIEW OF OTHER APPLICATIONS

305

(4)

9 Concluding Remarks

309

(2)

Bibliography

311

(14)

Index

325

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