Fuzzy Mathematical Programming: Methods and Applications

ISBN-13: 9783540560982 / Angielski / Miękka / 1992 / 306 str.

Young-Jou Lai; Ching-Lai Hwang

Opis

In the last 25 years, the fuzzy set theory has been applied in many disciplines such as operations research, management science, control theory, artificial intelligence/expert system, etc. In this volume, methods and applications of fuzzy mathematical programming and possibilistic mathematical programming are first systematically and thoroughly reviewed and classified. This state-of-the-art survey provides readers with a capsule look into the existing methods, and their characteristics and applicability to analysis of fuzzy and possibilistic programming problems. To realize practical fuzzy modelling, we present solutions for real-world problems including production/manufacturing, transportation, assignment, game, environmental management, resource allocation, project investment, banking/finance, and agricultural economics. To improve flexibility and robustness of fuzzy mathematical programming techniques, we also present our expert decision-making support system IFLP which considers and solves all possibilities of a specific domain of (fuzzy) linear programming problems. Basic fuzzy set theories, membership functions, fuzzy decisions, operators and fuzzy arithmetic are introduced with simple numerical examples in aneasy-to-read and easy-to-follow manner. An updated bibliographical listing of 60 books, monographs or conference proceedings, and about 300 selected papers, reports or theses is presented in the end of this study.

Spis treści

1 Introduction.- 1.1 Objectives of This Study.- 1.2 Fuzzy Mathematical Programming Problems.- 1.3 Classification of Fuzzy Mathematical Programming.- 1.4 Applications of Fuzzy Mathematical Programming.- 1.5 Literature Survey.- 2 Fuzzy Set Theory.- 2.1 Fuzzy Sets.- 2.2 Fuzzy Set Theory.- 2.2.1 Basic Terminology and Definition.- 2.2.1.1 Definition of Fuzzy Sets.- 2.2.1.2 Support.- 2.2.1.3 ?-level Set.- 2.2.1.4 Normality.- 2.2.1.5 Convexity and Concavity.- 2.2.1.6 Extension Principle.- 2.2.1.7 Compatibility of Extension Principle with ?-cuts.- 2.2.1.8 Relation.- 2.2.1.9 Decomposability.- 2.2.1.10 Decomposition Theorem.- 2.2.1.11 Probability of Fuzzy Events.- 2.2.1.12 Conditional Fuzzy Sets.- 2.2.2 Basic Operations.- 2.2.2.1 Inclusion.- 2.2.2.2 Equality.- 2.2.2.3 Complementation.- 2.2.2.4 Intersection.- 2.2.2.5 Union.- 2.2.2.6 Algebraic Product.- 2.2.2.7 Algebraic Sum.- 2.2.2.8 Difference.- 2.3 Membership Functions.- 2.3.1 A Survey of Functional Forms.- 2.3.2 Examples to Generate Membership Functions.- 2.3.2.1 Distance Approach.- 2.3.2.2 True-Valued Approach.- 2.3.2.3 Payoff Function.- 2.3.2.4 Other Examples.- 2.4 Fuzzy Decision and Operators.- 2.4.1 Fuzzy Decision.- 2.4.2 Max-Min Operator.- 2.4.3 Compensatory Operators.- 2.4.3.1 Numerical Example for Operators.- 2.5 Fuzzy Arithmetic.- 2.5.1 Addition of Fuzzy Numbers.- 2.5.2 Subtraction of Fuzzy Numbers.- 2.5.3 Multiplication of Fuzzy Numbers.- 2.5.4 Division of Fuzzy Numbers.- 2.5.5 Triangular and Trapezoid Fuzzy Numbers.- 2.6 Fuzzy Ranking.- 3 Fuzzy Mathematical Programming.- 3.1 Fuzzy Linear Programming Models.- 3.1.1 Linear Programming Problem with Fuzzy Resources.- 3.1.1.1 Verdegay’s Approach.- 3.1.1.1a Example 1: The Knox Production-Mix Selection Problem.- 3.1.1.1b Example 2: A Transportation Problem.- 3.1.1.2 Werners’s Approach.- 3.1.1.2a Example 1: The Knox Production-Mix Selection Problem.- 3.1.1.2b Example 2: An Air Pollution Regulation Problem.- 3.1.2 Linear Programming Problem with Fuzzy Resources and Objective.- 3.1.2.1 Zimmermann’s Approach.- 3.1.2.1a Example 1: The Knox Production-Mix Selection Problem.- 3.1.2.1b Example 2: A Regional Resource Allocation Problem.- 3.1.2.1c Example 3: A Fuzzy Resource Allocation Problem.- 3.1.2.2 Chanas’s Approach.- 3.1.2.2a Example 1: An Optimal System Design Problem.- 3.1.2.2b Example 2: An Aggregate Production Planning Problem.- 3.1.3 Linear Programming Problem with Fuzzy Parameters in the Objective Function.- 3.1.4 Linear Programming with All Fuzzy Coefficients.- 3.1.4.1 Example: A Production Scheduling Problem.- 3.2 Interactive Fuzzy Linear Programming.- 3.2.1 Introduction.- 3.2.2 Discussion of Zimmermann’s, Werners’s Chanas’s and Verdegay’s Approaches.- 3.2.3 Interactive Fuzzy Linear Programming — I.- 3.2.3.1 Problem Setting.- 3.2.3.2 The Algorithm of IFLP-I.- 3.2.3.3 Example: The Knox Production-Mix Selection Problem.- 3.2.4 Interactive Fuzzy Linear Programming — II.- 3.2.4.1 The Algorithm of IFLP-II.- 3.3 Some Extensions of Fuzzy Linear Programming Problems.- 3.3.1 Membership Functions.- 3.3.1.1 Example: A Truck Fleet Problem.- 3.3.2 Operators.- 3.3.3 Sensitivity Analysis and Dual Theory.- 3.3.4 Fuzzy Non-Linear Programming.- 3.3.4.1 Example: A Fuzzy Machining Economics Problem.- 3.3.5 Fuzzy Integer Programming.- 3.3.5.1 Fuzzy 0–1 Linear Programming.- 3.3.5.1a Example: A Fuzzy Location Problem.- 4 Possibilistic Programming.- 4.1 Possibilistic Linear Programming Models.- 4.1.1 Linear Programming with Imprecise Resources and Technological Coefficients.- 4.1.1.1 Ramik and Rimanek’s Approach.- 4.1.1.1a Example: A Profit Apportionment Problem.- 4.1.1.2 Tanaka, Ichihashi and Asai’s Approach.- 4.1.1.3 Dubois’s Approach.- 4.1.2 Linear Programming with Imprecise Objective Coefficients.- 4.1.2.1 Lai and Hwang’s Approach.- 4.1.2.1a Example: A Winston-Salem Development Management Problem.- 4.1.2.2 Rommelfanger, Hanuscheck and Wolf’s Approach.- 4.1.2.3 Delgado, Verdegay and Vila’s Approach.- 4.1.3 Linear Programming with Imprecise Objective and Technological Coefficients.- 4.1.4 Linear Programming with Imprecise Coefficients.- 4.1.4.1 Lai and Hwang’s Approach.- 4.1.4.2 Buckley’s Approach.- 4.1.4.2a Example: A Feed Mix (Diet) Problem.- 4.1.4.3 Negi’s Approach.- 4.1.4.4 Fuller’s Approach.- 4.1.5 Other Problems.- 4.2 Some Extensions of Possibilistic Linear Programming.- 4.2.1 Linear Programming with Imprecise Coefficients and Fuzzy Inequalities.- 4.2.1a Example: A Fuzzy Matrix Game Problem.- 4.2.2 Linear Programming with Imprecise Objective Coefficients and Fuzzy Resources.- 4.2.2a Example: A Bank Hedging Decision Problem.- 4.2.3 Stochastic Possibilistic Linear Programming.- 4.2.3a Example: A Bank Hedging Decision Problem.- 5 Concluding Remarks.- 5.1 Probability Theory versus Fuzzy Set Theory.- 5.2 Stochastic versus Possibilistic Programming.- 5.3 Future Research.- 5.4 Introduction of the Following Volume.- 5.5 Fuzzy Multiple Attribute Decision Making.- Books, Monographs and Conference Proceedings.- Journal Articles, Technical Reports and Theses.