The concept of fuzzy sets of type 2 has been defined by L. A. Zadeh as an extension of ordinary fuzzy sets. The fuzzy set of type 2 can be characterized by a fuzzy membership function the grade (or fuzzy grade) of which is a fuzzy set in the unit interval [0, 1] rather than a point in [0, 1]. This paper investigates the algebraic structures of fuzzy grades under the operations of join ⊔, meet ⊔, and negation ┐ which are defined by using the extension principle, and shows that convex fuzzy grades form a commutative semiring and normal convex fuzzy grades form a distributive lattice under ⊔ and ⊓. Moreover, the algebraic properties of fuzzy grades under the operations and which are slightly different from ⊔ and ⊓, respectively, are briefly discussed.

Fuzzy Logic and Its Application to Approximate Reasoning.

Some properties of fuzzy sets of type 2

Technological and Economic Development of Economy

This research was partially funded by NSF grant No. IRI-9309230 and grants from Intergraph Corporation. Additional support from NSF for the NCGIA under No. SBR-9204141 is gratefully acknowledged. Max J. Egenhofer University of Maine, National Center for Geographic Information and Analysis and Department of Surveying Engineering, Department of Computer Science, University of Maine, Orono, ME 04469-5711, max@mecan1.maine.edu

methods

Categorizing Binary Topological Relations Between Regions, Lines, and Points in Geographic Databases

The main research activities in economics during the last five years have significantly increased. The main research fields are operation research and sustainable development. The philosophy of decision making in economics is to assess and select the most preferable solution, implement it and to gain the biggest profit. Preferences are used in a lot of problem situations both in individual and organizational decision making processes. A number of effective decision making methods that support decisions under conditions of multiple criteria have appeared in the last decade. This paper presents a panorama of decision making methods in economics and summarizes the most important results and applications over the last five years. This paper considers decision making in light of the recent developments of multiple criteria decision making methods (because classical methods are overviewed in a lot of earlier publications). Authors of different approaches, pioneering studies and works are presented in s...

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Multiple criteria decision making (MCDM) methods in economics: an overview

Most known methods for spatial reasoning translate a spatial problem into an analytical formulation in order to solve it quantitatively. This paper describes a method for formal, qualitative reasoning about distances and cardinal directions in geographic space. The problem addressed is how to deduce the distance and direction from point A to C, given the distance and direction from A to B and B to C. We use an algebraic approach, discussing the manipulation of distance and direction symbols (e.g. ‘N’, ‘E’, ‘S’ and ‘W’, or ‘Far’ and ‘Close’) and define two operations, composition and inverse , applied to them. After a review of other approaches, the desirable properties of deduction rules for distance and direction values are analyzed. This includes an algebraic specification of the ‘path’ image schema, from which most of the properties of distance and direction manipulation follow. Specific systems for composition of distance are explored. For directions, a formalization of the well-known triangular concept of directions (here called cone-shaped directions) and an alternative projection-based concept are explored. The algebraic approach leads to the completion of distance or direction symbols with an identity element, standing for the direction or distance from a point to itself. The so completed axiom system allows deductions, at least ‘Euclidean-approximate’, for any combination of input values.

Fuzzy Logic and Its Application to Approximate Reasoning.

Citations

Some properties of fuzzy sets of type 2

Technological and Economic Development of Economy

Categorizing Binary Topological Relations Between Regions, Lines, and Points in Geographic Databases

Multiple criteria decision making (MCDM) methods in economics: an overview

Qualitative spatial reasoning about distances and directions in geographic space

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