On the geometry of join and meet calculations for general type-2 fuzzy sets

Mendel, Jerry M.

doi:10.1109/fuzzy.2011.6007309

Cited by 7 publications

(4 citation statements)

References 13 publications

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“…For comparison between two IT2F numbers we have used their rank values. Centroid ranking method (Mendel, 2011) has been used to find the value of rank.…”

Section: Uncertainty Handling With It2fsmentioning

confidence: 99%

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Bi-objective covering salesman problem with uncertainty

Tripathy

2023

J. Decis. Anal. Int. Comp.

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Humanitarian relief transportation and mass fatality management activities are the most strenuous tasks after a natural or artificial disaster. A feasible and realistic transport model is essential for accomplishing the tasks in a planned way. Covering Salesman Problem (CSP) is a variant of Traveling Salesman Problem (TSP) which has been used in many application areas, including disaster management. In this paper, we consider a bi-objective CSP in an uncertain environment where Interval Type 2 fuzzy numbers represent the costs of the edges. A new local search technique is introduced in the memetic algorithm, which has been used to solve the problem. A computational experiment on a set of instances indicates the effectiveness of the introduced local search technique along with the proposed methodology.

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“…For comparison between two IT2F numbers we have used their rank values. Centroid ranking method (Mendel, 2011) has been used to find the value of rank.…”

Section: Uncertainty Handling With It2fsmentioning

confidence: 99%

“…The actual figures are compared next. The literature has a variety of ranking techniques (Lee and Chen, 2008;Mendel, 2011;Chiao, 2012). The centroid ranking method (Karnik and Mendel, 2001) is employed in this work to compare the IT2FSs.…”

Section: Uncertainty Handling With It2fsmentioning

confidence: 99%

Bi-objective covering salesman problem with uncertainty

Tripathy

2023

J. Decis. Anal. Int. Comp.

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“…from the field of computational geometry; yet convexity remained a necessary condition. More recent works [5], [6], [7] studied the geometrical properties of some GT2FSs to find closed formulas or approximations for the join and meet operations in some specific cases. However, to the best of the authors knowledge, considering arbitrary secondary grades, which can be non-convex, has not been addressed to date.…”

Section: Introductionmentioning

confidence: 99%

Join and Meet Operations for Type-2 Fuzzy Sets With Nonconvex Secondary Memberships

Ruiz¹,

Hagras²,

Pomares³

et al. 2016

IEEE Trans. Fuzzy Syst.

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Abstract-In this paper we will present two theorems for the join and meet operations for general type-2 fuzzy sets with arbitrary secondary memberships, which can be non-convex and/or non-normal type-1 fuzzy sets. These results will be used to derive the join and meet operations of the more general descriptions of interval type-2 fuzzy sets presented in [1], where the secondary grades can be non-convex. Hence, this work will help to explore the potential of type-2 fuzzy logic systems which use the general forms of interval type-2 fuzzy sets which are not equivalent to interval valued fuzzy sets. Several examples for both general type-2 and the more general forms of interval type-2 fuzzy sets are presented.Index Terms-type-2 fuzzy logic, general type-2 fuzzy logic, non-convex fuzzy sets.

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“…Many papers have been published about different kinds of computations for a GT2 FS (e.g., operations[9],[13]-[15],[17],[22],[37],[41],[44],[48],[51],[52],[55],[57],[60],[63]; centroid/type-reduction[23],[30],[31],[37],[48],[70],[73],[75],[77]; defuzzification[18],[29],[63]; solving fuzzy equations[61]; uncertainty measures[76]; and, similarity[19]). For such studies, the nature of the representation of a GT2 FS is usually not a vital issue; therefore, when we discard a particular representation in this paper, we are in no way disparaging those earlier works.…”

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confidence: 99%

General Type-2 Fuzzy Logic Systems Made Simple: A Tutorial

Mendel

2014

IEEE Trans. Fuzzy Syst.

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266

109

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The purpose of this tutorial paper is to make general type-2 fuzzy logic systems (GT2 FLSs) more accessible to fuzzy logic researchers and practitioners, and to expedite their research, designs, and use. To accomplish this, the paper 1) explains four different mathematical representations for general type-2 fuzzy sets (GT2 FSs); 2) demonstrates that for the optimal design of a GT2 FLS, one should use the vertical-slice representation of its GT2 FSs because it is the only one of the four mathematical representations that is parsimonious; 3) shows how to obtain set theoretic and other operations for GT2 FSs using type-1 (T1) FS mathematics (α − cuts play a central role); 4) reviews Mamdani and TSK interval type-2 (IT2) FLSs so that their mathematical operations can be easily used in a GT2 FLS; 5) provides all of the formulas that describe both Mamdani and TSK GT2 FLSs; 6) explains why center-of sets type-reduction should be favored for a GT2 FLS over centroid type-reduction; 7) provides three simplified GT2 FLSs (two are for Mamdani GT2 FLSs and one is for a TSK GT2 FLS), all of which bypass type reduction and are generalizations from their IT2 FLS counterparts to GT2 FLSs; 8) explains why gradient-based optimization should not be used to optimally design a GT2 FLS; 9) explains how derivative-free optimization algorithms can be used to optimally design a GT2 FLS; and 10) provides a three-step approach for optimally designing FLSs in a progressive manner, from T1 to IT2 to GT2, each of which uses a quantum particle swarm optimization algorithm, by virtue of which the performance for the IT2 FLS cannot be worse than that of the T1 FLS, and the performance for the GT2 FLS cannot be worse than that of the IT2 FLS.Index Terms-Alpha-plane, general type-2 fuzzy logic system, horizontal slice, interval type-2 fuzzy logic system, vertical slice, zSlice.

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On the geometry of join and meet calculations for general type-2 fuzzy sets

Cited by 7 publications

References 13 publications

Bi-objective covering salesman problem with uncertainty

Bi-objective covering salesman problem with uncertainty

Join and Meet Operations for Type-2 Fuzzy Sets With Nonconvex Secondary Memberships

General Type-2 Fuzzy Logic Systems Made Simple: A Tutorial

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