MQ-Radial Basis Functions Center Nodes Selection with PROMETHEE Technique

Document Type: بنیادی - نظری

Authors

1 Phd of Operation Research Management, Allameh Tabataba'i University and Assistant professor, Imam Ali University, Tehran, Iran

2 Department of Applied Mathematics, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran

10.30473/coam.2019.46609.1117

Abstract

In this paper‎, ‎we decide to select the best center nodes‎ ‎of radial basis functions by applying the Multiple Criteria Decision‎ ‎Making (MCDM) techniques‎. ‎Two methods based on radial basis‎ ‎functions to approximate the solution of partial differential‎ ‎equation by using collocation method are applied‎. ‎The first is based‎ ‎on the Kansa's approach‎, ‎and the second is based on the Hermite‎ ‎interpolation‎. ‎In addition‎, ‎by choosing five sets of center nodes‎: ‎Uniform grid‎, ‎Cartesian‎, ‎Chebyshev‎, ‎Legendre and‎ ‎Legendre-Gauss-Lobato (LGL) as alternatives and achieving the error‎, ‎the condition number of interpolation matrix and memory time as‎ ‎criteria‎, ‎rating of cases with the help of PROMETHEE technique is‎ ‎obtained‎. ‎In the end‎, ‎the best center nodes and method is selected‎ ‎according to the rankings‎. ‎This ranking shows that Hermite‎ ‎interpolation by using non-uniform nodes as center nodes is more‎ ‎suitable than Kansa's approach with each center node. 

Keywords


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‎G.~D'Avignon‎, ‎B.~Mareschal‎, ‎An application of the {PROMETHEE and GAIA} methods‎, ‎Math‎. ‎Comput‎. ‎Model‎. ‎12 (1989) 1393--1400‎.

 

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‎P.~D‎. ‎Bois‎, ‎J.~P‎. ‎Brans‎, ‎F.~C.~B‎. ‎Mareschal‎, ‎{MEDICIS}‎: ‎An expert system for‎ ‎computer-aided diagnosis using the {PROMETHEE} multicriteria method‎. ‎case‎ ‎study‎: ‎{Iron and steel making industry}‎, ‎Eur‎. ‎J‎. ‎Oper‎. ‎Res‎. ‎39 (1989)‎

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‎C.~Macharis‎, ‎J.~S.~K‎. ‎de~Brucker A.~Verbeke‎, ‎{PROMETHEE and AHP‎: ‎the design of‎ ‎operational synergies in multicriteria analysis‎, ‎Strengthening PROMETHEE with‎ ‎ideas of AHP}‎, ‎Eur‎. ‎J‎. ‎Oper‎. ‎Res‎. ‎153 (2004) 307--317‎.

 

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‎Z.~Wu‎, ‎Hermite-{B}irkhoff interpolation of scattered data by radial basis‎ ‎functions‎, ‎Approx‎. ‎Theory 8 (1992) 1--11‎.

 

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‎M.~Goumas‎, ‎V.~Lygerou‎, ‎An extension of the {PROMETHEE} method for decision‎ ‎making in fuzzy environment‎: ‎{R}anking of alternative energy exploitation‎ ‎projects‎, ‎Eur‎. ‎J‎. ‎Oper‎. ‎Res‎. ‎123 (2000) 606--613‎.

 

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