MQ-Radial Basis Functions Center Nodes Selection with PROMETHEE Technique

Document Type: بنیادی - نظری


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



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. 



‎R.~Franke‎, ‎Scattered data interpolation‎: ‎test of some methods‎, ‎Math‎. ‎Comput‎. ‎38‎ ‎(1982) 181--200‎.



‎E.~J‎. ‎Kansa‎, ‎Multiquadrics-{A} scattered data approximation scheme with‎ ‎applications to computational fluid-dynamics-{I} surface approximations and‎ ‎partial derivative estimates‎, ‎Comput‎. ‎Math‎. ‎Appl‎. ‎19 (1990) 127--145‎.



‎E.~J‎. ‎Kansa‎, ‎{Multiquadrics-A scattered data approximation scheme with‎ ‎applications to computational fluid-dynamics-II‎. ‎Solutions to parabolic‎, ‎hyperbolic and elliptic partial differential equations}‎, ‎Comput‎. ‎Math‎. ‎Appl‎. ‎19 (1990) 147--161‎.



‎M.~Sharan‎, ‎E.~J‎. ‎Kansa‎, ‎S.~Gupta‎, ‎Application of the multiquadric method for‎ ‎numerical solution of elliptic partial differential equations‎, ‎Appl‎. ‎Math‎. ‎Comput‎. ‎84 (1997) 275--302‎.



‎H.~P‎. ‎M.~Zerroukat‎, ‎C.~Chen‎, ‎A numerical method for heat transfer problems‎ ‎using collocation and radial basis functions‎, ‎Int‎. ‎J‎. ‎Numer‎. ‎Meth‎. ‎Eng‎. ‎42‎ ‎(1998) 1263--1278‎.



‎N.~Mai-Duy‎, ‎T.~Tran-Cong‎, ‎Numerical solution of differential equations using‎ ‎multiquadric radial basis function networks‎, ‎Neural Networks 14 (2001)‎ ‎185--199‎.



‎M.~Tatari‎, ‎M.~Dehghan‎, ‎A method for solving partial differential equations via‎ ‎radial basis functions‎: ‎{A}pplication to the heat equation‎, ‎Eng‎. ‎Anal‎. ‎Bound‎. ‎Elem‎. ‎34 (2010) 206--212‎.



‎M.~Dehghan‎, ‎A.~Shokri‎, ‎{Numerical solution of the nonlinear Klein-Gordon‎ ‎equation using radial basis functions}‎, ‎J‎. ‎Comput‎. ‎Appl‎. ‎Math‎. ‎230 (2009)‎ ‎400--410‎.



‎M.~Mai-Duy‎, ‎T.~Tran-Cong‎, ‎An integrated-{RBF} technique based on {G}alerkin‎ ‎formulation for elliptic differential equations‎, ‎Eng‎. ‎Anal‎. ‎Bound‎. ‎Elem‎. ‎33‎ ‎(2009) 191--199‎.



‎N.~Mai-Duy‎, ‎Solving high order ordinary different al equations with radial‎ ‎basis function networks‎, ‎Int‎. ‎J‎. ‎Numer‎. ‎Meth‎. ‎Eng‎. ‎62 (2005) 824--852‎. 


‎M.~Dehghan‎, ‎M.~Tatari‎, ‎Use of radial basis functions for solving the‎ ‎second-order parabolic equation with nonlocal boundary conditions‎, ‎Numer‎. ‎Meth‎. ‎Part‎. ‎D‎. ‎E‎. ‎24 (2008) 924--938‎.



‎S.~Kazem‎, ‎J.~A‎. ‎Rad‎, ‎Radial basis functions method for solving of a non-local‎ ‎boundary value problem with neumann's boundary conditions‎, ‎Appl‎. ‎Math‎. ‎Modell‎. ‎36 (2012) 2360--2369‎.



‎S.~Kazem‎, ‎J.~A‎. ‎Rad‎, ‎K.~Parand‎, ‎Radial basis functions methods for solving‎ ‎{F}okker-{P}lanck equation‎, ‎Eng‎. ‎Anal‎. ‎Bound‎. ‎Elem‎. ‎36 (2012) 181--189‎.  


‎J.~A‎. ‎Rad‎, ‎S.~Kazem‎, ‎K.~Parand‎, ‎Radial basis functions approach on optimal‎ ‎control problems‎: ‎a numerical investigation‎, ‎Journal of Vibration and Control‎ ‎20~(9) (2014) 1394--1416‎.



‎J.~A‎. ‎Rad‎, ‎K.~Parand‎, ‎S.~Kazem‎, ‎A numerical investigation to viscous flow over‎ ‎nonlinearly stretching sheet with chemical reaction‎, ‎heat transfer and‎ ‎magnetic field‎, ‎International Journal of Applied and Computational‎ ‎Mathematics 3~(2) (2017) 919--935‎.



‎S.~Kazem‎, ‎J.~Rad‎, ‎K.~Parand‎, ‎M.~Shaban‎, ‎H.~Saberi‎, ‎{The numerical study on the‎ ‎unsteady flow of gas in a semi-infinite porous medium using an RBF‎ ‎collocation method}‎, ‎International Journal of Computer Mathematics 89~(16)‎ ‎(2012) 2240--2258‎.



‎K.~Parand‎, ‎S.~Abbasbandy‎, ‎S.~Kazem‎, ‎J.~A‎. ‎Rad‎, ‎A novel application of radial‎ ‎basis functions for solving a model of first-order integro-ordinary‎ ‎differential equation‎, ‎Commun‎. ‎Nonlinear Sci‎. ‎Numer‎. ‎Simulat‎. ‎16 (2011)‎ ‎4250--4258‎.



‎S.~Kazem‎, ‎J.~A‎. ‎Rad‎, ‎K.~Parand‎, ‎{A meshless method on non-Fickian flows with‎ ‎mixing length growth in porous media based on radial basis functions‎: ‎A‎ ‎comparative study}‎, ‎Comput‎. ‎Math‎. ‎Appl‎. ‎64 (2012) 399--412‎.



‎J.~A‎. ‎Rad‎, ‎S.~Kazem‎, ‎K.~Parand‎, ‎{A numerical solution of the nonlinear‎ ‎controlled duffing oscillator by radial basis functions}‎, ‎Comput‎. ‎Math‎. ‎Appl‎. ‎64 (2012) 2049--2065‎.



‎M.~Dehghan‎, ‎A.~Ghesmati‎, ‎Solution of the second-order one-dimensional‎ ‎hyperbolic telegraph equation by using the dual reciprocity boundary integral‎ ‎equation ({DRBIE}) method‎, ‎Eng‎. ‎Anal‎. ‎Bound‎. ‎Elem‎. ‎34 (2010) 51--59‎.



‎N.~Mai-Duy‎, ‎D.~Ho-Minh‎, ‎T.~Tran-Cong‎, ‎A galerkin approach incorporating‎ ‎integrated radial basis function networks for the solution of 2d biharmonic‎ ‎equations‎, ‎Int‎. ‎J‎. ‎Com‎. ‎Math‎. ‎86 (2009) 1746--1759‎.



‎K.~Parand‎, ‎S.~Abbasbandy‎, ‎S.~Kazem‎, ‎A.~Rezaei‎, ‎Comparison between two common‎ ‎collocation approaches based on radial basis functions for the case of heat‎ ‎transfer equations arising in porous medium‎, ‎Commun‎. ‎Nonlinear‎. ‎Sci‎. ‎Numer‎. ‎Simul‎. ‎16 (2011) 1396--1407‎.



‎S.~Kazem‎, ‎F.~Hadinejad‎, ‎{PROMETHEE technique to select the best radial basis‎ ‎functions for solving the 2-dimensional heat equations based on Hermite‎ ‎interpolation}‎, ‎Eng‎. ‎Anal‎. ‎Bound‎. ‎Elem‎. ‎50 (2015) 29--38‎.



‎S.~Kazem‎, ‎A.~Hatam‎, ‎{A modification on strictly positive definite RBF-DQ method‎ ‎based on matrix decomposition}‎, ‎Engineering Analysis with Boundary Elements‎ ‎76 (2017) 90--98‎.



‎S.~Kazem‎, ‎E.~A‎. ‎Chadwick‎, ‎A.~Hatam‎, ‎Making conditionally negative definite‎ ‎radial basis function interpolation well-conditioned by adding cardinal basis‎ ‎functions‎, ‎Ain Shams Engineering Journal‎.



‎R.~Hardy‎, ‎Multiquadric equations of topography and other irregular surfaces‎, ‎J‎. ‎Geophys‎. ‎Res‎. ‎176 (1971) 1905--1915‎.



‎W.~R‎. ‎Madych‎, ‎S.~A‎. ‎Nelson‎, ‎Multivariate interpolation and conditionally‎ ‎positive definite functions‎, ‎Approx‎. ‎Theory Appl‎. ‎4 (1988) 77--89‎.



‎M.~D‎. ‎Buhmann‎, ‎Radial basis functions‎, ‎Acta Numerica (2000) 1--38‎.



‎M.~D‎. ‎Buhmann‎, ‎Radial Basis Functions‎: ‎Theory and Implementations‎, ‎Cambridge‎ ‎University Press‎, ‎New York‎, ‎2004‎.



‎S.~Rippa‎, ‎An algorithm for selecting a good parameter c in radial basis‎ ‎function interpolation‎, ‎Advan‎. ‎Comp‎. ‎Math‎. ‎11 (1999) 193--210‎.



‎A.~H.~D‎. ‎Cheng‎, ‎M.~A‎. ‎Golberg‎, ‎E.~J‎. ‎Kansa‎, ‎Q.~Zammito‎, ‎Exponential convergence‎ ‎and {H}-c multiquadric collocation method for partial differential equations‎, ‎Numer‎. ‎Meth‎. ‎Part‎. ‎D‎. ‎E‎. ‎19 (2003) 571--594‎.



‎R.~E‎. ‎Carlson‎, ‎T.~A‎. ‎Foley‎, ‎The parameter ${R}^2$ in multiquadric‎ ‎interpolation‎, ‎Comput‎. ‎Math‎. ‎Appl‎. ‎21 (1991) 29--42‎.



‎A.~E‎. ‎Tarwater‎, ‎A parameter study of Hardy's multiquadric method for scattered‎ ‎data interpolation‎, ‎Report UCRL-53670‎, ‎Lawrence Livermore National‎ ‎Laboratory‎, ‎1985‎.



‎G.~Fasshauer‎, ‎J.~Zhang‎, ‎On choosing ''optimal'' shape parameters for {RBF}‎ ‎approximation‎, ‎Numer‎. ‎Algorithms 45 (2007) 346--368‎.



‎H.~Wendland‎, ‎Scattered Data Approximation‎, ‎Cambridge University Press‎, ‎New‎ ‎York‎, ‎2005‎.



‎Y.~Zhang‎, ‎Z.~P‎. ‎Fan‎, ‎Y.~Liu‎, ‎A method based on stochastic dominance degrees for‎ ‎stochastic multiple criteria decision making‎, ‎Comput‎. ‎Ind‎. ‎Eng‎. ‎58 (2010)‎ ‎544--552‎.



‎C.~B‎. ‎Cheng‎, ‎Solving a sealed-bid reverse auction problem by multiple criterion‎ ‎decision-making methods‎, ‎Comput‎. ‎Math‎. ‎Appl‎. ‎56 (2008) 3261--3274‎.



‎T.~H‎. ‎Chang‎, ‎T.~C‎. ‎Wang‎, ‎Using the fuzzy multi-criteria decision-making‎ ‎approach for measuring the possibility of successful knowledge management‎, ‎Inform‎. ‎Sciences‎. ‎179 (2009) 355--370‎.



‎V.~F‎. ‎Yu‎, ‎K.~J‎. ‎Hu‎, ‎An integrated fuzzy multi-criteria approach for the‎ ‎performance evaluation of multiple manufacturing plants‎, ‎Comput‎. ‎Ind‎. ‎Eng‎. ‎58‎ ‎(2010) 269--277‎.



‎R.~U‎. ‎Bilsel‎, ‎G.~B$ddot{u}$y$ddot{u}$k$ddot{o}$zkan‎, ‎D.~Ruan‎, ‎A fuzzy‎ ‎preference-ranking model for a quality evaluation of hospital web sites‎, ‎Int‎. ‎J‎. ‎Intell‎. ‎Syst‎. ‎21 (2006) 1181--1197‎.



‎D.~Bouyssou‎, ‎remarks on the notion of compensation in {MCDM}‎, ‎Eur‎. ‎J‎. ‎Oper‎. ‎Res‎. ‎26 (1986) 150--160‎.



‎T.~Gal‎, ‎T.~Hanne‎, ‎Nonessential objectives within a network approach for {MCDM}‎, ‎Eur‎. ‎J‎. ‎Oper‎. ‎Res‎. ‎168 (2006) 584--592‎.



‎R.~Narasimhan‎, ‎S.~K‎. ‎Vickery‎, ‎An experimental evaluation of articulation of‎ ‎preferences in multiple criterion decision-making {(MCDM)} methods‎, ‎Decision‎ ‎Sci‎. ‎19 (1988) 880--888‎.



‎H.~J‎. ‎Shyur‎, ‎H.~Shih‎, ‎A hybrid {MCDM} model for strategic vendor selection‎, ‎Math‎. ‎Comput‎. ‎Modell‎. ‎44 (2006) 749--761‎.



‎S.~Wadhwa‎, ‎J.~Madaan‎, ‎F.~T.~S‎. ‎Chan‎, ‎Flexible decision modeling of reverse ‎logistics system‎: ‎{A} value adding {MCDM} approach for alternative selection‎, ‎Robot‎. ‎Cim-Int‎. ‎Manuf‎. ‎25 (2009) 460--469‎.



‎J.~Ignatiusa‎, ‎S.~M.~H‎. ‎Motlagh‎, ‎M.~M‎. ‎Sepehri‎, ‎M.~Behzadian‎, ‎A.~Mustafa‎, ‎Hybrid‎ ‎models in decision making under uncertainty‎: ‎{T}he case of training provider‎ ‎evaluation‎, ‎J‎. ‎Intell‎. ‎Fuzzy‎. ‎Syst‎. ‎20 (2009) 1--16‎.



‎F.~Mirzazadeh‎, ‎F.~Hadinejad‎, ‎N.~A‎. ‎Roshan‎, ‎Investigating utility level of waste‎ ‎disposal methods using multicriteria decision-making techniques (case study‎: ‎Mazandaran-iran)‎, ‎Journal of Material Cycles and Waste Management 20~(1)‎

‎(2018) 505--515‎.



‎F.~Hadinejad‎, ‎M.~Ghasemi‎, ‎M.~Ahmadi‎, ‎Proposing a new model for the performance‎ ‎assessment of students through utilizing 360-degree feedback (case study‎: ‎Performance assessment of the students of Imam Ali military university)‎, ‎Military Management 16~(63) (2016) 1--33‎.



‎F.~Mirzazadeh‎, ‎F.~Hadinejad‎, ‎{Applying fuzzy AHP technique in waste disposal‎ ‎method selection (Case study‎: ‎Mazandaran province)‎, ‎waste management‎, ‎A‎ ‎glance at the world II‎, ‎38} (2015)‎.



‎M.~Moradian‎, ‎F.~Hadiynejad‎, ‎A.~Poormanafi‎, ‎Providing a model for assessing and‎ ‎analyzing the military power of countries‎, ‎Defence Studies 16~(4) (2018)‎ ‎169--200‎.



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



‎J.~Geldermann‎, ‎T.~Spengler‎, ‎O.~Rentz‎, ‎Fuzzy outranking for environmental‎ ‎assessment‎, ‎Fuzzy‎. ‎Set‎. ‎Syst‎. ‎115 (2000) 45--65‎.



‎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.



‎M.~Behzadian‎, ‎R.~B‎. ‎Kazemzadeh‎, ‎A.~Albadvi‎, ‎M.~Aghdasi‎, ‎{PROMETHEE‎: ‎A}‎ ‎comprehensive literature review on methodologies and applications‎, ‎Eur‎. ‎J‎. ‎Oper‎. ‎Res‎. ‎200 (2010) 198--215‎.



‎S.~Balli‎, ‎B.~Karasulu‎, ‎S.~Korukoglu‎, ‎En uygun otomobil se�imi problemi i�in‎ ‎bir bulanik {PROMETHE} y$ddot{o}$ntemi uygulamasi‎, ‎DE.$ddot{U}$ IIBF‎ ‎Dergisi 22 (2007) 139--147‎.



‎E.~Georgopoulou‎, ‎Y.~Sarafidis‎, ‎D.~Diakoulaki‎, ‎Design and implementation of a‎ ‎group {DSS} for sustaining renewable energies exploitation‎, ‎Eur‎. ‎J‎. ‎Oper‎. ‎Res‎. ‎109 (1998) 483--500‎.



‎J.~G‎. ‎Iniestra‎, ‎J.~G‎. ‎Guti�rrez‎, ‎Multicriteria decisions on interdependent‎ ‎infrastructure transportation projects using an evolutionary-based framework‎, ‎Appl‎. ‎Soft Comput‎. ‎9 (2009) 512--552‎.



‎H.~S‎. ‎Mohamadabadi‎, ‎G.~Tichkowsky‎, ‎A.~Kumar‎, ‎Development of a multicriteria‎ ‎assessment model for ranking of renewable and non-renewable transportation‎ ‎fuel vehicles‎, ‎Energy 34 (2009) 112--125‎.



‎T.~Briggs‎, ‎P.~L‎. ‎Kunsch‎, ‎B.~Mareschal‎, ‎Nuclear waste management‎: ‎{A}n‎ ‎application of the multicriteria {PROMETHEE} methods‎, ‎Eur‎. ‎J‎. ‎Oper‎. ‎Res‎. ‎44‎ ‎(1990) 1--10‎.



‎W.~C‎. ‎Chou‎, ‎W.~T‎. ‎Lin‎, ‎C.~Lin‎, ‎Application of fuzzy theory and {PROMETHEE}‎ ‎technique to evaluate suitable ecotechnology method‎: ‎{A} case study in‎ ‎{Shihmen Reservoir Watershed‎, ‎Taiwan}‎, ‎Ecol‎. ‎Eng‎. ‎31 (2007) 269--280‎.



‎A.~Albadvi‎, ‎S.~K‎. ‎Chaharsooghi‎, ‎A.~Esfahanipour‎, ‎Decision making in stock‎ ‎trading‎: ‎{An application of PROMETHEE}‎, ‎Eur‎. ‎J‎. ‎Oper‎. ‎Res‎. ‎177 (2007)‎ ‎673--683‎.



‎G.~Baourakis‎, ‎M.~Doumpos‎, ‎N.~Kalogeras‎, ‎C.~Zopounidis‎, ‎Multicriteria analysis‎ ‎and assessment of financial viability of agribusinesses‎: ‎The case of‎ ‎marketing co-operatives and juice-producing companies‎, ‎Agribusiness 18 (2002)‎ ‎543--558‎.



‎G.~Zhang‎, ‎Y.~Ni‎, ‎J.~Churchill‎, ‎S.~Kokot‎, ‎Authentication of vegetable oils on‎ ‎the basis of their physico-chemical properties with the aid of chemometrics‎, ‎Talanta 70 (2006) 293--300‎.



‎B.~Rekiek‎, ‎P.~de~Lit‎, ‎A.~Delchambre‎, ‎Hybrid assembly line design and user's‎ ‎preferences‎, ‎Int‎. ‎J‎. ‎Prod‎. ‎Res‎. ‎40 (2002) 1095--1111‎.



‎F.~Hadinejad‎, ‎A.~Ramzi‎, ‎M.~Khajeh‎, ‎Evaluation and comparison of multi-objective‎ ‎meta-innovative algorithms for solving the problem of optimization of the‎ ‎production design (case study‎: ‎Optimum design of military equipment)‎, ‎Military Management 17~(68) (2018) 128--163‎.



‎P.~Azimi‎, ‎F.~Hadinejad‎, ‎A multi objective optimization model for redundancy‎ ‎allocation problem in reparable systems using multi criteria decision making‎, ‎industrial management studies 14 (2016) 137--162‎.



‎K.~Hyde‎, ‎H.~Maier‎, ‎C.~Colby‎, ‎Incorporating uncertainty in the {PROMETHEE MCDA}‎ ‎method‎, ‎J‎. ‎Multi-Criteria Dec‎. ‎Anal‎. ‎12 (2003) 245--259‎.



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



‎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)‎




‎D.~L‎. ‎Olson‎, ‎Comparison of three multicriteria methods to predict known‎ ‎outcomes‎, ‎Eur‎. ‎J‎. ‎Oper‎. ‎Res‎. ‎130 (2001) 576--587‎.  


‎M.~Amiri‎, ‎F.~Hadinejad‎, ‎S.~a‎. Malekkhoyan‎, ‎Evaluation and prioritization of‎ ‎suppliers adopting a combined approach of entropy‎, ‎analytic hierarchy‎ ‎process‎, ‎and revised Prometheus (case study‎: ‎Youtab company)‎, ‎Journal of‎

‎Operational Research and Its Applications 14 (2017) 1--20‎.



‎F.~Hadinejad‎, ‎Designing a model for evaluation of information technology‎ ‎projects contractors‎, ‎Resource management in police 17 (2017) 236--264‎.  


‎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.



‎M.~A‎. ‎Golberg‎, ‎{Some recent results and proposals for the use of radial basis‎ ‎functions in the BEM}‎, ‎Eng‎. ‎Anal‎. ‎Bound‎. ‎Elem‎. ‎23 (1999) 285--296‎.



‎S.~Abbasbandi‎, ‎K.~Parand‎, ‎S.~Kazem‎, ‎A.~Sanaei~Kia‎, ‎A numerical approach on‎ ‎hiemenz flow problem using radial basis functions‎, ‎International Journal of‎ ‎Industrial Mathematics 5~(1) (2013) 65--73‎.



‎G.~E‎. ‎Fasshauer‎, ‎Solving partial differential equations by collocation with‎ ‎radial basis functions‎, ‎Vanderbilt University Press‎, ‎Nashville‎, ‎1997‎.



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



‎Z.~M‎. ‎Wu‎, ‎Radial basis function scattered data interpolation and the meshless‎ ‎method of numerical solution of {PDE}s‎, ‎J‎. ‎Eng‎. ‎Math‎. ‎19 (2002) 1--12‎.



‎R.~Schaback‎, ‎C.~Franke‎, ‎Covergence order estimates of meshless collocation‎ ‎methods using radial basis functions‎, ‎Adv‎. ‎Comput‎. ‎Math‎. ‎8 (1998) 381--399‎.



‎H.~Power‎, ‎V.~Barraco‎, ‎A comparison analysis between unsymmetric and symmetric‎ ‎radial basis function {C}ollocation methods for the numerical solution of‎ ‎partial differential equations‎, ‎Comput‎. ‎Math‎. ‎Appl‎. ‎43 (2002) 551--583‎.



‎J.~P‎. ‎Brans‎, ‎B.~H‎. ‎Vincke‎, ‎B.~Mareschal‎, ‎How to select and how to rank‎ ‎projects‎: ‎the {PROMETHEE} method‎, ‎Eur‎. ‎J‎. ‎Oper‎. ‎Res‎. ‎24 (1986) 228--238‎.  


‎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.



‎C.~Chareonsuk‎, ‎N.~Nagarur‎, ‎M.~T‎. ‎Tabucanon‎, ‎A multicriteria approach to the‎ ‎selection of preventive maintenance intervals‎, ‎Eur‎. ‎J‎. ‎Oper‎. ‎Res‎. ‎49 (1997)‎ ‎55--64‎.



‎M.~D‎. ‎deviren‎, ‎Decision making in equipment selection‎: ‎{An integrated approach‎ ‎with AHP and PROMETHEE}‎, ‎J‎. ‎Intell‎. ‎Manuf‎. ‎19 (2008) 397--406‎.



‎J.~P‎. ‎Brans‎, ‎B.~Mareschal‎, ‎How to decide with {PROMETHEE}‎, ‎J‎. ‎Intell‎. ‎Manuf‎. ‎19‎ ‎(1998) 397--406‎.


‎bibitem{a3Brans2}‎ ‎J.~P‎. ‎Brans‎, ‎P.~H‎. ‎Vincke‎, ‎A preference ranking organization method‎, ‎Manage‎. ‎Sci‎. ‎31 (1985) 647--656‎.