Abstract
The ordered fuzzy number (OFN) is determined as an ordered pair of fuzzy number (FN) and its orientation. FN is widely interpreted as imprecise number approximating real number. We interpret any OFN as an imprecise number equipped with additional information about the location of the approximated number. This additional information is given as orientation of OFN. The main goal of this paper is to determine the relation “greater than or equal to” on the space of all OFNs. This relation is unambiguously defined as an extension of analogous relations on the space of all FN. All properties of the introduced relation are investigated on the basis of the revised OFNs’ theory. It is shown here that this relation is a fuzzy one. The relations “greater than” and “equal to” also are considered. It is proven that the introduced relations are independent on the orientation of the compared OFNs. This result makes it easier to solve optimization tasks using OFNs.
Highlights
The concept of ordered fuzzy number (OFN) was intuitively introduced by Kosinski [1,2,3,4] as an extension of the notion of fuzzy number (FN) which is widely interpreted as imprecise approximation of real number
Since the notion of OFN is interpreted as an extension of the notion of FN, any formal model of order between OFNs should be consistent with the fixed ordering relation between FNs
In original Kosinki’s works “ordered fuzzy number” is defined without use of any ordering relation between FN. In each of these cases, “ordered fuzzy number” is defined as FN completed by orientation
Summary
The concept of ordered fuzzy number (OFN) was intuitively introduced by Kosinski [1,2,3,4] as an extension of the notion of fuzzy number (FN) which is widely interpreted as imprecise approximation of real number. If any alternatives are evaluated by OFNs their ranking leads to OFNs’ arrangement which is pre-given as an ordering relation “greater than or equal to” between OFNs. Since the notion of OFN is interpreted as an extension of the notion of FN, any formal model of order between OFNs should be consistent with the fixed ordering relation between FNs. Unlike in the case of real numbers, FNs have no natural order. It ntotthhhtaiaeottdnOOisoFFofNNrOisesFnaaNtrraeetaiaaonllndwwmaayryaisptshddimseefiefniitnntireceoddodpwuweciritetahdhtoo.iouuMtntthsouaurostseeneOoovOoFfefNFraa,NsnnsyosaymraooerrereadddlpeewirffrieaiennsyrgegesnnrrdteceeeleldafasittnbiinooeendtnSwebwbeceeteittitwnowhnFeoeeNue2nnt.s2uFa.FsNnINtedsisos.O.fIpIFnanoNnSiSnyseetcacoetrtdrieiodooenenxur2p2itn..l3h3ag,e,intrrtheeheeldeation between FNs thddhaieistsrooOerr.iFieIeNnnntstSaaetaticirotoeinnoanmmlw3aaaptphyiesissadiinunettftrhironoodedrduupccwreoedidvt.he.MsMotuohotrdareuetiosossvoevoermeoirref,e,nsasstonoaimmymtioepoenlrddedimifpefffrraeeioprnrpeeginensrcrciteenilsestasrbtboiaeeodrttnewuwfcbeeueeeldntfinw.lFlMFeeNNedonssrbeaFayonNnOvddserO.rOlo,IFnFsvNoNsSmksesycaea’trsrideoefiunefefxzxe2zppr.y3ella,anoiitcnrnhedeeesdedrbetween FNs and O dhhiosefeorrFereNi..eIn.InnItnaSStSeieoeccntctitioimoonnna3p34titthshheieenaataruuuottthdhhoouorrcreipnpdrtr.oroMovvdeeoussrcethethohedavartseetur.ss,coIosnhmmormSeeeelesacsittidmiimooifpnnpflele“3ergeptprnherrcoaeoetppseaeerubrrtettthhitieweaossnreaaeoprrnrreeoeFfvqfNuueullssfafiilaltllhtlneeoadd”t ObbsbeyoFytmNwOOeserrelalsonorivmevOssepkkFxylyNep’’slsspafwirfunuohzepzzidzecyyhrties are fulfilled b hoeoisrrdedc.eoernIrnosoiffsSFtFeeNcNnt.ti.oIIwnnniSSt3heectcOthtiireoolnonavu44sttkthhhyoee’rsaaupfuutrthzohzovoyrersioinnrttdthorreoarordtdd. The fuzzy set notion is applied for describing imprecise quantities
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