Information Sciences

(1)

Information Sciences 346–347 (2016) 302–317

ContentslistsavailableatScienceDirect

Information Sciences

journalhomepage:www.elsevier.com/locate/ins

A new method to rank fuzzy numbers using Dempster–Shafer theory with fuzzy targets

Kok Chin Chai

^a^,^∗

, Kai Meng Tay

^a

, Chee Peng Lim

^b

aUniversiti Malaysia Sarawak, Kota Samarahan, Sarawak, Malaysia

bCentre for Intelligent Systems Research, Deakin University, Australia

a rt i c l e i n f o

Article history:

Received 17 December 2014 Revised 1 October 2015 Accepted 2 January 2016 Available online 2 February 2016 Keywords:

Dempster–Shafer theory Fuzzy ranking Fuzzy targets

Murphy’s combination rule Transferable Belief Model

a b s t ra c t

In thispaper, an extendedranking methodfor fuzzynumbers,which isasynthesis of fuzzytargetsand theDempster–ShaferTheory(DST)ofevidence,isdevised.Theuseof fuzzytargetstoreflecthumanviewpointsinfuzzyrankingisnotnew.However,different fuzzytargetscanleadtocontradictoryfuzzyrankingresults;makingitdifficulttoreacha finaldecision.Inthispaper,theresultsfromdifferentviewpointsaretreatedasdifferent sourcesofevidence,andMurphy’scombinationruleisusedtoaggregatethefuzzyrank- ingresults.DSTallowsfuzzynumberstobecomparedandrankedwhilepreservingtheir uncertainandimprecisecharacteristics.Inaddition,ahybridmethodconsistingoffuzzy targetsandDSTwiththeTransferableBeliefModelisformulated,whichfulfilsanumber ofimportantorderingproperties.Aseriesofempiricalexperimentswithbenchmarkex- ampleshasbeenconductedandtheexperimentalresultsclearlyindicatetheusefulnessof theproposedmethod.

1. Introduction

Fuzzyrankingisaprocedureusedtocompareandorderasequenceoffuzzynumbers.Ithasbeenemployedinavariety ofapplicationdomains,such asdecisionmaking[4,6,7,12,15,39,40],riskassessment [29,30],andartiﬁcialintelligence[17]. Whilemanyfuzzyrankingmethodsareavailableintheliterature[1–9,12–16,21,23,37,39],agenericmethodthatcanprovide a satisfactory solution acrossa variety ofsituations has yetbeen developed[12]. Indeed,fuzzyranking is challengingas fuzzy numbers are represented by possibility distributions which can overlap with one another [15]. It is important to consider the overall possibility distribution of a fuzzy number, instead of converting a fuzzy number into a singlereal number (e.g., centroid-based method [9], minimizing andmaximizing set-based method [5], distance-based method [1], area-basedmethod[4,8]),asreducing theentireanalysisintoonenumbercan resultinthelossofinformationwhichcan affecttheaccuracyofcertaincalculations[11].

Anumberoffuzzyrankingmethodspertainingtotheprincipleofapossibilitydistribution[3,6,12,14,15,23,39]havebeen proposedintheliterature.Thesemethodscanbecategorisedaspossibilitydistribution-basedmethodswith:(i)noreference set[23];(ii)asinglereferenceset[14];(iii)multiplereferencesets[3,6,12,15,39].Thefocusofthispaperisonasynthesis ofpossibilitydistributionsandmultiplereferencesets(alsoknownasfuzzytargets[3,6,12,15,39]),whichallowsmorethan twofuzzynumberstoberankedsimultaneously.Fuzzytargets(i.e.,optimistic,neutral,andpessimistic)areadoptedtore- ﬂecthumanviewpointsonfuzzyranking[3,6,12,15,39].As anexample,aranking methodbasedonasatisfactionfunction

∗ Corresponding author. Tel.: + 60 146884403.

E-mail addresses: [email protected] (K.C. Chai), [email protected] (K.M. Tay), [email protected] (C.P. Lim).