Probabilistic linguistic TODIM approach for multiple attribute decision-making

Transcription

1 Graul. Cmput. (07) : 4 DOI 0.007/s ORIGINAL PAPER Prbabilistic liguistic TODIM apprach fr multiple attribute decisi-makig Peide Liu Xili Yu Received: 9 April 07 / Accepted: 5 July 07 / Published lie: 0 August 07 Ó Spriger Iteratial Publishig AG 07 Abstract Prbabilistic liguistic term sets (PLTSs) are a effective tl t express prefereces with differet weights fr differet liguistic terms, ad the TODIM methd is based prspect thery ad culd csider the decisi maker s cgitive behavir. I this paper, we exted the TODIM t slve the multi-attribute decisi-makig (MADM) prblems with PLTSs. First f all, the defiiti, peratis, cmparative methd, ad deviati degrees f PLTSs are itrduced; a ew stadardizati methd fr the attribute values was prpsed with respect t the situati i which the prbabilistic sum fr all liguistic terms is less tha. The, the bjective weights fr criteria ca be btaied by ifrmati etrpy thery ad the steps f the exteded TODIM methd fr PLTSs are prpsed. Fially, a example is t verify the develped apprach. Keywrds MADM PLTSs TODIM methd Itrducti I actual decisi-makig, there are a large umber f qualitative criteria which are hardly evaluated by accurate umerical values. Hece, hw t make the assessmet f alteratives precisely is pivtal. Hwever, fr the cmplex decisi-makig prblems, decisi makers usually prvide their piis by atural laguage, such as gd, fair, pr, ad ther similar liguistic terms (LTs) & Peide Liu Peide.liu@gmail.cm Schl f Maagemet Sciece ad Egieerig, Shadg Uiversity f Fiace ad Ecmics, Jia 5004, Shadg, Chia (Wu ad Xu 06). Nw, decisi-makig based LTs has becme a imprtat research aspect i the field f decisi aalysis (Xu ad Wag 06). Fr example, Xu (007) gave the gal prgrammig methd with LTs fr MADM prblems. Xu ad Wag (07) slved grup decisi-makig (GDM) prblem with multi-graularity liguistic mdel. Medel (06) prpsed three appraches t sythesizig a iterval type- fuzzy set mdel f LTs. I the traditial decisi-makig methds based LTs, the DMs ca express their prefereces ly by e LT. Hwever, smetimes, it is difficult t depict cmplex qualitative ifrmati ly by e LT. Fr istace, a DM may thik that it may be very gd, gd, r a little gd fr e bject, but he/she is t sure hw gd it is. I this situati, Rdriguez et al. (0) prpsed hesitat fuzzy LT sets (HFLTSs), which have several pssible LTs. The, Beg ad Rashid (0) ad Wei et al. (04) prpsed sme aggregati peratrs based HFLTSs. Zhu ad Xu (04) prpsed sme preferece relatis (HFLPRs) fr HFLTSs. Furthermre, Beg ad Rashid (0) prpsed a exteded TOPSIS methd fr the HFLTSs. Dg et al. (05) ad Rdríguez et al. (0) prpsed sme GDM methd with HFLTSs. Wag (05) prpsed the exteded HFLTSs (EHFLTSs) fr the ctiuus LTs. Liu ad Rdriguez (04) further develped the fuzzy evelpes f HFLTSs ad applied them t MADM prblems. Hwever, all pssible LTs give by the DMs i HFLTSs have the same imprtace. Obviusly, this is t realistic. I real decisi-makig, the DMs may prefer sme pssible LTs ad give sme differet imprtace degrees. I ther wrds, we ca give sme pssible LTs ad the give their imprtace degrees fr evaluatig a bject (Liu ad Rdríguez 04). This imprtace degree ca be regarded as prbabilistic distributi (Wu ad Xu 06), belief

2 4 Graul. Cmput. (07) : 4 degree (Yag 00; Yag ad Xu 00), ad s. T better depict such a situati, Pag et al. (06) prpsed the prbabilistic LT sets (PLTSs) which ca express differet imprtace degrees r weights f all the pssible LTs. Obviusly, PLTSs have the flexibility ad richess i expressig cmplex fuzzy liguistic ifrmati. T d a reasable ad feasible decisi-makig, the decisi-makig methds are w essetial ad a lt f effrts have bee made i the past few decades, such as the TOPSIS (Liu 009), the VIKOR (Chatterjee ad Samarjit 07), the GRA (Liu ad Liu 00), the PRO- METHEE (Liu ad Gua 009), the ELECTRE (Liu ad Zhag 0; Ry ad Bertier 97),ads.Nw, these methds have bee exteded fr differet attribute values, such as fuzzy umbers, LTs, ad s. Hwever, because the MADM prblems have becmig icreasigly cmplicated ver the years, there is a bvius shrtcmig i the existig methds, which suppses that the DMs are cmpletely ratial. Nw, may excellet researches ivlvig behavir experimets (Kahema ad Tversky 979; Tversky ad Kahema 99) have shw that the DMs are buded ratial i decisimakig prcess, ad the psychlgy ad behavir f the DMs are imprtat factrs which ca bviusly ifluece decisi results. Hece, Gmes ad Lima first preseted the TODIM i 99 (Gmes ad Lima 99), which is a valuable MADM methd csiderig the DMs psychlgy ad behavir based prspect thery (PT) (Kahema ad Tversky 979), ad it has bee applied t sme decisi-makig prblems (Pag et al. 06). Mrever, its sme ew extesis gave bee develped. Such as Lurezutti ad Krhlig (0) preseted a IF- RTODIM fr ituitiistic fuzzy umbers (IFNs). Wag et al. (06) prpsed a likelihd fucti f HFLTSs embedded it TODIM. Re et al. (06) develped a Pythagrea fuzzy TODIM apprach t MADM prblems. Obviusly, the TODIM methd is a valuable ad imprtat MADM tl csiderig the DMs psychlgy ad behavir, ad its extesis are als very effective i slvig the MADM prblems fr differet fuzzy evirmets, but there is research abut the TODIM methd fr PLTSs i the existig researches. Csequetly, csiderig the DMs psychlgy ad behavir, it will be a valuable research tpic abut hw t slve the MADM prblem with PLTSs. I additi, due t the icreasig cmplexity f the decisi-makig evirmet, it is usually difficult fr DMs t give the weight evaluati ifrmati cmpletely. Therefre, it is very ecessary t develp a methd t btai the bjective weight f each attribute ad the t prpse a exteded TODIM fr PLTSs, s that the mre cmprehesive ad reasable decisi ca be made i a itricate situati. Therefre, the aims f this paper are () t prpse the exteded TODIM methd t prcess the multi-attribute grup decisi-makig (MAGDM) prblems with PLTSs; () t explre the rmalizati methd f PLTSs with respect t the situati i which the prbabilistic sum fr all liguistic terms is less tha ; () t develp a weight determiati methd based etrpy; (4) t shw the effectiveess ad advatages f the prpsed apprach. T achieve this gal, the rest is itrduced as fllws: sme basic ccepts f the LTSs, HFLTSs, ad PLTSs, ad the TODIM were briefly itrduced i Sect.. Secti prpses the exteded TODIM t prcess PLTSs. I Sect. 4, a example is give t demstrate the validity ad advatages f ur methd. I Sect. 5, we cclude this paper. Prelimiaries. The LTSs ad HFLTSs The LTSs, which are fiite ad rdered, are regularly used t express DMs piis fr attributes f MADM prblems, ad they ca be defied as fllws (Herrera et al. 995): S ¼ fs a ja ¼ 0; ;...; sg ðþ where S a is called a liguistic variable; s is a psitive iteger. LTSs ca meet: () S a S b ; ifa [ b; () The egati peratr is: egðs a Þ ¼ S b, such that aþb ¼ s: T relieve the lss f ifrmati, a ctiuus LTS is btaied frm its discrete versi by Xu (0): S ¼ fs a ja ½0; sšg: ðþ Let S a ; S b S be ay tw LTs, ad the based the LTS S, the perati S a ad S b ca be defied by (Xu ad Wag 07): k S a k S b ¼ S k aþk b; ðþ where k ; k 0: Furthermre, we gave the defiiti f HFLTSs (Rdriguez et al. 0). Defiiti (Rdriguez et al. 0) Suppse that S ¼ S 0 ; S ;...; S g is a LTS, ad the, a HFLTS bs is defied as a subset f LTs S.

3 Graul. Cmput. (07) : 4 5 Example Let S be the fllwig LTS: S ¼ fs 0 ¼ extremely lw, S ¼ very lw, S ¼ lw, S ¼ slightly lw, S 4 ¼ fair, S 5 = slightly high, S 6 ¼ high, S 7 ¼ very high, S 8 = extremely highg: The, we give tw examples abut HFLTSs: b ¼ fs = very lw, S = lwg; b ¼ fs 5 = slightly high, S 6 = high, S 7 = very highg: Simplifyig the results abve, we get: b ¼ fs ; S g; b ¼ fs 5 ; S 6 ; S 7 g: Furthermre, Zhu ad Xu (04) gave a peratial defiiti f HFLTS. Defiiti (Zhu ad Xu 04) Suppse that b a ¼ b l a jl ¼ ; ;...;#b a ad bb ¼ b l b jl ¼ ; ;...;#b b are ay tw HFLTSs, such that #b a ¼ #b b, the b q a bq b ¼[ b qðþ l b qðþ l ð4þ b qðþ l ðþ kb q a ¼[ b qðþ l a where b q a ðþ l a b l a b l a ;bq l b l b b kb q a ðþ l a b ; k 0 ð5þ ad b qðþ l b are the lth LTs i b a ad b b, respectively, #b a ad #b b are the umbers f the LTs i b a ad b b, respectively.. PLTSs With respect t the shrtcmig f HFLTSs which cat express the prbability f pssible LTs, Pag et al. (06) prpsed PLTSs. Defiiti (Pag et al. 06) Suppse that S ¼ fs 0 ; S ;...; S s g is a LTS, ad the LTðpÞ ( X ) ¼ LT ðkþ jlt ðkþ S; 0; k ¼ ; ;...;; ð6þ is defied a PLTS, where LT ðkþ is the LT LT ðkþ with the prbability, ad is the umber f all differet LTs i LTðpÞ. We ca te that if P k¼ ¼, the the PLTS is with cmplete prbabilistic ifrmati; if P k¼ \, the the PLTS is with partial prbabilistic ifrmati; if P k¼ ¼ 0, the the PLTS is with cmpletely ukw prbabilistic ifrmati. k¼ Defiiti 4 (Pag et al. 06) Suppse that LTðpÞ ¼ LT ðkþ jk ¼ ; ;...; is a PLTS, ad r ðkþ is the subscript f LT LT ðkþ. If LT ðkþ ðk ¼ ; ;...;Þ are raked accrdig t the values f r ðkþ ðk ¼ ; ;...;Þ i descedig rder, the LTðpÞ is called a rdered PLTS, Example Suppse that the LTS S is the set used i Example, ad the, it ca be deted by the PLTS LTðpÞ ¼fS 4 ð0:þ; S 5 ð0:65þ; S 6 ð0:þg. We ca als calculate r ðkþ ðk ¼ ; ; Þ, ad get 4 0: ¼ 0:4; 5 0:65 ¼ :5; 6 0: ¼ :. Rerderig the LTs i LTðpÞ i descedig rder, we have LTðpÞ ¼fS 5 ð0:65þ; S 6 ð0:þ; S 4 ð0:þg: I the fllwig, Pag et al. (06) cme up with sme basic peratis: Defiiti 5 (Pag et al. 06) Let LT ðpþ ad LT ðpþ be tw rdered PLTSs, LT ðpþ ¼ LT ðkþ jk ¼ ; ;...;#LT ðpþg ad LT ðpþ ¼ LT ðkþ jk ¼ ; ;...;#LT ðpþg. The LT ðpþlt ðpþ ¼[ ðkþ LT LT ðpþ;lt ðkþ LT ðpþ ð7þ LTðkÞ LTðkÞ LT ðpþlt ðpþ ¼[ ðkþ LT LT ðpþ;lt ðkþ ( LT ðpþ ðkþ p ) ðkþ LT ðkþ p LT ðkþ ð8þ where LT ðkþ ad LT ðkþ are the k th LTs i LT ðpþ ad LT ðpþ, respectively, ad are the prbabilities f the k th LTs i LT ðpþ ad LT ðpþ, respectively. kltðpþ ¼[ LT LTðpÞk LT ðkþ ; k 0 ð9þ ðkþ kp ðltðpþþ k ðkþ ¼[ LT ðkþ LTðpÞ LT ðkþ : ð0þ. The cmparis fr tw PLTSs First, we itrduced the scre f PLTS which is defied by Pag et al. (06) as fllws. Defiiti 6 (Pag et al. 06) Let LTðpÞ ¼ LT ðkþ jk ¼ ; ;...; be a PLTS, ad r ðkþ is the subscript f LT LT ðkþ. The scre f LTðpÞ is give as fllws:

4 6 Graul. Cmput. (07) : 4 ELTðpÞ ð Þ ¼ S a ðþ where a ¼ P, P k¼ r ðkþ k¼ Fr ay tw PLTSs LT ðpþ ad LT ðpþ, if ELT ð ðpþþ\eðlt ðpþþ, the LT ðpþ LT ðpþ; if ELT ð ðpþþ[ ELT ð ðpþþ, the LT ðpþ LT ðpþ. Hwever, if ELT ð ðpþþ ¼ ELT ð ðpþþ, the tw PLTSs cat be cmpared by their scres. T slve this prblem, Pag et al. (06) further defied the deviati degree f a PLTS as fllws: Fr cveiece, let M ¼ f; ;...; mg ad N ¼ f; ;...; g. Step Idetify the decisi matrix X ¼ x ij m, where x ij is the jth attribute value with respect t the ith alterative, ad the rmalize X ¼ x ij m it G ¼ g ij m, ad x ij ad g ij are all crisp umbers, i M; j N. Step Calculate the relative weight w jr f the attribute C j t the referece attribute C r by: Defiiti 7 (Pag et al. 06) Suppse that LTðpÞ ¼ LT ðkþ jk ¼ ; ;...; is a PLTS, ad r ðkþ is the subscript f LT L ðkþ, ad ELTðpÞ ð Þ ¼ S a, where a ¼ P. P k¼ r ðkþ k¼. The deviati f LTðpÞ is: rðltðpþþ ¼ X k¼! =, X r ðkþ a k¼ ðþ Fr tw PLTSs LT ðpþ ad LT ðpþ, if ELT ð ðpþþ ¼ ELT ð ðpþþ ad r ðlt ðpþþ[ r ðlt ðpþþ, the LT ðpþ LT ðpþ; ad if r ðlt ðpþþ ¼ r ðlt ðpþþ, the LT ðpþ is idifferet t L ðpþ, deted by LT ðpþlt ðpþ. Therefre, there is the fllwig defiiti abut cmparis fr tw PLTSs. Defiiti 8 (Pag et al. 06) Give tw PLTSs LT ðpþ ad LT ðpþ, the If ELT ð ðpþþ[ ELT ð ðpþþ, the LT ðpþ LT ðpþ. else if ELT ð ðpþþ ¼ ELT ð ðpþþ, the (i) If rðlt ðpþþ[ rðlt ðpþþ, the LT ðpþ LT ðpþ. else if rðlt ðpþþ\rðlt ðpþþ, the LT ðpþ LT ðpþ. Defiiti 9 (Pag et al. 06) Let LT ðpþ ¼ LT ðkþ jk ¼ ; ;...;#LT ðpþ ad LT ðpþ ¼ LT ðkþ jk ¼ ; ;...;#LT ðpþ be ay tw PLTSs, #LT ðpþ ¼#LT ðpþ. The, the distace betwee LT ðpþ ad LT ðpþ is defied as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, u#lt X ðpþ dlt ð ðpþ; LT ðpþþ ¼ t rðkþ rðkþ #LT ðpþ: k¼.4 The traditial TODIM methd ðþ The TODIM is prpsed based prspect thery (Kahema ad Tversky 979), ad its mai advatage is the capability f capturig the DMs behavir. The steps f the traditial TODIM apprach are shw as fllws (Gmes ad Lima 99): Step Step 4 Step 5 w jr ¼ w j w r ; r; j N; ð4þ where w j is the weight f the attribute C j ad w r ¼ max w j jj N. Obtai the dmiace degree f alterative x i ver the alterative x t usig the fllwig expressi: # ðx i ; x t Þ ¼ X j¼ where / j ðx i ; x t Þ; 8ði; tþ; ð5þ / j ðx i ; x t Þ 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w jr ðg ij g tj Þ= P j¼ w jr; if g ij g tj [ 0; >< ¼ 0 if g ij g tj ¼ 0; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X h w >: j¼ jr g ij g tj =wjr if g ij g tj \0; ð6þ The fucti / j ðx i ; x t Þ is the ctributi f the attribute C j t # ðx i ; x t Þ.Theh is the DM atteuati parameter abut the lsses which is explaied by prspect thery (Kahema ad Tversky 979). I Eq. (6), three cases ca ccur: () if g ij g tj [ 0, the / j ðx i ; x t Þ represets a gai; () if g ij g tj ¼ 0, the / j ðx i ; x t Þ represets a il; () if g ij g tj \0, the / j ðx i ; x t Þ represets a lss. Get the verall prspect value f the alterative x i by P m t¼ dðx i Þ¼ # x P ð i;x t Þmi m i t¼ # ð x i;x t Þ P max m i t¼ # x P ð i;x t Þ m mii t¼ # x ; i M: ð i;x t Þ ð7þ Srt the alteratives by their verall prspect values dðx i Þði MÞ.

5 Graul. Cmput. (07) : 4 7 The exteded TODIM methd fr MADM prblems with PLTSs. Descripti f the MADM prblems Fr a MADM prblem with PLTSs, let x ¼ fx ; x ;...; x m g be a fiite set f alteratives ad C ¼ fc ; C ;...; C g be a set f attributes. Based the LTS S ¼ fs a ja ¼ 0; ;...; sg, the DMs evaluate the alteratives x i ði ¼ ; ;...; mþ fr the attributes C j ðj ¼ ; ;...; Þ, ad give the evaluati results by PLTSs LT ij ðpþ ¼ LT ðkþ ij ij jk ¼ ; ;...;#LT ij ðpþ, where LT ðkþ ij k ¼ ; ;...;#LT ij ðpþ are LTs with the crrespdig prbability ðkþ ij k ¼ ; ;...;#LT ij ðpþ, p ij [ 0, k ¼ ; ;...;#LT ij ðpþ, ad P #LT ij ðpþ k¼ ij. All the PLTSs LT ij ðpþði ¼ ; ;...; m; j ¼ ; ;...; Þ are used t build the decisi matrix R ¼ LT ij ðpþ, ad the gal is m t select the best alterative.. The rmalizati f PLTSs As metied i Sect.., there is existig partial prbabilistic ifrmati whe P k¼ \. T estimate the ukw part f prbabilistic ifrmati, we csider ctiuus re-distributi f the missig prbability by a recursive methd fr a PLTS LTðpÞ with P k¼ \. It will guaratee the uchaged preferece fr each expert by this way, ad the, the assciated steps are shw as fllwig: Fr cveiece, we use l t express the prbability f the LT L ðkþ after the lth iterati, ad use l l t express the igrace f prbabilistic ifrmati after the lth iterati. Let P k¼ ¼ a\, we ca first btai the assciated prbability f LT ðkþ ðk ¼ ; ;...;Þ, te that as ¼ þ ð aþ, the the ucertai prbability ca be calculated by f ¼ P ¼ P k¼ k¼ ð aþ. Next, we repeat the abve prcess i the fllwig: ¼ þ f ¼ ðþf Þ; f ¼ X ¼ X k¼ ¼ ðþf Þ X k¼ k¼ ðþf Þ ¼ ðþf Þð f Þ ¼ ðf Þ ; ¼ þ f ¼ þðf Þ ; f ¼ X ¼ X k¼ ðþf Þ X k¼ ¼ ðf Þ ¼ ðf Þ 4 ; k¼ ¼ þf ðþf Þ ¼ ð Þð f Þ 4 ¼ þ f ¼ þðf Þ 4 ; f 4 ¼ X 4 ¼ X k¼ ðþf Þ X k¼ k¼ ðþf Þ ¼ ¼ ðþf Þð f Þ ¼ ðf Þ ¼ ðf Þ 8 ; ¼ þ SðkÞ f ¼ þ ð f Þ Therefre, we trasfrm precedig frmula i the fllwig frm: ¼ ðþf Þ; ¼ ðþf Þ þðf Þ ; ¼ þf þ f ð Þ þðf Þ 4 ¼ ðþf Þ þðf Þ þðf Þ 4 ¼ þf þðf Þ þðf Þ þðf Þ 4 ¼ þf þðf Þ þðf Þ þðf Þ 4 þðf Þ 5 þðf Þ 6 þðf Þ 7 ; S ðkþ ¼ ðþf Þ þðf Þ þðf Þ 4 þðf ¼ þf þðf Þ þðf Þ þðf Þ 4 þðf Þ 5 þðf Þ 6 þðf Þ 7 þþðf Þ ¼ ð f Þ : f Because 0\f \; lim! ¼ lim! ð f Þ ¼ SðkÞ f ¼ P k¼ ¼ f P k¼ p : ðkþ Þ Frm the curse f calculability, we ca cclude the rmalized frm fr a PLTS LTðpÞ with P k¼ \ by ;

6 8 Graul. Cmput. (07) : 4 LT 0 ðpþ ¼ LT ðkþ p 0ðkÞ jk ¼ ; ;...; ð8þ where p 0ðkÞ ¼ = P k¼ 0pt P k¼. I additi, smetimes, the umbers f LTs i PLTSs are usually differet, it is ecessary t stadardize the cardiality f a PLTS fr the cveiece f cmputig. Defiiti 0 (Pag et al. 06) Let LT ðpþ ¼ LT ðkþ jk ¼ ; ;...;#LT ðpþ ad LT ðpþ ¼ LT ðkþ jk ¼ ; ;...;#LT ðpþ be ay tw PLTSs, ad let #LT ðpþ ad #LT ðpþ be the umbers f LTs i LT ðpþ ad LT ðpþ, respectively. If #LT ðpþ [ #LT ðpþ, the the #LT ðpþ#lt ðpþ LTs eed t be added t LT ðpþ ad make the umbers f LTs i LT ðpþ ad LT ðpþ equal. The added LTs are the smallest es i LT ðpþ, ad their prbabilities shuld be zer. Let LT ðpþ ¼ LT ðkþ jk ¼ ; ;...;#LT ðpþ ad LT ðpþ ¼ LT ðkþ jk ¼ ; ;...;#LT ðpþ be ay tw PLTSs, ad the, the rmalizati ca be perfrmed as fllws: () If P k¼ calculate LTi 0 \, the by the frmula (8), we ðpþ; i ¼ ;. () If #LT ðpþ 6¼ #LT ðpþ, the accrdig t Defiiti 0, we add sme elemets t the e with the smaller umber f elemets. i Example Let LT ðpþ ¼fS ð0:0þ; S ð0:0þ; S ð0:0þg ad LT ðpþ ¼fS ð0:60þ; S ð0:40þg be tw PLTSs, ad the: () accrdig t frmula (8), LT 0 ðpþ ¼ f S ð0:þ; S ð0:þ; S ð0:þg; () sice #LT ðpþ\#lt ðpþ, the we add the LT S t LT ðpþ ad make the umbers f LTs i LT ðpþ ad LT ðpþ idetical, ad thus, we have LT 0 ðpþ ¼ f S ð0:6þ; S ð0:4þ; S ð0þg.. Determiig bjective weights based etrpy measures It is imprtat t determie a reasable weight fr each attribute i the curse f decisi-makig. Because the DMs are usually iflueced by their kwledge structure, persal bias, ad familiarity with the decisi alteratives, it is ecessary t csider the MADM prblem with cmpletely ukw weights f criteria, ad we eed develp a weight determiati methd based etrpy uder prbabilistic liguistic evirmet. The steps are shw as fllws. First, trasfrmed decisi matrix R ¼ LT ij ðpþ m it Z ¼ LP ij ðpþ m, where LT ijðpþ ¼ P. #LT ij ðpþ k¼ r ðkþ #LT ij ðpþ: Next, calculate the etrpy fr attribute, the etrpy values fr the jth attribute are H j ¼ X m LT ij ðpþ l LTij ðpþ : ð9þ l m i¼ The, the weight f each attribute ca be defied by the fllwig: x j ¼ H j P j¼ H : ð0þ j.4 Prcedure fr prbabilistic liguistic TODIM methd I this sub-secti, we will give decisi-makig steps f the exteded TODIM methd fr the MAGDM prblems with the PLTSs. Step Stadardize decisi matrix I geeral, there are the beefit type ad cst type i the attributes. T keep all attributes cmpatible, we ca trasfrm the cst type it beefit e as fllws. ~LT ij ðpþ ¼ LT ijðpþ; fr beefit attribute c; ðþ C j LT ij ðpþ fr cst attribute Cj : c c¼ where LT ij ðpþ is cmplemet f LTij ðpþ, LT ij ðpþ egðlt ðkþ ij Þ ij jk ¼ ; ;...;#LT ij ðpþ. I additi, we eed t rmalize each attribute value accrdig t the abve steps. First, if P #LT ij ðpþ k¼ ij \, ad the, by the frmula (8), we calculate LTij 0 ðpþ. The, if the umbers f LTs i LT ij ðpþ are t equal, the we eed d a rmalizati accrdig t Defiiti 6. Step Obtai attribute weight vectr x j ¼ ðx ; x ;...; x Þ T f the fc ; C ;...; C g by Eqs. (9) ad (0). Step Obtai the relative weight w jr f the attribute C j t the referece C r by w jr ¼ w j w r ; r; j N; ðþ Step 4 where w j is the weight f the attribute C j ad w r ¼ max w j jj N : Obtai the dmiace f each alterative x i ver each alterative x t by

7 Graul. Cmput. (07) : 4 9 # ðx i ; x t Þ ¼ X j¼ where / j ðx i ; x t Þ; 8ði; tþ; ðþ / j ðx i ; x t Þ 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P w jr d ~LT ij ; ~LT tj = j¼ w jr; if ~LT ij ~LT tj >< ¼ 0; if ~LT ij ~LT tj rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X h w >: j¼ jr d ~LT ij ; ~LT tj =wjr ; if ~LT ij ~LT tj : ð4þ Step 5 Obtai the verall prspect value f the alterative x i by dðx i Þ P m t¼ ¼ # x P ð i; x t Þmi m i t¼ # ð x i; x t Þ P max m i t¼ # x P ð i; x t Þ m mii t¼ # x ; ð i; x t Þ i M: ð5þ S ¼ fs 0 ¼ e, S ¼ very lw, S ¼ lw, S ¼ medium, S 4 = high; S 5 = very high, S 6 ¼ perfectg t evaluate the prjects x i ði ¼ ; ; Þ by selectig a LT. The rigial ifrmati give five members is shw i Tables,,, 4, ad 5. Nte that the blaks f Tables,,, 4, ad 5 mea that the DM cat give the evaluati ifrmati. By directly sythesizig the ifrmati frm the five tables, we ca btai the grup s decisi matrix (Table 6) by the PLTSs; fr example, abut evaluati ifrmati f x with respect t C, because e f five members selects s, tw select s, e selects s 4, ad e cat give the evaluati ifrmati; this result ca be expressed by PLTS fs ð0:40þ; s 4 ð0:0þ; s ð0:0þg, ad because 0.4? 0.? 0. \, we ca rmalize it t fs ð0:50þ; s 4 ð0:5þ; s ð0:5þg. Usig frmulas (8) ad Defiiti 0, the rmalized matrix is shw i Table 7. Table Evaluati ifrmati frm the first member C C C C 4 x s s 4 s 4 s 5 x s s s s x s 4 s s 4 Step 6 Srt the alteratives by dðx i Þði MÞ. The bigger the dmiace degree dðx i Þ is, the better alterative A i is. Table Evaluati ifrmati frm the secd member C C C C 4 x s 4 s s 4 s 5 x s s s x s 4 s s 4 s 5 4 A example I this part, we cited a example frm Pag et al. (06) t shw the applicati ad the steps f the prpsed apprach. A cmpay wats t develp large prjects fr the future 5 years, ad five members are ivited t evaluate them. Three iitially selected prjects x i ði ¼ ; ; Þ are evaluated by fur attributes (suppse that all attributes are beefit type) ad iclude () C : ecmic ad scial perspective; () C : the service satisfacti perspective; () C : market perspective; (4) C 4 : grwth perspective. Suppse that their weight vectr is cmpletely ukw. The gal is t rak the three prjects. 4. The steps f the prbabilistic liguistic TODIM methd We ca slve this prblem by the prpsed prbabilistic liguistic TODIM methd, ad the steps are shw as fllws. Step The five members used the fllwig LTS: Table Evaluati ifrmati frm the third member Table 4 Evaluati ifrmati frm the furth member Table 5 Evaluati ifrmati frm the fifth member C C C C 4 x s 4 s 4 s 4 s x s 5 s s 4 x s s s 4 s 6 C C C C 4 x s 4 s 4 s 4 s x s s 4 s s x s s s 4 C C C C 4 x s s 4 s s 5 x s s s s x s s 4 s 4

8 40 Graul. Cmput. (07) : 4 If there is same weight fr the DMs, it ca be de as metied abve. Otherwise, we ca determie the weight f LT fr each PLTS accrdig t the weight vectr f DMs, respectively. Fr example, suppse that the weight vectr f DMs is x ¼ ð0:; 0:; 0:; 0:5; 0:5Þ T, abut evaluati ifrmati f x with respect t C, tw DMs select s with weights 0. ad 0.5, e DM selects s 4 with weight 0.5, ad last e DM selects s with weight 0., ad the, we ca get the result by PLTS fs ð0:45þ; s 4 ð0:5þ; s ð0:þg: Step Calculate the attribute weight vectr x j ¼ ðx ; x ;...; x Þ T by Eqs. (4) ad (5), ad we ca get H j ¼ ð0:457; 0:78; 0:6; :00Þ T Step Step 4 x j ¼ ð0:9; 0:0; 0:; 0:8Þ T : Obtai the relative weight w jr f the attribute C j t the referece attribute C r Sice w 4 ¼ maxfw ; w ; w ; w 4 g, the C 4 is the referece attribute ad the referece weight is w r ¼ 0:8. Therefre, the relative weights fr all the attributes C j ðj ¼ ; ; ; 4Þ are w r ¼ 0:79, w r ¼ 0:640, w r ¼ 0:680, ad w 4r ¼, respectively. Obtai the dmiace f each alterative x i ver each alterative x t by Eqs. () ad (4) (h ¼ ). Fr each attribute C j, we ca get the dmiace degree matrices accrdig t the Eq. (4) as fllws: x x x x 0 0:7 0: / ¼ x 0:69 0 0:4 5 ; x : :4 0 x x x x 0 0:48 0:7 / ¼ x :7 0 :57 5 x :76 0: 0 x x x x 0 0:5 :6 / ¼ x :40 0 :7 5 ; x 0:50 0:8 0 x x x x 0 0:7 0:59 / 4 ¼ x : 0 :5 5 : x 0:9 0:4 0 The, we ca get the verall dmiace degree betwee alteratives by Eq. (): x x x x # ¼ 0 :55 : x 6:49 0 4:0 5 x :8 0: 0 Step 5 Step 6 Obtai the verall prspect value f the alterative x i accrdig t the Eq. (5), ad the, we ca get the results dðx i Þði ¼ ; ; Þshw i Table 8. Srt the alteratives by their dðx i Þ. The bigger dðx i Þ is, the better alterative A i is, we get x x x : Therefre, the best chice is x : 4. Effect frm the atteuati factr f the lsses Kahema ad Tversky (979) suggested that the parameter h ca get the value frm.0 t.5, s we ca rak the three prjects accrdig t the differet value h by step 0., ad the rakig results are listed i Table 9. I Table 9, we ca tice the values f h frm t.5 by addig 0. fr each simulati ad the recrd the rakig results. As ca be see frm the results, the chage f the h frm h ¼ th ¼ :5 has effect the rakig results. I ther wrds, the rakig results are usual csistet with all the chage f the atteuati idex f lsses h. 4. Further discussis fr the case T verify the effective ad explai the advatages f the prpsed methd, we ca cmpare with the existig methds. 4.. Cmpare with the methds based prbabilistic liguistic ifrmati prpsed by Pag et al. (06) This example gt frm referece (Pag et al. 06), s we ca directly cmpare with it. Frm the rakig results f the alteratives, there is a rakig result x x x i (Pag et al. 06), s we ca get the same rakig result frm these tw methds. This will shw the effectiveess f the prpsed methd. The advatage f the methd prpsed by Pag et al. (06) is that it ca csider the prefereces i qualitative settig, amely, express the attributes with several pssible LTs. Naturally, the prpsed methd i this paper remais the same advatage. Furthermre, the prpsed methd ca csider the DMs psychlgy ad behavir, ad ca prduce mre reasable rakig result, while the methd

9 Graul. Cmput. (07) : 4 4 Table 6 Grup decisi matrix i PLTSs C C C C 4 x fs ð0:40þ; s 4 ð0:60þg fs ð0:0þ; s 4 ð0:80þg fs ð0:0þ; s 4 ð0:80þg fs ð0:40þ; s 5 ð0:60þg x fs 5 ð0:0þ; s ð0:80þg fs ð0:0þ; s ð0:40þ; s 4 ð0:0þg fs ð0:0þ; s ð0:40þ; s ð0:0þg fs 4 ð0:0þ; s ð0:80þg x fs ð0:40þ; s 4 ð0:60þg fs ð0:60þ; s 4 ð0:0þg fs ð0:0þ; s 4 ð0:0þ; s 5 ð0:0þg fs 4 ð0:80þ; s 6 ð0:0þg Table 7 Nrmalized grup decisi matrix i PLTSs C C C C 4 x fs 4 ð0:60þ; s ð0:40þ; s ð0:00þg fs 4 ð0:80þ; s ð0:0þ; s ð0:00þg fs 4 ð0:80þ; s ð0:0þ; s ð0:00þg fs 5 ð0:60þ; s ð0:40þ; s ð0:00þg x fs ð0:80þ; s 5 ð0:0þ; s ð0:00þg fs ð0:50þ; s 4 ð0:5þ; s ð0:5þg fs ð0:50þ; s ð0:5þ; s ð0:5þg fs ð0:80þ; s 4 ð0:0þ; s ð0:00þg x fs 4 ð0:60þ; s ð0:40þ; s ð0:00þg fs ð0:75þ; s 4 ð0:5þ; s ð0:00þg fs ð0:þ; s 4 ð0:þ; s 5 ð0:þg fs 4 ð0:80þ; s 6 ð0:0þ; s 4 ð0:00þg Table 8 Overall prspect values fr all alteratives x x x dðx i Þ prpsed by Pag et al. (06) has t this characteristic. Obviusly, ur prpsed methd is mre reasable ad ca als get a better decisi result, because the prpsed methd ca effectively csider the DMs psychlgy ad behavir. 4.. Cmpare with the TOPSIS methd based the traditial HFLTSs prpsed by Pag et al. (06) The rakig f the prpsed methd by Pag et al. (06) is x x x. Obviusly, it is differet frm the result prduced by the prpse methd. The mai reas is that the prpsed TOPSIS methd based the traditial HFLTSs ca te use the rigial prbabilistic ifrmati i the PLTSs, s it ca prduce the distrti f decisi results. Hwever, the prpsed methd i this paper ca give the cmprehesive values f each alterative by fully usig prbabilistic ifrmati, ad further give the rakig results. I additi, ur prpsed methd ca csider the DMs psychlgy ad behavir. Furthermre, this methd is able t capture the lss ad gai uder ucertaity frm the view f referece pit. Especially, whe the DM is mre sesitive t the lss, the prpsed methd ca be regarded as a useful buded ratiality behaviral decisi-makig methd. Table 9 Ifluece f the parameter h the rakig results f this example h ¼ :0 h ¼ : h ¼ : h ¼ : h ¼ :4 h ¼ :5 d Order d Order d Order d Order d Order d Order x x x h ¼ :6 h ¼ :7 h ¼ :8 h ¼ :9 h ¼ :0 h ¼ : d Order d Order d rder d rder d rder d rder x x x h ¼ : h ¼ : h ¼ :4 h ¼ :5 d Order d Order d Order d Order x x x

10 4 Graul. Cmput. (07) : 4 5 Cclusi I this paper, we explre a exteded TODIM methd t prcess the ifrmati f PLTSs. We first itrduced the sme basic kwledge f PLTSs ad the TODIM methd. The, we prpsed prbabilistic liguistic TODIM methd fr MADM ad describe the peratial prcesses i detail. Fially, a example is give t describe the decisi steps f develped methd ad t verify its effectiveess. Its prmiet characteristic is that it ca csider the decisi maker s psychlgical behavir. Therefre, it is mre flexible fr prcessig prbabilistic liguistic MAGDM prblems. Because the DMs are mre sesitive t the lss ad their buded ratiality, there is urget eed abut the prbabilistic liguistic TODIM methd t slve the related MADM prblems. I the further research, it is ecessary ad meaigful t exted sme ew methds based the PLTSs, because the PLTSs are a effective mathematical apprach f depictig prefereces with differet weights i qualitative settig; fr example, the VIKOR methd r GRA methd is exteded t prcess the PLTSs. Meawhile, we ca further study MADM prblems ifrmati aggregati peratrs with PLTSs r iterval-valued PLTS evirmets. Ackwledgemets This paper is supprted by the Natial Natural Sciece Fudati f Chia (Ns ad 774), Shadg Prvicial Scial Sciece Plaig Prject (Ns. 5BGLJ06, 6CGLJ ad 6CKJJ7), the Special Fuds f Taisha Schlars Prject f Shadg Prvice (N. ts05045), the Teachig Refrm Research Prject f Udergraduate Clleges ad Uiversities i Shadg Prvice (05Z057), ad Key research ad develpmet prgram f Shadg Prvice (06GNC006). Refereces Beg I, Rashid T (0) TOPSIS fr Hesitat fuzzy liguistic term sets. It J Itell Syst 8():6 7 Chatterjee K, Samarjit K (07) Uified Graular-umber-based AHP-VIKOR multi-criteria decisi framewrk. Graular Cmputig, pp Dg YC, Che X, Herrera F (05) Miimizig adjusted simple terms i the csesus reachig prcess with hesitat liguistic assessmets i grup decisi makig. If Sci 97:95 7 Gmes LFAM, Lima MMPP (99) TODIM: basics ad applicati t multicriteria rakig f prjects with evirmetal impacts. 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