Expert Systems wt Applcatos 40 (03) 484 4847 Cotets lsts avalable at ScVerse SceceD rect Expert Systems wt Applc atos oural omepage: www.elsever.com/locate/eswa A drect terval exteso of TOPSIS metod Ludmla Dymova, Pavel Sevastaov, Aa Tkoeko Isttute of Comp. & Iformato Sc., Teccal Uversty of Czestocowa, Dabrowskego 73, 4 0 Czestocowa, Polad artcle Keywords: TOPSIS Iterval exteso fo abstract Te TOPSIS metod s a tecque for order preferece by smlar ty to deal soluto. Ts tecque curretly s oe of te most popular metods for Multple Crtera Decso Makg (MCDM). Te TOPSIS metod was prmary developed for dealg wt oly real-valued data. I may cases, t s ard to preset precsely te exact ratgs of alteratves wt respect to local crtera ad as a result tese ratgs are cosdered as tervals. Tere are some papers devoted to te terval extesos of TOPSIS metod, but tese extesos are based o dfferet eurstc approaces to defto of postve ad egatve deal solutos. Tese deal solutos are preseted by real values or tervals, wc are ot attaable a decso matrx. Sce ts s cotradcto wt bascs of classcal TOPSIS metod, ts paper we propose a ew drect approac to terval exteso of TOPSIS metod wc s free of eurstc assumptos ad lmtatos of kow metods. Usg umercal examples we sow tat drect terval exteso of TOPSIS metod may provde te fal rakg of alteratves wc s substatally dfferet from te results obtaed usg kow metods. Ó 03 Elsever Ltd. All rgts reserved.. Itroducto Te tecque for order performac e by smlarty to deal soluto (TOPSIS) (La, Lu, & Hwag, 994 ) s oe of kow classcal MCDM metod. It was frst developed by Hwag ad Yoo (98) for solvg MCDM problems. Te basc prcple of te TOPSIS metod s tat te cose alteratve sould ave te sortest dstace from te postve deal soluto ad te fartest dstace from te egatve deal soluto. Tere exst a large amout of lterature volvg TOPSIS teory ad applcatos. Is was sow by Garca-Cascal es ad Lamata (0) ad Wag ad Luo (009) tat Oe of te problems attrbutable to TOPSIS s tat t ca cause te peomeo kow as rak reversal. I ts peomeo te alteratve s order of preferece cages we a alteratve s added to or removed from te decso problem. I some cases ts may lead to wat s called total rak reversal, were te order of prefereces s totally verted, tat s to say, tat te alteratve cosdered te best, wt te cluso or removal of a alteratve from te process, te becomes te worst. Suc a peomeo may cases may ot be acceptable. Wag ad Luo (009) sowed tat rak reversal peomeo occurs ot oly te TOPSIS metod, but may oter decso makg approaces suc as Aalytc Herarcy Process (AHP), te Borda Kedall (BK) metod for aggregatg ordal prefereces, te smple Correspodg autor. Tel./fax: +48 34 350 589. E-mal address: sevast@cs.pcz.pl (P. Sevastaov). addtve wegtg (SAW) metod, ad te cross-effcecy evaluato metod data evelopmet aalyss (DEA). Terefore, we ca say tat ts problem s typcal for kow metod of MCDM. I Garca-Cas cales ad Lamata (0), te autors proposed a ew metod for te soluto of ts problem te framework of TOPSIS metod. Neverteless t was poted out Garca-Cascal es ad Lamata (0) tat te two metods te classcal ad te ew do ot ave to gve te same order. Ts s especally so te case of evaluatg alteratves wc are very close. I oter words, te classcal ad te ew metods may provde dfferet results based o te same decso matrx. Hece, t s ot obvous tat te ew metod performs better ta classcal oe f tere s o eed to add or remove a alteratve from te decso problem. Terefore, ereafter we sall cosder oly classcal TOPSIS metod ad ts terval exteso. I classcal MCDM metods, te ratgs ad wegts of crtera are kow precsely. A survey of tese metods s preseted Hwag ad Yoo (98). I te classcal TOPSIS metod, te ratgs of alteratves ad te wegts of crtera are preseted by real values. Nevertel ess, sometmes t s dffcult to determe precsely te real values of ratgs of alteratves wt respect to local crtera, ad as a result, tese ratgs are preseted by tervals. Jaasa lo, Hossezade, ad Izadka (006) ad Jaasaloo, Hossezade Lotf, ad Davood (009) exteded te cocept of TOPSIS metod to develop a metodology for solvg MCDM problem wt terval data. Te ma lmtato of ts approac s tat te deal solutos are presete d by real values, ot by 0957-474/$ - see frot matter Ó 03 Elsever Ltd. All rgts reserved. ttp://dx.do.org/0.06/.eswa.03.0.0
484 L. Dymova et al. / Expert Systems wt Applcatos 40 (03) 484 4847 tervals. Te smlar approac to determg deal solutos s used Yue (0) ad Sayad, Heydar, ad Saaag (009) ts approac s used cotext of te so-called VIKOR metod wc s based o te measure of closees s to te deal solutos, too. I ts paper, we sow tat tese extesos may lead to te wrog results especally te case of tersect o of some tervals represetg te ratgs of alteratves. I Ce (0), Jaasaloo et al. (009), Jaasalo o, Kodabaks, Hossez ade Lotf, ad Moazam Goudarz (0), Tsaur (0), Ye ad L (009) ad Yue (0), te dfferet deftos of terval postve ad egatve deal solutos are proposed. Tey wll be aalysed te ext secto, but ter commo lmtato s tat tey are based o te eurstc assumptos (usually wtout ay aalyss ad clear ustfcato) ad provde terval deal solutos tat are ot always attaable te tervalvalued decso matrx. Sce ts s cotradcto wt bascs of classcal TOPSIS metod, ts paper, we propose a ew drect approac to terval exteso of TOPSIS metod wc s free eurstc assumptos ad lmtatos of kow metods. Ts approac makes t possble to obta te postve ad egatve deal solutos te terval form suc tat (opposte to te kow metods) tese terval-valued solutos are always attaable o te tal terval-valued decso matrx. Sce ts approac s based o te terval comparso, a ew smple, but well-ustfed metod for terval comparso s develope d ad preseted te specal secto. It s wort otg tat te most geeral approac to te soluto of MCDM problems te fuzzy settg s te preseta to of all fuzzy values by correspod g sets of a-cuts. Tere are o restrctos o te sape of membersp fuctos of fuzzy values ts approac ad te fuzzy TOPSIS metod s reduced to te soluto of MCDM problems usg terval exteded TOPSIS metod o te correspodg a-cuts (Wag & Elag, 006 ). Terefore, te developmet of a relable terval exteso of TOPSIS metod may be cosdered as a frst step te soluto of MCDM problems usg te fuzzy TOPSIS metod. Te rest of te paper s set out as follows. I Secto, we preset te bascs of TOPSIS metod ad aalyse ts kow terval extesos. Secto 3 presets te drect terval exteso of TOP- SIS metod ad te metod for terval comparso wc s eeded to develop ts exteso. Te results obtaed usg te metod proposed Jaasalo et al. (006) ad Jaasalo o et al. (009) are compare d wt tose obtaed by te develope d ew metod. Secto 4 cocludes wt some remarks.. Te bascs of TOPSIS metod ad kow approaces to ts terval exteso Te classcal TOPSIS metod s based o te dea tat te best alteratve sould ave te sortest dstace from te postve deal soluto ad te fartest dstace from te egatve deal soluto. It s assumed tat f eac local crtero s mootocally creasg or decreasg, te t s easy to defe a deal soluto. Te postve deal soluto s compose d of all te best acevable values of local crtera, wle te egatve deal soluto s composed of all te worst acevable values of local crtera. Suppose a MCDM problem s based o m alteratves A,A,...,A m ad crtera C, C,...,C. Eac alteratve s evaluated wt respect to te crtera. All te ratgs are assged to alteratves ad preseted te decso matrx D[x ] m, were x s te ratg of alteratve A wt respect to te crtero C. Let W =(w,w,...,w ) be te vector of local crtera wegts satsfyg P ¼ w ¼. Te TOPSIS metod cossts of te followg steps:. Normalz e te decso matrx: x P m k¼ x k r ¼ q ffffffffffffffffffffffffffffffff ; ¼ ;...; m; ¼ ;...; : ðþ Multpl y te colums of ormalzed decso matrx by te assocated wegts: v ¼ w r ; ¼ ;...; m; ¼ ;...; : ðþ. Determe te postve deal ad egatve deal solutos, respectvel y, as follows: A þ ¼ v þ þ ;...; vþ ¼fðmax v K b Þðmv K c Þg; ð3þ A ¼ v ;...; v ¼fðm v K b Þðmaxv K c Þg; ð4þ were K b s te set of beeft crtera ad K c s te set of cost crtera. 3. Obta te dstaces of te exstg alteratv es from te postve deal ad egatve deal solutos: two Eucldea dstaces for eac alteratves are, respectvel y, calculated as follows: vffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ux ¼ t ; ¼ ;...; m; S þ S ¼ ðv v þ Þ vffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ux ¼ t ; ¼ ;...; m: ¼ ðv v Þ 4. Calculate te relatve closeess to te deal alteratv es: RC ¼ S þ S þ S ð5þ ; ¼ ; ;...; m; 0 6 RC 6 : ð6þ 5. Rak te alteratv es accord g to te relatve closeess to te deal alteratves: te bgger s te RC, te better s te alteratve A. I Jaasa lo et al. (006) ad Jaasalo o et al. (009), a terval exteso of classcal TOPSIS metod was proposed. Ts approac may be descrbed as follows. Let ½x Š¼½x L ; xu Š be a terval value of t crtero for t alteratve (x L ad x U are te lower ad upper bouds of terval, respectvel y), W =(w,w,...,w ) be te wegt vector satsfyg P ¼ w ¼. Te D½½x L ; xu ŠŠ m s te terval-valued decso matrx. Te metod proposed Jaasalo et al. (006) ad Jaasaloo et al. (009) cossts of te followg steps:. Normalz g te decso matrx usg te followg expressos : r L ¼ r U ¼ P m k¼ P m k¼ x L k x L k x L þ x U x U k þ x U k ; ¼ ;...;m; ¼ ;...;; ð7þ ; ¼ ;...;m; ¼ ;...;: ð8þ. Takg to accout te mportace of crtera, te wegted ormalzed terval-valued decso matrx s obtaed usg te followg expressos: v L ¼ w r L ; vu ¼ w r U ; ¼ ;...; m; ¼ ;...; :
L. Dymova et al. / Expert Systems wt Applcatos 40 (03) 484 4847 4843 3. Te postve ad egatve deal solutos are obtaed as follows: A þ ¼ v þ ; vþ ;...þ ¼ maxv U K b ; mv L K c ; ð9þ A ¼ v ; v ;... ¼ mv L K b ; maxv U K c : ð0þ 4. Te separato of eac alteratve from te postve deal soluto s calculated usg te -dmesoal Eucldea dstace: ( ) S þ ¼ X K b v L v þ X þ K c v U v þ ; ¼ ;...; m: ðþ Smlarly, te separato from te egatve deal soluto s calculated as follows: ( ) S ¼ X K b v U v X þ K c v L v ; ¼ ;...; m: 5. Calculate te relatve closeess to te deal alteratves : RC ¼ S þ S þ S ðþ ; ¼ ; ;...; m; 0 6 RC 6 : ð3þ 6. Rak te alteratves accordg to te relatve closeess to te deal alteratv es: te bgger s te RC, te better s te alteratve A. I Jaasaloo et al. (009) ad Jaasa loo et al. (0), te terval exteso of TOPSIS metod based o terval-val ued deal soluto was proposed. Ts metod s based o te assumpto tat te deal ad egatve-deal solutos are caged for eac alteratve. Neverteles s, aalysg ts approac we ca see tat obtaed terval deal solutos are ot always attaable te terval decso matrx. I Ye ad L (009), a terval TOPSIS metod s used te framework of group mult-attrbute decso model. For te decso maker k, te postve ad egatve terval-valued solutos are preseted Ye ad L (009) ( our otato) as follows: A þk ¼ m þku ; m þku ;...; m þku ¼ max m kl ; max A k ¼ m þku ; m þku ;...; m þku ¼ m m kl ; m m ku m ku ; : Tere s o ay ustfcato of ts approac Ye ad L (009) ad oly wat we ca say about t s tat te case of oly beeft crtera te upper boud of A +k may be obtaed from (9) ad te lower boud of A k may be obtaed from (0). It s also mportat tat terval-v alued deal soluto s obtaed usg te metod proposed by Ye ad L (009) are ot always attaabl e te terval -valued decso matrx. Te smlar problem may be foud Ce (0). A more complcated approac to te defto of terval-valued deal solutos was proposed by Tsaur (0), were te ¼½m _ þl autor wrote Teoretca lly, a for pvot value _ þ m ; _ þu m Š for crtero te postve deal soluto, we kow tat bot of _ m þl ad _ þu m mgt be obtaed from dfferet alteratves.. Terefore, te deal solutos obtaed usg te metod proposed by Tsaur (0) are ot always attaable te terval-valued decso matrx. I Yue (0), te group decso makg problem was solved usg te modfed terval exteso of TOPSIS metod. I ts approac, te postve ad egatve deal solutos were preseted by terval-valued matrces. For example, te egatve deal soluto was preseted as follows A ¼ð½m L ; m U ŠÞ m, were m L ¼ m k m kl ; m U ¼ max k m ku (k s a umber of decso maker). It s easy to see tat obtaed A s ot always attaable te terval decso matrces provded by decso makers. Summars g, we ca say tat te commo lmtato of kow approac es to terval exteso of TOPSIS metod s tat tey (based o te dfferet assumptos) provde terval-valued deal solutos wc are ot always attaable correspod g terval-value d decso matrces. Ts s cotradcto wt bascs of classcal TOPSIS metod ad s a cosequece of eurstc assumpt os wc are ot usually ustfed eoug. I te most of aalysed approaces, te upper boud of postve terval-valued soluto s calculated as te expresso (9) ad te lower boud of egatve terval soluto s calculated as (0). Hece, we ca say tat te approac developed Jaasalo et al. (006) ad Jaasa loo et al. (009) provdg real-valued deal solutos attaable te terval-valued decso matrx seems to be more ustfed ta te oter aalysed ere approaces provdg terval-val ued deal solutos, wc are ot always attaable te terval-valued decso matrx. Terefore, to compare a drect terval exteso of TOPSIS metod we propose ts paper wt oter kow approac es, t seems to be eoug to compare te results obtaed by our metod wt tose obtaed usg te metod develope d Jaasalo et al. (006) ad Jaasa loo et al. (009) (see expressos (7) (3)). 3. A ew approac to te terval exteso of TOPSIS metod 3.. Te problem formulato Terefore, a more correct ad stragtforw ard approac to calculato of deal solutos s represetg tem te terval form usg te expressos : A þ ¼ ¼ v þl A ¼ ¼ max v L m þu ; v þl v L ; vu U ; v L v L U ; vþu K b ; v U K b ;...; v þl ; vþu ; m v L ; vu ;...; v L ; v U ; max v L ; vu K c K c ; ð4þ : ð5þ As tere are o ay type reducto s (represetato of terv als by real values) ad addt oal assumpt os cocered wt expressos (4) ad (5), we call our approac Drect Iterval Exteso of TOPSIS metod. It s easy to see tat expressos (4) ad (5) provde te postve ad egatve terval-valued deal solutos wc are always attaable te correspodg terval-valued decso matrx. To perform te dfferece of proposed approac from kow oes t s eoug to compare t wt te metod developed Jaasalo et al. (006) ad Jaasa loo et al. (009) (see explaato at te ed of prevous secto). Table Decso matrx. C C A [5,7] [,3] A [0,0] [5,7]
4844 L. Dymova et al. / Expert Systems wt Applcatos 40 (03) 484 4847 Suppose we deal wt te terval-valued decso matrx preseted Table, were [x ] = [5,7], [x ] = [,3], [x ] = [0,0] ad [x ] = [5,7] represet te ratgs of alteratves A ad A wt respect to te beeft crtera C ad C. Sce we deal wt te oly beeft crtera C ad C, te expresso (9) our case s reduced to A þ ¼fm þ ; mþ g¼max ðx U Þ. Usg ts expresso, from te frst colum of Table we get m þ ¼ 0 ad from te secod colum mþ ¼ 3. Terefore A þ ¼fm þ ; mþ g¼f0; 3g. Smlarly, from (0) we get A ¼fm ; m g¼m ðx L Þ¼f0; g. O te oter ad, usg ay metod for terval comparso (see Sevastaov (007), Wag & Kerre (00) ad subsecto 3.) we obta tat [x ]<[x ], [x ]<[x ] ad for te postve ad egatve terval deal solutos we get: ½A þ Š¼f½mŠ þ ; ½mŠþ g¼ f½5; 7Š; ½; 3Šg ad ½A Š¼f½mŠ ; ½mŠ g¼f½0; 0Š; ½5; 7Šg. It s easy to see tat fm þ ; mþ g¼f0; 3g s ot cluded f½mš þ ; ½mŠþ g¼f½5; 7Š; ½; 3Šg ad fm ; m g¼f0; g s ot cluded f½mš ; ½mŠ g¼f½0; 0Š; ½5; 7Šg. Tus, we ca say tat te cases we some tervals te decso matrx tersect, te approac proposed Jaasalo et al. (006) ad Jaasa loo et al. (009) may lead to wrog results. It s see tat our metod provdes te terval-valued deal solutos wc are strogly attaable te cosdered decso matrx (see Table.). Sce te oter kow metods aalysed prevous secto may produce terval-valued deal solutos wc are ot attaable te cosdered decso matrx (see Table.), tey may produce te wrog results too. We use ere te words wrog results to empasze tat oly our approac based o te expressos (4), (5) guaratees tat obtaed terval-valued deal solutos wll be always attaable te cosdered terval-valued decso matrx ad ts s complace wt bascs of classcal TOPSIS metod. As (4) ad (5) te mmal ad maxmal tervals must be cose, te ma dffculty te mplemetat o of te above metod s te problem of terval comparso. 3.. Te metods for terval comparso Te problem of terval comparso s of pereal terest, because of ts drect relevace practcal modelg ad optmzato of real-world processes. To compare tervals, usually te quattatve dces are used (see revews Sevastaov (007) ad Wag & Kerre (00)). Wag, Yag, ad Xu (005) proposed a smple eurstc metod wc provdes te degree of possblty tat a terval s greater/lesser ta aoter oe. For tervals B =[b L,b U ], A =[a L,a U ], te possbltes of B P A ad A P B are defed Wag et al. (005) Wag, Yag, ad Xu (005) as follows: PðB P AÞ ¼ maxf0; bu a L g maxf0; b L a U g a U a L þ b U b L ; ð6þ PðA P BÞ ¼ maxf0; au b L g maxf0; a L b U g a U a L þ b U b L : ð7þ Te smlar expresso s were proposed earler by Faccett, Rcc, ad Muzzol (998) ad by Xu ad Da (00). Xu ad Ce (008) sowed tat te expressos proposed Facce tt et al. (998), Wag et al. (005)<br/>ad Xu ad Da (00) are equvalet oes. A separate group of metods s based o te so-called probablstc approac to te terval compars o (see revew Sevastaov (007)). Te dea to use te probablty terpretato of terval s ot a ovel oe. Neverteles s, oly Sevastaov (007) te complete cosstet set of terval ad fuzzy terval relatos volvg separated equalty ad equalt y relatos develope d te framewor k of probablty approac s preseted. Nevertel ess, te results of terval comparso obtaed usg expressos (6) ad (7) geerally are smlar to tose obtaed wt te use of probablstc approac to te terval comparso. Te ma lmtatos of descrbed above metods s tat tey provde a extet to wc a terval s greater/l esser ta aoter oe f tey ave a commo area (te tersecto ad cluso cases sould be cosdered separately (Sevastaov (007))). If tere are o tersectos of compared tervals, te extet to wc a terval s greater/l esser ta aoter oe s equal to 0 or regardles s of te dstace betwee tervals. For example, Let A = [,], B = [3,4] ad C = [00,00]. Te usg descrbed above approac es we obta: P(C > A)=P(B > A)=,P(A > B)=0. Tus, we ca say tat te case of overlappg tervals te above metods provde te possblty (or probabl ty) tat a terval s greater/lesser ta aoter oe ad ts possblty (or probabl ty) ca be treated as te stregt of equalty or ( some sese) as te dstace betwee compared tervals. O te oter ad, te above metods ca ot provde te measure of tervals equalty (dstace) we tey ave o a commo area. Of course, te Hammg dstace d H ¼ ðal b L þa U b U Þ: or Eucldea dstace d E ¼ ððal b L Þ þða U b U Þ Þ ð8þ ð9þ ca be used as te dstace betwee tervals, but tese dstaces gve o formato about wc terval s greater /lesser. It ca be see tat tey ca ot be used drectly for terval compars o especally we a terval s cluded to aoter oe. Terefore, ere we propose to use drectly te operato of terval subtract o (Moore, 966 ) stead of Hammg ad Eucldea dstaces. Ts metod makes t possble to calculate te possblt y (or probabl ty) tat a terval s greater/l esser tat aoter oe we tey ave a commo area ad we tey do ot tersect. So for tervals A =[a L,a U ] ad B =[b L,b U ], te result of subtracto s te terval C = A B =[c L,c U ]; c L = a L b U, c U = a U b L.Its easy to see tat te case of overlapp g tervals A ad B, we always obta a egatve left boud of terval C ad a postve rgt boud. Terefore, to get a measure of dstace betwee tervals wc addtoal ly dcate wc terval s greater/l esser, we propose ere to use te followg value: D A B ¼ ððal b U Þþða U b L ÞÞ: ð0þ It s easy to prove tat for tervals wt commo ceter, D A B s always equal to 0. Really, expresso (0) may be rewrtte as follows: D A B ¼ ðal þ a U Þ ðbu þ b L Þ : ðþ We ca see tat expresso () represe ts te dstace betwee te ceters of compared tervals A ad B. Ts s ot a surprsg result as Wag et al. (005) oted tat most of te proposed metods for terval comparso are totally based o te mdpot s of terv al umbers. It easy to see tat te result of subtrac to of terv als wt commo ceters s a terval cetered aroud 0. I te framewo rk of terval aalyss, suc terval s treated as te terv al 0.
L. Dymova et al. / Expert Systems wt Applcatos 40 (03) 484 4847 4845 More strctly, f a s a real value, te 0 ca be defed as a a. Smlarly, f A s a terval, te terval zero may be defed as a terval A A =[a L a U,a U a L ] wc s cetered aroud 0. Terefore, te value of D A B equal to 0 for A ad B avg a commo ceter may be treated as a real-valued represetat o of terval zero. Te smlar stuato we ave statstcs. Let A ad B be samples of measure mets wt correspodg uform probablty dstrbutos suc tat tey ave a commo mea (mea A = mea B ), but dfferet varaces (r A > r B ). Te usg statstcal metods t s mpossbl e to prove tat te sample B s greater ta te sample A or tat te sample A s greater ta te sample B. Takg to accout te above cosderat o we ca say tat te terval comparso based o te assumpto tat tervals avg a commo ceter are equal oes seems to be ustfed ad reasoable. Obvously, te comparso of tervals based o comparso of ter ceters seems to be too smple. Neverteles s, as t s sow above, ts approac s based drectly o covetoal operato of terval subtracto. Terefore t s ot a eurstc oe. Moreover ts metod cocdes better wt commo sese ta more complcated kow approac es. Let us cosder two tervals A = [3, 5] ad B = [,4]. Sce a U > b U ad a L > b L, te accordg to te Moore (966) ad commo sese we ave A > B. Sce A ad B are ot detcal ad ave o a commo ceter tere s o cace for A ad B to be equal oes. Fally, accordg to commo sese ts case te possblty of A < B sould be equal to 0. Tere s o cace for B to be greater ta A as a wole, altoug some pot belogg to B te commo area of A ad B may be greater ta te pots of A ts area. Neverteles s, our case from (6) ad (7) we get PðB P AÞ ¼ 5 ad PðA P BÞ ¼ 4. Tus, we ca see tat te kow approaces may 5 provde coutertu tve results. I Table, we preset te values of P(A P B), P(B P A) (see expressos (6), (7)), te Hamm g d H ad Eucldea d H dstaces (see expressos (8), (9)) betwee A ad B, ad D A B for tervals A = [4,7], A = [5,8], A 3 = [8,], A 4 = [3,6], A 5 = [8,], A 6 = [,4], A 7 = [,5] ad B = [7,] placed as t s sow Fg.. Te umbers te frst row Table correspod to te umbers of tervals A, = to 7. (see Table ). We ca see tat te values of D A B are egatve we A 6 B ad become postve for A P B. Tese estmates cocde (at least qualtatvely) wt P(A P B) ad P(B P A ). So we ca say tat te sg of D A B dcates wc terval s greater/lesser ad te values of absðd A BÞ may be treated as te dstaces betwee tervals sce tese values are close te to te values of d E ad d H bot cases: Table Results of terval comparso. Metod 3 4 5 6 7 P(A P B) 0 0.06 0. 0.5 0.78 P(A 6 B) 0.94 0.78 0.5 0. 0 0 d E 0.8 0 7.8 6 7.8 0 0.8 d H 9 8 6 6 6 8 9 D A B 9 8 5 0 5 8 9 we tervals ave a commo area ad we te tere s o suc a area. 3.3. Te comparso of te drect terval exteso of TOPSIS metod wt te kow metod Usg D A B, t s easy to obta from (4), (5) te deal terval solutos A þ ¼f½v þl A ¼f½v L þu Š; ½vþL U Š; ½v L ; vþu ; v U Š;...; ½vþL Š;...; ½v L þu U As te framewo rk of our approac te dstace betwee terval s A ad B s preseted by te value of D A B, tere s o eed to use Hammg or Eucldea dstaces for calculato ofs þ ad S. Sce D A B s te subtracto of te mdpots of A ad B, te values of S þ ad S may be calculated as follows: S þ ¼ X ððv þl þ v þu Þ ðv L þ v U ÞÞ þ X ððv L þ v U Þ K B K C ðv þl þ v þu ÞÞ: ðþ Šg; Šg: S ¼ X ððv L þ vu Þ ðv L þ v U ÞÞ þ X K B ðv L þ vu ÞÞ: ððv L K C þ v U Þ ð3þ Fally, usg expresso (3) we obta te relatv e closeess RC to te deal alteratv e. Let us cosder some llustratve examples. Example. Suppose we deal wt tree alteratves A, = to 3 ad four local crtera C, = to 4 preseted by tervals Table 3, were C ad C are beeft crtera, C 3 ad C 4 are cost crtera. Suppose W = (0.5, 0.5, 0.5, 0.5). To stress te advatag es of our metod, ts example may tervals represet g te values of ratgs tersect. Te usg te kow metod for terval exteso of TOPSIS metod Jaasalo et al. (006) ad Jaasalo o et al. (009) (expressos (7) (3)) we obta R = 0.53, R = 0.6378, R 3 = 0.390 ad terefore R > R > R 3, wereas wt te use of our metod (expressos 7, 8,, 3 ad 3) we get R = 0.7688, R = 0.758, R 3 = 0.077 ad terefore R > R > R 3. We ca see tat tere s a cosderable dfferece betwee te fal rakg obtaed by te kow metod ad usg our metod based o te drect exteso of TOPSIS metod. Ts ca be explaed by te fact tat te metod proposed Jaasa lo et al. (006) ad Jaasalo o et al. (009) as some lmtato s Table 3 Decso matrx. C C C 3 C 4 Fg.. Compared tervals. A [6,] [0,5] [6,] [8,0] A [5,8] [8,] [0,30] [9,8] A 3 [9,3],7] [4,48] [40,49]
4846 L. Dymova et al. / Expert Systems wt Applcatos 40 (03) 484 4847 Table 4 Decso matrx. C C C 3 C 4 A [6,] [0,5] [3,9] [40,48] A [3,4] [7,] [0,30] [,8] A 3 [5,8] [8,0] [4,48] [8,0] Tus, we ca see tat eve te cage of oly oe elemet a decso matrx wc leads to te appearace of tersectg tervals te correspod g colum, may lead to te sgfcat cages te results obtaed by our metod, wereas te kow metod (Jaasalo et al. (006) ad Jaasa loo et al. (009)) does ot provde dfferet fal ratgs of compared alteratves. cocered wt te presetato of tervals by real values te calculato of deal solutos ad usg te Eucldea dstace we tervals tersect. Example. I ts example, tere are o tersectg tervals te colums of decso matrx (see Table 4). As te prevous example, C ad C are beeft crtera, C 3 ad C 4 are cost crtera, W = (0.5, 0.5, 0.5, 0.5). Te usg te kow metod proposed Jaasalo et al. (006) ad Jaasaloo et al. (009), ts example we get R = 0.485, R = 0.4984, R 3 = 0.543 ad terefore R 3 > R > R. Wt te use of our metod we obta R = 0.567, R = 0.4675, R 3 = 0.4488 ad terefore R > R > R 3. Hece, we ca coclude tat eve te case we tere are o tersectg tervals te colums of terval valued decso table, te kow metod proposed Jaasalo et al. (006) ad Jaasaloo et al. (009) ad our metods may provde very dfferet fal rakgs of alteratves. Tat may be explaed by te fact tat te metod proposed Jaasalo et al. (006) ad Jaasalo o et al. (009), te real-valued deal solutos are used, wereas our metod tey are preseted by tervals attaable te terval-valued decso table. Neverteles s, we tere are o tersectg tervals te colums of terval-valued decso table, te fal ratgs obtaed by te metod proposed Jaasalo et al. (006) ad Jaasalo o et al. (009) may cocde wt tose obtaed usg our metod. Cosder te llustratve examples. Example 3. I ts example we wll use te decso table preseted Table 4, were C ad C are beeft crtera, C 3 ad C 4 are cost crtera, but W = (0.5,0.,0.5,0.5). Te usg te metod from Jaasalo et al. (006) ad Jaasalo o et al. (009) we obta R = 0.48, R = 0.3, R 3 = 0.5798 ad terefore R 3 > R > R. Usg our metod we get R = 0.6653, R = 0.758, R 3 = 0.6363 ad terefore R 3 > R > R. Tus, ts case two cosdered metods provde cocded ratgs. Example 4. Let us cosder te decso matrx presete d Table 5, wc dffers from Table 4 by oly oe elemet [x 3 ]so tat [x 3 ] tersects wt [x ]. Tere are o oter tersecto s te colums of Table 5. Usg, as prevous case W = (0.5,0.,0.5,0.5), ad te metod from Jaasalo et al. (006) ad Jaasa loo et al. (009) we get R = 0.48, R = 0.53, R 3 = 0.5798 ad terefore R 3 > R > R as te prevous example, wereas wt te use of our metod we obta R = 0.6653, R = 0.758, R 3 = 0.6363 ad terefore R > R 3 > R. Table 5 Decso matrx. C C C 3 C 4 A [6,] [0,5] [3,9] [40,48] A [3,4] [7,] [0,30] [,8] A 3 [,8] [8,0] [4,48] [8,0] 4. Cocluso Te crtcal aalyss of kow approac es to te terval exteso of TOPSIS metod s presete d. It s sow tat tese extesos are based o dfferet eurstc approaces to defto of postve ad egatve deal solutos. Tese deal solutos are preseted by real values or tervals, wc are ot attaable a decso matrx. Sce ts s cotradcto wt bascs of classcal TOPSIS metod, a ew approac to te soluto of MCDM problems wt te use of TOPSIS metod te terval settg s proposed. Ts metod called drect terval exteso of TOPSIS metod s free of eurstc lmtatos of kow metods cocered wt te defto of postve ad egatve deal solutos ad usg te Eucldea dstace we tervals a decso matrx tersect. Te ma advatage of te proposed metod s tat (opposte to te kow metods) t provdes terval-valued postve ad egatve deal solutos wc complace wt te bascs of classcal TOPSIS metod are always attaable te terval-valued decso matrx. It s sow tat te use of kow metods may lead to te wrog results as well as te use of te Eucldea dstace we tervals represet g te values of local crtera tersect. 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