Evaluating Consulting Firms Using a Centroid Ranking Approach based Fuzzy MCDM Method

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1 EUSFAT-FA y Ax-es-Bas Frace Evaat Cost Frms Us a Cetro a Approach base Fzzy MCDM Metho Ta-Ch Ch Departmet o Maaemet a ormato Techooy Sother Tawa Uversty Tawa.O.C. E-ma: [email protected] Abstract The prpose o ths paper s to evaate a seect cost rms by sest a cetro ra approach base zzy mtpe crtera ecso ma (MCDM) metho rats o ateratves verss qatatve crtera a the eret mportace wehts amo crtera are assesse stc vaes represete by zzy mbers. Crtera are cateorze to qatatve beet qattatve a cost qattatve oes. Membershp ctos o the a zzy evaato vaes ca be eveope throh α-cts a terva arthmetc. The ra approach o Ecea stace base o cetro pots s appe to eterme the ra orer o a ateratves. Formas or the ra procere ca be ceary erve. Fay a merca exampe s se to emostrate the comptatoa process o the seste moe. Keywors: Cost rms Fzzy MCDM Cetro a. trocto To srvve tht competto toay s bsess wor a compay say eveops a ew proct whch s eret rom or better tha that o ts compettors. A crca actor sch as prc mst be eterme whe troc a ew proct to the maret becase t s very sestve to cstomers. Wro prc stratey or a ew proct eveope rom a heavy vestmet ca ea a compay to oss or eve barptcy. However eterm the best prc stratey or a ew proct s ct a may actors mst be cosere. To catosy ea wth ths probem may compaes say as or cost rm servce. Ths evaat a seect a stabe cost rm becomes a mportat sse. A cost rm s a rm o experts prov proessoa avce to a orazato or a ee. A cost rm cossts o costats who are experts ther e. For some oba cost rms ther empoyees represet rom may atoaty. Usay a cost rm proves ts servce whch s core bsess scpe rom maret to operatos; bt there are cost rms whch ot oy prove bsess servce bt potcs as we. May crtera mst be cosere whe evaat cost rms some o them are qatatve sch as reptato some are qattatve sch as rm sze; moreover crtera may have eret mportace. Thereore how to comprehesvey areate these crtera a mportace wehts becomes a crtca sse eectvey evaat cost rms. Some reevat wors have bee ste the evaato o cost rms. Crepets et a. [3] () aayze theoretcay a emprcay the ereces betwee costats a experts the ramewor o the owee-base ecoomy orer to troce the cetra cocepts o epstemc commty a commty o practce. However they ot ta eta abot the other crtera that are sppose to be cosere by a cost rm sch as the mpemetato cost a ts owee. Wa a Che [] (6) presete how hma pts (top maaemet sers a extera costats) are e to commcato eectveess a coct resoto the EP cost process as we as the eects o these actors o the qaty o the system mpemete. Ther s cate that top maaemet spport recty ehaces EP system qaty throh ts postve eect o coct resoto the cost process. The rests aso showe that hh ser spport ehaces commcato eectveess; however commcato eectveess oes ot ece coct resoto a EP system qaty. Ths paper acs ormas or compet ecso ma. Atma [3] (8) cate what sho compay coser a ve ees choos the rht costacy. Ths artce ves cear ersta abot what s a cost rm bt s short o qattatve process areat the eee crtera. 9 Sarem et a. [9] se Noma rop Techqe (NT) ec crtera or seect the best costat rm. They hepe compay to choose the best costat or TM seecto er zzy TOPSS qatatve a qattatve crtera are cosere. Bt a crtera eve mpemete cost are reare as qatatve oe. Cebec a a [7] (7) prove a aaytca too to seect the best qaty costat prov the most cstomer satsacto zzy aaytc herarchy process was appe to compare these costacy rms. Bt a the attrbtes are qatatve. To resove the above probems evaat cost rms ths paper sests a cetro ra approach base zzy MCDM metho both sbectve (or qatatve) a obectve (or qattatve) crtera are cosere. the propose moe rats o ateratves verss qatatve crtera a the mportace wehts o a the crtera are assesse. The athors - Pbshe by Atats Press

2 stc vaes represete by zzy mbers. May methos have bee propose to sove zzy MCDM probems. A revew a comparso o may o these methos ca be o Che a Hwa [9] (99) Carsso a Fèr [6] (996) bero [8] (996) a Trataphyo a [] (996). Some recet appcatos ca be o Che [8] () Cho [] (7) Cho et a. [] (6) a Ö üt et a. [7] (9). most zzy MCDM probems the a evaato vaes o ateratves are st zzy mbers a these zzy mbers ee a proper ra approach to ezzy them to crsp vaes or ecso ma. May approaches or ra zzy mbers have bee ste. A revew o may o these approaches ca be o Bortoa a Dea [5] (985) Che a Ha [9] (99) a Wa a Kerre [] (). Some recet methos ca be see Abbasbay a Asay [] (6) Abbasbay a Haar [] (9) Asay [4] () a Ha [6] (5) Wa a ee [3] (8) a o et a. [5] (6). spte o avataes some o these methos are comptatoa compex a ct to mpemet a oe o them ca satsactory ra zzy mbers a cases. Moreover the above wors o ot prove coecto by ormae betwee the a evaato vaes a the ra procere mt ther appcabty a comptato ececy. To resove these probems ths paper sests appy the ra approach o cetro rom Wa et a. [4] (6) to obta the Ecea stace to ezzy a the a zzy evaato vaes orer to compete the moe. Formae ca be ceary eveope to preset the ra procere. Fay a merca exampe emostrates the comptatoa process o the seste moe.. Fzzy set theory Fzzy set theory was rst troce by Zaeh [7] 965. t s to ea wth probems vov zzy pheomea. Aso t has bee cosere as a se too o moe aae represete by zzy mbers to approxmate system whch zzy pheomea exsts. Some basc cocepts zzy set theory are brey troce the oow sectos whch ca aso be see the athor s prevos wor []... Fzzy set The zzy set A ca be expresse as [5] : x A x A x U () U s the verse o scorse x s a eemet A x s the membershp U A s a zzy set U cto o A at x. The arer A x the stroer the rae o membershp or x A... Fzzy mbers A rea zzy mber A s escrbe as ay zzy sbset o the rea e wth membershp cto whch A possesses the oow propertes [4] : (a) A s a cotos mapp rom to []; x x a ; (b) A ] (c) A s strcty creas o a b ; () A x x b c; (e) s strcty ecreas o c ; A () x x [ ) A ; a b c The membershp cto ca aso be expresse as: A ca be eote as a b c x. o the zzy mber A A a x b b x c A x () A x c x otherwse A x a A x are et a rht membershp ctos o A respectvey..3. α-ct The α-cts o zzy mber A ca be ee as [5] : x x A A (3) A A s a o-empty boe cose terva A A A cotae a ca be eote by A a respectvey..4. Fzzy arthmetc operatos ve zzy mbers A a B A are ts ower a pper bos A B the α-cts o A a B are A a B A A B B respectvey. By terva arthmetc some ma operatos o A a B ca be expresse as oows [5] : A B A B A B A B A B A B A B A B A B A A A B B B A r A r A r r.5. stc vaes (4) A stc varabe s a varabe whose vaes are expresse stc terms. stc varabe s a very hep cocept or ea wth statos whch are too compex or ot we-ee to be reasoaby escrbe by tratoa qattatve expressos [6]. For exampe mportace s a stc varabe whose vaes ce U (mportat) (ess mportat) M 3

3 (mportat) M (more mportat) a V (very mportat). These stc vaes ca be rther represete by traar zzy mbers sch as V=(...5) =(..5.5) M=( ) H=(.5.75.) a VH=(.75..). 3. Moe estabshmet Assme that a commttee o ecso-maers ( D t=~) s resposbe or evaat m ateratves ( A =~m) er crtera a crtera are cateorze to sbectve ( C =~h) a obectve ( C =h+~) obectve/qattatve crtera are rther casse to beet (B) a cost (C). 3.. Averae rats o ateratves verss sbectve crtera et xt ( ot pt qt ) xt = h t =. x x... x t x = m o t p t q o p q t t t be the rat asse to A by D t or C. the averae rat o A er C. t t (5) x t x s 3.. Normaze perormace o ateratves verss obectve crtera Here qattatve vaes o ateratves verss obectve crtera are assme to be certa a ca be expresse traar zzy mbers. These vaes have eret ts a mst be ormaze. The ormazato s compete by a seste approach rom Ch a [] (9) whch preserves the property the raes o ormaze traar zzy mbers beo to [ ]. Sppose r ( e ) s the perormace o ateratve verss obectve crtero the ormaze vae ca be eote as: e e e e x x B e x x C e m e max (6) (7) e ~ m h ~. Beet (B) crtero has the characterstcs o the arer the better; whe cost (C) crtero has the characterstcs o the smaer the better Averae wehts et wt ( a t bt c t ) wt = ; t =. w w w... w (8) a t b t c t a b c w t t t t be the weht asse by D t to C. w s the averae weht o C assesse by ecso-maers Areate the wehte rats The areato o wehte rats or each ateratve s mpemete by the cocept o smpe atve weht as: et o p q x w = m =. e e e e B e C (9) () By appy Eqs. (4) a () Eq. (9) ca be eveope va arthmetc operato o zzy mbers as: x w p o b a o b a a p o oa p q b c q b c c p q q c et p o b a o b a a p o () 4

4 p q b c q b c c p q oa pb Z qc. By appy the above assmpto to Eqs () we have two eqatos to sove: x () Z x (3) The et a rht membershp ctos o ca be eveope as: / x 4 x x / ; x 4 x Z x Z. (4) (5) For coveece Eqs. (4) a (5) o ca be eote as Eq. (6) as oows: ( Z ; ; ) (6) The verse ctos o the above two eqatos (4) a (5) ca be proce as: y y y y y y y y () The we cacate the Ecea staces rom ther cetro pots to the or to ezzy zzy mbers. The arer the stace s the hher prorty zzy mber s rae. x y () Formas or cetros o x-axs a y-axs ca be eveope as oows: x x / x 4 x 4 et / x 4 x t / 4 x x () t 4 4 x t x t tt 4 4 / / 4 4 y y y y y y y Z y 3.5. Obta ra (7) (8) The cetro ra metho rom Wa et a. [4] (6) s appe to ra the ateratves a etae terpretato o cetro eqatos ca be see Wa et a. paper. The cetro o x-axs a y-axs ca be show as oows: x ( ) x ( x) ( x) x ( x) x ( x) ( x) ( x) Z Z (9) / 4 4 ' Assme / x x 4 ' t 4 tt t '5 5 ' we obta: x x / '5 5 ' Smary Z x x Z / x 4 x Z (3) (4) 5

5 5 "5 3 "3 3 4Z Z (5) x / 4 x '3 3 (6) Z x Z / 4 x Z 3 "3 Z (7) By appy Eqs. ()-(7) Eq. (9) ca be proce as: x '5 5 '3 3 5 "5 3 "3 Z Z '3 3 3 "3 Z Smary Eq. () ca be expresse as: y y y y Z y y y y y Z y y y Z 4 3 Z 3 (8) (9) The Eqs. (8)-(9) are appe to Eq. () to proce the Ecea staces o each zzy mber. Fay we ca ra the prorty o semet ateratves base o vaes o Ecea staces. Compay s seste to choose ateratve caate whch has hhest prorty as ts cost rm. 4. Nmerca exampe Ths secto ses a artca merca exampe to emostrate the comptatoa process o the seste moe. Sppose a maactr compay mst seect a cost rm to hep eterme the prce or ts ew proct. Ater premary scree or ateratve cost rms A A A 3 a A 4 are chose or rther evaato. A commttee o ve ecso-maers D D D 3 D 4 a D 5 s orme to coct the evaato a seecto o the or ateratve cost rms. Frther sppose three beet qatatve crtera sch as reptato (C ) techca ss (C ) a owee o bsess (C 3 ) as we as three beet qattatve crtera sch as compay sze (C 4 # o empoyees t: hre) poteta prot (C 5 t: mo $) a expecte rowth (C 6 t: %) a oe cost qattatve crtero sch as cost (C 7 t: thosa $) are cosere. Moreover sppose that the ecso-maers se the stc rat set a stc rat set W={V M H VH} V=Very ow=(..3) =ow=(..3.5) M=Mem=(.3.5.7) H=Hh =(.5.7.9) a VH=Very Hh=(.7.9) to evaate the stabty o cost rms er each o the qatatve crtera. ato the ecso-maers empoy a stc weht set W={U M M V} U=Umportat=(...5) =ess mportat=(..5.5) M=mportat=( ) M=More mportat=(.5.75.) a V=Very mportat=(.75..) to assess the mportace o a the crtera. Moreover sppose that the stabty rats o cost rms verss the ve qatatve crtera rom the three ecso-maers are presete Tabe vaes o cost rms er qattatve crtera are presete Tabe 3 a the mportace wehts o a crtera are presete Tabe 6. By Eq. (5) the averae stabty rats o cost rm A er each qatatve crtero C rom the ecso-ma commttee ca be obtae as show Tabe. By Eqs. (6)-(7) ormaze vaes o cost rm A verss each qattatve crtero ca be obtae a are spaye Tabes 4-5. Throh Eq. (8) the averae wehts o crtera rom the ecso-ma commttee ca be obtae as presete Tabe 7. Throh Eqs. (9)-(6) the membershp ctos o the a zzy evaato vaes =~4 ca be proce Tabe 8. Throh Eqs. (7)-(9) cetro pots ca be proce as show Tabe 9 a the Ecea staces or the ateratve maret semets ca the be obtae as show Tabe.. Tabe. ats o cost rms verss qatatve crtera A. C. D D D 3 D 4 D 5 A C H H VH H H C H M H H VH C 3 H H M H A C VH H VH VH H C M VH H H C 3 V M M H A 3 A 4 C H M V M M C M M M C 3 H V V C VH V M V C M M V M C 3 V V V 6

6 Tabe. Averae rats o cost rms verss qatatve crtera A. C. Averae rats A C ( ) C ( ) C 3 ( ) A C ( ) C (.4.6.8) C 3 (.4.4.6) C ( ) A 3 C (..4.6) C 3 (.4.3.5) C ( ) A 4 C ( ) C 3 ( ) Tabe 3. Nmerca vaes o qattatve crtera C. A A A 3 A 4 C 4 (5337.5) (89.3) ( ) ( ) C 5 (36) (6.58) (.845) (.5.5) C 6 (..5) (.5 3) (.533.5) (.5) C 7 (579) (68) (35) () Tabe 8. Fa zzy evaato vaes Vaes A Ateratves A A 3 A Z Tabe 9. Cetro pots Vaes A Ateratves A A 3 A 4 x y Tabe. Ecea stace Ateratves Vaes A A A 3 A 4 Dstace Tabe 4. Normaze vaes o qattatve crtera C. A A C 4 ( ) (.6.86.) C 5 ( ) ( ) C 6 ( ) ( ) C 7 (..3.5) (...4) Tabe 5. Normaze vaes o qattatve crtera (cot.) C. A 3 A 4 C 4 ( ) ( ) C 5 ( ) ( ) C 6 ( ) ( ) C 7 (.5.7.8) (.8.9.) Tabe 6. mportace wehts o crtera C. D D D 3 D 4 D 5 C V M C V M M V V C 3 V M M V C 4 U U C 5 V V V V C 6 U M C 7 V V M Tabe 7. Averae mportace wehts C. Averae wehts C ( ) C (.65.9.) C 3 ( ) C 4 (..5.4) C 5 ( ) C 6 (..4.65) C 7 ( ) Base o Tabe the ateratve caate (A ) has hhest score; thereore compay sho choose A as ts cost rm. Ths merca exampe has ceary emostrate the comptatoa procere o the seste moe. A merca comparso as we as sestvty aayss may be cocte the exteso o ths wor the tre or rther stcato. 5. Cocsos A cetro ra approach base zzy MCDM metho s propose to evaate a seect cost rms crtera are casse to qatatve beet qattatve a cost qattatve oes. the propose moe rats o ateratves verss qatatve crtera a the mportace wehts o a crtera are assesse stc vaes represete by zzy mbers. Membershp ctos o the a evaato vaes ca be eveope. The Ecea stace base o cetro pots s appe to ezzy a the a zzy evaato vaes to eterme the ra orer o ateratves orer to compete the moe. Formas or the two cetro pots o horzota a vertca axes ca be ceary eveope. Fay a merca exampe has emostrate the comptatoa process o the seste moe. The seste moe ca aso be appe to sove may other zzy maaemet probems a mtpe crtera evromet. However whe appe the oow probems are worth coser examato: () The otcome co be eret mber o ecso maers stc rats o varos ateratves er eret qatatve crtera stc wehts or crtera ormazato orma... etc. are eret. () A case sty s eee to better test the 7

7 eectveess o the seste moe. ato the mber o crtera ca be aste po eret cases eee. (3) the seste moe the crtera strctre s mte to oe eve. Ths may ot be eoh. some paret crtera cota severa sb-crtera a some o these sb-crtera rther ce severa sb-sb-crtera a herarchca strctre mst be eee to escrbe ths stato. Acowemets The athor wo e to tha aoymos reerees or ther shts a sestos whch mae a better presetato o ths wor. Speca thas to Eat a Nye Khoa T Uye or ther ata coect a typ. Ths wor was spporte part by Natoa Scece Coc Tawa.O.C. er rat NSC 98-4-H-8-8-M. eereces [] S. Abbasbay a B. Asay a o zzy mbers by s stace ormato Sceces 76 (6) [] S. Abbasbay a T. Haar A ew approach or ra o trapezoa zzy mbers Compters a Mathematcs wth Appcatos [3] W. Atma What s the pot o maaemet costats? Eeer & Techooy 3 () [4] B. Asay The revse metho o ra zzy mber base o evato eree Expert Systems wth Appcatos [5]. Bortoa a. Dea A revew o some methos or ra zzy mbers Fzzy Sets a Systems 5 () [6] C. Carsso a. Fèr Fzzy mtpe crtera ecso ma: ecet eveopmets Fzzy Sets a Systems 78 () [7] U. Cebec a D. a A mt-attrbte comparso o Trsh qaty costats by zzy AHP teratoa ora o ormato Techooy & Decso Ma 6 () [8] C.T. Che A zzy approach to seect the ocato o the strbto ceter Fzzy Sets a Systems 8 () [9] S.. Che a C.. Hwa Fzzy Mtpe Attrbte Decso Ma Sprer Ber 99. [] C.C. Cho A zzy MCDM metho or sov mare trasshpmet cotaer port seecto probems Appe Mathematcs a Comptato 86 () [] T.. Cho S.T. Cho a.h. Tze Evaat T/S vestmets: A zzy mt-crtera ecso moe approach Eropea ora o Operatoa esearch 73 (3) [] T.C. Ch a.c. A exteso to zzy MCDM Compters & Mathematcs wth Appcato 7 (3) [3] F. Crepet D. Dpoet F. Kem B. Mehmapazr a F. Mer Costats a experts maaemet cost rms esearch Pocy 3 (9) [4] D. Dbos a H. Prae Operatos o zzy mbers teratoa ora o Systems Scece 9 (6) [5] A. Kama a M.M. pta M.M. trocto to Fzzy Arthmetc: Theory a Appcato Va Nostra eho New or 99. [6] X.W. a S.. Ha a zzy mbers wth preerece weht cto expectatos Compters a Mathematcs wth Appcatos 49 (-) [7] S. Ö üt S.S. Kara a E. ş o term spper seecto s a combe zzy MCDM approach: A case sty or a teecommcato compay Expert Systems wth Appcatos [8].A. bero Fzzy mtpe attrbte ecso ma: A revew a ew preerece ectato techqes Fzzy Sets a Systems 78 () [9] M. Sarem S.F. Mosa a A. Saaye TM costat seecto SMEs wth TOPSS er zzy evromet Expert Systems wth Appcato 36 () [] E. Trataphyo a C.T. Deveopmet a evaato o ve zzy mt-attrbte ecso-ma methos teratoa ora o Approxmate easo 4 (4) [] T.. Wa a H.F. Che Eects o tera spport a costat qaty o the cost process a EP system qaty Decso Spport Systems 4 () [] X. Wa a E.E. Kerre easoabe propertes or the orer o zzy qattes () & () Fzzy Sets a Systems 8 (3) [3].. Wa a H.S. ee The revse metho o ra zzy mbers wth a area betwee the cetro a ora pots Compters & Mathematcs wth Appcatos 55 (9) [4].M. Wa.B. a D.. X a K.S. Ch 6. O the cetros o zzy mbers Fzzy Sets a Systems 57 (7) [5] D. o Z. Zh a.. a zzy mbers wth a area metho s ras o yrato Compters a Mathematcs wth Appcatos 5 (6-7) [6].A. Zaeh The cocept o a stc varabe a ts appcato to approxmate reaso part a 3 ormato Sceces 8 (3/4) 99-49/ ; 9 () [7].A. Zaeh.A. Fzzy set ormato a cotro 8 (3)