Ranking Bank Branches with Interval Data By IAHP and TOPSIS

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1 Rag Ba Braches wth terval Data By HP ad TPSS Tayebeh Rezaetazaa Departmet of Mathematcs, slamc zad Uversty, Badar bbas Brach, Badar bbas, ra Mahaz Barhordarahmad Departmet of Mathematcs, slamc zad Uversty, Badar bbas Brach, Badar bbas, ra BSTRCT Ths paper proposes a method for rag decso mag uts (DMUs) usg some of the multple crtera decso mag / multple attrbute decso mag (MCDM /MDM) techques, amely, terval aalytc herarchy process (HP) ad the techque for order preferece by smlarty to a deal soluto (TPSS). Sce the effcecy score of uty s assged to the effcet uts, we determe the effcet uts by stadard DE models, ad calculate the weghts of the crtera usg HP. t should be metoed that the judgmets are made crsp the terval parwse comparso matr by the Mote Carlo smulato. the ed, we utlze TPSS usg HP to ra ba braches ra. Keywords terval aalytc herarchy process, Mote Carlo smulato, TPSS, Rag. decso mag techque developed by Hwag ad Yoo [], receved a lot of atteto, sce t has a strog logc ad cosders both deal ad o-deal solutos. Recetly HP ad TPSS have bee used the fuzzy evromet by Öüt ad Soer []. order to solve a problem, TPSS requres the weghts of the crtera. Thus, amog the techques for calculatg weghts, HP s more relable because of havg advatages such as parwse comparso, usg terval judgmets rather tha judgmets wth precse values, ad computg the compatblty rate. The terval data ad terval parwse comparso are tured to crsp data by usg Mote Carlo (MC) smulato. Ths paper s orgazed as follows. The proposed model s descrbed secto. The theoretc descrptos for HP ad TPSS methods are preseted secto. ad.. secto three, the results of the proposed method are elaborated o usg a umercal eample. Fally, the paper s cocluded secto.. troducto DE, proposed by Chares et al. [], s a method for evaluatg DMUs. The frst ad secod basc DE models were proposed by Chares et al. [] ad Baer et al. [], respectvely. basc DE models, effcet uts receve the effcecy score of uty ad effcet oes receve scores less tha oe. Effcet uts wll fally be cluded the rag lst. basc DE models, the data are assumed to be defte. But dfferet applcato of DE, the data ca be probablstc, terval, or fuzzy. Thus, ths paper, the case whch the data are terval s cosdered. HP has bee recetly proposed by Jablosy [] to measure the relatve effcecy of producto uts. TPSS, a multple attrbute. The propose method ths secto, HP ad TPSS are troduced. s the data obtaed from the data collecto may ot be precse practcal tass, we cosder the data as tervals; these are tured to crsp data by usg MC smulato. Moreover, order for the judgmets to be more realstc, the judgmet tervals are determed by the decso maer, whch ca be computed wth the possblty of computg the decso compatblty rate HP, to mae sure of the compatblty of the decso. The, usg Saaty s [] egevector problem, the relatve weght ca be obtaed. The weghts of the crtera are the calculated by HP. Fally, TPSS uses the weghts of the crtera obtaed by HP ad performs the fal rag of the choces by presetg teratoal Joural of formato, Securty ad System Maagemet, 0, Vol., No., pp. 0-

2 JSSM, 0, (): 0- the best suggested soluto. the followg subsectos, HP ad TPSS are revewed... HP HP s a multple attrbute decso mag techque whch both quattatve ad qualtatve crtera ca be used. the herarchy dagram, goals hold the hghest posto, crtera ad subcrtera are placed the et level, ad choces are the lowest posto. order to use HP for rag, the parwse comparso matr s obtaed from the prefereces derved from Saaty s [] e-pot scale; that s as follows. testy of mportace Table : Scale of mportace Defto Equal mportace Wea mportace of oe over aother Essetal or strog mportace Very strog or demostrated mportace bsolute mportace,,, termedate values betwee adjacet scale values The weghts derved from comparso betwee crtera ad goals ad betwee choces ad crtera are combed to obta the fal weght. Cosderg the ucertaty ad complety of real-world problems, t s more realstc for the decso maer to use oral epressos rather tha accurate comparsos ad terval judgmets. geeral, terval parwse comparso matrces s as follows. () [ a, a] [ a, a ] [ a, a ] [ a, a ] [ aj ],, j,..., [ a, a] [ a, a ] for the elemets l a s the lower boud ad j u a s the upper boud j a, ad they reflect the valuato of ut j over ut j. free cells are fll by recprocal codto as follows: a l j u a j (),, j,...,. We coduct 000 smulato epermets for trasformg the terval comparsos to crsp. The, we calculate the weghts usg methods for obtag weghts, such as Saaty s [] egevector problem. () w. w,,...,, w, ma s the largest egevalue of ad w s the ma ormalzed egevector belogg to. e of the ma advatages of HP s ts ablty to determe the compatblty of the decso. () ma R, R Saaty [], R s called radom de of matrces. Thus, as stated by Saaty [], f R<0., the the compatblty of the decso s acceptable; otherwse, the decso maer had better revse the decso... TPSS TPSS s oe of the multple attrbute decso mag techques for presetg the best suggested soluto. The selected choce should have the shortest dstace from the postve deal soluto ad, meawhle, the logest dstace from the egatve deal soluto. The attrbutes proft ad cost are cosdered as the postve ad egatve deal solutos, respectvely. TPSS smultaeously eames the dstace of the choce from both postve ad egatve deal solutos, by calculatg the relatve closeess to the postve deal soluto. t also evaluates the decso matr cotag m alteratves ad attrbutes. m () C C C C m.. j m j j j mj m,,..., m are the alteratves ad j,,..., m, j,..., s the umercal value obtaed from the th alteratve ad the jth attrbute. The stages of usg TPSS are as follows

3 JSSM, 0, (): 0-. Normalzg the decso matr, j j m j r (),,..., m, j,...,. Weghtg the ormalzed decso matr: v j w () w r,,..., m, j,...,,, j w are the weghts obtaed by HP for the attrbutes,. Determg the postve ad egatve deal solutos: we defe two vrtual alteratves ad the least effectve choces, respectvely, Subject to: {(ma v j () j ),(m v () j as the most ad j J),..., } { j,..., s assocated wth beeft crtera.}, J { j,..., J s assocated wth cos t crtera.}. Calculatg the dstaces: the dstace of each choce ca be measured by the Eucldea method, j (0) d ( vj v j ),,..., () d ( vj v j ),,..., j. Calculatg the relatve closeess of to, () d R,,..., d d. Numercal eample We cosder 0 ba braches wth e attrbutes wth terval data. The frst thrd crtera were tae as puts ad the remag oes as outputs of the model.. terest pad. Worg hour of Persoel. Demad. terest-free savg accout. Curret accout. Short-term. Log-term. Loas. terest receved The terval put ad output of ths study are gve tables ad Table : terval puts of ba braches. W.H. uts terest pad Demad Persoel 0 0 [00.,000.] [00.,000.] [00.0,000.0] [0.,00.] [00.,000.] [000.0,00.0] [00.,0.] [00.,00.] [00.0,0.0] [00.0,000.0] [00.,000.] [00.,000.] [000.,00.] [00.,000.] [00.,000.] [000,000] [000.,00.] [00.,0.] [00.,000.] [00.,0.] [.,.] [.,.] [.,.] [.,.] [.,.] [.,.] [.,.] [.,.] [0.,.] [.,.] [0.,.] [.,.] [0.,.] [0.,.] [.,.] [.,.] [.,.] [.,.] [.,.] [.,.] [000,000] [0000,000] [0000,000] [000,00] [000,00] [000,0000] [00,000] [00,000] [000,0000] [00,000] [000,00] [000,00] [00,00] [00,000] [0000,000] [00,000] [00,000] [000,00] [00,000] [0000,000]. Rag the choces: the alteratves ca be raed descedg order of R.

4 JSSM, 0, (): 0- Table : terval outputs of ba braches. Log-term uts savg accout Curret accout Short-term Loas terest receved [0000,0000] [000,000] [00,00] [00,000] [00000,00000] [000.,0000.] [00,000] [000,000] [000,00] [000,000] [000,000] [00.,00.] [000,000] [000,0000] [00,00] [00,00] [00000,00000] [00.,00.] [000,00] [000,000] [00,00] [00,00] [000,000] [00.,00.] [000,00] [000,000] [00,00] [00,000] [000,000] [00.,000.] [000,000] [0000,0000] [000,000] [000,0000] [00000,00000] [000.,000.] [00,000] [000,000] [0,0] [00,00] [000,000] [000.,00.] [00,000] [000,0000] [000,00] [000,000] [000000,000000] [00.,00.] [00,00] [000,000] [000,000] [000,000] [00000,00000] [000.,000.] 0 [00,000] [000,000] [00,00] [00,00] [00000,000000] [000.,000.] [000,0000] [0000,0000] [00,00] [00,00] [00000,00000] [000.,000.] [0000,000] [000,000] [0,0] [000,000] [0000,000] [000.,000.] [000,00] [000,000] [00,00] [00,00] [00000,00000] [000.,000.] [00,00] [000,000] [000,000] [00,00] [000,000] [00.0,00.0] [00,000] [000,0000] [000,000] [000,00] [0000,0000] [00.,00.] [000,000] [000,0000] [000,000] [00,00] [000,0000] [00.,00.] [000,00] [000,000] [00,00] [0000,000] [000000,00000] [0000.,000.] [00,000] [000,000] [00,00] [00,00] [000,000] [00.,00.] [000,000] [000,000] [000,000] [00,00] [00000,00000] [00.,000.] 0 [000,00] [000,00] [00,00] [00,000] [0000,00000] [00.,00.] The terval data are tured to crsp data by MC smulato, the they are ormalzed. The crsp state of data Table, are used as the puts ad outputs of the BCC model, ad effcet DMUs are detfed. Now we mae terval judgmets for BCC-effcet alteratves usg HP ad we ca used HP process ths stage. [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] Table : Crsp parwse comparsos ad calculatos. w w ma R 0.0

5 JSSM, 0, (): 0- s stated by Saaty [], cosderg the compatblty rate of the system < 0., system compatblty s acceptable; other words, HP computatos are compatble. s the crtero matr has bee ormalzed HP, ow we weght the ormalzed decso matr usg the weghts obtaed from HP, ad the determe the postve ad egatve deal solutos. The results obtaed from calculatg the dstaces ad rag are preseted the table. uts 0 0 Table : The effcecy scores ad rag. d d R Ra BCC Refereces ) R.D. Baer,. Chares ad W.W. Cooper, Some models for estmatg tech- cal ad scale effceces data evelopmet aalyss, Maagemet Scece, 0 (), 0-0. ). Chares, W.W. Cooper ad E. Rhodes, Measurg the effcecy of decso mag uts, Europea Joural of peratoal Research, (), -. ) C.L. Hwag ad K. Yoo, Multple ttrbute Decso Mag Methods ad pplcato: State of the rt Survey. Sprger, New Yor,. ) J. Jablosy, Measurg the effcecy of producto uts by HP models, Mathematcal ad Computer Modellg, (00), 0-0. ) S. Öüt ad S. Soer, Trasshpmet ste selecto usg the HP ad TPSS approaches uder fuzzy evromet, Waste Maagemet (00), press. ) T.L. Saaty, The alytc Herarchy Process, McGrow-Hll, 0 s ca be see from the above table, the frst alteratve holds the frst ra ad the th alteratve has the last ra.. Cocluso The am of ths paper was to ra ba braches usg MCDM / MDM techques, TPSS ad HP. t ca be observed from the results of the evaluato of 0 ba braches, usg HP ad TPSS, that the frst ad the th braches have the best ad the worst ras, respectvely, the rag lst, as compared to the ras of other caddates. Ths paper utlzed TPSS to ra ba braches usg HP. Sce terval judgmets are more assurg, more fleble ad more realstc, HP has bee used to determe the weghts of the crtera the our proposed method.