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Ewa Roszowsa MULTI-CRITERIA DECISION MAING MODELS BY APPLYING THE TOPSIS METHOD TO CRISP AND INTERVAL DATA Abstract I ths paper, of the mult-crtera moels mag ecso, a Techque for Orer Preferece by Smlarty
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Ewa Roszowsa MULTI-CRITERIA DECISION MAING MODELS BY APPLYING THE TOPSIS METHOD TO CRISP AND INTERVAL DATA Abstract I ths paper, of the mult-crtera moels mag ecso, a Techque for Orer Preferece by Smlarty to a Ieal Soluto (TOPSIS), s escrbe. Some of the avatages of TOPSIS methos are: smplcty, ratoalty, comprehesblty, goo computatoal effcecy a ablty to measure the relatve performace for each alteratve a smple mathematcal form. The paper has a revew character. It systematses the owlege wth the scope of techques of ecso tag wth the use of the TOPSIS metho. Smple umercal eamples that referece real stuatos show practcal applcatos of fferet aspects of ths metho. The paper s orgaze as follows. The Itroucto presets a short overvew of the ecso mag steps as well as MCDM techques. Secto presets matr represetato of the MCDM problem. Secto escrbes the TOPSIS proceure for crsp ata, a Secto 3 for terval ata. The TOPSIS algorthm group ecso evromet the case of crsp a terval ata s also presete. I Secto 4 the problem of qualtatve ata TOPSIS moel s scusse. The umercal eamples showg applcatos of those techques the egotato process are presete Secto 5. Fally, coclusos a some coclug remars are mae last secto. eywors TOPSIS metho, umercal ata, terval ata, postve eal soluto, egatve eal soluto. Itroucto Mult-crtera ecso mag (MCDM) refers to mag choce of the best alteratve from amog a fte set of ecso alteratves terms of multple, usually coflctg crtera. The ma steps mult-crtera ecso mag are the followg [Hwag, Yoo, 98; Jahashahloo, Hossezaeh, Loft, Izahah, 006a]: MULTI-CRITERIA DECISION MAING MODELS... 0 establsh system evaluato crtera that relate system capabltes to goals, evelop alteratve systems for attag the goals (geeratg alteratves), evaluate alteratves terms of crtera, apply of the ormatve multple crtera aalyss methos, accept alteratve as optmal (preferre), f the fal soluto s ot accepte, gather ew formato a go to the et terato of multple crtera optmzato. Mult-crtera ecso mag techques are useful tools to help ecso maer(s) to select optos the case of screte problems. Especally, wth the help of computers, those methos have become easer for the users, so they have fou great acceptace may areas of ecso mag processes ecoomy or maagemet. Amog may mult-crtera techques, MAXMIN, MAXMAX, SAW, AHP, TOPSIS, SMART, ELECTRE are the most frequetly use methos [Che, Hwag, 99]. The ature of the recommeatos of of those methos epes o the problem beg aresse: choosg, rag or sortg. The selecto of moels/techques ca be also base o such evaluato crtera as: teral cosstecy a logcal souess, trasparecy, ease of use, ata requremets are cosstet wth the mportace of the ssue beg cosere, realstc tme a mapower resource requremets for the aalytcal process, ablty to prove a aut tral, software avalablty, where eee. The classfcato methos ca be categorze by the type of formato from the ecso maer (o formato, formato o attrbutes or formato o alteratves), ata type or by soluto ame at [Che, Hwag, 99, p.6-5]. The MAXMIN techque assume that the overall performace of a alteratve s eterme by ts weaest attrbute, the MAXMAX techque a alteratve s selecte by ts best attrbute value. The SAW (Smple Atve Weghtg) metho multples the ormalze value of the crtera for the alteratves wth the mportace of the crtera a the alteratve wth the hghest score s selecte as the preferre. The TOPSIS (Techque for Orer Preferece by Smlarty to the Ieal Soluto) selects the alteratve closest to the eal soluto a farthest from the egatve eal alteratve. The classcal TOPSIS metho s base o formato o attrbute from ecso maer, umercal ata; the soluto s ame at evaluatg, prortzg a selectg a the oly subjectve puts are weghts. The AHP (The Aaly- 0 Ewa Roszowsa tcal Herarchy Process) uses a herarchcal structure a parwse comparsos. A AHP herarchy has at least three levels: the ma objectve of the problem at the top, multple crtera that efe alteratves the mle a competg alteratves at the bottom. The major weaesses of TOPSIS are that t oes ot prove for weght elctato, a cosstecy checg for jugmets; o the other ha, the use of AHP has bee sgfcatly restrae by the huma capacty for the formato process. From ths pot of vew, TOPSIS allevates the requremet of pare comparsos a the capacty lmtato mght ot sgfcatly omate the process. Hece, t woul be sutable for cases wth a large umber of crtera a alteratves, a especally where objectve or quattatve ata are gve [Shh, Shyur, Lee, 007]. SMART (The Smple Mult Attrbute Ratg Techque) s smlar to AHP, a herarchcal structure s create to assst efg a problem a orgazg crtera. However, there are some sgfcat ffereces betwee those techques: SMART uses a fferet termology. For eample, SMART the lowest level of crtera the value tree (or objectve herarchy) are calle attrbutes rather tha sub-crtera a the values of the staarze scores assge to the attrbutes erve from value fuctos are calle ratgs. The fferece betwee a value tree SMART a a herarchy AHP s that the value tree has a true tree structure, allowg attrbute or sub-crtero to be coecte to oly hgher level crtero. SMART oes ot use a relatve metho for staarzg raw scores to a ormalze scale. Istea, a value fucto eplctly efes how each value s trasforme to the commo moel scale. The value fucto mathematcally trasforms ratgs to a cosstet teral scale wth lower lmt 0 a upper lmt. The ELECTRE (Elmato a Choce Epressg Realty) metho was to choose the best acto(s) from a gve set of actos, but t ca also be apple to three ma problems: choosg, rag a sortg. There are two ma parts to a ELECTRE applcato: frst, the costructo of or several outrag relatos, whch ams at comparg a comprehesve way each par of actos; seco, a eplotato proceure that elaborates o the recommeatos obtae the frst phase. Ths paper s focuse o the TOPSIS metho, whch was presete by Hwag a Yoo [98] a evelope later by may authors [Jahashahloo, Loft, Izahah, 006a; 006b; Zavasas,Turss, Tamosatee, 008; Hug, Che, 009]. The acroym TOPSIS stas for Techque for Orer Preferece by Smlarty to the Ieal Soluto. It s worth otg that the TOPSIS metho correspos to the Hellwg taoomc metho of orerg objects [Hellwg, 968]. The ma avatages of ths metho are the followg [Hug, Cheg, 009]: smple, ratoal, comprehesble cocept, tutve a clear logc that represet the ratoale of huma choce, ease of computato a goo computatoal effcecy, MULTI-CRITERIA DECISION MAING MODELS a scalar value that accouts for both the best a worst alteratves ablty to measure the relatve performace for each alteratve a smple mathematcal form, possblty for vsualzato. I geeral, the process for the TOPSIS algorthm starts wth formg the ecso matr represetg the satsfacto value of each crtero wth each alteratve. Net, the matr s ormalze wth a esre ormalzg scheme, a the values are multple by the crtera weghts. Subsequetly, the postve-eal a egatve-eal solutos are calculate, a the stace of each alteratve to these solutos s calculate wth a stace measure. Fally, the alteratves are rae base o ther relatve closeess to the eal soluto. The TOPSIS techque s helpful for ecso maers to structure the problems to be solve, couct aalyses, comparsos a rag of the alteratves. The classcal TOPSIS metho solves problems whch all ecso ata are ow a represete by crsp umbers. Most real-worl problems, however, have a more complcate structure. Base o the orgal TOPSIS metho, may other etesos have bee propose, provg support for terval or fuzzy crtera, terval or fuzzy weghts to moele mprecso, ucertaty, lac of formato or vagueess. I ths paper, the classcal TOPSIS algorthms for crsp, as well as terval ata are escrbe. Iterval aalyss s a smple a tutve way to trouce ata, ucertaty for comple ecso problems, a ca be use for may practcal applcatos. A eteso of the TOPSIS techque to a group ecso evromet s also vestgate. The cotet of mult-crtera group ecso mag both crsp a terval ata are escrbe. Fally, stuatos where crtera a ther weght are subjectvely epresse by lgustc varables are cosere. The practcal applcatos of the TOPSIS techque estmatg offers, for stace, buyer-seller echage are also propose.. The matr represetato of the MCDM problem The MCDM problems ca be ve to two s. Oe s the classcal MCDM set of problems amog whch the ratgs a the weghts of crtera are measure crsp umbers. Aother s the multple crtera ecso-mag set of problems where the ratgs a the weghts of crtera evaluate o complete formato, mprecso, subjectve jugmet a vagueess are usually epresse by terval umbers, lgustc terms, fuzzy umbers or tutve fuzzy umbers. I the classcal MCDM moel, we assume eact ata, objectve a precse formato, but ths s ofte aequate to moel real lfe stuatos. Huma jugmets are ofte vague uer may cotos. The soco-ecoomc 04 Ewa Roszowsa evromet becomes more comple, the preferece formato prove by ecso-maers s usually mprecse, a ca create hestato or ucertaty about prefereces. A ecso may have to be mae uer tme pressure a lac of owlege or ata, or the ecso-maers have lmte atteto a formato processg capactes. Most put formato s ot ow precsely, so that the values of may crtera are epresse subjectve or ucerta terms. The crtera, as well as ther weght, coul be subjectvely epresse by lgustc varables. Thus, may researchers etee the MCDM approach for ecso mag problems wth subjectve crtera, terval ata or fuzzy evromet usg grey system theory or fuzzy set theory. The grey system theory, evelope by Deg [98, 988] s base upo the cocept that formato s sometmes complete or uow [Ja, Hog, Frouz, Yusuff, 008; Lu, L, 006]. Eactly, the theory s base o the egree of formato ow whch s moele by tervals. If the system formato s uow, t s calle a blac system, f the formato s fully ow, t s calle a whte system. A a system wth formato ow partally s calle a grey system. The fuzzy set theory caot hale complete ata a formato, but s aequate to eal wth ucerta a mprecse ata [ahrama, 008; Che, Hwag, 99]. The avatage of the grey theory over the fuzzy theory s that the grey theory taes to accout the coto of the fuzzess; that s, the grey theory ca eal flebly wth the fuzzess stuato. We ca also coser sgle ecso mag a group ecso mag. Group ecso mag s more comple tha sgle ecso mag because t volves may cotractg factors, such as: coflctg vual goals, effcet owlege, valty of formato, vual motvato, persoal opo, power. I both mult-crtera ecso mag (MCDM) a group ecso mag (GDM), there are two steps: aggregato a eplotato. I MCDM, aggregato cossts combg satsfacto over fferet crtera whle GDM problem cossts combg the eperts opos to a group collectve. Group ecso mag ca be approache from two pots of vew. I the frst approach, vual mult-crtera moels are evelope base o vuals prefereces. Each ecso maer formulates a mult- -crtera problem efg the parameters accorg to these prefereces a solves the problem gettg a vual soluto set. Net, the separate solutos are aggregate by aggregato of operatos resultg the group soluto. I the seco approach, each ecso maer proves a set of parameters whch are aggregate by approprate operators, provg fally a set of group parameters. Upo ths set the mult-crtera metho s apple a the soluto epresses group preferece [Rgopoulos, Psarras, Asous, 008]. MULTI-CRITERIA DECISION MAING MODELS Solvg of each mult-crtera problem (vual or group ecso) begs wth the costructo of a ecso mag matr (or matrces). I such matres, values of the crtera for alteratves may be real, tervals umbers, fuzzy umbers or qualtatve labels. Let us eote by D = {,,..., } a set of ecso maers or eperts. The mult-crtera problem ca be epresse matr format the followg way: where: A C C C A A m m m m A, A,..., Am are possble alteratves that ecso maers have to choose from, C,C,..., C are the crtera for whch the alteratve performace s measure, j s the ecso maer ratg of alteratve A wth respect to the crtero C ( s umercal, terval ata or fuzzy umber). where j j I ths way for m alteratves a crtera we have matr X = ( ) j s value of alteratve wth respect to j crtero for ecso maer, j =,,...,, =,,...,. The relatve mportace of each crtero s gve by a set of weghts whch are ormalze to sum to. Let us eote by W = [ w,w,..., w ] a weght vector for ecso maer, where w j R s the ecso maer weght of crtero C j a w + w w =. I the case of ecso maer we wrte j, w j, X, respectvely. Mult-crtera aalyss focuses maly o three types of ecso problems: choce select the most approprate (best) alteratve, rag raw a complete orer of the alteratves from the best to the worst, a sortg select the best alteratves from the lst. j 06 Ewa Roszowsa. The classcal TOPSIS metho I the classcal TOPSIS metho we assume that the ratgs of alteratves a weghts are represete by umercal ata a the problem s solve by a sgle ecso maer. Complety arses whe there are more tha ecso maers because the preferre soluto must be agree o by terest groups who usually have fferet goals. The classcal TOPSIS algorthm for a sgle ecso maer a for group ecso mag s systematcally escrbe Secto. a Secto., respectvely... The classcal TOPSIS metho for a sgle ecso maer The ea of classcal TOPSIS proceure ca be epresse a seres of followg steps [Che, Hwag, 99; Jahashahloo, Loft, Izahah, 006a]. Step. Costruct the ecso matr a eterme the weght of crtera. Let ( ) j X = be a ecso matr a W = [ w, w,..., w ] a weght vector, where R, w R a w w w. j j + = Crtera of the fuctos ca be: beeft fuctos (more s better) or cost fuctos (less s better). Step. Calculate the ormalze ecso matr. Ths step trasforms varous attrbute mesos to o-mesoal attrbutes whch allows comparsos across crtera. Because varous crtera are usually measure varous uts, the scores the evaluato matr X have to be trasforme to a ormalze scale. The ormalzato of values ca be carre out by of the several ow staarze formulas. Some of the most frequetly use methos of calculatg the ormalze value j are the followg: j = m j = j, (.) j j =, (.*) ma j MULTI-CRITERIA DECISION MAING MODELS j m j f C s a beeft crtero ma j m j = (.**) j ma j j f C s a cos t crtero ma j m j for =,, m; j =,,. Step 3. Calculate the weghte ormalze ecso matr. The weghte ormalze value v j s calculate the followg way: v = w for =,, m; j =,,. (.) j j j where w s the weght of the j-th crtero, w =. j j= j Step 4. Determe the postve eal a egatve eal solutos. Ietfy the postve eal alteratve (etreme performace o each crtero) a etfy the egatve eal alteratve (reverse etreme performace o each crtero). The eal postve soluto s the soluto that mamzes the beeft crtera a mmzes the cost crtera whereas the egatve eal soluto mamzes the cost crtera a mmzes the beeft crtera. + Postve eal soluto A has the form: ( v v,..., v ) = mav j I, mv j J. + A =, j j (.3) Negatve eal soluto A has the form: A = ( v, v,..., v ) = mv j j I, mav j j J (.4) where I s assocate wth beeft crtera a J wth the cost crtera, =,, m; j =,,. 08 Ewa Roszowsa Step 5. Calculate the separato measures from the postve eal soluto a the egatve eal soluto. I the TOPSIS metho a umber of stace metrcs ca be apple *. The separato of each alteratve from the postve eal soluto s gve as + = / p p + ( vj v j ), =,,, m. (.5) j= The separato of each alteratve from the egatve eal soluto s gve as = / p p ( vj v j ), j= =,,, m. (.6) Where p. For p = we have the most use tratoal -mesoal Euclea metrc. + = ( + vj v j ) j=, =,,, m, (.5*) = ( vj v j ) j=, =,,, m. (.6*) Step 6. Calculate the relatve closeess to the postve eal soluto. The relatve closeess of the -th alteratve A j wth respect to A + s efe as where 0 R, =,,, m. R = + +, (.7) Step 7. Ra the preferece orer or select the alteratve closest to. A set of alteratves ow ca be rae by the esceg orer of the value of. R * Possble metrcs the frst power metrc (the least absolute value terms), Tchebychev metrc or others [see ahrama, Buyuoza, Ates, 007; Olso 004]. MULTI-CRITERIA DECISION MAING MODELS The classcal TOPSIS metho for group ecso mag I ths part we epla the etale TOPSIS proceure for group ecso mag base o the Shh, Shyur a Lee proposto [Shh, Shyur, Lee, 007]. Step. Costruct the ecso matres a eterme the weghts of crtera for -ecso maers. Let = ( ) X be a ecso matr, W = [ w,w,...,w ] weght vector j for ecso maer or epert, where j R, w j R, w + w w = for =,,...,. Step. Calculate the ormalze ecso matr for each ecso maer. I ths step some of the earler escrbe methos of ormalzato ca be use. Let us assume that we use j r = m = j ( ). j (.8) I ths proceure weghts are mapulate the et step. Step 3. Determe the postve eal a egatve eal solutos for each ecso maer. The postve eal soluto + A for ecso maer has the form {,,..., } ( ) A = r r r = ma r, ( ). j j I m r j j J (.9) The egatve eal soluto A for - ecso maer has the form: {,,..., } m( ) A = r r r = r, ma( ), j j I r j j J (.0) where I s assocate wth the beeft crtera a J wth the cost crtera. 0 Ewa Roszowsa Step 4. Calculate the separato measures from the postve eal soluto a the egatve eal soluto. Step 5.. Calculate the separatato measure for vuals. The separato of -th alteratve A from the postve eal soluto for each ecso maer s gve as + A p + ( ), m + p = wj rj rj =,,, m. (.) j= The separato of -th alteratve A from the egatve eal soluto for each ecso maer s gve as A p m p ( ) =, w j rj rj =,,, m, (.) j= where p. For p = we have the Euclea metrc. Step 5.. Calculate the separato measure for the group. The aggregato for measure for the group measures of the postve eal * a egatve eal soluto for the -th alteratve A s gve by of the operators: arthmetc mea: *+ + * + = = a (.3) * = = geometrc mea: or * + + = = * a =. (.3*) = Step 6. Calculate the relatve closeess to the postve eal soluto. The relatve closeess of the alteratve A to the postve eal soluto s efe as MULTI-CRITERIA DECISION MAING MODELS... * * R = * * + + for =,,, m (.4) where 0 R *. The larger the e value, the better the evaluato of the alteratve. Step 7. Ra the preferece orer or select the alteratve closest to. A set of alteratves ca ow be rae by the esceg orer of the value * of R. 3. The TOPSIS metho wth crtera values eterme as terval I some cases etermg the eact value of crtera s ffcult a ecso maers are usually more comfortable provg tervals to specfy moel put parameters. A terval umber ata formulato s a smple a tutve way to represet ucertaty, whch s typcal of real ecso problems. Here, the TOPSIS metho usg terval as the bass for evaluatg value alteratves s escrbe. However, we ca also coser a terval weghts escrpto [Ja, Hog, Frouz, Yusuff, 008]. 3.. The TOPSIS metho wth attrbute values eterme as terval for a sgle ecso maer A algorthmc metho whch etes TOPSIS for ecso-mag problems wth terval ata was propose by Jahashahloo, Loft, Izahah. Ths proceure ca be escrbe the followg steps [Jahashahloo, Loft, Izahah, 006a]. Step. Costruct the ecso matr a eterme the weght of crtera. Let = ( ) X be a ecso matr a W = w, w,..., w ] a weght vector, j where = [ j ] j j [,,, R, w R a w + w w. j j j = Step. Calculate the ormalze terval ecso matr. The ormalze values, j are calculate the followg way: j = m = j j ( j ) + ( j ) ) for =,, m; j =,,. (3.) Ewa Roszowsa = j j ( ) ( ) ) m j + j = The terval [ j, j ] s ormalze value of terval [ j j ] for =,, m; j =,,. (3.),. Step 3. Calculate the weghte ormalze terval ecso matr. The weghte ormalze values v j a vj are calculate the followg way: v = w for =,, m; j =,,, (3.3) j j j v j = w j j for =,, m; j =,,, (3.4) where w s the weght of the j-th crtero, w =. j j= j Step 4. Determe the postve eal a egatve eal solutos. The postve eal soluto has the form + A : A = ( v, v,..., v ) = mavj j I, mvj j J. (3.5) The egatve eal soluto has the form A : A = ( v, v,..., v ) = mv,, j j I mavj j J (3.6) where I s assocate wth beeft crtera a J wth cost crtera. Step 5. Calculate the separato measures from the postve eal soluto a the egatve eal soluto. The separato of each alteratve from the postve eal soluto s gve as * : + = ( + ) ( + vj v j j ) j + v v j= j=, =,,, m. (3.7) * Tratoal TOPSIS apple to Euclea orm s presete here. However, we ca also use other metrcs. MULTI-CRITERIA DECISION MAING MODELS... 3 The separato of each alteratve from the egatve eal soluto s gve as: = ( vj v j ) + ( vj v j ) j= j=, =,,, m. (3.8) Step 6. Calculate the relatv
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