A Multiplicative Approach to Derive Weights in the Interval Analytic Hierarchy Process

Transcription

1 Iteratoal Joural of Fuzzy Systems, Vol. 3, No. 3, September A Multplcatve Approach to Derve Weghts the Iterval Aalytc Herarchy Process Jg Rug Yu, Yu-We Hsao, ad Her-Ju Sheu Abstract Ths paper proposes a multplcatve approach to solve defectve ssues lecographc goal programmg of the aalytc herarchy process (LGPAHP). Although the LGPAHP ca hadle cosstet terval comparso matrces, t geerates dfferet weghts the upper ad lower tragular terval udgmets. Istead of adoptg addtve costrats, the proposed method uses multplcatve costrats to cope wth defceces heret the LGPAHP. Moreover, sce two devato varables are used to cope wth cosstet cases, weghts ca be flebly derved, wthout specfyg ay advace tolerace parameters. Four eamples are preseted to llustrate the proposed method more detal. Keywords: Iterval Aalytc Herarchy Process, Multplcatve Aalytc Herarchy Process, Multple Crtera Decso Makg, Icosstecy.. Itroducto Multple crtera decso makg (MCDM) s becomg creasgly mportat the curret dyamc evromet [6, 23, 3], as decsos formed upo thoughts wth multple crtera ca satsfy a varety of ssues ad reach compromses that result greater harmoy for both socety ad the partes volved. The aalytc herarchy process (AHP) plays a promet role MCDM [29]. The aalytc herarchy process (AHP) was orgally developed by Satty 980 [2], ad s a well-kow method used by decso makers to deal wth multple crtera decso-makg problems. Sce the, may Correspodg Author: Jg Rug Yu s wth the Departmet of Iformato Maagemet, Natoal Ch Na Uversty, 470 Uversty Rd., Pul Natou 545, Tawa. E-mal: efer@cu.edu.tw Yu-We Hsao E-mal: s32353@cu.edu.tw Her-Ju Sheu s wth the Departmet of Bakg ad Face ad the Departmet of Iformato Maagemet, Natoal Ch Na Uversty, 470 Uversty Rd., Pul Natou 545, Tawa. E-mal: hsheu@cu.edu.tw Mauscrpt receved July 2008; revsed August 200; accepted May 20. applcatos have bee publshed based o AHP [2, 3, 5, 7, 26]. The covetoal AHP calculates crsp parwse udgmets to derve weghts wthout cosderg the ucertaty of huma tuto. Satty ad Vargas [22] proposed the cocept of a terval AHP, whch ca vary crsp udgmets wth terval udgmets usg the Mote Carlo smulato approach, order to derve weghts from terval udgmets. However, for terval udgmets, t s dffcult to measure cossteces to geerate weghts [27]. Arbel [2], ad Arbel ad Vargas [3] poted out that terval udgmets ca be regarded as costrats o weghts. Kress [6] suggested that Arbel s method caot obta the prortes f terval udgmets are cosstet. To address ths ssue, Mkhalov ad Sgh [8, 9] proposed the fuzzy preferece programmg method (FPP), whch derves crsp prortes from terval or crsp comparso matrces. The FPP s used to geerate weghts, ad cosders both cosstet ad cosstet terval comparso matrces by troducg tolerace parameters. However, the FPP requres the decso makers to predeterme the values of all tolerace parameters for all udgmets, whle dealg wth cossteces. Meawhle, to address the cosstecy problems the terval AHP, Islam et al. [4], proposed the Lecographc Goal Programmg for AHP (LGPAHP), whch geerates weghts va terval parwse comparso matrces usg devato varables, especally from cosstet matrces. To mmze the summato of devato varables the obectve fucto, the cosstet matr s detfed va the postve obectve value, wthout pre-specfyg ay parameter. However, the above terval AHP methods cota addtve costrats. Barzla ad Gola [5] clarfed the rak reversal problem occurrg the AHP, suggestg the reaso for ths occurrece s the erroeous use of addtve aggregato methods. They proposed that multplcatve methods should be used place of aggregato methods, order to preserve rato scale propertes. Therefore, multplcatve AHP (MAHP) has bee wdely developed ad dscussed [4, 7, 9, 0,, 20, 24, 25]. Rak reversal wll ot occur MAHP oce the alteratves are removed [24]. I aother approach, logarthmc goal programmg (LGP) by Bryso geerates a prorty vector, but oly for 20 TFSA

2 226 Iteratoal Joural of Fuzzy Systems, Vol. 3, No. 3, September 20 crsp comparso matrces, ad ot for terval comparso matrces [7, 25]. Bryso ad Joseph [8] proposed a tegrated logarthmc goal programmg techque for geeratg a group cosesus prorty vector group decso-makg cotets. Meawhle, the logarthmc least-squares method (LLS) [0], the logarthmc least absolute values method (LLAV) [], ad stochastc decso-makg usg a multplcatve AHP [20] are also used for crsp comparso matrces. Chadra et al. [9] appled a two-stage lear programmg (LP) model to determe weghts, usg logarthms for crsp ad terval cosstet matrces. Ther approach adopts the geometrc mea for terval udgmets, however, t has too may costrats wth regard to the umber of weghts. Wag [27] dcated ad proved the drawbacks of the LGPAHP, showg t obtas dfferet prortes ad rakgs from the upper ad lower tragular udgmets, whch result from the addtve costrats for geeratg weghts. Hece, the LGPAHP s uable to detfy whch weght vector ad rakg s more accurate [27]. However, wth the ecepto of Wag et al. [28], o soluto for ths problem s provded Wag s paper [27]. Wag et al. [28] used a two-stage logarthmc goal programmg method to dvdually geerate terval weghts, whch esures that the same terval weghts are geerated from both the upper ad lower tragular udgmets of a terval comparso matr. However, whe more terval weghts are obtaed, loger processg tmes are requred, ad the process becomes comple. To overcome ths defect, ths paper proposes a multplcatve techque wth the obectve of mmzg the summato of devato varables. By usg multplcatve costrats rather tha addtve costrats, the proposed method ca hadle the upper tragular udgmets as well as lower tragular udgmets the terval parwse comparso matr. I other words, the proposed method ca address the cosstecy problems ad overcome the defceces of LGPAHP poted out by Wag [27]. The remader of ths paper s orgazed as follows. Secto 2 troduces the LGPAHP method. Secto 3 presets the proposed method, a modfcato of the LGPAHP. Comparatve results of the LGPAHP ad the proposed method, wth four eamples, are demostrated Secto 4. The paper s cocluded Secto The LGPAHP model The LGPAHP deals wth terval comparso matrces, especally the hadlg of cosstet terval matrces. Suppose that a terval parwse comparso matr A wth terval compoets s provded by a decso maker. Takg to accout the terval udgmets from the upper tragular udgmets of the followg matr A, we have the followg [24]: [ l 2, u2] [ l3, u3] [ l, u ] [ l2, u2] [ l23, u23] A= [ l, ] [, ] 3 u3 l32 u32 () [ l, u ] [ l 2, u 2] [ l 3, u 3] where l u, l = / u ad u = / l. Let weght vector, W = (w,, w ) ad w =. Each terval udgmet = matr A ca be defed as the followg equalty (2) or (3): w l u w, (2) or w wu 0, w + wl 0, (3) =, 2,...,, = +, + 2,...,, >. For equalty (2), the terval rato w / w s raged from l to u. The cosstet cases satsfy the above two equalty costrats, whereas the cosstet cases are ot teable. Thus, the devato varables are troduced to allow the equalty costrats to have valdty cosstet udgmets, where equalty (3) s trasformed to the followg equalty (4): lw p w uw + q, (4) =, 2,...,, = +, + 2,...,, >. where the devato varables p ad q are both oegatve real umbers, however, both caot be postve at the same tme, amely p q = 0. Iequalty (4) dcates that, there are two codtos: () whe the udgmets are cosstet, both p ad q are foud to be 0; () whe the udgmets are cosstet, ether p or q must be greater tha 0. Codto () s equvalet to equalty (2). Sce the obectve fucto s to mmze the cosstecy of the terval comparso matr, a summato of all devato varables of p ad q s requred to be as small as possble ts equalty (4). By usg the above metoed epressos, the lecographc goal programmg (LGPAHP) model [4, 27], s as follows: M Z = ( p + q ) (5) = = + s.t. w w u q 0, (6) w w l + p 0, (7)

3 Jg Rug Yu et al.: A Multplcatve Approach to Derve Weghts the Iterval Aalytc Herarchy Process 227 w = =, (8) w 0, (9) p 0, (0) q 0, () =, 2,...,, = +, + 2,...,, >. The LGPAHP utlzes equatos (5)-(7) to detfy ad deal wth the cosstecy va mmzg the summato of devato varables. I other words, the model ca flebly hadle cosstet cases of the terval comparso udgmet, thus, elmatg loadg by decso makers. I cotrast to the FPP, where all tolerace parameters for the cosstet case the FPP are assged by a decso maker. However, the drawback of the LGPAHP s to obta two dfferet weghts ad rakgs, whch are geerated from the upper ad lower tragular udgmets of the cosstet terval parwse comparso matr, respectvely [27, 28]. 3. The proposed method Sce the LGPAHP would lead to defectve prortes ad rakgs due to the covetoal addtve costrat (8). Istead of employg addtve costrats, ths study uses multplcatve costrats [28], ad corporates the advatage of the LGPAHP, amely, by usg devato varables to solve the mperfecto proved by Wag [27]. The proposed method adopts the pros of the LGPAHP ad multplcatve costrats poted out by Barzla ad Gola [5]. Takg logarthms of terval udgmets, we ca geerate ratoal weghts ad the same prorty rakgs by processg the upper ad lower tragular parts of the terval comparso matr. Sce usg the multplcatve costrats to obta weghts dffers from usg the addtve costrats, the weght vector s deoted as W = (v,, v ). The multplcatve type of the codto s, v =, ad the = logarthmc equato s, lv = 0. The multplcatve = costrats am to resolve the ssues of dfferet weghts, whch may cause dfferet prorty rakgs from upper ad lower tragular udgmets of the same matr. For the upper tragular udgmets, ths study uses a loga- v rthmc cocept to trasform equalty l u v to the followg epresso: ll p lv lv l u + q, =, 2,...,, = +, + 2,...,, >. (2) The devato varables p ad q are used for the codto of logarthmc costrats to cope wth cosstecy ssues. Whe both varables p ad q are equal to 0, the matr s cosstet. O the other had, f the matr s cosstet, oe of the devato varables should be greater tha 0. The decso varable s troduced the proposed model: = l (v ). Note that, = 0.e. v =, ad all fal weghts ca be ormalzed to the sum of, ad all decso varables of are urestrcted. The equalty (2) s reformulated as the followg: + p l l, q lu, =, 2,...,, = +, + 2,...,, >. (3) The the model for upper tragular udgmets s as follows: < The model for upper tragular udgmets > M Z = ( p + q ) (4) = = + s.t. + p ll, (5) q lu, (6) = 0, (7) p 0, q 0, (8) =, 2,...,, = +, + 2,...,, urestrcted. For the lower recprocal tragular udgmets, the equalty v s trasformed to the followg u v l epresso, whe a cosstecy occurs: v l( ) p l l( ) + q, u v l =, 2,...,, = +, + 2,...,, >. (9) Let = l (v ). + q l l, p l u, =, 2,...,, = +, + 2,...,, >. (20) The, the model for lower tragular udgmets s as follows: < The model for lower tragular udgmets > = 2 M Z = ( p + q ) (2) s.t. = + q l, (22) l p u, (23) l = 0, (24)

4 228 Iteratoal Joural of Fuzzy Systems, Vol. 3, No. 3, September 20 p 0, q 0, (25) = 2, 3,...,, =, 2,...,. urestrcted. Theorem : The models for upper ad lower tragular udgmets are equvalet. Proof: From equaltes (3) ad (20), we ca see p = q ad q = p, ad equaltes (3) ad (20) are equvalet. Therefore, the summato of all recprocal pars of devato varables ( p + q ) s equal to ( p = = + + q = 2 = ). Hece, these two models are completely equvalet. Accordg to Theorem, both the above multplcatve models are detcal, ad thus, ca geerate the same weghts ad prorty rakgs. The devato varables are used to allow the cossteces occurrg, ad the hadle them well wthout specfyg ay advace tolerace parameters. Hece, ths study, the defcecy of the LGPAHP, as poted out by Wag [27], s solved. I the et secto, four umercal eamples are employed to llustrate the proposed method more detal. vector of the upper tragular matr s W = (0.303, 0.455, 0.5, 0.09) T, ad the rakg s w 2 > w > w 3 > w 4. The weght result for the lower tragular matr s W = (0.364, 0.364, 0.82, 0.090) T, ad the prorty rakg s w = w 2 > w 3 > w 4. Whe all the udgmets of the matr A are employed, W = (0.370, 0.370, 0.85, 0.074) T, ad the prorty rakg s w = w 2 > w 3 > w 4. By usg the LGPAHP o the same matr, three dfferet prorty vectors, ad two dfferet prorty rakgs, are geerated. I fact, the lower tragular udgmets of the matr A provde the same formato as the upper tragular udgmets, thus, a detcal soluto should be obtaed. However, the LGPAHP rases a questo, whch weght soluto or rakg s more relable. O the other had, by usg the proposed method, the same prorty ad rakg from the upper, lower tragular, or all udgmets of the matr are realzed, amely the geerated weght vector W = (0.28, 0.469, 0.56, 0.094) T, ad the rakg s w 2 > w > w 3 > w 4. Thus, the proposed method overcomes the drawback of the LGPAHP very well. Table. The weght comparso of Eample. 4. The umercal eamples Wag [27] eamed three cosstecy eamples to hghlght how the LGPAHP geerates dfferet weghts resultg dfferet rakgs, whe the upper or lower tragular udgmets of both parts of a terval comparso matr are adopted. Therefore, these three eamples are demostrated to show that the proposed method ca overcome ths defcecy, ad are compared wth the results of Wag [27]. The, a vestg case [4] s also preseted eample 4. Eample : Ths followg terval comparso matr s from Kress [6], Islam et al. [4], ad Wag [27]. [, 2] [, 2] [2, 3] [, ] [3, 5] [4, 5] 2 A = [, ] [, ] [6, 8] [, ] [, ] [, ] The upper, lower tragular udgmets, ad the whole matr are used to geerate weghts by the proposed method ad the LGPAHP, respectvely. The weghts of comparso are show Table. Apparetly, the prorty vectors of the LGPAHP are dfferet for the upper ad lower tragular, or the whole matr. The prorty Eample 2: Ths eample s from Saaty ad Vargas [22], ad s dscussed by Wag [27]. It dsplays the defectve ature of the LGPAHP. The matr s show as follows: [2, 4] [3, 5] [3, 5] [, ] [, ] [2, 5] A = [, ] [, 2] [, ] [, ] [, ] [, 3] The geerated weghts for comparso are show Table 2. By usg the LGPAHP, the weght vector of the upper tragular matr s W = (0.57, 0.207, 0.72, 0.03) T, ad the rakg s w > w 2 > w 3 > w 4. The weght vector for the lower tragular matr s W = (0.57, 0.43, 0.43, 0.43) T, ad the prorty rakg s w > w 2 = w 3 = w 4. For all the udgmets of matr A, the geerated weght vector s W = (0.588, 0.47, 0.47, 0.8) T, ad the prorty rakg s w > w 2 = w 3 > w 4. From the same matr, three dfferet prorty vectors ad three dfferet prorty rakgs have bee obtaed,

5 Jg Rug Yu et al.: A Multplcatve Approach to Derve Weghts the Iterval Aalytc Herarchy Process 229 however, o gudace s provded to show whch oe s suggested. By usg the proposed method, the same weghts ad rakg from the upper, lower tragular, ad all udgmets of the matr are yelded. The weght vector s W = (0.46, 0.23, 0.54, 0.54) T, ad the rakg s w > w 2 > w 3 = w 4. beefts (Tb) ad lqudty (L). The herarchy structure s show Fg.. There are fve terval comparso matrces [4, 30]. Tables 4 ad 5 depct the local ad global weghts geerated by the proposed method ad LGPAHP respectvely. Table 4. The weghts geerated by the proposed method. Table 2. The weght comparso of Eample 2. Eample 3: The followg eample s from Wag [27]. [, 2] [3, 4] [, ] [, 2] [6, 7] [, ] A = [, ] [, ] [, 2] [2, 3] [3, 4] [, ] 2 The geerated weghts of comparso are show Table 3. The LGPAHP obtas the weght vector of the upper tragular matr, as W = (0.2, 0.36, 0.053, 0.42) T, ad the rakg s w 4 > w 2 > w > w 3. The resultat weght vector for the lower tragular matr s W = (0.250, 0.67, 0.083, 0.500) T, ad the prortes rakg s w 4 > w > w 2 > w 3. For all the udgmets of the matr A, the geerated W = (0.94, 0.94, 0.032, 0.58) T, ad the prortes rakg s w 4 > w = w 2 > w 3. Oce aga, three dfferet prorty vectors ad rakg have bee obtaed, wth o gudace to decde whch s the most sutable. By usg the proposed method, the same weghts ad rakg from the upper, lower tragular, ad all udgmets of the matr are yelded. The weght vector s W = (0.88, 0.375, 0.063, 0.375) T, ad the rakg s w 4 = w 2 > w > w 3. Table 5. The weghts geerated by the LGPAHP. Usg upper tragular udgmets Portfolo Re R Tb L Global weght BD DB GB SH Usg lower tragular udgmets Portfolo Re R Tb L Global weght BD DB GB SH Usg all udgmets Portfolo Re R Tb L Global weght BD DB GB SH Table 3. The weght comparso of Eample 3. Eample 4: Ths eample s from Islam et al. [4]. A perso would lke to vest hs moey to ay oe of these portflos: bak depost (BD), debetures (DB), govermet bods (GB), or shares (SH). He ca select oly oe based o four crtera: retur (Re), rsk (R), ta Fg.. Portfolo selectg structure. The LGPAHP geerates the weght vectors of the upper tragular, the lower tragular ad all udgmets of the matrces are W = (0.245, 0.75, 0.4, 0.466) T, W = (0.237, 0.65, 0.47, 0.450) T, ad W = (0.238, 0.66,

6 230 Iteratoal Joural of Fuzzy Systems, Vol. 3, No. 3, September , 0.439) T respectvely. Three dfferet prorty vectors have bee obtaed, but wth o gudace to decde whch s the most sutable. However, the same prortes W = (0.256, 0.288, 0.23, 0.333) T from the upper tragular, the lower tragular, ad all udgmets of the matrces are yelded. Accordg to both methods, the best choce for ths vestor s to vest hs moey shares (SH). However, the problem of dfferet weght vectors geerated from the upper tragular, the lower tragular ad all udgmets of the matrces stll est. Oce aga, the same prortes are obtaed usg the proposed method. 5. Coclusos By takg advatage of the devato varables to mmze the cossteces, the proposed method hadled the cosstet parwse comparso matr well, as show the eamples. I addto, order to cosder the same prorty rakgs for the upper ad lower tragular matrces, multplcatve costrats were employed to geerate weghts, rather tha usg addtve costrats. Moreover, the weghts could be more flebly obtaed wthout predetermg the tolerace parameter. Hece, the four eamples, the dffereces betwee the proposed method ad the LGPAHP method were compared. The results demostrated that the LGPAHP method obtaed feror prortes ad rakgs for the lower ad upper tragular udgmets. O the cotrary, detcal prortes ad rakgs were obtaed usg the proposed method. Ths study has solved the defect poted out by Wag [27], whch s that the LGPAHP method obtas defectve prortes ad rakgs for the lower ad upper tragular parts of a terval matr. The proposed approach ca also be eteded to solve the same problem of the AHP alteratves wth addtve costrats. I realty, the fuzzy umbers ca be trasformed to terval data by meas of α-cut [, 30]. For future research, the proposed method wll be eteded to hadle fuzzy udgmets, ad develop ew aalytc etwork processes for terval/fuzzy udgmets. Ackowledgemets Ths work was supported by the Natoal Scece Coucl, Tawa (NSC No H ). Refereces [] M. Amr, M. Zadeh, R. Solta, ad B. Vahda, A hybrd mult-crtera decso-makg model for frms competece evaluato, Epert Systems wth Applcatos, vol. 36, o. 0, pp , [2] A. Arbel, Appromate artculato of preferece ad prorty devato, Europea Joural of Operatoal Research, vol. 43, o. 3, pp , 989. [3] A. Arbel ad L. G. Vargas, Preferece smulato ad preferece programmg: Robustess ssues prorty dervato, Europea Joural of Operatoal Research, vol. 69, o. 2, pp , 993. [4] J. Barzla, Dervg weghts from parwse comparso matrces, Joural of the Operatoal Research Socety, vol. 48, o. 2, pp , 997. [5] J. Barzla ad B. Golay, AHP rak reversal, ormalzato ad aggregato rules, INFOR, vol. 32, pp , 994. [6] J. P. Bras, The maagemet of the future: Ethcs OR: Respect, mult-crtera maagemet, happess, Europea Joural of Operatoal Research, vol. 53, o. 2, pp , [7] N. Bryso, A goal programmg method for geeratg prorty vectors, Joural of the Operatoal Research Socety, vol. 46, o. 5, pp , 995. [8] N. Bryso ad A. Joseph, Geeratg cosesus prorty terval vectors for group decso makg the AHP, Joural of Mult-Crtera Decso Aalyss, vol. 9, o. 4, pp , [9] B. Chadra, B. Golde, ad E. Wasl, Lear programmg models for estmatg weghts the aalytc herarchy process, Computers & Operatos Research, vol. 32, o. 9, pp , [0] G. Crawford ad C. Wllams, A ote o the aalyss of subectve udgmet matrces, Joural of Mathematcal Psychology, vol. 29, o. 4, pp , 985. [] W. D. Cook ad M. Kress, Dervg weghts from parwse comparso rato matrces: A aomatc approach, Europea Joural of Operatoal Research, vol. 37, o. 3, pp , 988. [2] Y. C. Eresal, T. Öca, ad M. L. Demrca, Determg key capabltes techology maagemet usg fuzzy aalytc herarchy process: A case study of Turkey, Iformato Sceces, vol. 76, o. 8, pp , [3] T. Ertay, D. Rua, ad U. R. Tuzkaya, Itegratg data evelopmet aalyss ad aalytc herarchy for the faclty layout desg maufacturg systems, Iformato Sceces, vol. 76, o. 3, pp , [4] R. Islam, M. P. Bswal, ad S. S. Alam, Preferece programmg ad cosstet terval udgmets, Europea Joural of Operatoal Research,

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