Management Science Letters

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1 Maagemet Scece Letters 2 (202) Cotets lsts avalable at GrowgScece Maagemet Scece Letters homepage: A goal programmg method for dervg fuzzy prortes of crtera from cosstet fuzzy comparso matrces Mohammad Izadkhah * Departmet of Mathematcs, Islamc Azad Uversty, Arak Brach, Arak, Ira A R T I C L E I N F O A B S T R A C T Artcle hstory: Receved July 20, 20 Receved Revsed form September, 22, 20 Accepted 4 October 20 Avalable ole 7 October 20 Keywords: Tragular fuzzy umber Fuzzy par-wse comparso matrx Goal programmg Rakg fucto Decso makg problem s the process of fdg the best opto from all of the feasble alteratves. Oe of the most mportat cocepts decso makg process s to detfy the weghts of crtera. I real-world stuato, because of complete or o-obtaable formato, the data (attrbutes) are ofte ot determstc ad ca be treated forms of fuzzy umbers. Ths paper vestgates a method for dervg the weghts of crtera from the parwse comparso matrx wth fuzzy elemets. Fdg the weghts of crtera has bee oe of the most mportat ssues the feld of decso-makg ad the preset method uses goal programmg to solve the resulted model. I addto, usg a rakg fucto we covert each obtaed fuzzy weght to a crsp oe, whch makes t possble to compare the crtera. The proposed model of ths paper s supported by several examples ad a case study. 202 Growg Scece Ltd. All rghts reserved.. Itroducto I evaluatg competg alteratves,, uder a gve crtero, t s atural to usethe framework of parwse comparsos represeted by a square matrx from whch a set of preferece values for the alteratves s derved. Because of ease of uderstadg ad applcato, parwse comparsos play a mportat role assessg the prorty weghts of decso crtera. Geoffro's gradet search method (972), Hames' surrogate worth tradeoff method (980), Zots- Walleus' method (976), Saaty's aalytc herarchy process (980), Cogger ad Yu's egevector method (985), Takeda, Cogger ad Yu's GEM (987), ad the logarthmc least square method (Crawford ad Wllams (985)) are ust some methods whch are prmarly based o parwse comparsos. Parwse comparso matrces are ofte used Mult-attrbute Decso Makg for weghtg the attrbutes or for the evaluato of the alteratves wth respect to crtera. Determg crtera weghts s a cetral problem mult-crtera decso makg (MCDM). Weghts are used to express the relatve mportace of crtera MCDM. The determato of weghts are requred whe applyg MCDM methods such as goal programmg, the aalytc herarchy process (AHP), ad the weghted score method. I practce, t s dffcult for decso maker * Correspodg author. Tel: E-mal addresses: m-zadkhah@au-arak.ac.r, m_zadkhah@yahoo.com (M. Izadkhah) 202 Growg Scece Ltd. All rghts reserved. do: /.msl

2 30 to supply relatve umercal weghts of dfferet decso crtera. Qute ofte, decso makers are much more comfortable smply assgg ordal raks to the dfferet crtera uder cosderato. I such cases, relatve crtera weghts ca be derved from crtera raks suppled by decso makers. The classcal par-wse comparso matrx requres the decso maker (DM) to express hs/her prefereces the form of a precse rato matrx ecodg a valued preferece relato. However t ca ofte be dffcult for the DM to express exact estmates of the ratos of mportace ad therefore express hs/her estmates as fuzzy umbers. The theory of fuzzy umbers s based o the theory of fuzzy sets, whch Zadeh troduced 965. Frst, Bellma ad Zadeh (970) corporated the cocept of fuzzy umbers to decso aalyss. The methodology preseted ths paper s useful assstg decso makers to determe crtera fuzzy weghts from crtera, ad t s helpful alteratve selecto whe these fuzzy weghts are used wth oe of the techques of MCDM. To dervg the weghts of crtera from ths fuzzy par-wse comparso matrx s a mportat problem. Islam et al. (997) ad Wag (2006) developed a lexcographc goal programmg to geerate weghts from cosstet par-wse terval comparso matrces. May methods for estmatg the preferece values from the parwse comparso matrx have bee proposed ad ther effectveess comparatvely evaluated. Some of the proposed estmatg methods presume tervalscaled preferece values (Barzla, (997) ad Salo, (997)). I ths paper we apply the goal programmg method to derve fuzzy weghts of crtera. Goal programmg was orgally proposed by Chares ad Cooper (96), ad s a mportat techque for DMs to cosder smultaeously several obectves fdg a set of acceptable soluto. Also order to compare the crtera we use the rakg fucto proposed by Asady ad Zedeham (2007). Techque for order performace by smlarty to deal soluto (TOPSIS), oe of kow classcal MCDM method, was frst developed by Che ad Hwag (992), wth referece to Hwag ad Yoo (987), for solvg a MCDM problem. TOPSIS, kow as oe of the most classcal MCDM methods, s based o the dea, that the chose alteratve should have the shortest dstace from the postve deal soluto ad o the other sde the farthest dstace of the egatve deal soluto. Recetly, some research, the TOPSIS method s cosdered for exteso. For example, Che (2000) exteded the cocept of TOPSIS to develop a methodology for solvg mult-perso mult-crtera decsomakg problems a fuzzy evromet. Abo-Sa et al. (2005) exteded the TOPSIS method to solve mult-obectve olear programmg problems. Also, Jahashahloo et al. (2006,a,b) ad Izadkhah (2009) exteded the TOPSIS method for decso makg problems wth terval ad fuzzy data. Ths paper also exteds the cocept of TOPSIS to develop a methodology for solvg mult-crtera decso-makg problems wth fuzzy data. For ths task, we use the fuzzy weghts obtaed by the proposed method. The structure of the rest of ths paper s followg: The followg secto provdes some requred prelmares. The thrd secto of the paper gves a goal programmg approach for dervg weghts of crtera. Two examples ad a case study are preseted secto 4. The paper eds wth cocluso. 2. Prelmares I ths secto we revew some basc deftos about fuzzy umbers, fuzzy par-wse comparso matrx ad goal programmg method.

3 M. Izadkhah/ Maagemet Scece Letters 2 (202) Fuzzy umbers Fuzzy umbers are oe way to descrbe the vagueess ad lack of precso of data. The theory of fuzzy umbers s based o the theory of fuzzy sets, whch Zadeh troduced Some basc deftos of fuzzy umbers Defto. A fuzzy umber s a fuzzy set lke μ A : R I = [0,] whch satsfes: μ A s cotuous, μ ( x A ) = 0 outsde some terval [a,d], There are real umbers b, c such that a b c d ad. μ A( x ) s creasg o [a,b], 2. μ A( x ) s decreasg o [c,d], 3. μ A( x ) =, b x c. We deote the set of all fuzzy umbers by F(R). Parametrc form of fuzzy umbers s defed Asady ad Zedeham (2007) as follows: Defto 2. A fuzzy umber A parametrc form s a par ( A( r), A( r)) of fuctos A(), r A() r, 0 r, whch satsfes the followg requremets:. A() r s a bouded creasg cotuous fucto, 2. A() r s a bouded decreasg cotuous fucto, 3. Ar () Ar (), 0 r. A crsp umber λ s smply represeted by Ar () = Ar () = λ, 0 r. Defto 3. ( -level set or -cut). The -cut of a fuzzy set A s a crsp subset of X ad s deoted by: [ A ] α = { x μ ( x) α}, () A where μ ( x ) A s the membershp fucto of A ad α [0,]. Defto 4. A tragular fuzzy umber s deoted as A = ( abc,, ), see Fg.. Membershp fucto Fg.. The tragular fuzzy The membershp fucto of a tragular fuzzy umber s express as

4 32 x a, a x b b a c x μ A( x) =, b x c c b 0, Otherwse Defto 5. A fuzzy umber A = ( abc,, ) s set to be o-egatve fuzzy umber, f ad oly f a 0. Corollary.The parametrc form of tragular fuzzy umber A = ( abc,, ) s obtaed as: A() r = a+ ( b a) r A = ( A( r), A( r)) = (3) A() r = c ( c b) r Defto 6. (Multplcato of tragular fuzzy umbers) Suppose that we have two tragular fuzzy umbers A ad B such that A = ( a, a2, a3) ad B = ( b, b, b ), the, the multplcato of the fuzzy 2 3 umbers A ad B sdefed as follows: A. B = ( ab, a b, a b ) (4) Defto 7. Let A = ( a, a2, a3) ad B = ( b, b, b ) be two tragular fuzzy umbers, the the dstace 2 3 betwee them usg vertex method s defed as d( A, B ) = (( a b) + ( a2 b2) + ( a3 b3) ) (5) 3 (2) Comparso betwee two fuzzy umbers I ths subsecto, order to compare two fuzzy umbers, we use the cocept of rakg fucto. A rakg fucto s a fucto g : F( R) R, whch maps each fuzzy umber to the real le, where a atural order exsts. Asady ad Zedeham proposed a defuzzfcato usg mmzer of the dstace betwee two the fuzzy umber. They troduced dstace mmzato of a fuzzy umber A that deoted by M ( A ) whch was defed as follows: M ( A ) = ( A( r) + A( r)) dr 2 (6) 0 Ths rakg fucto have the followg propertes: Property. If A ad B be two fuzzy umbers the: M( A ) > M( B ) ff A B, M( A ) < M( B ) ff A B, (7) M( A ) = M( B ) ff A B, Property 2. If A ad B be two fuzzy umbers the: M( A B ) = M( A ) +M( B ) (8)

5 M. Izadkhah/ Maagemet Scece Letters 2 (202) 33 Property 3.If A = ( abc,, ) be a tragular fuzzy umber, the we have: M ( A ) = { a+ 2b+ c} (9) Fuzzy par-wse comparso matrx Suppose the decso maker provdes fuzzy udgmets stead of precse udgmets for a par-wse comparso. Wthout loss of geeralty we assume that we deal wth par-wse comparso matrx wth tragular fuzzy umbers beg the elemets of the matrx. We cosder a par- wse comparso matrx where all ts elemets are tragular fuzzy umbers as follows L M U L M U ( a, a, a ) ( a, a, a ) A = L M U L M U ( a, a, a ) ( a, a, a ) Where a ( L, M, U = a a a ) s a tragular fuzzy umber, see Che et al. (992). We say that A s recprocal, f the followg codto s satsfed (Ramk & Korvy, 200): L M U a = ( a, a, a ) mples a = (,, ) U M L for all, =,...,. a a a 2.3 Goal programmg Cosder the followg problem: max { f ( x),..., f ( x)} subect to x X k (0) where f,..., fk are obectve fuctos ad X s oempty feasble rego. Model (0) s called multple obectve programmg. Goal programmg s ow a mportat area of multple crtera optmzato. The dea of goal programmg s to establsh a goal level of achevemet for each crtero. I goal programmg method requres the decso maker to set goals for each obectve that he/she wshes to obta. A preferred soluto s the defed as the oe, whch mmzes the devatos from the set goals. The GP ca be formulated as the followg achevemet fucto. k + m ( d + d ) = subect to + f( x) + d d = b, =,..., k, x X, + d d = 0, =,..., k, + d, d 0, =,..., k, ()

6 34 The DMs for ther goals set some acceptable asprato levels, b ( =,..., k), for these goals, ad try to acheve a set of goals as closely as possble. The purpose of GP s to mmze the devatos betwee the achevemet of goals, f ( x ), ad these acceptable asprato levels, b ( =,..., k). Also, d + ad d are, respectvely, over- ad uder-achevemet of the th goal. 3. Fuzzy TOPSIS method I ths secto, we revew the exteded TOPSIS method fuzzy evromet proposed by Jahashahloo et al. (2006a).The approach to exted the TOPSIS method to the fuzzy data s as followg steps: Frst step s, detfcato the evaluato crtera. Step 2 s, geeratg alteratves. Step 3 s, evaluatg alteratves terms of crtera. Step 4. Costruct the fuzzy decso matrx. Step 5 s, detfyg the weght of crtera. Step 6. Calculate the ormalzed fuzzy decso matrx as follows: Frst, for each fuzzy umber, we calculate the set of -cut as,, 0, Therefore, each fuzzy umber s trasform to a terval, ow by a approach proposed Jahashahloo et al. (2006,b) we ca trasform ths terval to ormalzed terval as follows: m 2 2 ( ) [ ] = [ x ] ([ x ] ) + ([ x ] ), =,... m; =,...,, L L L U α α α α = m U U L 2 U α α α α 2 = L U α α ( ) [ ] = [ x ] ([ x ] ) + ([ x ] ), =,... m; =,...,, L U ow, terval [ ],[ ] s ormalzed of terval [ ],[ ] x α x α. We ca trasform ths ormalzed terval to a tragular fuzzy umber such as N = ( a, b, c ) such that, b s obtaed whe α =, also by settg α = 0 we have a [ ] L = α = 0 ad c [ ] U = α = 0 ad therefore N s a ormalzed postve tragular fuzzy umber. Step 7. costruct the weghted ormalzed fuzzy decso matrx By cosderg the dfferet mportace of each crtero, we ca costruct the weghted ormalzed fuzzy decso matrx as: v = N. w where w s the weght of th attrbute or crtero. Step 8. Idetfy the fuzzy postve deal ad fuzzy egatve deal solutos Now, each v s ormalzed fuzzy umbers ad ther rages s belog to [0,]. So, we ca detfy The fuzzy postve deal soluto ad fuzzy egatve deal soluto as: A = ( v,..., v ), A = ( v,..., v ), where v + = (,,) ad v = (0,0,0), =,...,, for each crtera. Step 9. Calculate the separato of each alteratve

7 M. Izadkhah/ Maagemet Scece Letters 2 (202) 35 The separato of each alteratve from the fuzzy postve deal soluto, usg the dstace measuremet betwee two fuzzy umber (see Defto 7) ca be curretly calculated as: d + = d( v, v + ), =,..., m, = Smlarly, the separato from the fuzzy egatve deal soluto ca be calculated as: d = d( v, v ), =,..., m, = Step 0. Calculate the closeess coeffcet. A closeess coeffcet s defed to determe the rakg order of all alteratves oce the d + ad d of each alteratve A has bee calculated. The relatve closeess of the alteratve A wth respect to A + ad A sdefed as: R + = d ( d + d ), =,..., m, Obvously, a alteratve A s closer to the A + ad farther from A as R approaches to. Therefore, accordg to the closeess coeffcet, we ca determe the rakg order of all alteratves ad select the best oe from amog a set of feasble alteratves. 4. Dervg the fuzzy weghts of crtera I the covetoal case, f a par-wse comparso matrx A be recprocal ad cosstet the the weghts of each crtero s smply calculated as a w =, =,...,. (2) a k = k where {,..., } s the dex of a arbtrary colum. That s w a = or equvaletly aw w = 0. w I the case of cosstet matrx, the relato (2) s o loger holds. I the case of fuzzy matrx, we L M must obta the fuzzy mportace weghts ( L, M, U L w M w w = w w w ), =,...,, such that a =, a L = M w w U U w ad a =. From relato () ths s equvalet to fd w ( L, M, U ),,...,, U = w w w = such that w L L L M M M aw w = 0, a w w = 0 ad a U U U w w = 0. Therefore the case of ucertaty, for dervg the fuzzy weghts of crtera from cosstet fuzzy L L M M U U comparso matrx for, =,..., we troduce devato varables p, q, p, q ad p, q, where devato varables are oegatve real umbers, but ca t be postve at the same tme,.e. L L M M U U pq = 0, p q = 0 ad p q = 0. Now we apply the goal programmg method. It s desrable that the devato varables are kept to be small as possble, whch leads to the goal programmg model (3). L M U By solvg model (3) the optmal fuzzy weght vector w = ( w, w, w ), =,...,, whch show the fuzzy mportace of each crtero wll be obtaed. We ca use these weghts the process of solvg a multple crtera decso makg problem. Also, these weghts show that whch crtero s more mportat tha others.

8 36 * L L M M U U d = m ( p + q + p + q + p + q ) subect to = = L L L L L a w w + p q = 0,, =,..., M a w w + p q = 0,, =,..., U U U U U a w w + p q = 0, = L M U ( w + w + w ) =,, =,..., M L w w 0, =,..., U M w w 0, =,..., L L L M M U U w, p, q, p, q, p, q 0,, =,..., (3) Remark.Ths method s also useable, eve f all data of comparso matrx be exact form. I such case we obta the crsp weghts for crtera. Theorem.The model (3) s always feasble. Proof. Cosder the vector Wˆ = ( wˆ ˆ,..., w ), where w =( w L, w M, w U ) s such that 3 L M U ( w + w + w ) =, = M L w w 0, =, 2,3, U M w w 0, =, 2,3, L w 0, =, 2,3. The we defe L L L L p = max{ ( a w w ),0}, L L L L q = max{( a w w ),0}, p = max{ ( a w w ),0}, q = max{( a w w ),0}, U U U U p = max{ ( a w w ),0}, U U U U q = max{( a w w ),0}, It s clear that ( W, p L, q L, p M, q M, p U, q U ) s a feasble soluto for model (3). Remark 2. I order to rakg of these crtera, we assg the rak to the crtero wth the maxmal value of M( w ), etc., a decreasg order of M( w ). Specal Case: The case of matrx wth crsp elemets. I the case of matrx wth crsp data, order to dervg the weghts of crtera from the cosstet par-wse comparso matrx, the goal programmg model (3) ca be coverted to the followg model:

9 M. Izadkhah/ Maagemet Scece Letters 2 (202) 37 * d = p + q = = = m ( ) subect to aw w+ p q = 0,, =,..., w =, w, p, q 0,, =,..., where p ad q are devato varables. By solvg model (4) the optmal weght vector w, =,...,, whch show the mportace of each crtero wll be obtaed. Theorem 2.I the case of crsp data, the parwse comparso matrx A s cosstet f ad oly f * d = 0. Proof. Let us frst prove that, f Sce * d = 0 we have p q 0 * d = 0 the matrx A s cosstet. (4) w = =. Therefore aw w = 0 ad hece a =. Ths gves w aa k = ak, ad we coclude that matrx A s cosstet. Coversely, suppose that matrx A s cosstet. That s aa k = ak,, k, =,..., Now, f we defe a k w =, =,..., a t= tk p = q = 0, the t s easy to check that ( W, p, q ) s feasble for model (4). Sce model (4) has mmzato * form, we coclude that d = 0. Theorem 3. Model (4) s always feasble. Proof. By Theorem, proof s evdet. 5. Illustratg examples I ths secto we preset some llustratg example showg that the proposed approach s a coveet tool ot oly for calculatg the fuzzy weghts of crtera from a par-wse comparso matrces wth fuzzy elemets, but also for calculatg the weghts of crtera of crsp par-wse comparso matrces. 5. Example : Matrx wth crsp elemets Cosder 3 3recprocal matrx A wth crsp elemets 2 4 A =

10 38 We ca easly check that the parwse comparso matrx A s recprocal but t s cosstet. Now, for dervg the weghts of crtera we apply a goal programmg model (4) to matrx A. Therefore we must solve the followg goal programmg model * d = m p2 + q2 + p3 + q3 + p2 + q2 + p23 + q23 + p3 + q3 + p32 + q32 subect to 0.50w2 w + p2 q2 = 0, 0.25w3 w + p3 q3 = 0, 2.00w w2 + p2 q2 = 0, (5) 0.25w3 w2 + p23 q23 = 0, 4.00w w3 + p3 q3 = 0, 4.00w2 w3+ p32 q32 = 0, w+ w2+ w3 =, w, p, q 0,, 3. By solvg model (5), we obta the optmal vector W = ( w, w2, w3). We assg the rak to the crtera wth the maxmal value of w, etc., a decreasg order of w. The result s show Table *. The optmal obectve of model (5) s d = 0.249, whch shows that the parwse comparso matrx A s cosstet by Theorem 2. Table The result of proposed method for example. Crtera The obtaed weghts Rak of crtera w = w 2 = w 3 = I ths example the rak order of these crtera s as ~ The results of rakg these crtera are show last colum of Table Example 2: Matrx wth fuzzy elemets Cosder the followg 3 3recprocal matrx A wth tragular fuzzy elemets: (,,) (2,3, 4) (4,5,6) A = (,, ) (,,) (3, 4,5) (,, ) (,, ) (,,) For dervg the fuzzy weghts of crtera, smlar to model (5), we costruct the goal programmg model (6) as

11 M. Izadkhah/ Maagemet Scece Letters 2 (202) L L M M U U m ( p + q + p + q + p + q ) = = subect to w w + p q = 0, L L L L w w + p q = 0, L L L L w w + p q = 0, L L L L w w + p q = 0, 0.667w L L L L w L L 3 + p q = 0, L L w w + p q = 0, L L L L w w + p q = 0, w w + p q = 0, w w + p q = 0, w w + p q = 0, w w + p q = 0, The results of the model (6) are show Table w w + p q = 0, w w + p q = 0, w w + p q = 0, U U U U w w + p q = 0, U U U U w w + p p = 0, U U U U w w + p q = 0, U U U U U U U U w w + p q = 0, 3 = w w + p q = 0, U U U U L M U ( w + w + w ) =, L M U w w w, =, 2,3, L L L M M U U w, p, q, p, q, p, q 0,, 3. (6) Table 2 The fuzzy weghts for example 2 Crtera The obtaed fuzzy weghts M ( w ) Rak of Crtera w = (0.47, , 0.294) w = (0.0735, , ) w = (0.0245, 0.044, ) It ca be see that the above example we derve the fuzzy weghts of crtera whe the elemets of ts par-wse comparso matrx are the form of tragular fuzzy umbers. I order to compare ad rak these crtera, we use the rakg fucto M(.). Results are show the two last colums of Table A case study Suppose that a software compay desres to hre a system aalyss egeer. After prelmary screeg, three caddates, ad rema for further evaluato. The compay asks a decso-maker, to coduct the tervew ad to select the most sutable caddate. Tree beeft crtera are cosdered: ) persoalty ( ), 2) past experece ( ), 3) self-cofdece ( ). The herarchcal structure of ths decso problem s show as Fg. 2. Fg. 2. The herarchal structure The decso maker provdes a fuzzy par-wse comparso matrx betwee crtera as Table 3.

12 40 Table 3 The fuzzy par-wse comparso matrx (,,) (0.,0.25,0.2) (0.5,0.75,) (5,8,0) (,,) (,2,5) (,.333,2) (0.2,0.5,) (,,) The decso-maker uses the tragular fuzzy ratg varables to evaluate the ratg of alteratves wth respect to each crtero ad preset t Table 4. Table 4 The fuzzy ratgs of the three caddates by decso maker uder all crtera (5,7,9) (4,7,0) (5,6,7) (3,4,6) (9,9,0) (7,8,9) (7,9,0) (3,3,5) (3,4,7) The proposed method s curretly appled to solve ths problem ad the computatoal procedure s summarzed as follows: Utl ow, four steps of algorthm have bee doe. Step 5: Determe the mportace of the crtera. By usg the formato of Table 3 ad the proposed goal programmg method the fuzzy weghts of crtera are obtaed as Table 5. Table 5 The fuzzy weghts for crtera Crtera Fuzzy weght (0.0256,0.053) (0,0.205,0.528) (0,0.026,0.026) Step 6. Calculate the ormalzed fuzzy decso matrx. Step 7. Calculate the weghted ormalzed fuzzy decso matrx. Normalzed fuzzy decso matrx ad weghted ormalzed fuzzy decso matrx are gve Tables 6 ad 7, respectvely. Table 6 The ormalzed fuzzy decso matrx (0.289,0.40,0.520) (0.220,0.420,0.550) (0.309,0.368,0.432) (0.73,0.234,0.346) (0.495,0.540,0.550) (0.432,0.490,0.556) (0.404,0.527,0.577) (0.65,0.80,0.275) (0.85,0.245,0.432) Table 7 The weghted ormalzed fuzzy decso matrx (0,0.005,0.0267) (0,0.086,0.2820) (0,0.0378,0.0443) (0,0.0060,0.078) (0,0.08,0.2820) (0,0.0503,0.0570) (0,0.035,0.0296) (0,0.0369,0.40) (0,0.025,0.0443) Step 8. Idetfy the fuzzy postve deal ad fuzzy egatve deal solutos. Step 9. Calculate the separato of each alteratve.

13 M. Izadkhah/ Maagemet Scece Letters 2 (202) 4 The separato measures are show Table 8. Table 8 The closeess coeffcet ad rakg of crtera Rak Step 0. Calculate the closeess coeffcet. The closeess coeffcets, whch are defed to determe the rakg order of all alteratves by calculatg the dstace to both the fuzzy postve-deal soluto ad the fuzzy egatve-deal soluto smultaeously, are gve fourth colum of Table 8. Now a preferece order ca be raked accordg to the order of R. Therefore, the best alteratve s the oe wth the shortest dstace to the fuzzy postve deal soluto ad wth the logest dstace to the fuzzy egatve deal soluto. Accordg to the closeess coeffcet, rakg the preferece order of these alteratves s as Table Cocluso Fdg the weghts of crtera has bee oe of the most mportat ssues the feld of decso makg. I ths paper, we have vestgated the problem of dervg the fuzzy weghts of crtera from the par-wse comparso matrx wth fuzzy elemets. I the preseted method the goal programmg method have bee put forward. We drve the fuzzy weghts of crtera by usg the goal programmg method. The proposed method s appled for two par-wse comparso matrces wth crsp ad fuzzy elemets, respectvely. I addto, we appled the proposed method to solve mult-crtera decso problem based o TOPSIS method. I order to compare the crtera we use the rakg fucto proposed by Asady ad Zedeham. The approach s llustrated by usg some examples. I addto, the proposed method ca be appled to solve other mult-crtera decso problems. Refereces Abo-Sa, M.A., & Amer, A.H. (2005). Extesos of TOPSIS for mult-obectve large-scale olear programmg problems. Appled Mathematcs ad Computato, 62, Asady, B., & Zedeham, M. (2007). Rakg of fuzzy umbers by mmze dstace, Appled Mathematcal Modelg,3, Bellma, R.E., & Zadeh, L.A. (970). Decso makg a fuzzy evromet. Maagemet Scece, 7, Barzla, J. (997). Dervg weghts from parwse comparso matrces. Joural of Operatoal Research Socety, 48, Chares, A., & Cooper, W.W. (96). Maagemet Model ad Idustral Applcato of Lear Programmg, st ed., Wley, New York. Che, C.T. (2000). Extesos of the TOPSIS for group decso-makg uder fuzzy evromet. Fuzzy Sets ad Systems, 4, -9. Che, S. J., & Hwag C. L. (992). Fuzzy Multple Attrbute Decso Makg: Methods ad Applcatos, Sprger-Verlag. Berl. Cogger, K.O., & Yu, P.L. (985). Ege weght vectors ad least dstace approxmato for revealed preferece parwse weght ratos. Joural of Optmzato Theory ad Applcatos, 46,

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