Models for Selecting an ERP System with Intuitionistic Trapezoidal Fuzzy Information

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1 JOURNAL OF SOFWARE, VOL 5, NO 3, MARCH Models for Selectg a ERP System wth Itutostc rapezodal Fuzzy Iformato Guwu We, Ru L Departmet of Ecoomcs ad Maagemet, Chogqg Uversty of Arts ad Sceces, Yogchua, Chogqg, 060, PRCha E-mal: weguwu@63com Abstract o solve multple attrbute decso makg problems wth tutostc trapezodal fuzzy formato, two optmzato models based o the maxmzg devato method, by whch the attrbute weghts ca be determed, s establshed We utlze the tutostc trapezodal fuzzy weghted geometrc mea (IFWGM operator to aggregate the tutostc trapezodal fuzzy formato correspodg to each alteratve ad get the overall value of the alteratves, the rak the alteratves ad select the most desrable oe(s accordg to the dstace betwee the overall value of the alteratves ad deal soluto Fally, a llustratve example about selectg a ERP system s gve Idex erms Multple Attrbute Decso Makg; Itutostc rapezodal Fuzzy Numbers; Operatoal Laws; Itutostc rapezodal Fuzzy Weghted Geometrc Mea (IFWGM Operator; Attrbute Weght I INRODUCION I today s dyamc ad upredctable busess evromet, compaes face the tremedous challege of expadg markets ad rsg customer expectatos hs compels them to lower total costs the etre supply cha, shorte throughput tmes, reduce vetores, expad product choce, provde more relable delvery dates ad better customer servce, mprove qualty, ad effcetly coordate globe demad, supply ad producto [,] I order to accomplsh these obectves, more ad more compaes are turg to the eterprse resource plag systems (ERP A ERP s a packaged eterprse-wde formato system that tegrates all ecessary busess fuctos, such as product plag, purchasg, vetory cotrol, sales, facal ad huma resources, to a sgle system wth a shared database [3, ] A eterprse resource plag (ERP s a eterprse-wde applcato software package that tegrates all ecessary busess fuctos to a sgle system wth a commo database I order to mplemet a ERP proect successfully a orgazato, t s ecessary to select a sutable ERP system hs paper presets a ew model, whch s based o the tutostc trapezodal fuzzy formato processg, for dealg wth the problem of ERP system selecto he remader of ths paper s set out as follows I the ext secto, we troduce some basc cocepts related to tutostc trapezodal fuzzy sets I Secto 3 we troduce the MADM problem deal wth selectg a ERP system wth tutostc trapezodal fuzzy formato, whch the formato about attrbute weghts s completely kow, ad the attrbute values take the form of tutostc trapezodal fuzzy umbers We utlze tutostc trapezodal fuzzy weghted geometrc mea (IFWGM operator to aggregate the tutostc trapezodal fuzzy formato correspodg to each alteratve, ad the rak the alteratves ad select the most desrable oe(s I Secto, a llustratve example s poted out I Secto 5 we coclude the paper ad gve some remarks II PRELIMINARIES I the followg, we shall troduce some basc cocepts related to tutostc trapezodal fuzzy umbers Defto Let s a tutostc trapezodal fuzzy umber, ts membershp fucto s[-6]: x a, a x< b; b a, b x c; ( x ( d x a, c x d; d c < 0, others ts o-membershp fucto s: ( x ( b x x a b a, b x c; x c d x d c 0, others (, a x< b;, c< x d ; where 0 ;0 ad a a ; abcd,,, R ( he a ([ abcd,,, ],([ a, bcd,, ] ; s called a tutostc trapezodal fuzzy umber abcd,,,, For coveece, let ([ ] Defto Let a [ a, b, c, d ], a [ a, b, c, d ], ( ( a a ad be two tutostc trapezodal fuzzy umber, ad 0, the[6] ( ([ a a b b c c d d],,,, ; ; 00 ACADEMY PUBLISHER do:030/sw

2 76 JOURNAL OF SOFWARE, VOL 5, NO 3, MARCH 00 ( ([ a a, b b, c c, d d] ;, ; (3 a [ a b c d] ( a (,,, ;, a (,,,, ; ( a b c d ( Defto 3 Let a ([ a, b, c, d ], a ([ a, b, c, d ], ad be two tutostc trapezodal fuzzy umber, the the ormalzed Hammg dstace betwee ad s defed as follows[6]: (, d a a ( ( ( a a ( ( b b ( ( c c ( ( d d Defto For a ormalzed tutostc trapezodal fuzzy decso makg matrx R ( ( a, b, c, d,, where m m 0 a b c d, 0,, 0, the tutostc trapezodal fuzzy postve deal soluto ad tutostc trapezodal fuzzy egatve deal soluto are defed as follows: ( a, b, c, d, ([,,, ];,0 r ( a, b, c, d, ([ 0,0,0,0 ];0,,,, ;, a ( a b c d ( a a Defto 5 Let [ ] ad a [ a, b, c, d ], be two tutostc trapezodal fuzzy umber, the the a, b, c, d,, ( ( a a dstace betwee [ ] a [ a, b, c, d ], ad tutostc trapezodal fuzzy postve deal soluto are deoted as d ( a, r ad d ( a, r f d(, < d(,, the >, I the followg, we shall develop a ew operator called tutostc trapezodal fuzzy weghted geometrc mea (IFWGM operator a, b, c, d, Defto 6 Let ( a a (,,, (3 L be a collecto of tutostc Q Q, trapezodal fuzzy umbers, ad let IFWGM: f IFWGM (,,, a ( a L ( where (,,, L be the weght vector of (,,, L, ad > 0,, the IFWGM s called the tutostc trapezodal fuzzy weghted geometrc mea (IFWGM operator We ca derve the heorem from Defto a, b, c, d, heorem Let ( a a (,,, L be a collecto of tutostc trapezodal fuzzy umbers, the ther aggregated value by usg the IFWGM operator s also a tutostc trapezodal fuzzy umbers, ad IFWGM,, L, ( a, b, c, d ; (, ( where (,,, (5 L be the weght vector of (,,, L, ad > 0, III MODELS FOR SELECING ERP SYSEMS WIH INERVAL-VALUED INUIIONISIC FUZZY INFORMAION he followg assumptos or otatos are used to represet the tutostc trapezodal fuzzy MADM problems wth complete weght formato: ( he alteratves are kow Let A { A, A, L, Am } be a dscrete set of alteratves; ( he attrbutes are kow Let G { G, G, L, G} be a set of attrbutes; (3 he formato about attrbute weghts s completely kow Let w ( w, w, L, w Hbe the weght vector of attrbutes, where w 0,,, L,, w, H s a set of the kow weght formato, whch ca be costructed by the followg forms [7-0], for : Form A weak rakg: w w ; Form A strct rakg: w w α, α > 0 ;Form 3 A rakg of dffereces: w w wk wl, for k l;form 00 ACADEMY PUBLISHER

3 JOURNAL OF SOFWARE, VOL 5, NO 3, MARCH A rakg wth multples: w β w, 0 β ; Form 5 A terval form: α w α ε,0 α < α ε Suppose that R ( (,,, ;, a b c d m m s the tutostc trapezodal fuzzy decso matrx, where [ 0,], [ 0,],,,, L, m,,, L, he maxmzg devato method [] s selected here to compute the dffereces of the performace values of each alteratve For the attrbuteg G, the devato of alteratve A to all the other alteratves ca be defed as follows: Let m ( ( D w d r, r w,,, L, m,,, L, k k m m m ( ( (,,,, L, D w D w w k k d(, k ( ( a ( a ( b ( k k bk ( c ( k k ck ( d ( k k dk he D ( k k k w represet the devato value of all alteratves to other alteratves for the attrbute G G Based o the above aalyss, we have to choose the weght vector w to maxmze all devato values for all the attrbutes o do so, we ca costruct a o-lear programmg model as follows ( M- : m m m max Dw ( D ( w dr (, k w k Subect to w H, w, 0,,,, w L By solvg the model ( soluto w ( w w w M-, we get the optmal,, L,, whch ca be used as the weght vector of attrbutes If the formato about attrbute weghts s completely ukow, we ca establsh aother programmg model( M- : m m m max Dw ( D( w dr (, w st w w k k, 0,,, L, By solvg( M-, we get a smple ad exact formula for determg the attrbute weghts as follows: m m m m (, (, w d d (6 k k k k Based o the above models, we develop a model for selectg a ERP system based o IFWG operator uder tutostc trapezodal fuzzy evromet he method volves the followg steps: Step If the formato about the attrbute weghts s partly kow, the we solve the model (M- to obta the attrbute weghts If the attrbute weght s completely ukow, the we ca obta the attrbute weghts by usg Eq (6 Step Utlze the weght vector w ad IFWG operator, we obta the overall values of the A,, L, m alteratve ( ( a, b, c, d, IFWGM r, r,, r,,, L, m ( L w Step 3 Calculate the dstaces betwee overall values ad tutostc trapezodal fuzzy postve deal soluto r d, ( ( ( ( a a ( ( b ( b ( d ( d ( d ( d Step Rak all the alteratves A (,, L, m ad select the best oe(s accordace wth d ( r, r he smaller d(, r, the better the alteratves A Step 5 Ed IV ILLUSRAIVE EXAMPLE (7 hs secto presets a umercal example to llustrate the method proposed ths paper Suppose a orgazato plas to mplemet ERP systems here s a pael wth fve possble ERP systems to select he compay employs some professoal orgazatos (or experts to ad ths decso-makg he Proect team selects four attrbute to evaluate the alteratves: G s the fucto ad techology; G s strategc ftess;3g 3 s the vedor s ablty;g s the vedor s reputato he fve possble alteratves A (,, L,5 are to be evaluated usg the tutostc trapezodal fuzzy formato by the decso maker uder the above 00 ACADEMY PUBLISHER

4 7 JOURNAL OF SOFWARE, VOL 5, NO 3, MARCH 00 four attrbutes, as lsted the followg matrx R ( [ 05,06,07,0 ];05,0 ([ 0,0,03,0 ];06,03 ( [ 06,07,0,09 ];07,03([ 05,06,07,0 ];07,0 ([ 0,0,0,05 ];06,0([ 0,03,05,06 ];05,0 ([ 03,0,05,06 ];0,0 0, ([ 03,0,05 ];06,03 ( [ 0,03,0,05 ];06,0([ 03,0,05,06 ];0,03 ([ 05,06,0,09 ];03,06([ 0,05,06,07 ];0,07 ([ 0,05,07,0 ];07,0([ 05,06,07,09 ];0, 05 ([ 05,06,07,0 ];05,03([ 03,05,07,09 ];0,03 ([ 0,03,05,07 ];03,0([ 06,07,0,09 ];0,06 ([ 0,03,0,05 ];07,0([ 05,06,07,0 ];0,03 he, we utlze the approach developed to get the most desrable ERP system (s he, we utlze the approach developed to get the most desrable alteratve(s Case : If the formato about the attrbute weghts s partly kow ad the kow weght formato s gve as follows: { H 05 w 0, 00 w 05, 0 w 05,05 w 030, 3 w 0,,,3,, w he, we utlze the approach developed to get the most desrable alteratve(s Step Utlze the model (M- to establsh the followg sgle-obectve programmg model: ( 3 max Dw 500w 0w 300w 65w st w H Solvg ths model, we get the weght vector of attrbutes: w ( Step Utlze the weght vector w ad by Eq (7, we obta the overall values of the A,,3,,5 alteratve ( [ ] [ ] 3 [ ] [ ] [ 055,039,0 ] 5 ( 0339,0560,050,0667 ;035,0503 ( 0500,060,0767,0566 ;059,03307 ( 076,03636,05595,0660 ;03997,0399 ( 03,093,05506,06759 ;006,036 ( 97,0597 ;033,030 } Step 3 Calculate the dstaces betwee overall values ([ a, b, c, d ], ad tutostc trapezodal fuzzy postve deal soluto ( [ ],,, ;, (,,, ;,0 (, 07907, d(, ( 3, 07590, d(, 0766 ( 5, r a b c d Step Rak all the alteratves (,,3,,5 accordace wth the dstaces ( values r ([ a, b, c, d ], A d, r betwee overall ad tutostc trapezodal fuzzy postve deal soluto: A f A f A f A f A, ad thus the most desrable 5 3 alteratve s A Case : If the formato about the attrbute weghts s completely ukow, we utlze aother approach developed to get the most desrable alteratve(s Step Utlze the Eq (6 to get the weght vector of attrbutes: w ( Step Utlze the decso formato gve matrx R, ad the IFWGM operator, we obta the overall preferece values of the ERP A,, L, 5 system ( ([ 0375,060,0579,0679 ];03650,0575 ([ 099,05997,0765,059 ];06066, ([ 076,03603,0559,06776 ];06,0355 ([ 07,0037,05360,066 ];060, ([ 0773,03,0 56,057 ];0359,06 Step 3 Calculate the dstaces betwee overall values ([ a, b, c, d ], ad tutostc trapezodal fuzzy postve deal soluto ( [ ],,, ;, (,,, ;,0 (, 07907, d(, ( 3, 07590, d(, 0766 ( 5, r a b c d Step Rak all the alteratves (,,3,,5 accordace wth the dstaces ( A d, r betwee overall 00 ACADEMY PUBLISHER

5 JOURNAL OF SOFWARE, VOL 5, NO 3, MARCH values r ([ a, b, c, d ], ad tutostc trapezodal fuzzy postve deal soluto: A f A f A f A f A, ad thus the most desrable 5 3 ERP system s A Step 5 Rak all the ERP systems A (,,3,,5 accordace wth the scores S( (,, L,5 of the overall preferece values r (,, L,5 : A f A5 f A3 f A f A, ad thus the most desrable ERP system s A V CONCLUSION I ths paper, we have vestgated the problem of MADM wth completely kow formato o attrbute weghts to whch the attrbute values are gve terms of tutostc trapezodal fuzzy umbers wo optmzato models based o the maxmzg devato method are establshed to derve the attrbute weghts he, we utlze the tutostc trapezodal fuzzy weghted geometrc mea (IFWGM operator to aggregate the tutostc trapezodal fuzzy formato correspodg to each alteratve, ad the rak the alteratves ad select the most desrable oe(s Fally, a llustratve example about selectg a ERP system s gve ACKNOWLEDGMEN hs research was supported by the Scece ad echology Research Foudato of Chogqg Educato Commsso uder Grat KJ090 REFERENCE [] Y L, X Lao, HZ Le, A kowledge maagemet system for ERP mplemetato, Systems Research ad Behavoral Scece 3 ( [] E Umble, R Haft, M Umble, Eterprse resource plag: mplemetato procedures ad crtcal success factors, Europea Joural of Operatoal Research 6 ( [3] A Al-Mudmgh, M Al-Mashar, ERP software mplemetato: a tegratve framework, Europea Joural of Iformato System 0( ( [] P Soffee, B Golay, D Dor, ERP modelg: a comprehesve approach, Iformato Systems ( [5] K Ataassov, Itutostc fuzzy sets, Fuzzy Sets ad Systems, Vol 0, pp7-96, 96 [6] K Ataassov, More o tutostc fuzzy sets, Fuzzy Sets ad Systems, Vol33, pp37-6, 99 [7] L A Zadeh, Fuzzy sets, Iformato ad Cotrol, Vol, pp33-356, 965 [] Z S Xu ad R R Yager, Some geometrc aggregato operators based o tutostc fuzzy sets, Iteratoal Joural of Geeral System, Vol35, pp7-33, 006 [9] K Ataassov ad G Gargov, Iterval-valued tutostc fuzzy sets, Fuzzy Sets ad Systems, Vol3, pp33-39, 99 [0] K Ataassov, Operators over terval-valued tutostc fuzzy sets, Fuzzy Sets ad Systems, Vol6, No, pp59-7, 99 [] Z S Xu, Methods for aggregatg terval-valued tutostc fuzzy formato ad ther applcato to decso makg, Cotrol ad decso, Vol, No, pp5-9, 007 [] ZSXu, Models for multple attrbute decso-makg wth tutostc fuzzy formato, Iteratoal Joural of Ucertaty, Fuzzess ad Kowledge-Based Systems, Vol5, No3, pp5-97, 007 [3] G W We, Maxmzg devato method for multple attrbute decso makg tutostc fuzzy settg, Kowledge-based Systems, Vol, No, pp33-36, 00 []JQWag, Overvew o fuzzy mult-crtera decso-makg approach, Cotrol ad decso, Vol3, No6, , 00 [5]JQ Wag, Zh Zhag, Programmg method of mult-crtera decso-makg based o tutostc fuzzy umber wth complete certa formato, Cotrol ad decso, Vol3, No0, 5-, 00 [6]JQ Wag, Zh Zhag, mult-crtera decso-makg method wth complete certa formato based o tutostc fuzzy umber, Cotrol ad decso, Vol, No, 6-30, 009 [7] KS Park, SH Km, ools for teractve mult-attrbute decso makg wth completely detfed formato, Europea Joural of Operatoal Research, Vol 9, -3, 997 [] SH Km, SH Cho, JK Km, A teractve procedure for multple attrbute group decso makg wth complete formato: rage based approach, Europea Joural of Operatoal Research, Vol, 39-5, 999 [9] SH Km, BS Ah, Iteractve group decso makg procedure uder complete formato, Europea Joural of Operatoal Research, Vol 6, 9-507, 999 [0] KS Park, Mathematcal programmg models for characterg domace ad potetal optmalty whe multcrtera alteratve values ad weghts are smultaeously complete, IEEE trasactos o systems, ma, ad cyberetcs-part A, Systems ad Humas, Vol 3, 60-6, 00 []YM Wag, Usg the method of maxmzg devatos to make decso for mult-dces System Egeerg ad Electrocs 7(99-6, 3 [] G W We, Some geometrc aggregato fuctos ad ther applcato to dyamc multple attrbute decso makg tutostc fuzzy settg, Iteratoal Joural of Ucertaty, Fuzzess ad Kowledge-Based Systems, Vol7, No, pp79-96, 009 Guwu We receved the BE, ME ad DE degree maagemet scece ad egeerg from Cha West Normal Uversty, SouthWest Petroleum Uversty ad SouthWest Jaotog Uversty, Cha, respectvely Hs research terests are decso makg aalyss, compute wth words ad formato aggregatg operator Dr Guwu We has worked for Departmet of Ecoomcs ad Maagemet, Chogqg Uversty of Arts ad Sceces, Cha as assocate professor sce ACADEMY PUBLISHER