(I) NSC E

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1 (I) NSC

2 (I) A Iterval Arthmetc Approach or WA based uzzy MCDM Model ad Its Applcato to Mauacturg Capablty valuato (I) * -mal: mrchu@mal.stut.edu.tw

3 Abstract 9/08/0~9/07/3 Ths s a three-year project (9/08/0~9/07/3) suggestg a uzzy weghted average (WA) based uzzy multple crtera decso mag (uzzy ( 9/08/0~9/07/3) MCDM) model va terval arthmetc (rst year 9/08/0~9/07/3). A ( smulato system wll be developed to very 9/08/0~93/07/3) the cosstecy o rag alteratves ( betwee the avalable WA methods ad the 93/08/0~9/07/3) proposed model (secod year [-] 9/08/0~93/07/3). Moreover the proposed model wll be appled to evaluate the actory mauacturg capablty to demostrate ts applcablty (thrd year [-7] 93/08/0~9/07/3). I uzzy MCDM [-] the cocept o smple addtve weghtg s usually appled to solve the model. The WA s also a very mportat cocept uzzy decso mag ad has bee [-7] vestgated by may wors [-7]. owever uzzy MCDM model ratgs o alteratves versus crtera are usually ormalzed beore weghted. I the cocept o WA s urther appled to the above model the calculato process becomes very complcated whch maes the avalable WA methods [-7] dcult eecuto. To resolve the above problems we propose terval arthmetc to establsh a WA based uzzy MCDM model where weghts o all crtera ad the ratgs o alteratve versus subjectve crtera are assessed lgustc terms represeted by uzzy umbers. I the proposed model each crtero weght s produced rom the average o the weghts assged by decso-maers ad the alteratve ratg versus each crtero s also produced rom the average o the ratgs gve by decso-maers. The average ratgs o alteratves versus crtera [8] must be ormalzed to a comparable scale beore eecutg WA. sg terval arthmetc ca the develop the membershp ucto o the al total uzzy evaluato value o each alteratve. A rag method ca be easly appled to deuzzy all the above al uzzy umbers or decso mag. Ad the a smulato system wll be developed to testy the cosstecy o

4 rag the al uzzy evaluato values betwee the avalable WA methods ad the proposed oe order to demostrate the eectveess o the proposed model. ally the proposed model wll be appled to evaluate the actory mauacturg capablty [8] or umercal aalyss order to very ts easblty ad applcablty. The proposed method maes WA cocept more sutable or applcato uzzy MCDM ad more ecet eecuto. Moreover the proposed method eteds the research elds o both the uzzy MCDM ad WA. Keywords: uzzy MCDM WA Smulato Mauacturg Capablty a commttee o decso-maers (.e. D t t ~ ) s resposble or selectg m alteratves (.e. A ~ m ) also assume that there are selecto crtera (.e. C j j ~ ). urthermore assume that the perormace ratgs uder each o the crtera as well as the mportace weghts o the crtera are assessed lgustc terms represeted by postve trapezodal uzzy umbers.. Aggregate the mportace weghts et jt a jt b jt c jt d jt w jt j... t... be the mportace weght gve by decso-maer D t to (uzzy MCDM) crtero C j. The aggregated mportace [-] weght w (smple addtve weghtg) j ( a j b j c j d j ) o crtero C j assessed by the commttee o (membershp ucto) decso-maers ca be evaluated as [30]: (uzzy w j ( / ) ( w j w j w j ) weghted average WA) algorthm where a j a jt / b j b jt / t t weghted average crsp MCDM c j c jt / d j d jt /. WA uzzy MCDM t t Dog ad Wog [] [5-7]. Aggregate the perormace ratgs WA et jt ( o jt pjt qjt rjt ) jt WA uzzy MCDM... m j... t... be the lgustc perormace ratg assged to (ormalzato) alteratve A by decso-maer D t or crtero C j. The aggregated perormace ratg (9/08/0~9/07/3) (terval j ( oj pj qj rj ) o alteratve arthmetc) WA (uzzy A uder crtero C j assessed by the weghted average) commttee o decso-maers ca be uzzy MCDM evaluated as [30]: j ( / ) ( j j j ) where o j ojt / pj pjt / t t w

5 q j t q jt / rj r jt t /. 3. Develop membershp ucto The membershp ucto or WA ca be developed as ollows [39-0]: y y ( w ) ( w ) ( w ) y l w [ y y ] l u j w j j w ( b j a j )( pj oj ) [ a j ( pj oj ) oj ( b j a j )] a joj j j j ( c j d j ) d j j j y u ( c j d j )( qj rj ) [ d j ( qj rj ) rj( c j d j )] d jrj j j j ( b j a j ) a j j j et a joj j c j qj j j j a d j j Y b j pj V j d jrj j ( c j d j ) j b j c j 3 j j ( b j a j ) j ( b j a j )( pj oj ) j [ a j ( pj oj ) oj ( b j a j )] j ( c j d j )( qj rj ) j [ d j ( qj rj ) rj ( c j d j )] j We ow have two shorter equatos to solve: ( ) 0 ( ) 0. Oly roots [0] wll be retaed the above two equatos. ( ) y ad () o y y ca be developed as: y y () () [ ] I J V 3 [ ] I J Y. where I / / J I J 0 ad 0. Proposto. () 0. Proposto. () Proposto 3. () Proposto. () 0 y y V 3 y Y y.... WA uzzy MCDM. WA uzzy MCDM 3. ( 9/08/0~9/07/3) WA based uzzy MCDM ag alteratves va 3

6 mamzg set ad mmzg set based uzzy MCDM approach accepted by Joural o the Chese Isttute o geers (003) Solvg uzzy MCDM by Dvdg Cost Crtera to Beet Crtera Iteratoal Mathematcal Joural V.3 pp.7-3 (003) A Iterval Arthmetc Method or valuatg Weapo System der uzzy vromet Joural o Iormato & Optmzato Sceces V. pp (003). [] C. Carlsso ad. uller uzzy multple crtera decso mag: ecet developmets uzzy Sets ad Systems 78 (996) [] S.J. Che ad C.. wag uzzy Multple Attrbute Decso Mag Sprger Berl (99). [3] T.C. Chu A uzzy Number Iterval Arthmetc Based uzzy MCDM Algorthm Iteratoal Joural o uzzy Systems (00) [] W.M. Dog ad.s. Wog uzzy weghted averages ad mplemetato o the eteso prcple uzzy Sets ad Systems (987) [5] Y.Y. uh C.C. o ad.s. ee uzzy weghted average: The lear programmg approach va Chares ad Cooper s rule uzzy Sets ad Systems 7 (00) [6] Y.Y. uh C.C. o K.M. Wag ad.s. ee uzzy weghted average: A ma-m pared elmato method Computers & Mathematcs wth Applcatos 3 (996) 5-3. [7] C. Kao ad S.T. u ractoal programmg approach to uzzy weghted average uzzy Sets ad Systems 0 (00) 35-. [8] C. Kao ad S.T. u Compettveess o mauacturg rms: a applcato o uzzy weghted average I Trasactos o Systems Ma ad Cyberetcs Part A: Systems ad umas 9 (999) [9] Kauma A. ad upta M. M. Itroducto to uzzy Arthmetc: Theory ad Applcato VaNostrad ehold New Yor (99). [0].S. ag uzzy MCDM based o deal ad at-deal cocepts uropea Joural o Operatoal esearch (999) [] T.S. ou ad M.-J.J. Wag uzzy weghted average: A mproved algorthm uzzy Sets ad Systems 9 (99) Proposto. () 0 Proo. () y y. [ ] I J / [ ( ) ] ( Q J ) I [ ] [( ) ( )] /. / /

7 5 Obvously () 0 y. Proposto. () y 3 V. Proo. y / / / 3 whe V Q /. Proposto 3. () y Y. Proo. () / y J I / J I Q / /

8 6 whe Y Q. / Proposto. () 0 y. Proo. () y / / /. Obvously 0 y.