A Fuzzy Inventory System with Deteriorating Items under Supplier Credits Linked to Ordering Quantity

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1 JOURNAL OF INFORMAION SCIENCE AND ENGINEERING 6, 3-53 () A Fuzzy Ivtory Syst with Dtrioratig Its udr Supplir Crdits Likd to Ordrig Quatity LIANG-YUH OUYANG, JINN-SAIR ENG AND MEI-CHUAN CHENG 3 Dpartt of Maagt Scics ad Dcisio Makig 3 Graduat Istitut of Maagt Scics akag Uivrsity asui, aipi Hsi, 5 aiwa Dpartt of Marktig ad Maagt Scics h Willia Patrso Uivrsity of Nw Jrsy Way, Nw Jrsy 747-3, U.S.A. h ivtory probl associatd with trad crdit is a popular topic i which itrst ico ad itrst payts ar iportat issus. Most studis rlatd to trad crdit assu that th itrst rat is both fixd ad prdtrid. Howvr, i th ral arkt, ay factors such as fiacial policy, otary policy ad iflatio, ay affct th itrst rat. Morovr, withi th virot of rchadis storag, so distictiv factors aris which ultiatly affct th quality of products such as tpratur, huidity, ad storag quipt. hus, th rat of itrst chargs, th rat of itrst ard, ad th dtrioratio rat i a ral ivtory probl ay b fuzzy. I this papr, w dal with ths thr iprcis paratrs i ivtory odlig by utilizig th fuzzy st thory. W dvlop th fuzzy ivtory odl basd o Chag t al. s [] odl by fuzzifyig th rat of itrst chargs, th rat of itrst ard, ad th dtrioratio rat ito th triagular fuzzy ubr. Subsqutly, w discuss how to dtri th optial ordrig policy so that th total rlvat ivtory cost, i th fuzzy ss, is iial. Furthror, w show that Chag t al. s [] odl (th crisp odl) is a spcial cas of our odl (th fuzzy odl). Fially, urical xapls ar providd to illustrat ths rsults. Kywords: ivtory, dtrioratig it, dlay payts, fuzzy st, sigd distac. INRODUCION h fuzzy st thory is dvlopd for solvig th phoo of fuzziss prvalt i th ral world. Up to this poit, th fuzzy st thory has b widly applid i ay filds, such as applid scic, dici ad ivtory aagt. h applicatio of fuzzy st cocpts i ivtory odls hav b proposd by ay rsarchrs. Prtrovic ad Swy [] fuzzifid th dad, lad ti ad ivtory lvl ito triagular fuzzy ubrs i a ivtory cotrol odl, ad th dtrid th ordr quatity with th fuzzy propositios thod. Yao t al. [3] ivstigatd th Ecooic Lot Schdulig Probl (ELSP) with fuzzy dads. hy usd th Idpdt Solutio as wll as th Coo Cycl approach to solv th fuzzy ELSP probl. Yao t al. [4] prstd a fuzzy ivtory syst without th backordr odl i which both th ordr quatity ad th total dad wr fuzzifid as th triagular fuzzy ubrs. Chag [5] discussd th Ecooic Ordr Quatity (EOQ) odl with iprfct quality Rcivd Dcbr 5, 7; rvisd May 8 & August & Octobr 7, 8; accptd Novbr, 8. Couicatd by Chi-g Li. 3

2 3 LIANG-YUH OUYANG, JINN-SAIR ENG AND MEI-CHUAN CHENG its by applyig th fuzzy sts thory, ad proposd th odl with both a fuzzy dfctiv rat ad a fuzzy aual dad. Chag t al. [6] cosidrd th ixtur ivtory odl ivolvig variabl lad ti with backordrs ad lost sals. hy fuzzifid th rado lad-ti dad to b a fuzzy rado variabl ad th total dad to b th triagular fuzzy ubr. Basd o th ctroid thod of dfuzzificatio, thy drivd a stiat of th total cost i th fuzzy ss. Ch t al. [7] itroducd a fuzzy cooic productio quatity odl with dfctiv products i which thy cosidrd a fuzzy opportuity cost, trapzoidal fuzzy cost ad quatitis i th cotxt of th traditioal productio ivtory odl. Maiti [8] dvlopd a ulti-it ivtory odl with stock-dpdt dad ad two-storag facilitis i a fuzzy virot (whr purchas cost, ivstt aout ad storhous capacity ar iprcis) udr iflatio ad icorporatig th ti valu of oy. Othr rlatd articls o this topic ca b foud i work by Ch ad Wag [9], Vujosvic t al. [], G t al. [], Roy ad Maiti [], Ishii ad Koo [3], L ad Yao [4], Yao ad L [5], Chag t al. [6], Chag t al. [7], Ouyag t al.[8], Yao t al. [9]. I today s busiss virot, trad crdit plays a iportat rol ad th ivtory probl associatd with trad crdit has bco a popular topic i th ivtory fild. A supplir usually prits th rtailr to dlay i sttlig th total aout owd to th for a fixd priod of ti. Usually, itrst dos ot bgi accruig for th outstadig aout providd that it is paid withi th prissibl dlay priod. hrfor, th rtailr ca ar itrst o th accuulatd rvu rcivd by dfrrig th payt util th last ot of this prissibl priod. Goyal [] dvlopd a EOQ odl udr coditios of prissibl dlay i payts, i which h calculatd itrst ico basd o th purchasig cost of goods sold withi th prissibl dlay priod. Byod th prissibl priod, itrst payts ar calculatd basd o th purchasig cost of th goods ot yt sold. g [] add Goyal s [] odl by calculatig itrst ard basd o th sllig pric of goods sold. I Chag ad g [], th supplirs offr cash discouts or dlay payt to rtailrs. Withi th prissibl dlay priod, rtailrs ar itrst o sals rvu. Byod th prissibl priod, itrst is chargd for th outstadig aout. Chag t al. [] stablishd a EOQ odl for dtrioratig its, i which th supplir provids a prissibl dlay to th purchasr if th ordr quatity is gratr tha or qual to a prdtrid quatity. h rtailr ca obtai itrst ico withi th prissibl dlay priod. Byod th prissibl priod, itrst payts accru for th goods ot yt sold. hr ar ay itrstig ad rlvat articls rlatd to trad crdit, such as Davis ad Gaithr [3], Ouyag t al. [4, 5] ad g t al. [6]. Wh discussig trad crdit, itrst ico ad itrst payts ar iportat issus. h abov tiod studis o trad crdit assud that th rat of itrst chargs, th rat of itrst ard, ad th dtrioratio rat ar fixd ad prdtrid. Howvr, i th ral arkt, ay factors ay caus fluctuatios i itrst rats. For istac, th growth rat of th cooy ay affct th itrst rat. Wh th cooy is dvlopig at a fastr rat, th dad for capital is strog, which i rtur pushs up th itrst rat. Covrsly, durig cooic dowturs, dad for capital dclis ad itrst rats fall. Aothr factor iflucig itrst rats is th supply of oy i th cooy. Wh th supply of oy is high, itrst rats fall ad wh supply is low, itrst rats ris. h third factor affctig itrst rats is th itrst rats i othr

3 A FUZZY INVENORY SYSEM UNDER SUPPLIER CREDIS 33 coutris. Sic oy flows btw atios, th diffrig itrst rats btw coutris utually ifluc ach othr. hrfor, th rat of itrst chargs ad th rat of itrst ard ar ulikly to rai costat du to various ucrtaitis. Morovr, withi th virot of rchadis storag, so distictiv factors aris which ultiatly affct th quality of products such as tpratur, huidity, ad storag quipt. hrfor, i th ral world, th ivtory dtrioratio rat is ot kow with crtaity. Aogst xtdd studis o this topic, oly Ch ad Ouyag [7] hav tratd itrst rats as variabl, whri th authors xtdd th odl of Jaal t al. [8] by fuzzifyig th carryig cost rat, itrst paid rat ad itrst ard rat siultaously. Suarizig th abov, w obsrv that th ivtory probl associatd with trad crdit has yt to b fully xplord ad udrstood spcially wh itrst rats fluctuat. I ordr to fill this gap, this study tris to rcast Chag t al. s [] odl by furthr fuzzifyig th rat of itrst chargs, th rat of itrst ard, ad th dtrioratio rat ito th triagular fuzzy ubr. W costruct thr diffrt itrvals to iclud th rat of itrst chargs, th rat of itrst ard, ad th dtrioratio rat, thus drivig th fuzzy total rlvat ivtory cost. By th sigd distac thod of dfuzzificatio, w driv th stiat of th total rlvat ivtory cost i th fuzzy ss. Furthr, w discuss how to dtri th optial ordrig policy such that th total rlvat ivtory cost i th fuzzy ss is iial. Fially, urical xapls ar giv to illustrat th solutio procdur. his articl is orgaizd as follows. I sctio, w provid th prliiaris of fuzzy athatics. I sctio 3, w itroduc th ivtory probl. A brif rviw of Chag t al. s [] odl is icludd i sctio 3. ad th fuzzy ivtory odl is providd i sctio 3.. W th us th sigd distac thod of dfuzzificatio to driv th stiat of th total rlvat ivtory cost i th fuzzy ss. I sctio 4, w obtai th optial rplisht ti itrval ad th optial ordr quatity by iiizig th stiat of th total rlvat ivtory cost i th fuzzy ss. Svral urical xapls ar giv to illustrat th rsults i sctio 5. I sctio 6, w discuss two probls of th proposd odl. Fially, sctio 7 draws coclusios ad suggsts pottial dirctios for futur rsarch.. PRELIMINARIES I this sctio, so cocpts of th fuzzy st thory ar rviwd. W itroduc thr dfiitios, dcopositio thor ad o proprty which w will us throughout this articl. Dfiitio For α ad p < q, th fuzzy st [p, q; α] o R is calld a lvl α fuzzy itrval if th brship fuctio of [p, q; α] is giv by μ [ pq, ; α ] α, p x q; ( x) (), othrwis. Dcopositio hor (s, Kaufaa ad Gupta [9]) Lt D b a fuzzy st o R ad D F s, α. h α-cut of D is D(α) [D L (α),

4 34 LIANG-YUH OUYANG, JINN-SAIR ENG AND MEI-CHUAN CHENG D U (α)]. h, w hav or D α D( α), () α μ ( ) V ( ), (3) D x α C D( α ) x α whr (i) αd(α) is a fuzzy st with brship fuctio μ αd( α) α, x D( α); ( x), othrwis. (ii) C D(α) (x) is a charactristic fuctio of D(α), that is, C D( α ), x D( α); ( x), x D( α). Fro th Dcopositio hor ad Eq. (), w obtai or (4) D α D( α) [ D ( α), D ( α); α], α α μd ( x) α C ( x) μ ( x). L D( α) [ DL( α), DU ( α); α] α α U V V (5) For ay a, b, c, d, k R, whr a < b ad c < d, th followig dfiitios of th itrval opratios ca b foud i [9]. (i) [a, b](+)[c, d] [a + c, b + d]. (ii) [a, b]( )[c, d] [a d, b c]. [ ka, kb], k > ; (iii) k()[ a, b] [ kb, ka], k <. (6) If a < b ad c < d, th (iv) [a, b]( )[c, d] [ac, bd]. If a < b ad < c < d, th (v) [a, b]( )[c, d] [a/d, b/c]. Nxt, siilar as Yao ad Wu [3], w itroduc th cocpt of th sigd distac which will b dd latr. W first cosidr th sigd distac o R.

5 A FUZZY INVENORY SYSEM UNDER SUPPLIER CREDIS 35 Dfiitio For ay a ad R, dfi th sigd distac of a to as d (a, ) a. If a >, iplis that a is o th right-had sid of origi with distac a d (a, ); ad if a <, iplis that a is o th lft-had sid of origi with distac a d (a, ). So, w calld d (a, ) a is th sigd distac of a to. If D F s, fro Eq. (4), w hav D [ D ( α), D ( α); α]. (7) α L U Ad for vry α [, ], thr is a o-to-o appig btw th lvl α fuzzy itrval [D L (α), D U (α); α] ad ral itrval [D L (α), D U (α)], that is, th followig corrspodc is o-to-o appig: [D L (α), D U (α)] [D L (α), D U (α); α]. (8) W shall us this rlatio latr. Fro Dfiitio, th sigd distac of th lft d poit D L (α) of th α-cut [D L (α), D U (α)] of D to th origi is D L (α), ad th sigd distac of th right d poit D U (α) to th origi is D U (α). hir avrag, /[D L (α) + D U (α)], is dfid as th sigd distac of α-cut [D L (α), D U (α)] to, that is, w dfi th sigd distac of th itrval [D L (α), D U (α)] to as: d([ D ( ), ( )], ) [ L α DU α d( DL( α), ) + d( DU( α), )] [ DL( α) + DU( α)]. (9) Furthr, fro Eqs. (8) ad (9), th sigd distac of lvl α fuzzy itrval [D L (α), D U (α); α] to th fuzzy poit ca b dfid as : d([ D ( ), ( ); ], ) L α DU α α d([ DL( α), DU( α)], ) [ DL( α) + DU( α)]. () hus, fro Eqs. (7) ad (), w ca dfi th sigd distac of a fuzzy st D F s to as follows. Dfiitio 3 For D F s, dfi th sigd distac of D to as dd (, ) d([ D( ), ( ); ], ) [ ( ) ( )]. L α DU α α D L α + DU α Lt D, E F s, fro Eq. (7), w hav D [ D ( α), D ( α); α], E [ E ( α), L U L α α E U (α); α], ad fro Eq. (8), w hav th followig o-to-o appig: For vry α [, ], [D L (α), D U (α)] [D L (α), D U (α); α], [E L (α), E U (α)] [E L (α), E U (α); α].

6 36 LIANG-YUH OUYANG, JINN-SAIR ENG AND MEI-CHUAN CHENG h, fro Eq. (6), w hav [D L (α), D U (α)](+)[e L (α), E U (α)] [D L (α) + E L (α), D U (α) + E U (α)] [D L (α) + E L (α), D U (α) + E U (α); α]. hrfor, fro th Dcopositio hor, w ca gt D ( + ) E [ D ( α) + E ( α), D ( α) + E ( α); α]. () α Siilarly, w hav L L U U [ kdl( α), kdu( α); α], k > ; k α () D [ kdu( α), kdl( α); α], k <. α () Fro th abov discussio, w obtai th followig proprty. Proprty For D, E F s, ad k R, (i) d( D ( + ) E, ) d( D, ) + d( E, ). (ii) dk ( ( ) D, ) kdd (, ). (3) Proof: h proof ca b asily obtaid fro Eq. () ad Dfiitio HE INVENORY PROBLEM o dvlop th proposd odl, w adopt th followig otatio ad assuptios usd i Chag t al. []. Notatio: D: th dad pr yar h: th uit holdig cost pr yar xcludig itrst chargs p: th sllig pric pr uit c: th uit purchasig cost, with c < p I c : th itrst chargs pr $ i stocks pr yar by th supplir I d : th itrst ard pr $ pr yar S: th ordrig cost pr ordr M: th prissibl dlay i sttlig accout (i.., th trad crdit priod) Q: th ordr quatity Q d : th iiu ordr quatity at which th dlay i payts is prittd d : th ti itrval that Q d uits ar dpltd to zro du to both dad ad dtrioratio : th costat dtrioratio rat, whr < I(t): th lvl of ivtory at ti t, t : th rplisht ti itrval Z(): th total rlvat ivtory cost pr yar

7 A FUZZY INVENORY SYSEM UNDER SUPPLIER CREDIS 37 Assuptios: () h dad for th it is costat with ti. () Shortags ar ot allowd. (3) Rplisht is istataous. (4) If th ordr quatity is lss tha Q d, th th payt for th its rcivd ust b ad idiatly. (5) If th ordr quatity is gratr tha or qual to Q d, th th dlay i payts up to M is prittd. Durig th trad crdit priod th accout is ot sttld, gratd sals rvu is dpositd i a itrst barig accout. At th d of th prissibl dlay, th custor pays off all uits ordrd, ad starts payig for th itrst chargs o th its i stocks. (6) i horizo is ifiit. h total rlvat ivtory cost cosists of (a) cost of placig ordrs, (b) cost of dtrioratd uits, (c) cost of carryig ivtory (xcludig itrst chargs), (d) cost of itrst chargs for usold its at th iitial ti or aftr th prissibl dlay M, ad () itrst ard fro sals rvu durig th prissibl priod. 3. Rviw of Chag t al. s Modl Udr th abov otatio ad assuptios, Chag t al. [] cosidrd th four cass: () < < d, () d < M, (3) d M ad (4) M d ad obtaid th total rlvat ivtory cost pr yar as follows: whr Z( ), if < < d ; Z( ), if d < M; Z ( ) Z3( ), if d M ; Z4( ), if M d. ( ) ( ) ( ) S Dh+ c + cic Dh+ cic Z cd+ ( ), (4) ( ) ( ) S Dh+ c Z cd+ ( ) hd pi ( /), d D M (5) ( ) ( ) 3( ) S Dh+ c ( ) hd cicd M Z cd+ + [ ] cicd pidd ( M) M, (6) ad Z 4 () is th sa as Z 3 (), i.., ( ) ( ) 4( ) S Dh+ c ( ) hd cicd M Z cd+ + [ ] cicd pidd ( M) M. (7)

8 38 LIANG-YUH OUYANG, JINN-SAIR ENG AND MEI-CHUAN CHENG 3. h Fuzzy Ivtory Modl I Chag t al. s [] odl, th dtrioratio rat, th rat of itrst chargs, ad th rat of itrst ard ar assud to b costat. I a ral-lif ivtory probl, howvr, it is ot always asy to dtri thir valus xactly. I ost cass, it is likly thy will hav a littl disturbac du to various ucrtaitis durig th ivtory priod. First, w cosidr th dtrioratio rat ar ad fuzzify it as a triagular fuzzy ubr. Fro Yao t al. [9], w hav th followig cocpt: lt Δ < < + Δ whr is a ay fixd poit i [ Δ, + Δ ]. Corrspodig to th itrval [ Δ, + Δ ], ca b cosidrd for fuzzificatio as th triagular fuzzy ubr i th followig: a dcisio akr taks a poit fro th itrval [ Δ, + Δ ], if th poit is, th rror btw th poit ad fixd poit is zro. Basd o th cofidc lvl cocpt; if th rror is zro, th th cofidc lvl is th axiu valu ad st to. If th poit is tak fro th itrval [ Δ, ), wh th poit ovs away fro, th th rror btw th poit ad bcos largr, i.., th cofidc lvl bcos sallr. Morovr, if th poit is qual to Δ, th cofidc lvl attais th iiu valu ad is thus st to. Siilarly, if th poit is tak fro th itrval (, + Δ ], wh th poit ovs away fro, th cofidc lvl bcos sallr. Morovr, if th poit is qual to + Δ, th cofidc lvl rachs. hrfor, corrspodig to th itrval [ Δ, + Δ ], th followig triagular fuzzy ubr is st. ( Δ,, + Δ ), (8) whr < Δ < ad < Δ. hrfor, w obtai th brship fuctio of x +Δ, Δ x ; Δ μ ( x) x +Δ (9), x +Δ; Δ, othrwis, ad th lft ad right d poits of th α-cut of, α, ar L (α) ( α)δ >, ad U (α) + ( α)δ >, () rspctivly. Lt y g(x) x, ad by xtsio pricipl i fuzzy st (s, Zira [3]), w gt th brship fuctio of fuzzy st g( ) as follows: l y + Δ ( Δ), y ; Δ μ ( y) sup ( x) (l y/ ) ( ) l y g μ μ + Δ ( +Δ ) () x y, y ; Δ, othrwis,

9 A FUZZY INVENORY SYSEM UNDER SUPPLIER CREDIS 39 ad th lft ad right d poits of th a-cut of, α, ar ( ) L (α) [-(-α)δ ], ad ( ) U (α) [+(-α)δ ], () rspctivly. W also fuzzify th rat of itrst chargs I c as th followig triagular fuzzy ubr Ĩ c (I c Δ 3, I c, I c + Δ 4 ), (3) whr < Δ 3 < I c ad < Δ 4. h lft ad right d poits of th α-cut of Ĩ c, α, ar (I c ) L (α) I c ( α)δ 3 >, ad (I c ) U (α) I c + ( α)δ 4 >, (4) rspctivly. Furthr, w fuzzify th rat of itrst ard I d as th followig triagular fuzzy ubr Ĩ d (I d Δ 5, I d, I d + Δ 6 ), (5) whr < Δ 5 < I d ad < Δ 6. h lft ad right d poits of th α-cut of Ĩ d, α, ar (I d ) L (α) I d ( α)δ 5 >, ad (I d ) U (α) I d + ( α)δ 6 >, (6) rspctivly. For covic, w lt a S cd, a Dh, a3 Dc, a Dc 4 Dh+, a5 Dc, a 6 pd M, c ( M) D pdm c a, a, P ( ), P ( ), P I () ( ), P4 ( ), P5 ( ), P6 Ic( ), P7 Ic( ), P 8 I ( M) d ad P 9 I c () ( ) ad lt ã j b a fuzzy poit at ral ubr a j, j,,, 8. (7) h, cotrast to Eqs. (4)-(7), w hav th total rlvat ivtory costs pr yar i th crisp cas ad th corrspodig fuzzy cas as follows: Cas : < < d Fro Eq. (4), w lt g ( I, ; ) Z ( ) c I c S cd Dh Dc Dc Dh Dh Dc Dc Ic Ic Dc I c Ic Ic a + a + a + a a a a a. (8)

10 4 LIANG-YUH OUYANG, JINN-SAIR ENG AND MEI-CHUAN CHENG hus, th fuzzy total rlvat ivtory cost pr yar is g( I, c ; ) a ( + )( a () P )( + )( a 3() P )( + )( a 3() P 3)( )( a () P 4)( )( a 4() P 5) ( )( a () P )( )( a () P ). (9) Cas : d < M Fro Eq. (5), w lt g( Id, ; ) Z( ) S cd Dh Dc Dh Dc Dh pd( M ) + + I + d a+ a + a 3 a a 4 a6id. (3) h th fuzzy total rlvat ivtory cost pr yar is g ( I, ; ) a ( + )( a () P )( + )( a () P )( )( a () P )( )( a () P )( )( a () P ). (3) d Cas 3: d M Fro Eq. (6), w lt g ( I, I, ; ) Z ( ) 3 c d 3 ( M) S Dh Dc Dh Dc Dc I c cd Dh Dc Ic c ( M) D Ic pdm I d ( M) I c Ic Ic a + a + a a a + a a a a Id. (3) h th fuzzy total rlvat ivtory cost pr yar is g3( I c, I, d ; ) a ( + )( a ( ) P )( + )( a 3( ) P )( )( a ( ) P 4)( )( a 4( ) P 5)( + )( a 3( ) P 9) ( )( a () P )( )( a () P )( )( a () P ). (33) Cas 4: M d Cas 4 is th siilar to cas 3. hrfor, th fuzzy total rlvat ivtory cost pr yar is g4( I c, I, d ; ) a ( + )( a ( ) P )( + )( a 3( ) P )( )( a ( ) P 4)( )( a 4( ) P 5)( + )( a 3( ) P 9) ( )( a () P )( )( a () P )( )( a () P ). (34) Fro Eqs. (), (), (4), (6) ad (6), w obtai th lft ad right d poits of th α-cut ( α ) of fuzzy sts P (j,,, 9) i Eq. (7) rspctivly as follows: j

11 A FUZZY INVENORY SYSEM UNDER SUPPLIER CREDIS 4 ( ) ( ) [ ( ) ] ( ) ( ) [ ( ) ] L α α Δ U α + α Δ L α P U α [ U( α)] [ + ( α) Δ] [ L( α)] [ ( α) Δ] ( P) ( ),( ) ( ), ( ) ( ) [ ( ) ] ( ) ( ) [ ( ) ] L α α Δ U α + α Δ L α P U α U( α) ( α) L( α) ( α) ( P ) ( ),( ) ( ), + Δ Δ (36) (35) [ ( α) Δ] ( Ic) L( α)( ) L( α) ( Ic ( α) Δ3) 3 L α [ U ( α)] [ + ( α) Δ ] ( P ) ( ), [ + ( α) Δ ] ( Ic) U( α)( ) U( α) [ Ic+ ( α) Δ4 ] 3 U α [ L ( α)] [ ( α) Δ] ( P ) ( ), (37) ( P4) L( α),( P 4) U( α), [ U( α)] [ + ( α) Δ] [ L( α)] [ ( α) Δ ] (38) ( P ) ( α),( P ) ( α), 5 L 5 U U( α) + ( α) Δ L( α) ( α) Δ (39) ( Ic) L( α) Ic ( α) Δ 3 ( Ic) U( α) Ic + ( α) Δ4 ( P6) L( α),( P 6) U( α), [ U( α)] [ + ( α) Δ] [ L( α)] [ ( α) Δ ] (4) ( I ) ( α) I ( α) Δ ( I ) ( α) I + ( α) Δ ( P ) ( ),( ) ( ), c L c 3 c U c 4 7 L α P7 U α U( α) + ( α) Δ L( α) ( α) Δ (4) ad ( P ) ( α) ( I ) ( α) I ( α) Δ,( P ) ( α) ( I ) ( α) I + ( α) Δ, (4) 8 L d L d 5 8 U d U d 6 ( M) [ ( α) Δ]( M) ( Ic) L( α)[ ] L( α) [ Ic ( α) Δ3] 9 L α [ U ( α)] [ + ( α) Δ] ( P ) ( ), ( M) [ + ( α) Δ ]( M) ( Ic) U( α)[ ] U( α) [ Ic+ ( α) Δ4 ] 9 U α [ L ( α)] [ ( α) Δ] ( P ) ( ). I ordr to calculat th sigd distac of th fuzzy st P j (j,,, 9) i Eq. (7), w cosidr th followig fuctios ad lt Z w + u( α), whr u, (43) ( w+ u( α)) w+ u w u α w + K ( w, u, w, u; ) d ( w u Z) / ZdZ w+ u ( α) u u u ( ) + (44) r r (( ) ) ( ) l ( ) r u ( ) r w u w w w u w+ u w + w u,for, u u w r u u r r r ; r!( r)! whr! ( )

12 4 LIANG-YUH OUYANG, JINN-SAIR ENG AND MEI-CHUAN CHENG ( w+ u( α)) w+ u u (,,, ; ) ( w α + ) / w K w u w u d w u Z Z dz ( w+ u ( α)) u u u ( w ) ( ) ( w u w u w u) ( )l w+ u u + u w w+ u u u w u r r (( ) ) ( )( w r r w+ u w + r w u) ( ),for, u u r (45) r ( w+ u( α))( w + u( α)) K3( w, u, w, u, w, u; ) ( w+ u ( α)) w+ u u u ( w w u Z)( w w u Z) / Z dz u + + w u u u u ( w w u) K( w, u, w, u; ) u K( w, u, w, u; ), u + u for. (46) Furthr, by usig aylor s forula, w hav! ax N a x, as ax is ough sall, whr N is th positiv itgr which is dcidd by th dcisio-akr appropriatly. hrfor, fro Eqs. (35)-(46) ad usig aylor s forula, w gt th sigd distac of th fuzzy sts P (j,,, 9) (i Eq. (7)) as follows. j dp (,) [( P ) L( α) + ( P) U( α)] + ( α) Δ ( α) Δ ( + ( α) Δ) ( ( α) Δ) Δ Δ +!!! N N N ( ) ( ( ) ) α (( α) ) d α ( ( ) ) + α Δ ( ( α) Δ) N N N ( ) (,,, ; ) (,,, ; ), K K! Δ Δ + Δ Δ!! (47) ( α) Δ ( α) Δ ( dp,) [( P ) L( ) ( P) U( )] d α α α ( α) Δ ( α) Δ N N N ( ) ( ( ) ) α Δ (( α) Δ) d d!! α + α + ( α) Δ! ( α) Δ N N N ( ) (,,, ; ) (,,, ; ), K K! Δ Δ + Δ Δ!! (48)

13 A FUZZY INVENORY SYSEM UNDER SUPPLIER CREDIS 43 dp ( 3,) [( P 3) L( α) + ( P3) U( α)] + ( ( ) ) ( ( ) ) ( α) Δ ( α) Δ ( Ic ( α) Δ 3) ( Ic + ( α) Δ4) + α Δ α Δ!! ( ( ) ) + N ( ) ( c ( α) 3)( ( α) ) + α Δ N 3 I Δ Δ N3 ( Ic + ( α) Δ4)(( α) Δ)! ( ( α) Δ) N3 N ( ) 3( c, 3,,,, ; ) K I! Δ Δ Δ! N + K I Δ Δ Δ! 3 3( c, 4,,,, ; ), (49) dp ( 4, ) [( P 4) L( α) + ( P4) U( α)] ( + ( α) Δ) ( ( α) Δ) + K(,,, Δ ; ) + K(,,, Δ ; ), (5) dp ( 5,) [( P 5) L( α) + ( P5) U( α)] + ( α) Δ ( α) Δ + K(,,, Δ ; ) + K(,,, Δ ; ), (5) dp ( 6, ) [( 6) ( ) ( 6) ( )] P L α + P U α I ( α) Δ I + ( α) Δ + c 3 c 4 ( + ( α) Δ) ( ( α) Δ) K( I c, Δ3,, Δ ;) + K ( Ic, Δ4,, Δ ;), (5) dp ( 7, ) [( 7) ( ) ( 7) ( )] P L α + P U α I ( α) Δ I + ( α) Δ + ( ) ( ) c 3 c 4 + α Δ α Δ K( I c, Δ3,, Δ ;) + K( Ic, Δ4,, Δ ;), (53)

14 44 LIANG-YUH OUYANG, JINN-SAIR ENG AND MEI-CHUAN CHENG ad dp ( 8, ) [( P 8) L( α) + ( P8) U( α)] I d ( ) 5d I ( ) d 6d Id ( 6 5), α Δ α + + α Δ α + Δ Δ 4 (54) dp ( 9,) [( P 9) L( α) + ( P9) U( α)] + ( ( ) ) ( ( ) ) ( α) Δ( M) ( α) Δ( M) ( M) ( Ic ( α) Δ 3) ( Ic + ( α) Δ4) + α Δ α Δ N4 N [ ( M)] ( M) ( Ic ( α) Δ )( ( ) Δ )!! ( + ( ) Δ ) 3 α α N4 ( M) ( Ic + ( α) Δ4)(( α) Δ) + d! α ( ( α) Δ) N N4 [ ( M)] ( M) K 3( I c, 3,,,, ; )! Δ Δ Δ! N ( M) + K I Δ Δ Δ! 4 3( c, 4,,,, ; ). (55) I Eqs. (47)-(49) ad (55), N ad N ij (i,, 3, 4 ad j, ) ar dcidd by th dcisio-akr appropriatly. Now, usig Eqs. (47)-(55) ad th proprty i sctio, w ca dfuzzify g (Ĩ c, ; ), g (Ĩ d, ; ), g 3 (Ĩ c, Ĩ d, ; ) ad g 4 (Ĩ c, Ĩ d, ; ) i Eqs. (9), (3), (33) ad (34), rspctivly ad gt th stiats of th total rlvat ivtory cost pr yar i th fuzzy ss as follows: Cas : < < d * Z( ) d( g( I, c ; ),) a+ ad( P, ) + a3d( P, ) + a3d( P 3, ) ad( P 4, ) a4d( P 5, ) ad( P, ) ad( P, ) (56) Cas : d < M * Z( ) d( g( I, d ; ),) a + a d( P, ) + a d ( P, ) a d( P, ) a d( P, ) + a d( P, ) (57) Cas 3: d M * Z3( ) d( g3( I c, I, d ; ),) a+ ad ( P, ) + a3d( P, ) ad( P 4, ) a4d( P 5, ) + a3d( P 9, ) ad( P, ) ad( P, ) ad( P, ) (58)

15 A FUZZY INVENORY SYSEM UNDER SUPPLIER CREDIS 45 Cas 4: M d Z ( ) d( g ( I, I, ; ),) Z ( ). (59) * * 4 4 c d 3 4. HE OPIMAL SOLUION I this sctio our objctiv is to dtri th optial valus of which iiiz th total rlvat ivtory cost pr yar Z * (), Z * (), Z 3 * () ad Z 4 * () for cass -4, rspctivly. Cas : < < d * Solvig Z ( ), w gt, ad th obtai Q D ( ). Furthror, w d to chck whthr < < d holds. If it dos, th (, Q) is idd th optial solutio to cas. W lt ( *, Q * ) (, Q), ad hc obtai Z * fro Eq. (56). Othrwis, thr is o fasibl solutio for cas. Cas : d < M * Solvig Z ( ), w gt, ad th obtai Q D ( ). Furthror, w d to chck whthr d < M holds. If it dos, th (, Q) is idd th optial solutio to cas. W lt ( *, Q * ) (, Q), ad hc obtai Z * fro Eq. (57). Othrwis, thr is o fasibl solutio for cas. Cas 3: d M * Solvig Z 3 ( ), w gt, ad th obtai Q D ( ). Furthror, w d to chck whthr d M holds. If it dos, th (, Q) is idd th optial solutio to cas 3. W lt ( *, Q * ) (, Q), ad hc obtai Z * 3 fro Eq. (58). Othrwis, thr is o fasibl solutio for cas 3. Cas 4: M d * Solvig Z 4 ( ), w gt, ad th obtai Q D ( ). Furthror, w d to chck whthr M d holds. If it dos, th (, Q) is idd th optial solutio to cas 4. W lt ( *, Q * ) (, Q), ad hc obtai Z * 4 fro Eq. (59). Othrwis, thr is o fasibl solutio for cas NUMERICAL EXAMPLES Svral urical xapls ar giv to illustrat th abov solutio procdur. I ordr to copar th rsults with thos obtaid fro th crisp cas, w cosidr th data usd i Chag t al. []. Exapl : Giv D uits/yar, h $4/uit/yar, I c.9/$/yar, I d.6/$/ yar, c $ pr uit, p $3 pr uit,.3, M 3 days (.89 yars) ad Q d 7 uits. I additio, w lt Δ Δ Δ 3 Δ 4 Δ 5 Δ 6.5. W also cosidr S, ad 3 pr ordr ad th gt th coputatioal rsults as show i abl.

16 46 LIANG-YUH OUYANG, JINN-SAIR ENG AND MEI-CHUAN CHENG abl. h optial solutios of xapl. Ordrig Cost Rplisht Cycl Ecooic Ordr Quatity otal Rlvat Cost Icrt S * Q(*) pr Yar Z(*) δ (%) Z * () Z * () Z * 3 () fuzzy aual total cost crisp aual total cost Not: δ % crisp aual total cost abl. h optial solutios of xapl. Mi Ordr Quatity Rplisht Cycl Ecooic Ordr Quatity otal Rlvat Cost Icrt Q d * Q(*) pr Yar Z(*) δ (%) Z * 3 () Z * () Z * () fuzzy aual total cost crisp aual total cost Not: δ % crisp aual total cost abl 3. h optial solutios of xapl 3. Crdit Priod Rplisht Cycl Ecooic Ordr Quatity otal Rlvat Cost Icrt M * Q(*) pr Yar Z(*) δ (%) Z * 4 () Z * 3 () Z * () fuzzy aual total cost crisp aual total cost Not: δ % crisp aual total cost Exapl : Giv D uits/yar, h $4/uit/yar, I c.9/$/yar, I d.6/$/ yar, c $3 pr uit, p $4 pr uit,.3, M 3 days (.89 yars) ad S 3 pr ordr. I additio, w lt Δ Δ Δ 3 Δ 4 Δ 5 Δ 6.5. If Q d 8, 9 or uits, th d.7994, or.9985 yar. W hav th coputatioal rsults as show i abl. Exapl 3: Giv D uits/yar, h $4/uit/yar, I c.9/$/yar, I d.6/$/ yar, c $ pr uit, p $35 pr uit,.3, S 5 pr ordr ad Q d 8 uits. I additio, w lt Δ Δ Δ 3 Δ 4 Δ 5 Δ 6.5. Cosqutly, w obtai d.7994 yar. If M, 3 or 4 days, th w ca asily obtai th optial solutios as show i abl 3. Exapl 4: I this xapl, w lt Δ * Δ Δ 3 Δ 5, Δ ** Δ Δ 4 Δ 6 ad S ad cosidr svral valus of diffrt (Δ *, Δ ** ). h raiig ar th sa as th valus i Exapl ad th optial solutios ar suarizd i abl 4.

17 A FUZZY INVENORY SYSEM UNDER SUPPLIER CREDIS 47 abl 4. h optial solutios of xapl 4 for diffrt valus of (Δ *, Δ ** ). Δ * ** Rplisht Δ Cycl * Ecooic Ordr Quatity Q(*) otal Rlvat Cost pr Yar Z(*) Icrt δ (%) Z * () Z * () Z * () Z * () Z * () Z * () Z * () Z * () Z * () Z * () Z * () Z * () Z * () Z * () fuzzy aual total cost crisp aual total cost Not: δ % crisp aual total cost h coputatioal rsults i abl rval that a highr valu of ordrig cost iplis a lowr valu for th diffrc btw th total rlvat ivtory cost pr yar i th crisp ad fuzzy odl, but highr valus of ordr quatity ad rplisht cycl. abl shows that a highr valu of th iiu ordr quatity at which th dlay i payts is prittd causs highr valus of th diffrc btw th total rlvat ivtory cost pr yar i th crisp ad fuzzy odl ad th total rlvat ivtory cost, ordr quatity ad rplisht cycl. abl 3 rvals that a highr valu of th trad crdit priod iplis a lowr valu of th total rlvat ivtory cost. Fro abl 4, w fid that wh Δ * Δ **, th total rlvat ivtory cost i th fuzzy ss is gttig closr to th crisp total rlvat ivtory cost i Chag t al. []. his phoo is discussd i sctio DISCUSSION 6. h Rasoig Bhid th Choic to Dfuzzify th Fuzzy otal Rlvat Ivtory Cost i Cass -4 by Usig th Sigd Distac Mthod Istad of th Ctroid Mthod If w us th ctroid thod to dfuzzify th fuzzy st Ã, w obtai C( A ) x μ ( x) dx μ ( x) dx A A which iplis w hav to fid th brship fuctio of fuzzy st à first. Howvr, i this papr, sic th fuzzy sts of th fuzzy total rlvat ivtory cost i Eqs. (9), (3), (33), ad (34) ar obtaid through coplicatd coputatioal oprators icludig (+), ( ), ( ), ( ), thus it is difficult to fid thir brship fuctio by usig th xtsio pricipl. hrfor w apply th sigd distac thod to dfuzzify th fuzzy total rlvat ivtory cost.

18 48 LIANG-YUH OUYANG, JINN-SAIR ENG AND MEI-CHUAN CHENG 6. h Rasoig Bhid th Choic to Apply th Itrval Opratios Istad of th Stadard Fuzzy Arithtic Opratios Multipl-occurrc of sa fuzzy paratrs i a fuzzy fuctio will caus a fuzzy ubr b ovrstiatd or illgal wh th stadard fuzzy arithtic is applid to solv th probl (s Chag [3]). I this papr, istad of th stadard fuzzy arithtic opratios, w applid th itrval opratios to dal with our odl. h itrval opratios ar adoptd by ay rsarchrs to hadl coplicatd coputatio btw fuzzy ubrs (s Chag [5], Ouyag t al. [8] ad Yao [9]). 6.3 h Rlatioship btw th Fuzzy Cas ad th Crisp Cas Fro Eq. (44), w obtai K (, b,, b; ) l( + b ) as b, ad for, b r ( r ) r r ( b) K(, b,, b; ) l b ( ). b br Du to + b li l ad b b b b r r ( + b) r li, w gt br ( r ) r li K(, b,, b; ) ( ) ( ). (6) r Siilarly, fro Eqs. (45) ad (46), w hav ad b r ( ) r li K(, b,, b; ) ( ), (6) r K3( Ic, b,, b,, b;) K( Ic, b,, b;) ( )( ) l b ( ) Ic Ic + + Ic + asb, b + b rspctivly. hrfor, by Eqs. (6) ad (6), w hav li K ( I, b,, b,, b; ) b 3 c ( I + ) li K (, b,, b; ) li K (, b,, b; ), for. (6) c b b Furthror, wh Δ Δ b, fro Eqs. (47) ad (6), (48) ad (6), (5) ad (6), (5) ad (6), w obtai r r

19 A FUZZY INVENORY SYSEM UNDER SUPPLIER CREDIS 49 ad li d( P, ), (63) b b li d( P, ), (64) li dp ( 4, ), (65) b li dp ( 5, ). (66) b Wh Δ Δ Δ 3 Δ 4 b, fro Eqs. (49), (55) ad (6), w gt ad b I c 3 li d( P, ), (67) b ( M) Ic 9 li d( P, ). (68) ad Fro Eqs. (44) ad (45), w hav ( Ic ) Ic K( Ic, b,, b;) ( Ic )l b b asb b K ( I, b,, b;) c ( Ic ) ( ) l Ic I b + c + + asb. b + b (69) (7) Wh Δ Δ Δ 3 Δ 4 b, fro Eqs. (5), (53) ad (69), (7), w obtai ad Ic li dp ( 6, ) (7) b Ic li dp ( 7, ). (7) b Wh Δ 5 Δ 6, fro Eq. (54), w gt (,). (73) d P8 I d Cobig th Eqs. (63)-(68), (7)-(73), (56)-(59), (8), (3) ad (3), w gt th followig rsult. Wh Δ Δ Δ 3 Δ 4 b ad Δ 5 Δ 6, w hav b * j li Z ( ) Z ( ), j,, 3, 4. (74) j

20 5 LIANG-YUH OUYANG, JINN-SAIR ENG AND MEI-CHUAN CHENG hat is, wh Δ Δ Δ 3 Δ 4 b ad Δ 5 Δ 6, th fuzzy cas is th sa as th crisp cas. hus, a crisp cas is th spcial situatio of a fuzzy cas. 7. CONCLUSION Du to th various ucrtaitis that ay occur i th cotxt of th ral-world ivtory probl, th rat of itrst chargs, th rat of itrst ard, ad th dtrioratio rat ay ot rai costat. o dal with ths ucrtaitis, w apply th fuzzy st thory basd o Chag t al. s [] odl. W costruct thr diffrt itrvals to iclud th rat of itrst chargs, th rat of itrst ard, ad dtrioratio rat ad subsqutly driv th fuzzy total rlvat ivtory cost. By utilizig th sigd distac thod of dfuzzificatio, w driv th stiat of th total rlvat ivtory cost i th fuzzy ss. W also discuss how to dtri th optial ordrig policy so that th total rlvat ivtory cost i th fuzzy ss is iial. h optial ordrig policy is dtrid i a fuzzy virot which provids th dcisio akr with a dpr isight ito th probl. Fially, so urical xapls ar providd to illustrat th solutio procdur. abl 4 shows that as th itrval variatio of th rat of itrst chargs, th rat of itrst ard ad th dtrioratio rat bcos sufficitly sall, th fuzzy total rlvat ivtory cost pr yar bcos clos to th crisp total rlvat ivtory cost pr yar. hat is, th crisp odl ca b viwd as a spcial cas of th fuzzy odl. Futur rsarch o this probl ay iclud additioal sourcs of ucrtaity i th fuzzy odls, such as a ucrtai dad rat. ACKNOWLEDGEMENS h authors would lik to gratfully xprss thir apprciatio to Profssor Jig- Shig Yao, Eritus Profssor of Natioal aiwa Uivrsity, ad Profssor Yu-Hwa Chiag, Profssor of Mig Chua Uivrsity for thir valuabl ad costructiv cots to iprov this papr ad to th aoyous rfrs for thir valuabl ad hlpful isights ad suggstios rgardig a arlir vrsio of th papr REFERENCES. C.. Chag, L. Y. Ouyag, ad J.. g, A EOQ odl for dtrioratig its udr supplir crdits likd to ordrig quatity, Applid Mathatical Modllig, Vol. 7, 3, pp D. Prtrovic ad E. Swy, Fuzzy kowldg-basd approach to tratig ucrtaity i ivtory cotrol, Coputr Itgratd Maufacturig Systs, Vol. 7, 994, pp M. J. Yao, P.. Chag, ad S. F. Huag, O th cooic lot schdulig probl with fuzzy dads, Itratioal Joural of Opratios Rsarch, Vol., 5, pp J. S. Yao, S. C. Chag, ad J. S. Su, Fuzzy ivtory without backordr for fuzzy quatity ad fuzzy total dad quatity, Coputrs ad Opratios Rsarch, Vol.

21 A FUZZY INVENORY SYSEM UNDER SUPPLIER CREDIS 5 7,, pp H. C. Chag, A applicatio of fuzzy sts thory to th EOQ odl with iprfct quality its, Coputrs ad Opratios Rsarch, Vol. 3, 4, pp H. C. Chag, J. S. Yao, ad L. Y. Ouyag, Fuzzy ixtur ivtory odl ivolvig fuzzy rado variabl lad-ti dad ad fuzzy total dad, Europa Joural of Opratioal Rsarch, Vol. 69, 6, pp S. H. Ch, S.. Wag, ad S. M. Chag, Optiizatio of fuzzy productio ivtory odl with rpairabl dfctiv products udr crisp or fuzzy productio quatity, Itratioal Joural of Opratios Rsarch, Vol., 5, pp M. K. Maiti, Fuzzy ivtory odl with two warhouss udr possibility asur o fuzzy goal, Europa Joural of Opratioal Rsarch, Vol. 88, 8, pp S. H. Ch ad C. C. Wag, Backordr fuzzy ivtory odl udr fuctioal pricipl, Iforatio Scics, Vol. 95, 996, pp M. Vujosvic, D. Ptrovic, ad R. Ptrovic, EOQ forula wh ivtory cost is fuzzy, Itratioal Joural of Productio Ecooics, Vol. 45, 996, pp M. G, Y. sujiura, ad P. Z. Zhg, A applicatio of fuzzy st thory to ivtory cotrol odls, Coputrs ad Idustrial Egirig, Vol. 33, 997, pp K. Roy ad M. Maiti, A fuzzy EOQ odl with dad-dpdt uit cost udr liitd storag capacity, Europa Joural of Opratioal Rsarch, Vol. 99, 997, pp H. Ishii ad. Koo, A stochastic ivtory with fuzzy shortag cost, Europa Joural of Opratioal Rsarch, Vol. 6, 998, pp H. M. L ad J. S. Yao, Ecooic ordr quatity i fuzzy ss for ivtory without backordr odl, Fuzzy Sts ad Systs, Vol. 5, 999, pp J. S. Yao ad H. M. L, Fuzzy ivtory with or without backordr for fuzzy ordr quatity with trapzoid fuzzy ubr, Fuzzy Sts ad Systs, Vol. 5, 999, pp H. C. Chag, J. S. Yao, ad L. Y. Ouyag, Fuzzy ixtur ivtory odl with variabl lad-ti basd o probabilistic fuzzy st ad triagular fuzzy ubr, Mathatical ad Coputr Modlig, Vol. 39, 4, pp P.. Chag, M. J. Yao, S. F. Huag, ad C.. Ch, A gtic algorith for solvig a fuzzy cooic lot-siz schdulig probl, Itratioal Joural of Productio Ecooics, Vol., 6, pp L. Y. Ouyag, K. S. Wu, ad C. H. Ho, Aalysis of optial vdor-buyr itgratd ivtory policy ivolvig dfctiv its, Itratioal Joural of Advacd Maufacturig chology, Vol. 9, 6, pp J. S. Yao, W.. Huag, ad.. Huag, Fuzzy flxibility ad product varity i lot-sizig, Joural of Iforatio Scic ad Egirig, Vol. 3, 7, pp S. K. Goyal, Ecooic ordr quatity udr coditios of prissibl dlay i payts, Joural of th Opratioal Rsarch Socity, Vol. 36, 985, pp J.. g, O th cooic ordr quatity udr coditios of prissibl dlay i payts, Joural of th Opratioal Rsarch Socity, Vol. 53,, pp C.. Chag ad J.. g, Rtailr s optial ordrig policy udr supplir crd-

22 5 LIANG-YUH OUYANG, JINN-SAIR ENG AND MEI-CHUAN CHENG its, Mathatical Mthods of Opratios Rsarch, Vol. 6, 4, pp R. A. Davis ad N. Gaithr, Optial ordrig policis udr coditios of xtdd payt privilgs, Maagt Scic, Vol. 3, 985, pp L. Y. Ouyag, C.. Chag, ad J.. g, A EOQ odl for dtrioratig its udr trad crdits, Joural of th Opratioal Rsarch Socity, Vol. 56, 5, pp L. Y. Ouyag, J.. g, ad L. H. Ch, Optial ordrig policy for dtrioratig its with partial backloggig udr prissibl dlay i payts, Joural of Global Optiizatio, Vol. 34, 6, pp J.. g, C.. Chag, ad S. K. Goyal, Optial pricig ad ordrig policy udr prissibl dlay i payts, Itratioal Joural of Productio Ecooics, Vol. 75, 5, pp L. H. Ch ad L. Y. Ouyag, Fuzzy ivtory odl for dtrioratig its with prissibl dlay i payt, Applid Mathatics ad Coputatio, Vol. 8, 6, pp A. M. M. Jaal, B. R. Sarkr, ad S. Wag, A ordrig policy for dtrioratig its with allowabl shortag ad prissibl dlay i payt, Joural of Opratios Rsarch Socity, Vol. 48, 997, pp A. Kaufa ad M. M. Gupta, Itroductio to Fuzzy Arithtic: hory ad Applicatios, Va Nostrad Rihold, Nw York, 99, pp J. S. Yao ad K. Wu, Rakig fuzzy ubrs basd o dcopositio pricipl ad sigd distac, Fuzzy Sts ad Systs, Vol. 6,, pp H. J. Zira, Fuzzy St hory ad its Applicatios, 3rd d., Kluwr Acadic Publishrs, Dordrcht, P.. Chag, Fuzzy stratgic rplact aalysis, Europa Joural of Opratioal Rsarch, Vol. 6, 5, pp Liag-Yuh Ouyag ( ) is a Profssor i th Dpartt of Maagt Scics ad Dcisio Makig at akag Uivrsity i aiwa. H ard his B.S. i Mathatical Statistics, M.S. i Mathatics ad Ph.D. i Maagt Scics fro akag Uivrsity. His rsarch itrsts ar i th fild of productio/ivtory cotrol, probability ad statistics. H has publicatios i Joural of th Opratioal Rsarch Socity, Coputrs ad Opratios Rsarch, Europa Joural of Opratioal Rsarch, Coputrs ad Idustrial Egirig, Itratioal Joural of Productio Ecooics, IEEE rasactios o Rliability, Productio Plaig ad Cotrol, Joural of th Opratios Rsarch Socity of Japa, Itratioal Joural of Systs Scic, Mathatical ad Coputr Modllig ad Applid Mathatical Modllig.

23 A FUZZY INVENORY SYSEM UNDER SUPPLIER CREDIS 53 Ji-sair g ( ) rcivd a B.S. dgr i Mathatical Statistics fro akag Uivrsity, a M.S. dgr i Applid Mathatics fro Natioal sig Hua Uivrsity i aiwa, ad a Ph.D. i Idustrial Adiistratio fro Cargi Mllo Uivrsity i U.S.A. H joid th Dpartt of Marktig ad Maagt Scics at Willia Ptrso Uivrsity of Nw Jrsy i 99. His rsarch itrsts iclud supply chai aagt ad arktig rsarch. H has publishd rsarch articls i Maagt Scics, Marktig Scic, Joural of th Opratioal Rsarch Socity, Opratios Rsarch Lttrs, Naval Rsarch Logistics, Europa Joural of Opratioal Rsarch, Applid Mathatical Modllig, Joural of Global Optiizatio, Itratioal Joural of Productio Ecooics, ad othrs. Mi-Chua Chg ( ) is currtly a Ph.D. studt at akag Uivrsity i aiwa. Sh rcivd hr M.B.A. dgr i statistics fro akag Uivrsity ad a B.S. dgr i Statistics fro Fu J Catholic Uivrsity. Hr rsarch itrst lis i th fild o th aalysis of ivtory syst, probability ad statistics. Sh has publicatios i Itratioal Joural of Iforatio ad Maagt Scics, Yugoslav Joural of Opratios Rsarch, Itratioal Joural of Systs Scic ad OP a Official Joural of th Spaish Socity of Statistics ad Opratios Rsarch.