PERISHABLES INVENTORY CONTROL MODEL UNDER TIME- VARYING AND CONTINUOUS DEMAND

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1 PERISHABLES INVENTORY CONTROL MODEL UNDER TIME- VARYING AND CONTINUOUS DEMAND Xangyang Ren 1, Hucong L, Meln Ce ABSTRACT: Ts paper consders e yseress persable caracerscs and sorage amoun of delayed rae of persables compreensvely, and esablses a realer nvenory model sorage pon and meamorpsm pon. Persables sorage occurred afer meamorpsm, e realer nvenory cos n addon o e orderng cos, e nvenory oldng cos, e sorage losses cos nvolves also e cos of meamorpsm. Fnally, a numercal example s gven o analyze e proposed model and e parameer sensvy s dscussed. KEY WORDS: persables, nvenory conrol model, connuous demand, me-varyng 1 INTRODUCTION W e mprovemen of e level of producon modernzaon, e producon scale and nvenory scale gradually ncreasng, e nvenory managemen of persables as become one of e o ssues wc as araced by scolars n recen decades. Persables can be also called seasonal producs, persable producs and e sor lfe cycle producs ec. w e obvous caracerscs, suc as e sor sales cycle, long producon lead me, uncerany demands, low resdual value of unsold n end of e perod and ger processng cos ec Lu, Ln, & Cen, 212). Persables as caracerscs of g me requremen, erefore e researc on e yseress persables nvenory sraegy under me-varyng demand as a very mporan conrbuon. Some researces abou nvenory model consderng e deeroraon rae ave been presened. Moon suded e nvenory model w consan deeroraon rae Moon, Gr, & Ko, 25). Aggarwal esablsed e opmal orderng quany model of deeroraon producs were s assumed a e deeroraon rae s consan and e deferred paymen s allowed Aggarwal & Jagg, 1995). Cu proved e oal cos s a pecewse convex funcon n furer and proposed a smpler soluon process based on e researc by Aggarwal Cu, Cung, & Lan, 1998). Cang proposed e opmal economc order model a consdered e nflaon and allowed e deferred paymen w e deeroraon rae uncanged Cange, 24). Sa esablsed a bulk purcasng model for deerorang ems under wo scenaros, fxed 1 Scool of Economcs and Managemen, Hebe Unversy of Engneerng, Handan, Hebe Provnce, P.R.Cna Emal: boyrenxy@126.com perod and unfxed perod Sa & Sa, 1993; Sa, 1998), and a bulk purcasng model w a consan deeroraon rae s gven were e replensmen lead me s se o zero Sa & Sa, 2). Ln esablsed an nvenory replensmen polcy w e assumpon a deeroraon rae ncreases lnearly w me and n a fxed perod e nvenory updae speed and e servce level are equal Ln, Tan & Lee, 2). We esablsed a deeroraon producs nvenory model n wc e rae of deeroraon obeys wo-parameer Webull dsrbuon Wee, 1999). Cauur nroduced e wo-parameer and e ree-parameer Webull dsrbuon funcon o replace lnear and exponenal relaed deeroraon rae. And e woparameer and e ree-parameer Webull dsrbuon funcon makes nvenory model more ally w e acual suaon Cakrabary, Grl & Cauur, 1998). Cen Cen, 1998), Cang Cang & Dye, 2) suded and found e deeroraon rae of persable goods was a wo-parameer Webull dsrbuon, consderng e servce level, replensmen cycle and prce dscouns of e deeroraon producs nvenory model. Moon and Lee respecvely esablsed an nvenory replensmen model for persable goods referrng o exponenal dsrbuon and Gaussan dsrbuon Moon & Lee, 2). Guo suded e opmal economc order model wle e deeroraon rae of e deeroraon producs s Guo, 24). Huang esablsed an EOQ model for e deeroraon producs, akng e facors wc affecng e sorage me on e deeroraon rae no accouns Huang, Huang & Ca, 26). L esablsed an opmal orderng and nvenory model for e deeroraon producs based on reeparameer Webull dsrbuon funcon L, Huang & Luo, 24). Peng esablsed an opmal nvenory and prcng model for e deeroraon producs w e assumpon a deeroraon rae 36 ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 14, ISSUE 1/216

2 s an exponenal funcon and consderng e me value of capal and e effec of purcase quany dscouns Peng & Tan, 24). Te researc on backloggng rae of nvenory mode as been also suded. Heng esablsed an nvenory conrol model w e yseress producs supply rae cange as e wang me leng of cusomer and e sorage and orderng coss are dfferen a dfferen cycles Heng, Labban & lnn, 1991). Papacrsos researced on backloggng rae wc reduced w e cusomer wang me from e perspecve of manufacurer, and usng e deerorang ems nvenory model were quany dscouns ncrease w respec o e sales volume Papacrsos & Skour, 27). Cang esablsed e producs prof funcon based on wo suaons respecvely, for example e sorage volume s fully delayed and e sorage volume does no allow delayed Cang, 24). Cung consdered compreensvely e mpac of cas dscouns and me value of capal, en esablsed a deeroraon producs nvenory replensmen sraegy model Cung & Ln, 21). Zao suded e mpac caused by e cange of e nvenory cos and e nvenory level on e sellng rae w e purpose of e mnmum cos and maxmum prof Zao & Lu, 24). P suded vendor managemen opmal nvenory polcy of deeroraon producs under e Supply Can Managemen envronmen P, Meng & Huang, 21). Leng esablsed a nonlnear deeroraon producs nvenory model assumng a durng e sockou perod, cusomers amoun of los s a Gaussan dsrbuon and goods sorage permsson Leng, Lu & Huang, 24). Yang used e s,s) nvenory polcy o resock, esablsed a prof model for deeroraon producs assumng a e sock level affec e sales rae, sorage compleely delayed Yang & Huang, 25). Luo esablsed an EOQ model for e deeroraon producs based on e wo suaons a sorage compleely delayed and no sorage delayed Luo, Xong & Yang, 22). Ts paper esablses a new sock conrol model consderng e deeroraon rae and backloggng raes. In e proposed model, e amoun of e producs s no enoug o supply once e meamorpsm occurs, and e realer nvenory cos n addon o e orderng cos, e nvenory oldng cos, e cos of sorage losses nvolves also e cos of meamorpsm. 2 VARIABLE DEFINITIONS Te nvenory sysem esablsed s operang n a fne me orzon H. Realers demand per un me s a lnear funcon as e nvenory level, and le f ) denoes e demand rae a me s: D I ) I ) f) D I ) 1) were D s e cusomers demand rae a me, s an mpac coeffcen of e sock volumes on e sale rae, D, are consans D, ). I ) s e nvenory level a me assumng a e un me s very sor. Ts paper deals w e me varable approxmaely as a connuous varable M & Zou, 21). Te sorage appears wen e realer nvenory oldngs canno mee e cusomer demand. Te sorage exs parally n e nex cycle, bu n e end of cycle s no allowed, and e sorage volume delayed rae s relaed o e cusomer wang me. Se e proporonal funcon a cusomer s wllng o wa delvery durng e sorage b ) : b) e, b ) 1, T. s e scale facor, s e me leng a s no allowed sorage durng a cycle, and T s e duraon of an order cycle. Demand rae of realers wang for delvery a me s S ). Realer nvenory deeroraon rae s consans, and. Hyseress area were deeroraon does no occur s, e laer s meamorpc area Cao, Ce & Wu, 212). Realers ake e same cycle resock. Order number s n wn e nerval H, en e resock cycle s T H n, and e resock me pon s T 1) H / n, T H. Te orderng cos s C b. n Insananeous replensmen s consdered, a s o say e replensmen lead me s. Te nvenory oldng cos per un me and per un persables s d, e sorage loss cos per un me and per un persables s f, e deeroraon cos per un me and per un persables s g, e oal nvenory oldng cos over a cycle s C, e oal sorage losses cos s C f, e oal deeroraon d ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 7,ISSUE 1/29 37

3 cos s C g, and e oal nvenory cos over e nerval H s C. Accordng o e above assumpons, e delayed demand rae s b ) a me durng sorage,, T 1]. Terefore, e demand of cusomers wang S ) a e sorage me sould sasfy e followng dfferenal equaon Wang, 211): ds) d b ),, T 1 2) Te boundary condon s S,), and en we ave e soluon: e e S ) b u) du 3), 3 INVENTORY CONTROL MODE Accordng o assumpon 6), e cycle orzon H s dvded no n equal w e leng l H. n Te replensmen pon a e begnnng of cycle T l H 1) 1). Te las cycle s no n allowed o be ou of sock, so n Tn 1 H. I ncludes wo perods: non-deeroraon perod T, ) and deeroraon perod, T ] were,, 1, s e deeroraon pon of e cycle. Oer cycles T, T 1 nclude e followng perods: wen e sorage pon, occurs n e nondeeroraon of e,, namely wen sorage pon s e former. I s dvded no nvenory oldng and non-deeroraon perod T, ) and, sorage perod, ]. Te ou of sock pon, T 1, occurs n e deeroraon perod, wc mples a wen e deeroraon pon s e former. I s dvded no nvenory oldng and nondeeroraon perod T,,), nvenory oldng and deeroraon perod and ou of sock,,, perod,, T 1]., s e ou of sock pon of e cycle. Snce, s known a e demand ncreases gradually w e sock volumes dmnsed rapdly over one cycle, us e nvenory funcon wn a cycle s a downwardly convex decreasng curve. I ) denoes a e nvenory level of realer a durng replensmen cycle. 3.1 Invenory conrol model w sorage pon prevous n-1) cycles Te realer s no allowed o be sorage sae n e las cycle as menoned n e prevous secon, so we only dscusses e frs n 1) cycles. I ncludes wo perods: nvenory oldng wou deeroraon perod T, ) and sorage perod, ]., T 1, In e nerval T, ), e nvenory ems do, no urn deerorave and e nvenory level s posve. Te cange rae of nvenory level s equal o e demand rae of e ems wen I ). So w n T, ), e nsananeous sae of, nvenory level sould sasfy e followng condon: 4) di,1 ) D I d,1 ), T, Te boundary condon s I ), e nvenory of realer a me s:, 1, D, ) I,1 ) e 1] T,,) 5) In e nerval, ], e nvenory level s, T 1 negave and ere s no deeroraon. Te cange rae of nvenory level s equal o e demand rae of e ems wen I ). So durng e nerval, ], e nsananeous sae of, T 1 nvenory level sould sasfy e followng condon: di,2 d ) D, T 6), 1 Te boundary condon s I ), e nvenory of realer a s:, 2, I,2 ) D, ),, T 1] 7) Te orderng cos of e realer n e cycle sc b. Te oal nvenory oldng cos of e realer over a cycle s defned as:, D, Cd d I,1 ) d d e 2, 1) 8) T 38 ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 14, ISSUE 1/216

4 Te sorage cos of e realer n e cycle s gven by: T T ft f fe fe C f f S) d e e 9) 2 2 Te nvenory cos of e realer over a fne orzon H n addon o e las cycle) s: n1 C n 1) C C C ) 1) b d f 1 Opmal soluon analyss: from Eq. 1), e oal nvenory cos C s e connuous funcon of e ndependen varable,. By deermnng e value of,, e objecve funcon C can be obaned o e mnmum, wc corresponds o oban e maxmum value of C C ). Te d f necessary condons for C C ) o e maxmum are gven as: d C C ) d d, f d f 11), dd, e f, T 1) dde 12) Proposon 1: Eq. 12) as e only soluon. Proof: Frsly, n order o prove e exsence of zero soluon of Eq. 12), le:,, 1 g ) e f T ) dde, dd,, 13) Ten: dd lm ) 14) g, T 1 f dd lm ) lm ) dd, dde ], g, e f, T 1,, 15) So: Eq. 12) mus as a zero soluon. In e followng e unqueness s proved, and akng e dervave of e funcon g ) and,. dg ) d,,,, e f T 1,) e, f dde, 16) Tus: g ) s e ncreasng funcon of e, ndependen varable,, so Eq. 12) as e only soluon. Proof compleed. Proposon 2: Ta, mees Eq. 12) s e unque soluon o oban e mnmum of C C ). d f Proof: 2 d Cd C f 2 d, ), e f T 1,) f ], dde So: C C ) as e mnmum a,. d f Proof compleed. 17) 3.2 Invenory conrol model w meamorpsm pon prevous n-1) cycles Te realer s no allowed o be sorage n e las cycle, so s secon dscusses e frs n 1) cycles separaed from e n cycle. 1) Te nvenory level of realer n e frs 1) n cycles a) Wn e nerval, e nvenory ems do no urn deerorave and nvenory level s posve. Te cange rae of nvenory level s equal o e demand rae of e ems wen I ). So wn T, ], e nsananeous sae of, nvenory level sould sasfy: di,3) D I ), T d I ) canges connuously. So:,3, D ),, ) I,3, ) e 1 Te nvenory of realers a s: 18) 19), ) D D ),, ) D I,3 ) e e 1 ] T,, ] ) 2) b) Wn e nerval, ), nvenory,, ems urn deerorave and nvenory level s posve. Te cange rae of nvenory level s equal ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 7,ISSUE 1/29 39

5 o e demand rae and e deeroraon rae of e ems wen I ). So wn, ), e nsananeous sae of,, nvenory level sould sasfy e followng condon: di,4) D I,4 ) I,4 ),,, d 21) Te boundary condon s I ), and e nvenory of realer a s:, 4, D ), ) I,4 ) e 1],,, ) 22) c) Wn e nerval, ], e nvenory, T 1 level s negave. Te cange rae of nvenory level s equal o e demand rae of e ems wen I ). So wn e me duraon, ], e, T 1 nsananeous sae of nvenory level sould sasfy e followng condon: di,5 ) D,, T 1 d 23) Te boundary condon s I ), and e nvenory of realer a s:, 5, I, 5 ) D, ),, T 1] 24) Te orderng cos of e realer n e cycle s C. b Te oal nvenory oldng cos of e realer n e cycle s as follows: s: C d d T,, I, 3 ) d I,4 ) d], 1 D D ),, ) e d e 1) ] D d, dd e ) ),, ) 1,,, ) 25) Te sorage cos of e realer n e cycle ft T 1 1, C f f S) d e, T 1, f, fe fe, e ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 14, ISSUE 1/216 26) Te deeroraon cos of e realer n e cycle s:, C ) g g I,4 d, ),, ) gd e 1,, 27) Te nvenory cos of e realer over e me orzon H n addon o e las cycle) s: n1 C n 1) C C C C ) b d f g 1 28) 2) Te nvenory level of realer n e n cycle a) Wn e nerval T, ), e nvenory n n, ems do no urn deerorave. Te cange rae of nvenory level s equal o e demand rae of e ems wen I ). So wn T, ), e nsananeous sae of n n, nvenory level sould sasfy e followng dfferenal equaon: din,1 ) D I d n,1 ), T n n, 29) I n, 1 ) s connuous, so e nvenory of realers a s: ) D ) Tn 1 ) 1] D ) In,1 e e ) D T n, n, ) 3) b) Durng e me range, T ], e n, n1 nvenory ems urn o deeroraon. Te cange rae of nvenory level s equal o e demand rae and e deeroraon rae of e ems wen I ). So wn e nerval, T ], e n, n1 nsananeous sae of nvenory level sould sasfy e followng dfferenal equaon: din,2 ) D I d n,2 ) I n,2 ), n, T n1 31)

6 Te boundary condon s I ) n, 2 T n 1, e nvenory of realer a s: D ) Tn1 ) In,2 ) e 1] n,, Tn 1] ) 32) Te orderng cos of e realer n e las cycle s C. b Te oal nvenory oldng cos of e realer n e las cycle s as follows: n, Tn 1 d n,1 ) n,2 ) ] T n n, C d I d I d D D ) Tn1 ) e 1] ) D n, n, d e 1) ) Tn 1 n, ) D e 1 Tn 1 n, )] 33) Te deeroraon cos of e realer n e las cycle s: g Tn 1 C g I d n, ) Tn ) 1 n, ) n,2 gd e 1 Tn 1 n, ] 34) Te nvenory cos of e realer n e las cycle s: C Cb Cd Cg 35) Opmal soluon analyss: sows a e oal nvenory cos C s e connuous funcon of ndependen varable T n1. By deermnng e value of T n1, e objecve funcon C s obaned o e mnmum, and e necessary condon s sown as: dc dt n1 g d) D e g d) D e ) T n n ) T n n, ) ) 1, ) ) 1] Invenory conrol model analyss 1] 36) 1) Solve e gven nal value Te paper uses Malab o calculae e numercal example, and obans e prevous deeroraon pon, e opmal sorage pon of sorage me n deeroraon perod s 13.1 days, dc By dt ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 7,ISSUE 1/29 41 n1, e funcon C s a monoone decreasng funcon. Snce C s a bounded funcon, ere s T n 1 o make C o e mnmum. Proof compleed. 4 EMPIRICAL ANALYSIS 4.1 Te seng of e conrol parameers Te paper nvesgaed persable producons of a supermarke n Handan as long as possble; e specfc parameers are sown n Table 1. Table 1. Te Parameers Used n e Proposed Model Te symbols and meanngs of e Param parameers eer value Plannng perod H Te leng of an order cycle T Order mes n n plannng perod H Cusomer demand rae per un me D Te coeffcen of nvenory mpacng sales 9 days 15 days 6 mes 6 arcles.15 Te rao wllng o wa for delvery of cusomers.2 Deerorang rae.35 Order cos of realer eac me C b Invenory oldng cos per un me per un produc d Invenory sorage cos per un me per un produc f Deeroraon cos per un me per un produc g Tme leng wou deeroraon n a cycle, 2 RMB/me days and e realer s opmal nvenory cos s 1681 RMB. 2) Sensvy analyss of parameers a) Te nfluence caused by e cange of e cusomer demand rae per un me on e oal nvenory cos Keepng oer parameers uncanged, e nfluence caused by e cange of n e cusomer demand rae per un me on e opmal sorage

7 pon and e oal nvenory cos s analyzed. Te resuls are sown n Table 2. Table 2. Sensvy Analyss of e Cusomer Demand Rae D Per Un Tme D day) C RMB) Table 2 sows a n e sae of prevous produc deeroraon pon, e nfluence caused by e cange of e cusomers demand rae D per un me on e opmal sorage pon and e oal nvenory cos. W e ncrease of e cusomer demand rae D, e opmal sorage pon bascally remans uncanged and e nvenory cos gradually ncreases. Te cange of e cusomers demand rae D per un me as a less nfluence on e opmal sorage pon and e nvenory cos. b) Te nfluence caused by e cange of e nvenory affecng sale rae on e nvenory cos Keep oer parameers uncanged, e nfluence caused by e coeffcen of e nvenory mpacng sale rae on e opmal sorage pon and nvenory cos s suded. Te resuls are gven n Table 3. Table 3. Sensvy Analyss of e Coeffcen of e Invenory Impacng Sale Rae day) C RMB) Table 3 sows a n e sae of prevous produc deeroraon pon, e nfluence caused by e coeffcen of nvenory mpacng sale rae on e opmal sorage pon and e oal nvenory cos. W e ncrease of e coeffcen of nvenory mpacng sale rae, e opmal sorage pon gradually decreases, and e nvenory cos gradually ncreases. Te cange of e coeffcen of nvenory mpacng sales rae as a less nfluence on e opmal sorage pon and e nvenory cos. c) Te nfluence caused by e cange of cusomer s wllng o wa for delvery rae on e nvenory cos Keep oer parameers uncanged, analyss on e nfluence caused by e coeffcen of cusomer s wllng o wa for delvery rae on e opmal sorage pon and nvenory cos are done. Te calculaon resuls are sown n Table 4. Table 4. Sensvy Analyss of Cusomer s Wllng o Wa for Delvery Rae day) C RMB) Table 4 sows a n e sae of prevous produc deeroraon pon, e nfluence caused by e coeffcen of cusomer s wllng o wa for delvery rae on e opmal sorage pon and e oal nvenory cos. W e ncrease of e coeffcen of cusomers wllng o wa for delvery rae, e opmal sorage pon and e nvenory cos bascally remans uncanged. Te cange of e coeffcen of cusomer s wllng o wa for delvery rae as a less nfluence on e opmal sorage pon and e nvenory cos. d) Te nfluence caused by e cange of e deeroraon rae on e nvenory cos Keep oer parameers uncanged, analyss e nfluence caused by e canged deeroraon rae on e opmal sorage pon and nvenory cos are dscussed. Te calculaon resuls are sown n Table 5. Table 5. Sensvy Analyss of e Deeroraon Rae day) C RMB) Table 5 sows a n e sae of prevous produc deeroraon pon, e nfluence caused by e deeroraon rae on e opmal sorage pon and e oal nvenory cos. W e ncrease of e deeroraon rae, e opmal sorage pon gradually decreases, e nvenory cos gradually ncreases. Te cange of e deeroraon rae as a less nfluence on e opmal sorage pon and e nvenory cos. e) Te nfluence caused by e cange of nvenory oldng cos d on e nvenory cos Keep oer parameers uncanged, analyss of e nfluence caused by e canged nvenory oldng cos on e opmal sorage pon and 42 ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 14, ISSUE 1/216

8 nvenory cos are dscussed. Te calculaon resuls are as sown n Table 6. Table 6. Sensvy Analyss of e Invenory Holdng Cos d d da y) C R MB) Table 6 sows a n e sae of prevous produc deeroraon pon, e nfluence caused by e oldng cos d on e opmal sorage pon and e oal nvenory cos. W e ncrease of e oldng cos d, e opmal sorage pon gradually decreases, e nvenory cos gradually ncreases. Te cange of e oldng cos d as a less nfluence on e opmal sorage pon and e nvenory cos. f) Te nfluence caused by e cange of nvenory sorage cos f on e nvenory cos Keep oer parameers uncanged, e nfluence caused by e canged nvenory sorage cos on e opmal sorage pon and nvenory cos are gven. Te calculaon resuls are sown n Table 7. Table 7. Sensvy Analyss of e Invenory Sorage Cos f f day) C RMB) Table 7 sows a n e sae of prevous produc deeroraon pon, e nfluence caused by e nvenory sorage cos f on e opmal sorage pon and e oal nvenory cos. W e ncrease of e nvenory sorage cos f, e opmal sorage pon and e nvenory cos reman uncanged. Te cange of e nvenory sorage cos f as a less nfluence on e opmal sorage pon and e nvenory cos. g) Te nfluence caused by e cange of deeroraon cos g Keep oer parameers uncanged, e nfluence caused by e canged deeroraon cos on e opmal sorage pon and nvenory cos s analyzed fnally. Te calculaon resuls are sown n Table 8. Table 8. Sensvy Analyss of e Deeroraon Cos g g day) C RMB) Table 8 sows a n e sae of prevous produc deeroraon pon, e nfluence caused by e deeroraon cos g on e opmal sorage pon and e oal nvenory cos. W e ncrease of e deeroraon cos g, e opmal sorage pon and e nvenory cos ncrease. Te cange of e deeroraon cos g as a less nfluence on e opmal sorage pon and e nvenory cos. 5 CONCLUSIONS Troug e parameers sensvy analyss, sows a e cusomer demand rae D per un me and e coeffcen of nvenory mpacng sale rae ave a larger effec on e nvenory cos. Cusomer s wllng o wa for delvery rae and e deeroraon rae ave a less effec on e nvenory cos. In e praccal applcaons, w e cange of parameers, realers sould pay aenon o e nfluence caused by D and on e nvenory cos n order o adjus nvenory sraegy mely. Te furer researc focuses on e nvenory conrol model w dscree random varable consderng e sorage pon and meamorpsm pon. 6 ACKNOWLEDGEMENTS Ts work benefed from Naural Scence Foundaon of Hebe Provnce F214424; G ), Hebe Sof Scence Researc Program D; D), Grand Projec of Socal Scence of Hebe Educaon Deparmen ZD21442). 7 REFERENCES Aggarwal, S.P., Jagg, C.K., 1995, Orderng Polces of Deerorang Iems under Permssble Delay n Paymens. Journal of e Operaonal Researc Socey, 46/5: Cao, Q.K., Ce, M.L., Wu, X.R., 212, Researc on e Model of Delay Meamorpc Produc of Invenory w e Back Loggng Rae. Logscs Sc-Tec, 35/1: Cakrabary, T., Grl, B.C., Cauur, K.S., 1998, An EOQ Model for Iems w Webull Dsrbuon Deeroraon, Sorages and Trended Demand an ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 7,ISSUE 1/29 43

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