OPTIMAL REPLENISHMENT POLICY FOR DETERIORATING INVENTORY

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1 ach. Vl., N. ISSN Intnatinal Junal f Engining and Applid Scincs EAAS & ARF. All ights svd OPIAL REPLENISHEN POLICY FOR DEERIORAING INVENORY WIH RADE CREDI FINANCING, CAPACIY CONSRAINS AND SOCK-DEPENDEN DEAND Nita H. Shah and Apan D. Shah Dpatmnt f athmatics, Gujaat Univsity, Ahmdabad-89, India Cspnding Auth: Pf. Nita H. Shah addss: nitahshah@gmail.cm ABSRAC his sach aims at ptimizing th plnishmnt cycl tim whn units in invnty dtiat at a cnstant at und a pmissibl dlay in paymnts whn dmand is stck-dpndnt. It is assumd that wn wahus capacity is limitd and xcss invnty is std in ntd wahus. h unit hlding cst in ntd wahus is high than that in th wnd wahus. h dtiatin ats f itms in wnd wahus and ntd wahus a diffnt. A dcisin plicy is wkd ut t mak a chic f wn wahus hi th ntd wahus. h uniqunss f th ptimal slutin is divd. Finally, numical xampls a psntd t validat th ppsd pblm. Snsitivity analysis is caid ut t div managial insights. KEYWORDS: Invnty, EOQ, Stck-dpndnt dmand, w-wahus, Dtiatin, ad Cdit. INRODUCION In businss tansactin, th ff f allwabl cdit pid t sttl th accunts against th puchass mad is cnsidd t b an ffctiv pmtinal tl. his tl stimulats th dmand, attacts custms tc. Gyal (985 divd an EOQ mdl und th cnditin f a pmissibl dlay in paymnts. Huang (4 visitd Gyal (985 s mdl and btaind th fficint algithm t dtmin th ptimal cycl tim. anna and Chaudhui (5 studid ptimal ding plicy f dtiating itm und a nt cdit plicy using DCF appach. anna t al. (8 discussd invnty mdl whn th suppli ffs a fixd cdit pid t th tail and units in invnty dtiat with spct t tim. hy stablishd that th suppli incus m pfit whn cdit pid is gat that th plnishmnt cycl tim. Shah t al. ( gav an up-t-dat viw aticl cmpising f availabl litatu n tad cdit. Fm an cnmic pspctiv, cdit plicis a cnsidd t b an altnativ t pic discunt t fav lag ds, s th qustin is wh t stck this d. h mst f th fncs sitd in Shah t al. ( assumd that th tail wns a singl 7 wahus with unlimitd capacity. Hwv, any wahus has finit capacity. Hatly (976 dvlpd an invnty mdl cmpising f tw wahuss viz. wnd wahus (OW and ntd wahus (RW. h units m than th fixd capacity w std in th ntd wahus. h hlding cst f an itm in th RW was assumd t b high than OW. Sama (99 assumd infinit pductin at and analyzd ptimal statgis f tw wahuss. Gswami and Chaudhai (99 allwd shtags whn dmand is incasing linaly. hs mdls d nt cnsid dtiatin f itm, Sama (987 fist fmulatd a tw-wahus mdl f dtiating itms with an infinit plnishmnt at and shtags. Sm th lvant aticls by Pakkala and Achaya (99- a, 99-b,Hiaki and Ns (996, Bnkhuf (997, Yang (4, Zhu and yang (5, L (6, Yang (6. Shah and Shah ( discussd vaiants f tw-wahus invnty mdls in fuzzy nvinmnt. Lvin t al. (97 qutd that lag pils f gds attact m custms. his was tmd as stck-dpndnt dmand. Uban (5 btaind ptimal plicis whn dmand is

2 ach. Vl., N. ISSN Intnatinal Junal f Engining and Applid Scincs EAAS & ARF. All ights svd stck-dpndnt. Zhu and Yang (5 dvlpd a tw-wahus mdl with a stckdpndnt dmand and cnsidatin f tansptatin cst. Ry and Chaudhui (7 dvlpd an invnty mdl whn dmand is stck-dpndnt and planning hizn is finit. h ffcts f inflatin and tim valu f mny a incpatd t ptimiz bjctiv functin. hy allwd cmplt back-lgging. Ry and Chaudhui ( discussd an cnmic pductin lt siz mdl f pic-snsitiv stck-dpndnt dmand whn units in invnty a subjct t cnstant at f dtiatin. Lia and Huang ( analyzd plnishmnt plicy f dtiating itms with tw-stag facilitis and a pmissibl paymnt dlay. his study aims t dvlp ptimal ding plicy f dtiating itms with twwahus und stck-dpndnt dmand and cdit financing. h dtiatin ats f itms in tw-wahuss a diffnt. h unit hlding cst in ntd wahus (RW is high than that in wnd wahus (OW. h bjctiv is t maximiz th ttal pfit. h analytic sults a divd t stablish th xistnc and uniqunss f th cycl tim. h thms a dducd t dtmin th ptimal cycl tim. h numical xampls a givn t illustat ths thms. hs sults will hlp th dcisin mak t dcid Whth nt t nt RW? t stck m itms t maximiz annual pfits.. NOAIONS AND ASSUPIONS h fllwing ntatins and assumptins a adptd t dvlp ppsd mathmatical mdl.. Ntatins OW Ownd wahus RW Rntd wahus A h ding cst p d R( Ik ( t h stck dpndnt dmand at R( Ik ( t = Ik ( t,wh dnts scal dmand; dnts stck-dpndnt paamt; and I ( t dnts invnty lvl W P C h k at any instant f tim t. wh k h finit stag capacity f OW h slling pic p unit h puchas cst p unit, with C P h hlding cst p unit tim in OW h h hlding cst p unit tim in RW, h h h dtiatin at OW, h dtiatin at RW, Q and h cycl tim ( a dcisin vaiabl h puchas quantity (a dcisin vaiabl h tim at which invnty dplts t z in RW 8 ln I c I Z( w, tim at which invnty dplts t z in OW, wh h cdit pid ffd by th suppli Intst chagd p unit p annum Intst and p $ p annum h annual nt pfit p unit tim.. Assumptins ( h tw-wahus invnty systm stcks singl itm. ( h planning hizn is infinit. ( Lad-tim is z ngligibl. Shtags a nt allwd. (4 h OW has a fixd capacity f W- units. (5 h RW has unlimitd capacity. (6 h itms in RW a cnsumd fist. (7 h hlding cst and dtiatin cst in RW a high than ths in th OW. (8 h suppli ffs a cdit pid. h tail ans intst at th at I n th sals vnu duing allwabl cdit pid. At th nd f th cdit pid, th accunt is sttld. On th unsld stck, th tail has t pay intst chags at th at I c (Shah (99. (9 I ( t dnts th invnty lvl at tim t(, in th OW, in which th

3 ach. Vl., N. ISSN Intnatinal Junal f Engining and Applid Scincs EAAS & ARF. All ights svd invnty is dplting du t dtiatin f th itm and stckdpndnt paamt f invnty. I ( t dnts th invnty lvl at tim t(, in th RW, in which th invnty is dcasing t z du t stck-dpndnt dmand and dtiatin f th itm. It ( dnts th invnty lvl at tim t (, wh th invnty lvl dps t z du t dmand and dtiatin f th itm.. AHEAICAL ODEL h tail has t dcid whth it is advantagus RW t stck m itms t maximiz annual nt pfit nt whn ff tad cdit is availabl. Obviusly if Q W, thn th is n nd f RW. Othwis, W units a stckd in OW and maining in RW. his suggsts that w nd t discuss fllwing tw scnais: (A th singl-wahus invnty systm, and (B th tw-wahus invnty systm. If w dnt ln w, (FIG - th inquality Q W hlds if and nly if. Singl-wahus invnty mdl ( (Fig. h cycl stats with Q units in th invnty systm. Duing th pid (, th invnty lvl dplts in th OW du t stckdpndnt dmand and dtiatin. h at f chang f invnty lvl at any instant f tim t is gvnd by th diffntial quatin di ( t It (, t ( dt Lt Using bunday cnditin I ( ( is, th slutin f diffntial quatin I t t ( ( ( t, and th d quantity is Q I ( ( h diffnt cmpnnts f annual nt pfit p unit tim f th invnty systm a ( Sals vnu = ( P C Q ( Oding cst = A ( Hlding cst in th RW= bcaus w nd nly n wahus. (4 Hlding cst in th OW h h I ( t dt F intst and and intst chagd w nd t bsv th lngth and. w cass may ais Cas : H, all units a sld ff bf th allwabl cdit pid. S th intst chags a z and th intst and p unit tim is PI R( I ( ( t tdt Q PI Cas : In this cas, th tail slls th itms and pays at. S duing,, th intst PI and p unit tim is R( I ( t tdt and duing,, th intst paid at th at Ic n CIc th unsld stck is I ( t dt. h annual nt pfit p unit tim, Z( is sals vnu minus sum f ding cst, hlding cst in OW, intst chagd plus intst and. Cnsquntly, th nt pfit p unit tim is Z(, if Z( = Z(, if (4 Wh, Z ( A h ( P C PI 9

4 ach. Vl., N. ISSN Intnatinal Junal f Engining and Applid Scincs EAAS & ARF. All ights svd A Z ( ( P C h (5 PI ( ( PI CI c ( ( Sinc, Z( Z(, Z( is wll-dfind and cntinuus. h fist and scnd d divativ f Z ( Z ( a Z ' ( ( PI (6 h ( P C A ( PI (7 Z A Z " ( ' ( h ( P C PI ( PI ( (8 h ( P C A PI CI ( c ( PI ( ( ( (9 Z " ( A X ( ( CI c ( PI ( ( ( ( ( PI ( ( If, quatins (4-( a cnsistnt with ths givn in Lia and Huang (. Additinally, if P C, thn ths quatins a sm as givn in Hwang and Shinn (997. Using lmma and f Chung t al. (, w '' hav Z ( f all, and Z '' ( f all spctivly. Hnc, w hav fllwing thm. hm Lt. Z ( is cncav n (,. ( 6 4

5 ach. Vl., N. ISSN Intnatinal Junal f Engining and Applid Scincs EAAS & ARF. All ights svd Z ( is cncav n [,.. Z ( is cncav n (,. Als, lt h ( P C A PI PI Simila t thm f Chung t al. (, w hav fllwing thm. HEORE. If >, thn. If <, thn... If =, thn. hus, it is stablishd that f, th tail uss nly OW and d Q W.. w wahus invnty mdl ( (Fig. Q W indicats that tw wahus invnty systm is invlvd. Out f Q units civd in th bginning f th cycl, W units a kpt in th OW and maining QW units a stckd in th RW. Duing (, w fist cnsum itms fm RW and thn fm OW. Sinc bth wahus a having diffnt stcking facilitis. h at f chang f invnty lvl in RW duing (, can b dscibd by th diffntial quatin, di ( t I( t, t ( dt with bunday cnditins I (. H. h slutin f quatin ( is I t, t ( ( ( t Duing (, th at f chang f invnty lvl, I ( t in OW is gvnd by th diffntial quatin, di ( t I( t, t ( dt With initial cnditin I( W.h slutin f quatin ( is I ( t t W, t (4 Duing (,, th at f chang f invnty is gvnd by th diffntial quatin, 4 di( t It (, t 5 dt with bunday cnditin I (.h slutin f quatin (5 is ( I( t t, 6 t h d quantity duing th cycl tim is Q I ( I ( = W (7 Simila t sctin., th diffnt cst cmpnnts f th annual nt pfit p unit tim a. Sals vnu = ( P C Q. Oding cst = A. Hlding cst in RW= h h I (, t dt 4. Hlding cst in OW = h I ( t dt I( t dt ( h W ( 5. Intst and is sam as givn in singl wa hus cas f. 6. Intst chagd CI c = I( t dt I( t dt I( t dt ( ( CI c W ( ( Hnc, th annual nt pfit p unit tim is,

6 ach. Vl., N. ISSN Intnatinal Junal f Engining and Applid Scincs EAAS & ARF. All ights svd Q A h Z ( ( P C ( h W ( ( ( CIc W ( ( PI ( ( (8 Using cntinuity, w hav I ( I(, which givs t b a functin f as ln W Als, d d W (9 ( Substituting valu f fm quatin (9 int quatin (8, w btain annual nt pfit p unit tim t b a functin f. h ncssay cnditin f Z ( t b maximum is t st Z ' ( t b z. hus w btain dz( f ( ( d wh, d f( A ( P C W ( d h d ( d 4 W ( ( h d ( d W ( d d CI ( ( ch PI ( ( ( ( Wh h ( ( ( ( d d W d d ( ( ( ( d d Claly, bth f ( and Z ( hav th sam sign and dmain. Lt slutin f f (. thm. hm. If f (, if it xists b th hn w claim fllwing is th uniqu cycl tim which maximizs Z ( n [,.. If f (, thn Z ( is dcasing n [,. S,. Pf:. Pf fllws fm Shah (99 and hmas and Finny (996.. Pf is simila t that f Lia and Huang (. 4. Optimizatin f w-wahus mdl h annual nt pfit p unit tim is

7 ach. Vl., N. ISSN Intnatinal Junal f Engining and Applid Scincs EAAS & ARF. All ights svd Z(, if Z( Z(, if Z (, if ( Als, at and th capacity f th wahus is W w. Z ( cntinuus functin xcpt at. is ' ' W hav, Z ( Z ( h ( P C A ( PI PI (4 A ' Z ( h ( P C PI PI ( ( CI c (5 ' and Z( f (. (6 F simplicity, lt h A ( ( P C f ( PI PI 4 And h ( P C f( A PI PI ( ( CI c Sinc Z ( is cncav n ' [,, Z ( is dcasing n [, and f ( f (. Als, f( f( and f ( (7 f ( (8 ( f if and nly if (9 ( f if and nly if ( S w hav fllwing dcisin making abl f ptimum cycl tim. abl Optimum Cycl tim f ( f ( Optimal Cycl tim (Whichv f ( givs maximum pfit f ( > f ( f ( > Optimal Cycl tim (Whichv givs maximum pfit

8 ach. Vl., N. ISSN Intnatinal Junal f Engining and Applid Scincs EAAS & ARF. All ights svd > > %, 5%,. ya, P $5 /unit/ya, C $.5 /unit/ya, I 4%, I %, W 5 unit/ya. hn c f ( 56.9, Using tabl, w hav ptimal slutin in abl f diffnt valus f A. A f ( ( f Q Z ( In nxt sctin, w discuss tw xampls t illustat th ppsd pblm. Exampl Lt 4 units/ya, %, $. unit/ya, h $.5 unit/ya, h %, 5%,. ya, P $ /unit/ya, C $5 /unit/ya, I 5%, I %, W units/ya. hn C f ( 8.5. h ptimal slutin is givn in abl f diffnt valus f ding cst, A. abl Optimal slutin f Exampl A f ( f ( Q < < =.78 9 > < =.4 =.8 8 > > Z ( h bsvatins a (a Optimal cycl tim and puchas quantity a vy snsitiv t ding cst. h annual pfit p unit tim dcass with incas in A. It indicats that tail shuld hav lag d in a lag cycl tim. (b F A= and A=5, th d quantity is lw than th stag capacity W. s th tail shuld nt g f ntd wahus. It suggsts that lw ding cst will facilitat th tail t b with wn wahus. Exampl Lt D 6 units/ya, %, $4 /unit/ya, h $5 /unit/ya, h < < > > Obsvatins a simila t ths statd f xampl. Using data f xampl, w cay ut snsitivity analysis t find ut citical invnty paamts which fcs th tail t pt f ntd wahus. Fig.4, Fig.5and Fig. sspctivly f ptimum puchas quantity, ttal annual pfit p unit tim and ptimum cycl tim 7. Cnclusins his study aims t analyz th nd f ntd wahus whn wn wahus has limitd stag capacity and dmand is stck dpndnt. h itms in wahuss a subjct t diffnt dtiatin at. It is bsvd that ff f dlay paymnt by th suppli t th tail ncuags f a lag d. h dcisin plicy is suggstd t th tail t us th ntd wahus t maximiz pfit. h numical xampls and snsitivity analysis will hlp th dcisin mak t tak th favabl dcisin. h futu study shuld cnsid timdpndnt shtags, tim-dpndnt dtiatin f units tc. Rfncs. Bnkhuf, L., 997. A dtiatin d invnty mdl f dtiating itms with tw stag facilitis. Intnatinal Junal f Pductin Ecnmics, 48, Chung,K.J., Chang, S.L and Yang, W.D,. h ptimal cycl tim f xpnntially dtiating pducts und tad cdit financing. h Engining Ecnmist, 46, -4.

9 ach. Vl., N. ISSN Intnatinal Junal f Engining and Applid Scincs EAAS & ARF. All ights svd Gswami, A. and Chaudhai, K.S., 99. An cnmic d quantity mdl f itms with tw lvls f stag f a lina tnd in dmand. Junal f th Opatinal Rsach Scity, 4, Gyal, S.K., 985. Ecnmic d quantity und cnditins f pmissibl dlay in paymnts. Junal f th Opatinal Rsach Scity, 6, Hatly, V.R., 976 Opatins Rsach A managial mphasis. Santa nica, CA, Hiaki, Ishii and Ns,., 996. Pishabl invnty cntl with tw typs f custms and diffnt slling pics und th wahus capacity cnstaints. Intnatinal Junal f Pductin Ecnmics, 44, Huang, Y. F. (4. Rtail s plnishmnt plicis und cnditins f pmissibl dlay in paymnts. Yugslav Junal f Opatins Rsach, 4 (, Hwang, H. and Shinn, S.W., 997. Rtail s picing and lt sizing plicy f xpnntially dtiating pducts und th cnditin f pmissibl dlay in paymnts. Cmput and Opatins Rsach, 4, L, C.C, 6. w wahus invnty mdl with dtiatin und FIFO dispatching plicy. Eupan Junal f Opatinal Rsach, 74, Lvin, R.I., claughlin, C.P., Lamn, R.P. and Kttas, J.F., 97. Pductins/Opatins managmnt cntmpay plicy f managing pating systms, acgaw-hill, Nwyk, 7.. Lia, J.J. and Huang, K.N.,. Dtministic invnty mdl f dtiating itms with tad cdit financing and capacity cnstaints. Cmputs and Industial Engining, 59, anna, S. K., Chaudhui, K. S., 5. An EOQ mdl f a dtiating itm with nn-lina dmand und inflatin and a tad cdit plicy. Yugslav Junal f Opatins Rsach, 5 (, anna, S. K., Chaudhui, K. S. and Chinag, C., 8. Optimal picing and lt-sizing dcisins und wibull distibutin dtiatin and tad cdit plicy. Yugslav Junal f Opatins Rsach, 8 (, Pakkala,.P. and Achaya.K.K., 99 a.a dtministic invnty mdl f dtiating itms with tw wahuss and finit plnishmnt at. Eupan Junal f Opatinal Rsach, 57, Pakkala,.P. and Achaya.K.K., 99 b. Disct tim invnty mdl f dtiating itms with tw wahuss. OPSEARCH, 9, Ry,. and Chaudhui, K. S., 7. A finit tim-hizn dtministic EOQ mdl with stck-dpndnt dmand, ffcts f inflatin and tim valu f mny with shtags in all cycls. Yugslav Junal f Opatins Rsach, 7 (, Ry,. and Chaudhui, K. S.,. An EPLS mdl f a vaiabl pductin at with stck-pic snsitiv dmand and dtiatin. Yugslav Junal f Opatins Rsach, (, Sama, K.V.S., 98. A dtministic invnty mdl with tw lvl f stag and an ptimum las at. OPSEARCH,, Sama K.V.S, 987. A dtministic d lvl invnty mdl f dtiating itms with tw stag facilitis. Eupan Junal f Opatinal Rsach, 9, Shah, Nita H., 99.A lt siz mdl f xpnntially dcaying invnty whn dlay in paymnts is pmissibl. Cahis du CERO, 5, 5-.. Shah, Nita H., Sni, H. and Jaggi, C.K.,. Invnty mdls and tad cdit. Rviw.Cntl and cybntics, 9(, Shah, Nita H. and Shah, A.D.,.Optimal ding tansf plicy f dtiating invnty itms with fuzzy stck-dpndnt dmand. appa in xican Junal f Opatin Rsach.. hmas, C.B. and Finny, R.L, 996. Calculus with analytic gmty (9 th d

10 ach. Vl., N. ISSN Intnatinal Junal f Engining and Applid Scincs EAAS & ARF. All ights svd Addisn Wsly Publishing Cmpany, Inc. 4. Uban,.L., 5, Invnty mdls with invnty lvl dpndnt dmand. A cmphnsiv viw and unifying thy. Eupan Junal f Opatinal Rsach, 64(, Yang, H.L., 4. w-wahus invnty mdls f dtiating itms with shtags und inflatin, Eupan Junal f Opatinal Rsach, 57, Yang, H.L., 6, w-wahus patial backlgging invnty mdls f dtiating itms und inflatin. Intnatinal Junal f Pductin Ecnmics,, Zhu, Y.W and Yang, S.L., 5. A tw-wahus invnty mdl f itms with stck lvl dpndnt dmand at. Intnatinal Junal f Pductin Ecnmics, 95,

11 ach. Vl., N. ISSN Fig. INVENORY LEVEL Intnatinal Junal f Engining and Applid Scincs EAAS & ARF. All ights svd IE Fig. INVENORY LEVEL IE 47

12 Optimum Pfit p unit tim Optimal Q Optimum Cycl im ach. Vl., N. ISSN Intnatinal Junal f Engining and Applid Scincs EAAS & ARF. All ights svd Fig. Pcntag Chang in Invnty Paamts α β h h θ θ P C Fig. 4 Pcntag Chang in Invnty Paamts α β h h θ θ P Fig. 5 Pcntag Chang in Invnty Paamts α β h h θ θ P C 48