iate Part ABSTRACT 1. Introduction immediate part payment to Payment demandd rate, i.e. the by the is influenced Normally, the where the demand rate

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1 Jounal of Applid Matmatis and Pysis,, 0,, 5-0 ttp://dx.doi.og/0.46/jamp Publisd Onlin Otob 0 (ttp:// An EPQ-Basd Invntoy Modl fo Dtioating Itms und Stok-Dpndnt Dmand wit Immdi iat Pat Paymnt P. Majumd, U. K. Ba Dpatmntt of Matmatis, ational Institut of nology, Agatala, Bajala, Jiania, India pinki.mjmd@diffmail.om, ba_uttam@yaoo.o.in Rivd July 0 ABSRAC In tis pap, an EPQ-basd invntoy poliy fo an itm is psntd wit stok-dpndnt dmand duing two tad dit piods. In addition, t is a povision fo an immdiat pat paymnt to t wolsal, boowing som mony fom mony lnding sou fo t immdiat pat paymnt, suppli o wolsal offs a tad dit piod to is tail and tail also offs a tad dit piod to is ustom. Against t abov onjtus invnost. toy modl as bn fomulatd wit spt to t tail s point of viw fo minimizing t total invntoy non-lina optimization mtod-gnalizd Rdud Gadint (GRG mtod is usd to find t optimal solutions. Lastly umial xampls a st to illustat tis modl. Finally w us LIGO softwa to solv tis modl. Kywods: EPQ Modl; Immdiat Pat Paymnt. Intodution omally, t paymnt fo an od is mad by t - tail to t suppli immdiatly just aft t ipt of t onsignmnt. owadays, du to t stiff omptition in t makt to attat mo ustoms, a dit piod is offd by t suppli to t tail. Bfo t nd of t tad dit piod, t tail an sll t goods, aumulat vnu and an intst. Goyal [] fist x- plod a singl itm EOQ modl und pmissibl dlay in paymnts and Cung [] simplifid t sa of t optimal solution fo t poblm xplod by Goyal []. Howv, t pnomnon of ioation was ignod in t abov modls. Aggawal and Jaggi [] xtndd Goyal s modl to t as wit ioating itms. Jamal t al. [4] fut gnalizd t abov invntoy modl to allow fo sotags. Sak t al. [5] dvlopd a mod- l to min an optimal oding poliy fo ioat- and allowabl sotag. Cang [6] stablisd an EOQ modl fo ioating itms und inflation wn t suppli offs a pmissibl dlay to t puas if t od quantity is gat tan a pmind quantity. Cung and Huang [7] fut xtndd Goyal s modl to ing itms und inflation, pmissibl dlay in paymnts t as tat t units a plnisd at a finit at un- d dlay in paymnts. All t abov paps und tad dit finaning assumd tat t makt dmand was it onstant o mly dpndnt on t tailing pi Liao t al. [8] onsidd an initial-stok-lvl-dpndnt dmandd at, i.. t dmand at is inflund by t tail s od quantity. Mo ntly, Sana and Caudui [9] analyzd a kind of EOQ modl wit usuppli nt-stok-dpndnt dmand at w t givs a tail bot a dit piod and a pi disount on t puas of mandis. Soni and Sa [0] d- wn vlopd t optimal oding poliy fo tail dmandd is stok-dpndnt and wn suppli offs two pogssiv dit piods. But ts modls wit unt stok-dpndnt faild to onsid ioation pno- mnon. In t al lif, owv, ioation of itms is a ommon pnomnon. On t ot and, t nti abov modl assumd tat t suppli would off t tail a dlay piod but t tail would not off t tad dit piod to is/ ustoms. In most businss tansations, spially in supply ain, tis assumption is unalisti. Huang assumd tatt t tail will also adopt t tad dit poliy to is/ ustoms to potail s mot makt omptition and dvlopd t plnismnt modl tat is a two-lvl an invntoy modl wit tad dit. Min and Zao [] intodud stok-dpndnt dmand und two lvl of tad dit piod. owadays fo t spdy movmnt of apital, a wolsal tis to maximiz is/ makt toug sv- in al mans. Fo tis, vy oftn som onssions tms of unit pi, dit piod t. a offd to t tails against immdiat pat paymnt. o avail ts Copyigt 0 SiRs.

2 6 P. MAJUMDER, U. K. BERA bnfits, a tail is tmptd to as down a pat of t paymnt immdiatly vn making a loan fom mony lnding sou wi ags intst against tis loan. H an amount, boowd fom t mony lnding sou as a loan wit intst, is paid to t wolsal at t bginning on ipt of goods. In tun, t wolsal/suppli offs a laxd dit piod as pmissibl dlay in paymnt of st amount. onpt immdiat pat paymnt was fist intodud by M. Maiti []. Guia, Das, Mondal and Maiti ntodud an invntoy poliy fo an itm wit inflation indud puasing pi, slling pi and dmand wit immdiat pat paymnt. In tis pap, w dvlop a mo gnal invntoy modl wit dlay in paymnt. Fistly, t dmand at of t itms is assumd to b dpndnt on t tail s unt stok lvl. Sondly, t itms stat ioating fom t momnt ty a put into invntoy. idly not only would t suppli off a fixd dit piod to t tail, but t tail also adopts t tad dit poliy to is/ ustoms. Foutly t suppli must b givn an immdiat pat-paymnt by t tail. Lastly ts modls a illustatd wit numial xampls. Finally w us GRG mtod and LIGO softwa to solv tis modl.. otations and Assumptions following notations and assumptions a usd tougout t pap otations: I(t= Invntoy lvl at tim t. k =Podution at p ya. = unit aw matial ost. 4 = unit slling ost. 5 p = unit podution ost. W p= lk wk k 6 = oding ost p od. 7 = Stup ost. 8 l = Cost du to labou. 9 w = Cost du to wa and ta. 0 = Envionmntal pottion ost. = invntoy olding ost p ya xluding intst ags. A=Immdiat pat paymnt. M= Rtail s tad dit piod 4 =Custom s tad dit piod offd by t suppli. 5 I =Intst payabl p $ p ya by t tail to t suppli. 6 I =Intst and p $ p ya by t tail to t tail. 7 I b = Rat of intst p unit to b paid by t tail to mony lnd against immdiat pat paymnt A. 8 = Cyl tim in yas. 9 = Optimal yl tim. 0 Z( = otal invntoy ost p tim piod. Assumptions: dmand at R(t is a known funtion of tail s instantanous stok lvl I(t,wi is givn by R(t = D α I(t, w D and α a positiv onstants. Sotags a not allowd to ou. tim oizon of t invntoy systm is infinit. 4 lad tim is ngligibl. 5 fixd dit piod offd by t suppli to t tail is no lss tan t dit piod pmittd by t tail to is/ ustoms i.. M. 6 Wn M, t aount is sttld at t=m and t tail would pay fo t intst ags on itms in stok wit at I ov t intval [M,]. Wn M, t aount is also sttld at t = M and t tail dos not nd to pay any intst ag of itms in stok duing t wol yl.. Matmatial Fomulation of t Modl A onstant podution stats at t = 0 and ontinus up to t = t w t invntoy lvl as maximum lvl. Podution tn stops at t = t and t invntoy gadually dplts to zo at t nd of t podution yl t = du to ioation and onsumption. fo, duing t tim intval (0, t, t systm is subjt to t fft of podution, dmand and ioation. n t ang in t invntoy lvl an b dsibd by t following diffntial quation: dq ( t q( t = k D αq( t,0 t t ( Wit t initial ondition q ( 0 = 0 ( On t ot and, in t intval ( t,, t systm is fftd by t ombind fft of dmand and ioation. Hn, t ang in t invntoy lvl is govnd by t following diffntial quation: dq ( t q( t = D αq( t, t t ( Wit t nding ondition q = 0 (4 solution of t diffntial Equations ( and ( a sptivly psntd by ( k D t q ( t ( α = ; 0 t t (5 ( α D ( t q ( t ( α = ( α ; t t (6 In addition, using t bounday ondition q t = q t, w obtain t following quations: Copyigt 0 SiRs.

3 P. MAJUMDER, U. K. BERA 7 ( k D ( t α ( α D ( α α ( ( t = D and t = ln[ ( ] (α k annual total lvant ost Annual oding ost = Annual stok olding ost t = [ q ( t d( t q ( 0 t = ( t d( t ] k t D α Annual ost du to ioatd units = ( α 4 Annual Podution ost = k t D pkt 5 Annual St up ost = Dpnding upon M, and t ass ais: Cas-: M, Cas :- M, Cas :- M Aoding to givn assumption, t a t ass to ou in intst agd fo t itms kpt in stok p ya. Cas-. M Annual intst payabl I = [ q ( t AIb] M I D ( M =. [ ( α ( M ] AIb ( α Cas-. M In tis as total intst payabl = A I b Cas-. M In tis as total intst payabl = A I b (v Aoding to givn assumption, t ass will ou in intst and p ya. Cas-. M annual intst and by t tail I D M α D( M = { ( α ( α α D M ( A } ( α Cas -. M annual intst and by t tail (7 I D ( = { ( ( A} D ( (M ( α Cas-. M annual intst and by t tail I D α D = { ( ( M A} annual total ost inud by t tail Z( = Oding ost olding ost st up ost ioation ost podution ost intst payabl intst and Z,if M Z( = Z,if M (8 Z,if o< w Z = ( k D ( α pkt ( k D? ( α I D ( M. [ ( α ( ( α (9 I D( M M ] AIb { [ ( α α D( M ( α D M ] α α α A ( α α Z = ( kt D ( α pkt ( kt D ( α I D( AIb { [ α } ( D ( ( ( α ( (M ]] A} (0 Copyigt 0 SiRs.

4 8 P. MAJUMDER, U. K. BERA Z = ( k ( α α I D AIb { [ pkt α α ] M A} t -D ( Sin Z ( M = Z ( M and Z ( = Z ( fo Z( is ontinuous and wll dfind. All Z(,Z(,Z( a dfind on >0. Equations (9-( yild ' Z( t = ( kt D ( α ( α d ( α pk pkt ( α ( α ( α k D kt D d k D d I D ( M. ( α ( M ( α I D. ( M α ( α I D M α D M { [ ( α α α D M ] A} ( I α D M α D { [ α ( α M (( } ' Z( t = ( kt D ( α k D ( kt D d ( α pk pkt ( α ( α ( ( α d k D d I D α D ( { [ ( ( α ( α α D D ( α ( α k t -D ( ( I D ( M ] A} { [ ( α ( α ( α D ( α ( α ( α ( α D ( ( ]} M ( α ( α ' And Z( t = ( kt D ( α ( α d ( α pk pkt α ( M ( α ( α k D kt D d k D ( α d I D { ( ( α ( α I D ( M A} { [ ] } ' Z ( t = ( α ( α ow 0 ( (4 k kt d ( α pk pkt d α ( M ID( α ( α ( M ( M I D α α ( α D I{ [ ( α ( M ( M ( M α ] A} =0 I{ [ M α M α ]} objtiv of tis pap is to find an optimal yl tim to minimiz t annual total lvant ost fo t tail. Fo tis t optimal yl tim is obtaind by stting t Equation ( qual to zo; is t oot of t following quation ( α ( ( α d ( α pk pkt ( α d k kt Copyigt 0 SiRs.

5 P. MAJUMDER, U. K. BERA 9 ( M ( α ( α ( M α α ( α ID M ID D I{ [ ( α ( M ( M α α α M α α ] A} I { [ ( M ( α M α ] ( ( α α ( α ( ( A} I { [ D α α α α ( D( α ( α ( M ( α D ( α ]} (5 = 0 Equation ( 5 is t optimality ondition of (9 ' ow Z (t = 0 ( α ( ( α k kt d ( α pk pkt d α D I{ [ ( α ( ( ( α (6 D D ( M ] ( = 0 Equation ( 6 is t optimality ondition of (0 ' Again Z (t = 0 ( α ( ( α k kt d ( α pk pkt ( α d I { [ α D α ] ( M A} { ( α α ( α ( M } = 0 I D D Equation ( 7 is t optimality ondition of ( 4. umial Exampls (7 o illustat t sults of t poposd modl w solv t following numial xampls Exampl. Wn = Rs50 / unit, = Rs00 / od, = Rs50, M = 0.5 ya, = 0.0 ya, D = 500 units p ya, k = 000 units/ya, A = Rs 700, = Rs 5/unit, = Rs 4/unit, l = Rs00, w = Rs , = Rs 0., I = 0., I = 0.7, I b = 0.5, α = 0., = 0. tn t optimal valu of is = 5.65 and Z ( = Exampl. Wn = Rs50 / unit, = Rs00 / od, = Rs50, M =.5 ya, = 0.0 ya, D = 000 units p ya, k = 500 units/ya, A = Rs 000, = Rs 0/unit, = Rs 4/unit, l = Rs00, w = Rs 0.05, = Rs 0., I = 0., I = 0.7, I b = 0.5, α = 0., = 0. tn t optimal valu of is = and Z ( = Exampl. Wn = Rs6 / unit, = Rs00 / od, = Rs0, M=. ya, =.07 ya, D = 000 units p ya, k = 500 units/ya, A = Rs 800, = Rs 0/unit, = Rs 4/unit, l = Rs00, w = Rs 0.05, = Rs 0., I = 0.5, I = 0.0, I b = 0., α = 0., = 0. tn t optimal valu of is = and Z ( = Conlusions In tis pap, w dvlop an EPQ modl fo ioating itms und pmissibl dlay in paymnts. pimay diffn of tis pap as ompad to pvious studis is tat w intodud a gnalizd invntoy modl by laxing t taditional EOQ modl in t following svn ways: t dmand of t itms is dpndnt on t tail s unt stok lvl, t tail s slling pi p unit is ig tan its puas unit ost, many itms ioat ontinuously su as fuits and vgtabls, 4 t suppli not only would off a fixd dit piod to t tail, but t tail also adopts t tad dit poliy to pomot makt omptition, 5 suppli must b givn an immdiat pat paymnt by t tail aft ipt of goods, 6 minimizing invntoy ost is usd as t objtiv to find t optimal plnismnt poliy. numial xampls a st to illustat tis modl. Fo t fist xampl, is t optimal valu of, fo sond and tid, t optimal valu of is and sptivly. is psntd modl an b fut xtndd to som mo patial situations, su as w ould allow fo sotags. Also quantity disounts, tim valu of mony and inflation t. may b addd in tis pap. REFERECES [] S. K. Goyal, Eonomi Od Quantity und Conditions of Pmissibl Dlay in Paymnts, Jounal of t Copyigt 0 SiRs.

6 0 P. MAJUMDER, U. K. BERA Opational Rsa Soity, Vol. 6, o. 4, 985, pp [] K. J. Cung, A om on t Dtmination of Eonomi Od Quantity und Conditions of Pmissibl Dlay in Paymnts, Computs & Opations Rsa, Vol. 5, o., 998, pp [] S. P. Aggawal and C. K. Jaggi, Oding Poliis of Dtioating Itms und Pmissibl Dlay in Paymnts, Jounal of t Opational Rsa Soity, Vol. 46, o. 5, 995, pp [4] A. M. M. Jamal, B. R. Sak and S. Wang, An Oding Poliy fo Dtioating Itms wit Allowabl Sotags and Pmissibl Dlay in Paymnt, Jounal of t Opational Rsa Soity, Vol. 48, 997, pp [5] B. R. Sak, A. M. M. Jamal and S. Wang, Supply Cain Modl fo Pisabl Poduts und Inflation and Pmissibl Dlay in Paymnt, Computs & Opations Rsa, Vol. 7, o., 000, pp ttp://dx.doi.og/0.06/s ( [6] C.. Cang, An EOQ Modl wit Dtioating Itms und Inflation Wn Suppli Cdits Linkd to Od Quantity, Intnational Jounal of Podution Eonomis, Vol. 88, o., 004, pp ttp://dx.doi.og/0.06/s095-57( [7] K. J. Cung and Y. F. Huang, Optimal Cyl im fo EPQ Invntoy Modl und Pmissibl Dlay in Paymnts, Intnational Jounal of Podution Eonomis, Vol. 84, o., 00, pp ttp://dx.doi.og/0.06/s095-57( [8] H. C. Liao, C. H. sai and C.. Su, An Invntoy Modl wit Dtioating Itms und Inflation Wn a Dlay in Paymnt Is Pmissibl, Intnational Jounal of Podution Eonomis, Vol. 6, o., 000, pp ttp://dx.doi.og/0.06/s095-57( [9] S. S. Sana and K. S. Caudui, A Dtministi EOQ Modl wit Dlays in Paymnts and Pi-Disount Offs, Euopan Jounal of Opational Rsa, Vol. 84, o., 008, pp ttp://dx.doi.og/0.06/j.jo [0] H. Soni and. H. Sa, Optimal Oding Poliy fo Stok-Dpndnt Dmand und Pogssiv Paymnt Sm, Euopan Jounal of Opational Rsa, Vol. 84, o., 008, pp ttp://dx.doi.og/0.06/j.jo [] J. Min, Y.-W. Zou and J. Zao, An Invntoy Modl fo Dtioating Itms und Stok-Dpndnt Dmand and wo-lvl ad Cdit, Applid Matmatial Modlling, Vol. 4, o., 00, pp ttp://dx.doi.og/0.06/j.apm [] A. Guia, B. Das, S. Mondal and M. Maiti, Invntoy Poliy fo an Itm wit Inflation Indud Puasing Pi, Slling Pi and Dmand wit Immdiat Pat Paymnt, Applid Matmatial Modlling, Vol. 7, o. -, 0, pp Copyigt 0 SiRs.