Probabilistic inventory model for deteriorating items in presence of trade credit period using discounted cash flow approach

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1 Intrnational Journal of Scintific and Rsarch Publications, Volu, Issu, Fbruary ISSN Probabilistic invntory odl for riorating its in prsnc of trad crdit priod using discountd cash flow approach Ntu, Sapna Mahajan, Dr. Arun Kuar or 3 P. G. Dptt. of Mathatics, APJ Collg of Fin Arts, Jalandhar, India Dptt. of Mathatics, ARNI Univrsity, Kathgarh, H.P, India 3 P. G. Dptt. of Mathatics, SMDRSD Collg, Pathankot, India Abstract- In this papr, w ar using discountd cash flow approach for riorating its in th prsnc of trad crdit priod. Hr it is assud that th dand during th priod (, ) is a rando variabl x with continuous probability dnsity function f(x) and th dand rat is a powr dand pattrn. Shortags ar not allowd and rioration follows th Wibull distribution rat. Mathatical odls ar drivd for thr diffrnt cass: Instantanous cash flows, crdit only on units in stock, fixd crdit priod. Inflation and ti valu of ony is also considrd. Indx rs- crdit-priod, rioration, invntory, probabilistic M I. INRODUCION any authors hav considrd conoic ordr quantity odls for riorating its.ghar & Schradr (963) wr th first who studid invntory odls of riorating its. hy assud th constant arkt dand. With th passag of ti svral othr rsarchrs dvlopd th invntory odl for riorating its with ti dpndnt dand rat. Donaldson (997) drivd an optial algorith for solving classical invntory odl with no shortag analytically with linar trnd in dand ovr fixd ti horizon. Dutta and Paul (997) considrd th both rinistic and probabilistic vrsion of powr dand pattrn with variabl rat of rioration. Hariga (995) studid th ffcts of inflation and ti valu of ony on an invntory odl with ti dpndnt dand and shortags. Bhunia t al (998) dvlopd an invntory odl of riorating its with lot siz dpndnt rplnishnt cost and linar trnd in dand. Gnrally it is assud that th buyr ust pay for th its as soon as h rcivs th fro th supplir, but in rality supplir will allow a crtain fixd priod calld crdit priod, for sttling th aount th rtailr ows to hi for th its supplid. h crdit priod rducs th buyr s cost of holding stock bcaus it rduc th aount of capital invstd in stock for th duration of th prissibl priod. Chung (989) usd th discountd cash flows (DCF) approach for studying th optial invntory policy in th prsnc of th trad crdit, which prits an xplicit rcognition of th xact tiing of cash flows associatd with th invntory syst. A DCF approach prits a propr rcognition of th financial iplication of th opportunity cost and out of pockt costs in invntory syst. Aggarwal and Jaggi (994) analyzd th crdit financing in conoic ordring policis of riorating its in th prsnc of trad crdit using a DCF approach. Liao t al. () prsntd a odl with riorating its undr inflation, whn dlay in paynts is prissibl. Chang (4) prsntd an EOQ odl with riorating its undr inflation whn th supplir provids a prissibl dlay of paynts for a larg ordr that is gratr than or qual to th pr-rind quantity. Shah and Shah (998) prsntd a probabilistic invntory odl with cost in cas dlay in paynts is prissibl. Chang t al. () dvlopd a finit ti horizon invntory odl with both rioration and ontary ti valu whn paynt priods ar offrd. Huang and Chung (3) discussd rplnishnt and paynt policis to iniiz th total cost of cash discount and paynt dlays. Huang (3) considrd an EOQ odl in which supplir offrs a crdit priod to rtailr and rtailr offrs a crdit priod to th custors. Dand also dpnds on th rtailrs sals fforts. his situation was discussd by aylor (). H provd that coordination cannot b achivd with linar rbats and rturns or targt rbats alon. H providd a proprly dsignd targt rbat and rturns contracts to gt coordination. Krishanan t al. (4) analysd th coordination of contracts for dcntralizd supply chains with th rtailr prootional fforts. hy found that a buy back policy cannot b utilizd to coordinat th channls and thy providd thr contracts to achiv channl coordination. Liang t al. (5) discussd an invntory odl with non-instantanous rcipt undr trad crdit in which th supplir provids not only a prissibl dlay but also a cash discount to th rtailr and obtaind th optial ordr cycl and ordrs rcipt priod so that th total rlvant cost pr unit ti is iniizd. ripathi () prsntd th conoic ordring policis of ti dpndnt riorating its in prsnc of trad crdit using discountd cash flow approach. In th last odl w hav takn rinistic powr dand. But, sinc in ost of th practical situations, dand is not rinistic, rathr it varis fro cycl to cycl. Hnc, w ar considring th probabilistic vrsion of dand. Effcts of inflation and ti valu of ony is also considrd.

2 Intrnational Journal of Scintific and Rsarch Publications, Volu, Issu, Fbruary ISSN II. ASSUMPIONS. Dtrioration of its starts aftr a dfinit ti.. Dtrioration rat varis with ti and follows a two paratr Wibull distribution. 3. Rplnishnt is instantanous. 4. Lad ti is zro. 5. Shortags ar not allowd. 6. hr is no rpair or rplacnt of riorating its during th priod undr considration. 7. Inflation and ti valu of ony is considrd. 8. h dand during th priod (, ) is a rando variabl x with continuous probability dnsity function x (, ) f(x), and dand rat is powr dand pattrn, IV. MAHEMAICAL FORMULAION h lvl of invntory I(t) at ti t is dpltd du to both arkt dand and rioration. h diffrntial quation dscribing th invntory syst ovr (, ) is givn by di( t) x +α t I( t) = t, t< () I = with th boundary condition, h solution of quation () is givn by t x α αt I( t) = t + C x = t +α t + C ( ) D t x = t (, ), whr is th pattrn indx. III. NOAIONS. C is unit cost of th it.. Q is th ordr quantity. 3. D(t) is th dand rat at ti t. 4. i is Invntory holding cost fraction. 5. ic is th out-of-pockt invntory carrying cost pr unit ti. 6. R is constant rprsnting th diffrnc btwn th discount rat and inflation rat. 7. H is th ordring cost pr unit. 8. I(t) is th invntory lvl at ti t. 9. is th optial cycl ti for cas I.. is th optial cycl ti for cas II.. 3 is th optial cycl ti for cas III.. Z () is th prsnt valu of all futur cash-flows for cas I. 3. Z () is th prsnt valu of all futur cash-flows for cas II. 4. Z 3 () is th prsnt valu of all futur cash-flows for cas III. 5. Z ( ) is th optial valu of all futur cash-flows for cas I. 6. Z ( ) is th optial valu of all futur cash-flows for cas II. 7. Z 3 ( 3 ) is th optial valu of all futur cash-flows for cas III. 8. is th invntory cycl ti. x + t αt = + + C + α<, th approxiat solution is [Assuing a vry sall valu of α obtaind by nglcting th scond and highr ordr trs of α ] I = Now C I t x + α = + + x + + α t αt = Ordr quantity, α Q= I = x + + αt h nubr of riorating units during on cycl, D Q D t = x Q t = α αx x + x= + + () (3) (4) Using DCF approach, w hav thr cass on th trad crdit trs.

3 Intrnational Journal of Scintific and Rsarch Publications, Volu, Issu, Fbruary 3 ISSN Cas I: Instantanous cash-flows In this cas, w prsnt th DCF approach to th invntory odl of ti-dpndnt riorating its undr instantanous invntory holding cost. Hnc at th bginning of ach cycl, thr will b cash out flows of ordring cost and purchasing cost. Sinc, th invntory carrying cost is proportional to th valu of th invntory, th out-of-pockt invntory carrying cost pr unit ti at t is ici(t). Hnc, th prsnt valu of cash flow for th first ordr cycl z is Rt z = H+ CQ+ ic I t + + α x α t αt = H+ Cx + + ic + αt Rt α icx α t αt αt αt R t = H+ Cx αrt Rt αrt α α R αr = H+ Cx + + icx (5) Hnc th prsnt valu of all futur cash flows is nr z Z = z = R n= R Sinc R (approx.) R R = R Equation (5) bcos i α α R αr Z = H+ Cx icx R ( + )( ++ ) ( + ) ( ++ ) H + α α R αr = + Cx icx R h optial valu of can b found by solving (6) = Diff. quation (6) partially w.r.t., and quating to zro, w gt (7) H α α R αr + = + Cx + + icx R = and 3 Z H α( )( ) α ( ) αr( + ) = + Cx + + icx 3 3 > R + ( + )( ++ ) ( ++ ) hus optiu valu of can b found fro quation (7). Lt it b. hn, th optiu valu of ordr quantity α Q = x + + and iniu cost fro quation (6). Z can b found Cas II: Crdit only on th its in stock In this cas, paynt is connctd to th subsqunt us of its. Hr, thr xist a crdit priod M. During this priod, th custors ak paynt to th supplir idiatly aftr th us of th its and th raining balanc is paid by th custor on th last day of th crdit priod. Hr w hav two cass dpnding on th valu of and crdit priod M. Sub cas I: If M, thn th prsnt valu of cash flows for th first cycl is x Rt Rt = z H C t CD ic I t + + x αx x α t αt = H+ C { t ( Rt) } + C + ic t Rt + α R Cαx α R αr = H+ Cx icx ( + )( ++ ) ( + ) ( ++ ) h prsnt valu of all futur cash flows is z z Z = = R R H R Cαx Z = + Cx + R + + α R αr + icx + + ( + )( ++ ) ( + ) ( ++ ), M (8) h ncssary condition for Z = + H α ( ) C x R + to b iniu is = + Cx + α R α R + icx + = ( + )( ++ ) ( + ) ++

4 Intrnational Journal of Scintific and Rsarch Publications, Volu, Issu, Fbruary 4 ISSN H Cx α ( )( ) = RM R α ( ) αr( + ) ( )( ) ( ) 3 Z C x + + icx > Sub Cas II: If >M, thn th prsnt valu of cash flows for th first cycl is M x M Rt x Rt z = H+ C t + C Q t + ic I( t) + α x M RM x xm = H+ C + C x α R αr + icx + + ( + )( ++ ) ( + ) ( ++ ) h prsnt valu of all futur cash flows is Z z z = = R R + H Cx M RM x αx xm Z = + C + + R ( +) + α R αr + icx +, > M () Z to b iniu is h ncssary condition for = + C + ( +) Cx + αx + xm H M RM x R α R α R + icx + = ( + )( ++ ) ( + ) ++ () H + M RM x = + C R ( +) 3 Z C x x + + α xm + α αr + + icx > = and th optiu valu of can b found fro quation (9) and (). Hnc if th paynts to th supplir is don idiatly aftr th us of atrials and if th crdit priod (M) is longr than cycl lngth (), thn only out of pockt cost and th discountd cost of rioration ar rlvant in finding th optial cycl lngth. Whn M, thn thr would b no opportunity cost in th xprssion of total cash flows bcaus in this cas, th fir financs th invntory invstnt with th trad crdit offrd by its supplir. Cas III: Fixd Crdit Priod : I n this cas crdit priod is fixd and hnc, th custor pays th full purchas aount on th last day of th crdit priod. h prsnt valu of cash-flows for on cycl, z 3 () is Rt z3 = H+ CQ + ic I t + + α α R αr = H+ Cx + + icx ( + )( ++ ) ( + ) ( ++ ) h prsnt valu of all futur cash-flows is z3 z3 Z3 = = R R H + α α R αr Z3 = + Cx + + icx + R + + ( + )( ++ ) ( + ) ( ++ ) For optiu valu of, 3 = () H α( ) α R αr( + ) + Cx + + icx = R + ( + )( ++ ) ( + ) ( ++ ) H α ( )( ) (3) R + Z 3 3 = + Cx α αr + + icx > = Hnc th optiu valu of 3 can b obtaind fro quation (3) and th corrsponding optial ordr quantity is α3 Q= Q3 = x + +. h corrsponding optial prsnt valu of all futur cash-flows Z3 = Z3( 3) is obtaind fro quation (). Equation (3) contains trad crdit, th corrct opportunity cost and th cost of rioration, which ar th discountd cost of rioration. his rsults that ffctiv capital cost should b lss than that of th instantanous paynts. V. CONCLUSION In this papr, w hav considrd powr dand pattrn with variabl rat of rioration undr th ffct of inflation. Sinc it

5 Intrnational Journal of Scintific and Rsarch Publications, Volu, Issu, Fbruary 5 ISSN is not possibl to forcast xact dand in advanc, so w hav considrd probabilistic vrsion of dand. Hr, th dand pattrn is sa as rinistic but it dpnds on th valu of x which ay vary fro to. Discountd cash flow approach for riorating its in prsnc of trad crdit priod is usd for thr cass. h optial valu of all futur cash flows is found for thr cass : instantanous cash-flows, crdit only on units in stock, fixd crdit priod. REFERENCES [] Aggarwal, R., Rajput, D. and Varshny, N. K. 9. Intgratd invntory syst with th ffct of inflation and crdit priod. lntrnational Journal of Applid Enginring Rsarch, Vol.4, No. II, pp [] Aggarwal, S.P. and Jaggi, C.K Ordring policis of riorating its undr prissibl dlay in paynts. Journal of th Oprational Rsarch Socity, Vol. 46, pp [3] Chang, H.J. Dy, C.Y.. An invntory odl for riorating its with linar dand undr condition of prissibl dlay in paynts. Production planning and control, Vol., pp [4] Chung, K.J. and Hwang, Y.F. 3. h optional cycl ti for EOQ invntory odl undr prissibl dlay in paynts, Intrnational Journal of Production Econoics, Vol. 84, pp [5] Chung. K.J. and Hwang, Y.F., and Huang, C.K.. h rplnishnt dcision for EOQ invntory odl undr prissibl dlay in paynts. Op sarch, Vol. 39. No.5 & 6, pp [6] Covrt, R.P. and Philip, G.C An EOQ odl for riorating its with Wibull distributions rioration. AHE rans. Vol.5, pp [7] Db, M. and Chaudhuri, K.S An EOQ Modl for its with finit rat of production and variabl rat of. rioration: Opsarch, Vol.3, pp [8] Hng, K.J., Labban, J., Linn, R.J. 99. An ordr lvl lot siz invntory odl for riorating its with finit rplnishnt rat. Conputrs and Industrial Enginring, Vol., pp [9] Hou, K.L. and Lin, L.C. 9. A cash flow orintd EOQ Modl with riorating its undr prissibl dlay in paynts. Journal of Applid Scincs, Vol. 9, No.9, pp [] Huang, Y.F. 3. Optial Rtailr s ordring policis in th EOQ odl undr trad crdit financing. Journal of th Oprational Rsarch Socity, Vol.54, pp. -5. [] Huang, Y.F. 7. Econoic ordr Quantity undr conditionally prissibl dlay in paynts. Europan Journal of oprational Rsarch, 3. Vol.76, pp [] Hung. Y.F.. and Chung, K.J. 3. Optial rplnishnt and paynt policis in th EOQ odl undr cash discount and trad crdit, Asia- Pacific Journal of oprations Rsarch, Vol..pp [3] Hwang, H. and Shinn, SW Rtailr s pricing and lot- sizing policy for xponntially riorating product undr condition of prissibl dlay in paynts. Coputrs and Oprations Rsarch Vol.4. pp [4] Jaggi. C.K. and Aggarwal. S.P Crdit financing in conoic ordring policis of riorating Its. Intrnational Journal of Production Econoics, Vol.34. pp [5] Jaggi, C.K., Goyal S.K., and Gol, S.K. 8.Rtailr s optial rplnishnts dcisions with crdit-linkd dand undr prissibl dlay in paynts. Europan Journal of Oprational Rsarch. Vol. 9. pp [6] L, W.N. and Wu, J-W,. An EOQ odl for its with Wibull distributd rioration, shortags and Powr Dand Pattrns, Inforation and Managnt Scincs, Vol. 3, Nubr, pp [7] Liao, H.C., sai, C.H. and Su, C.. An invntory odl with riorating its undr inflation whn a dlay in paynts is prissibl, Intrnational Journal of production Econoic, Vol.63, pp [8] Misra, R.B A study of inflation ffcts on invntory syst. Logistic spctru. Vol.9. pp [9] Roy Choudhary, M. and Chaudhuri, KS. (983). An ordr lvl invntory odl for riorating its with finit rat of rplnishnt. Opsarch, Vol., pp [] Shah N.H Probabilistic ti schduling odl for an xponntially dcaying invntory whn dlay in paynts ar prissibl. Intrnational Journal of Production Econoics, Vol.3. pp [] ng, J.., and Chang, C..5. Econoic production quantity odls for riorating its with pric and stock dpndnt dand, Coputrs and Oprational Rsarch, Vol.3, pp [] ripathi, R.P. and Misra, S.S.. Crdit financing in conoic ordring policis of non-riorating its with ti-dpndnt dand rat. Intrnational Rviw of Businss and Financ Vol., No., pp [3] ripathi, R.P. and Kuar, Manoj,. Intrnational Journal of Businss, Managnt and Social Scincs, Vol., No. 3,, pp crdit financing in conoic ordring policis of ti-dpndnt riorating its. [4] Wn-Chuan,, Inforation and anagnt scincs, an EOQ odl for its with Wibull Distributd Dtrioration, Shortags and Powr Dand Pattrn. First Author Ntu (Asst. Profssor), PG Dptt. of Mathatics, APJ Collg of Fin Arts, Jalandhar. Eail : annat_7ind@yahoo.co.in Scond Author Sapna Mahajan (Asst. Profssor), Dptt. of Mathatics, ARNI Univrsity, Kathgarh, H.P., Eail : sapnaahajan7@gail.co 4. hird Author Dr. Arun Kuar or (Asst. Profssor) PG Dptt. of Mathatics, SMDRSD Collg, Pathankot. Eail : tor4@rdiffail.co