Time Dependent Quadratic Demand Inventory Models when Delay in Payments is Acceptable

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1 Intrntionl OPEN ACCESS Journl Of odrn Enginring Rsr IJER im Dndnt Qudrti Dmnd Invntory odls wn Dly in Pymnts is Atl R. Vnktswrlu,. S. Rddy GIA Sool of Intrntionl Businss, GIA Univrsity, Visktnm 0 0, Indi BVSR Enginring Collg, Cimkurty - 6, Indi Astrt: An EOQ modl is onstrutd for dtriorting itms wit tim dndnt qudrti dmnd rt. It is ssumd tt t dtriortion rt is onstnt nd t sulir offrs is rtilr t rdit riod to sttl t ount of t rourmnt units. o solv t modl it is furtr ssumd tt sortgs r not llowd nd t rlnismnt rt is instntnous. W v rsntd t modls undr two diffrnt snrios, viz., it offrd rdit riod is lss tn or qul to t yl tim nd ii t offrd rdit riod y t sulir to t rtilr for sttling t ount is grtr tn yl tim. ojtiv is to minimiz t rtilrs totl invntory ost. Slvg vlu is lso tkn to s its fft on t totl ost. A numril xml is givn to study t fft of llowl rdit riod nd t totl ost of t rtilr. Ky words: Qudrti dmnd, risl, onstnt dtriortion, trd rdit, olding ost. I. Introdution It is wll known tt trd rdit oliy is t most fftiv wy of sulir to nourg rtilr to uy mor goods nd to ttrt mor rtilrs. rd rdit n lso usd s multi-ftd mrkting mngmnt or rltionsi mngmnt tool wi givs som informtion to t mrkt or to uyr out t firm or its roduts or its futur lns. EOQ modl dvlod y Wilson ws sd on t ssumtion tt t rtilr will y for t itms s soon s it is rivd y t systm. In rlity t sulir my offr som rdit riod to t rtilr to sttl t ounts in rsonl tim riod. us t dly in ymnt n trtd s kind of ri disount to t rtilr. rltionsi twn invntory oliy nd rdit oliy in t ontxt of t lssil lot siz modl ws studid y Hly nd Higgins 97. Cmn t l. 98 dvlod n onomi ordr quntity modl wi onsidrs ossil rdit riods llowl y sulirs. is modl is sown to vry snsitiv to t lngt of t rmissil rdit riod nd to t rltionsi twn t rdit riod nd invntory lvl. Dvis nd Gitr 98 dvlod otiml ordr quntitis for firms tt r offrd on tim oortunity to dly ymnt for n ordr of ommodity. A mtmtil modl is dvlod y Goyl 98 wn sulir nnouns rdit riod in sttling t ount, so tt no intrst rgs r yl from t outstnding mount if t ount is sttld witin t llowl dly riod. S t l. 988 xtndd t ov modl y llowing sortgs. ndl nd Pujdr 989, v studid Goyl 98 modl y inluding intrst rnd from t sls rvnu on t stok rmining yond t sttlmnt riod. Crlson n d Roussu 989 xmind EOQ undr dt trms sulir rdit y rtitioning rrying ost into finnil ost nd vril olding osts. Cung nd Hung 00 xtndd Goyl 98 modl wn rlnismnt rt is finit. Dlln 986, 988, Wrd nd Cmn 987, Cmn nd Wrd 988 rgud tt t usul ssumtions s to t inidn nd t vlu of t invntory invstmnt oortunity ost md y t trditionl invntory tory r orrt nd lso stlisd tt if trd rdit surlus is tkn into ount, t otiml ordring quntitis drss rtr tn inrs. Cung 998 stlisd t onvxity of t totl nnul vril o s t funtion for otiml onomi ordr quntity undr onditions of rmissil dly in ymnts. Jml t l. 000 disussd t rolm in wi t rtilr n y t sulir itr t t nd of rdit riod or ltr inurring intrst rgs on t unid ln for t ovrdu riod. Srkr t l. 00 otind otiml ymnt tim undr rmissil dly in ymnts wn units in n invntory r sujt to dtriortion. Ad nd Jggi 00 onsidrd t sllr-uyr nnl in wi t nd dmnd is ri snsitiv nd t sulr offrs trd rdit to t uyr. Sinn nd Hwng 00 dlt wit t rolm of dtrmining t rtilr s otiml ri nd ordr siz simultnously undr t ondition of ordr siz dndnt dly in ymnts. It is ssumd tt t lngt of t rdit riod is funtion of t rtilr s ordr siz nd lso t dmnd rt is funtion of t slling ri. Cung t l. 00 dtrmind t IJER ISSN: Vol. Iss. r. 0

2 im Dndnt Qudrti Dmnd Invntory odls wn Dly in Pymnts is Atl onomi ordr quntity undr onditions of rmissil dly in ymnts wr t dly in ymnts dnds on t quntity ordrd wn t ordr quntity is lss tn t quntity t wi t dly in ymnts is rmittd, t ymnt for t itm must md immditly. Otrwis, t fixd rdit riod is llowd. Hung 007 xmind otiml rtilr s rlnismnt disions in t EOQ modl undr t w o lvls of trd rdit oliy y ssuming tt t sulir would offr t rtilr rtilly rmissil dly in ymnts wn t ordr quntity is smllr tn rdtrmind quntity. ng t. l. 007 drivd rtilr s otiml ordring oliis wit trd rdit finning. litrtur is rlt in t fild of trd-rdit. Prviously svrl onomi ordr quntity invntory modls wr dvlod wit trd-rdit nd vry fw rodution invntory modls wr dvlod undr llowl dly in ymnt. All ts works wr sd on t ssumtion tt t dmnd rt is itr linr or xonntil funtion of tim. Svrl utors rgud tt, in rlisti trms, t dmnd nd not follow itr linr or xonntil trnd. So, it is rsonl to ssum tt t dmnd rt, in rtin ommoditis, is du to ssonl vritions my follow qudrti funtion of tim [i.., Dt = + t + t ; 0, 0, 0 ]. funtionl form of tim-dndnt qudrti dmnd xlins t lrtd rtrdd growt/dlin in t dmnd ttrns wi my ris du to ssonl dmnd rt Knr nd Cuduri 00. W my xlin diffrnt tys of rlisti dmnd ttrns dnding on t signs of nd. Bndri nd Srm 000 v studid singl riod invntory rolm wit qudrti dmnd distriution undr t influn of mrkting oliis. Knr nd Cuduri 00 v disussd n ordrlvl invntory rolm wit t dmnd rt rrsntd y ontinuous qudrti funtion of tim. It is wll known tt t dmnd for sr rts of nw ro lns, omutr is of dvnd omutr mins, t. inrs vry ridly wil t dmnds for srs of t osolt ro lns, omutrs t. drs vry ridly wit tim. is ty of nomn n wll ddrssd y invntory modls wit qudrti dmnd rt. Sn nd Cuduri 00 v dvlod stok-rviw invntory modl for risl itms wit uniform rlnismnt rt nd stok-dndnt dmnd. Rntly, Gos nd Cuduri 00 v dvlod n invntory modl for dtriorting itm ving n instntnous suly, qudrti tim-vrying dmnd nd sortgs in invntory. y v usd two-rmtr Wiull distriution to rrsnt t tim to dtriortion. Vnktswrlu nd on 0 v dvlod invntory modls for dtriorting itms wit tim dndnt qudrti dmnd nd slvg vlu. Rntly Vnktswrlu nd on 0 studid invntory modl for tim vrying dtriortion nd ri dndnt qudrti dmnd wit slvg vlu. In litrtur w sldom find on t invntory modls wit trd rdit oliy for risl itms wit tim dndnt qudrti dmnd rt. us, in tis r, w wis to dvlo mtmtil modl wn t units in n invntory r sujtd to onstnt dtriortion rt nd t dmnd rt follows tim dndnt qudrti funtion. It is ssumd tt t sulir offrs rdit riod to t rtilr to sttl t ount. W v lso onsidrd t slvg vlu for dtriorting units of t invntory. Snsitivity nlysis is rsntd wit numril xml. II. Assumtions nd Nottions following ssumtions r usd to dvlo t modl: systm dl wit singl itm dmnd rt R is tim dndnt qudrti dmnd rlnismnt rt is infinit. ld tim is zro nd sortgs r not llowd. slvg vlu, 0 is ssoitd to dtriortd units during t yl tim. Hr is t urs ost of n itm. following nottions r usd to dvlo t modl: Dmnd rt R t t tim t is ssumd to R t t t 0, 0, 0. Hr is t initil rt of dmnd, is t initil rt of ng of t dmnd nd is t lrtion of dmnd rt. θ 0 is t onstnt rt of dtriortion. A is t ordring ost r ordr. S is t slling ri r itm S. Qt is t ordring quntity t tim t=0 is r unit olding ost xluding intrst rgs r unit r yr. I is t intrst rnd r yr. I is t intrst rgd r stoks r yr. IJER ISSN: Vol. Iss. r. 0 6

3 im Dndnt Qudrti Dmnd Invntory odls wn Dly in Pymnts is Atl is t rmissil dly in sttling in t ounts, 0<<. is t intrvl twn two sussiv ordrs K is t totl ost r unit tim. III. Formultion nd Solution of t odl ojtiv of t modl is to dtrmin t totl ost of t systm nd t dmnd rt of itms is tim dndnt qudrti funtion wit onstnt rt of dtriortion. Lt It t invntory lvl t tim t 0 t. invntory dlts du to dtriortion nd t dmnd, nd tn t diffrntil qution wi dsris t invntory lvl t tim t is givn y di t I t R t, 0 t dt wr R t t t nd i I 0 wn t, ii I 0 Q. solution of qution using t oundry ondition I 0 is givn y I t t t t t t t t wr w tkn sris xnsion nd ignord t sond nd igr owrs of θ s θ is smll. Sin I 0 Q, w otin Q t numr of dtriortd units D during on yl is givn y D Q R ost du to dtriortion is givn y CD D 6 slvg vlu of dtriortd units is SV CD 7 invntory olding ost during t yl is IHC 0 I t dt 0 0 Ordring ost is givn y 0 0 t OC A 9 Following Nit S nd Pndy 008, w v onsidrd t following two ss for intrst rgd nd t intrst rnd: Cs-: offrd rdit riod is lss tn or qul to t yl tim i..,. 8 Figur- IJER ISSN: Vol. Iss. r. 0 6

4 im Dndnt Qudrti Dmnd Invntory odls wn Dly in Pymnts is Atl Cs-: offrd rdit riod for sttling t ount is grtr tn t yl tim i..,. Figur-. Cs- rtilr n sl units during [0, ] t sl ri: S r unit wi n ut n intrst rt I r unit r nnum in n intrst ring ount. So t totl intrst rnd during [0, ] is IE R t tdt 0 0 Now, in t riod [, ], t sulir will rg t intrst to t rtilr on t rmining stok t t rt I r unit r nnum. Hn, totl intrst rgs yl y t rtilr during [, ] is IC I t dt Now, t totl ost K r tim unit is K OC IHC CD IE IC SV A Sin our ojtiv is to minimiz t totl ost K r unit tim, t nssry ondition for t totl ost to minimum is K 0 i.., K X X X X X X 6 wr X 0 X IJER ISSN: Vol. Iss. r. 0 6

5 im Dndnt Qudrti Dmnd Invntory odls wn Dly in Pymnts is Atl IJER ISSN: Vol. Iss. r A X X X X From qution, w v A W solv t ov qution for otiml using AHCAD. For tis otiml, t totl ost is minimum only if 0 K... Numril Exml o dmonstrt t fftivnss of t modls dvlod, numril xml is tkn wit t following vlus for t rmtrs: =00 = =0. = S= =0. I =0. I = =0 A=00 For t ov xml, it is found tt t otimlity onditions r stisfid in ll t following four ss for ll viz., i > 0, > 0 nd > 0 i.., lrtd growt modl ii > 0, < 0 nd > 0 i.., rtrdd growt modl iii > 0, < 0 nd < 0 i.., lrtd dlin modl iv > 0, > 0 nd < 0 i.., rtrdd dlin modl AHCAD outut is givn in l- troug tl- wi sows t vritions of t dtriortion rt, nd t dly riod,. From t outut sown in tl- to tl-, it is osrvd tt t uyr s totl ost drss wit t inrs in dly riod for fixd vlu of dtriortion rt. For xml, if t dtriortion rt is 0.0, t totl ost K drss wn t dly in ymnt inrss from dys to dys in ll t modls. W my ttriut tis du to t intrst rnd y uyr wo rns mor rvnu from t sold itms. Furtr, ross ll t modls, it n notid tt t uyr s totl ost inrss wn t rt of dtriortion inrss from 0.0 to 0.0.

6 im Dndnt Qudrti Dmnd Invntory odls wn Dly in Pymnts is Atl S.No. l-: > 0, > 0 nd > 0 i.., lrtd growt modl = dys =0 dys = dys = dys K K K K S.No. l-: > 0, < 0 nd > 0 i.., rtrdd growt modl = dys =0 dys = dys = dys K K K K S.No. l-: > 0, < 0 nd < 0 i.., lrtd dlin modl = dys =0 dys = dys = dys K K K K S.N o. l-: > 0, > 0 nd < 0 i.., rtrdd dlin modl = dys =0 dys = dys = dys K K K K IJER ISSN: Vol. Iss. r. 0 6

7 im Dndnt Qudrti Dmnd Invntory odls wn Dly in Pymnts is Atl IJER ISSN: Vol. Iss. r Cs- In tis s, t intrst rnd is R tdt t R IE 0 nd t intrst rgs is zro 0.,. IC i totl ost K r tim unit is SV IC IE CD IHC OC K A 6 Sin our ojt is to minimiz t totl ost K r unit tim, t nssry ondition for t totl ost to minimum is 0 K i.., 6 K 7 wr A 6 Now from qution 7, ,. A i 8 wi minimiss t K only if 0 K for ll vlus of.

8 im Dndnt Qudrti Dmnd Invntory odls wn Dly in Pymnts is Atl.. Numril Exml On gin w onsidr t vlus of t rmtrs s givn in... For ts vlus, t outut is rsntd in l- troug l-8. It n osrvd tt t viour of ts modls, in t s of >, is quit similr to t rsults otind s in t s of <. It is lso osrvd tt t totl ost K is lss tn K wn t dly riod is inrsd from dys to dys. us it n onludd tt t totl ost in ot t ss is lmost sm wn t dly riod inrss from dys to dys. For =, w v K K Z Z Z wr Z A 0 Z Z. IV. Snsitivity Anlysis > 0, > 0 nd > 0 i.., lrtd growt modl W now study t snsitivity of t modls dvlod in.. nd.. to xmin t imlitions of undrstimting nd ovrstimting t rmtrs,,, I, I, nd on otiml vlu of yl tim nd totl ost of t systm. Hr w v tkn t dtriortion rt s θ=0.0 nd t dly riod is 0 dys. snsitiv nlysis is rformd y nging of t rmtr y -0%, -0%, +0% nd +0% tking on rmtr t tim nd king t rmining rmtrs r unngd. rsults r sown in l-9 nd l-0. following intrsting osrvtions r md from t ov tls: Cs- i Inrs drs in rmtrs, I, nd drss inrss t yl tim wr s t totl ost inrss drss wit t inrs drs in ts rmtrs. Howvr t rt of inrs/drs is mor ronound in s of t ngs md in t rmtrs nd wi indit tt t otiml vlus of yl tim nd t totl ost r lss snsitiv to I nd. ii fft of t rmtrs, nd I on t otimum vlu of t yl tim nd t totl ost is similr ut t rt of ng is insignifint. iii slvg vlu of dtriortd itms is not sown mu fft on t otiml totl ost of t systm. Cs- iv Inrs drs in rmtrs, nd drss inrss t yl tim wr s t totl ost inrss drss wit t inrs drs in ts rmtrs. Howvr t rt of inrs/drs is mor ronound in s of t ngs md in t rmtrs nd nd lss snsitiv to t ngs in v nd K r lss snsitiv to t ngs md in t rmtr nd modrtly snsitiv to I. vi fft of slvg vlu is not so signifint on otiml oliis. V. Conlusions min ojtiv of tis study is t formultion of dtrministi invntory modl for itms wi v onstnt dtriortion rt nd follows tim dndnt qudrti dmnd rt wn sulir offrs sifi rdit riod. totl ost of t systm is lultd wn sortgs r not llowd. Slvg vlu is onsidrd wil lulting t totl ost of t systm. Snsitivity of t modls is lso disussd. IJER ISSN: Vol. Iss. r. 0 67

9 S.N o im Dndnt Qudrti Dmnd Invntory odls wn Dly in Pymnts is Atl l-: > 0, > 0 nd > 0 i.., lrtd growt modl = dys =0 dys = dys = dys K K K K S.No. l-6: > 0, < 0 nd > 0 i.., rtrdd growt modl = dys =0 dys = dys = dys K K K K S.No. l-7: > 0, < 0 nd < 0 i.., lrtd dlin modl = dys =0 dys = dys = dys K K K K S.No. l-8: > 0, > 0 nd < 0 i.., rtrdd dlin modl = dys =0 dys = dys = dys K K K K IJER ISSN: Vol. Iss. r. 0 68

10 im Dndnt Qudrti Dmnd Invntory odls wn Dly in Pymnts is Atl Cs- : l-9 S.No Prmtr I I γ % Cng % Cng in % Cng in K -0% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % IJER ISSN: Vol. Iss. r. 0 69

11 im Dndnt Qudrti Dmnd Invntory odls wn Dly in Pymnts is Atl Cs- : l-0 S.No Prmtr I I γ % Cng % Cng in % Cng in K -0% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % REFERENCES [] Hlly, C. G. & Higgins, R. C., 97. Invntory Poliy nd rd Crdit Finning. ngmnt Sin. 0, 6 7. [] Cmn, C. B., Wrd, S. C., Coor, D. F. & Pg,. J., 98. Crdit Poliy nd Invntory Control. Journl of t Ortionl Rsr Soity., [] Dvis, R. A. & Gitr, N. 98. Otiml Ordring Poliis Undr Conditions of Extndd Pymnt Privilgs, ngmnt Sins., [] Goyl, S. K. 98. Eonomi Ordr Quntity Undr Conditions of Prmissil Dly in Pymnt. Journl of t Ortionl Rsr Soity. 6, 8. [] S, V. R., Ptl, H. C. & S.. K., 988. Eonomi Ordring Quntity wn Dly in Pymnts of Ordrs nd Sortgs r Prmittd. Gujrt Sttistil Rviw., 6.6 [6] ndl, B. N. & Pujdr, S Som EOQ odls Undr Prmissil Dly in Pymnts. Intrntionl Journl of ngmnts Sin,, [7] ndl, B.N. & Pujdr, S An Invntory odl f or Dtriorting Itms nd Stok Dndnt Consumtion Rt. Journl of Ortionl Rsr Soity. 0, IJER ISSN: Vol. Iss. r. 0 70

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