Supply Chain Production Inventory Model: Innovative Study for Shortages Allowed With Partial Backlogging

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1 Inernaional Journal of odern Engineering Research (IJER Vol., Issue. 5, Sep.-Oc. pp ISS: Supply Chain Producion Inenory odel: Innoaie Sudy for Shorages Allowed Wih Parial Backlogging Jasinder Kaur, Rajendra Sharma *(Deparmen Of ahemaics, Graphic Era Uniersiy Dehradun,India ** (Deparmen Of ahemaics Graphic Era Uniersiy Dehradun, india Asrac: In his paper, we hae sried o comine all he aoe menioned facors ino a single prolem. We shall underake o explore a wo echelon supply chain, comprising of a endor a uyer. he whole enironmen of usiness dealings has een assumed o e progressie credi period, which conforms o he pracical marke siuaion. he whole cominaion is ery unique ery much pracical. he ariale holding cos ariale seup has een explored numerically as well; an opimal soluion has een reached. he final oucome shows ha he model is no only economically feasile, u sale also. Keywords: Inenory model, parial acklogging, Progressie permissile delay, Supply chain, Shorages, EOQ (Economics Order Quaniy. I. IRODUCIO Inenory represens one of he mos significan possessions ha mos usinesses possess. I is in direc ouch wih he user deparmen in is day oday aciiies. Inenory managemen is playing a key role in seing up efficien closed loop supply chains. A supply chain is a nework of faciliies disriuion opions ha performs he funcions of procuremen of maerials, ransformaion of hese maerials ino inermediae finished producs, he disriuion of hese finished producs o cusomers. I consiss of a nework of companies which are dependen on each oher while making independen decisions. he supply chain no only includes he manufacurer suppliers, u also ransporers, warehouses, reailers, cusomers hemseles. herefore, supply chain analysis ools mehodologies hae ecome more more imporan. I can e a source of grea efficiency cos-saings gains. Supply chain speed flexiiliy hae ecome key leers for compeiie differeniaion increased profiailiy. he faser he supply chain, he eer a company can respond o changing marke siuaion he less i needs inenory which resuling in higher reurn on capial employed. Supply chain managemen offers a large poenial or organizaions o reduce coss improe cusomer serice performance. In he exising lieraure, mos of he inenory models sudies only aimed a he deerminaion of he opimum soluions ha minimized cos or maximized profi from he endor s endor s side. Howeer, in he modern gloal compeiie marke, he uyer endor should e reaed as sraegic parners in he supply chain wih a long erm cooperaie relaionship. Recenly, many researchers hae considered he uyer endor as a uni o find he opimal EOQ in achieing he minimum oal cos. In oday's usiness ransacions, i is more more common o see ha he cusomers are allowed some grace Period efore hey sele he accoun wih he supplier. his proides an adanage o he cusomers, due o he fac ha hey do no hae o pay he supplier immediaely afer receiing he produc, u insead, can defer heir paymen unil he end of he allowed period. he cusomer pays no ineres during he fixed period hey are supposed o sele he accoun; u if he paymen is delayed eyond ha period, ineres will e charged. he cusomer can sar o accumulae reenues on he sale or use of he produc earn ineres on ha reenue. So i is o he adanage of he cusomer o offer he paymen o he supplier unil he end of he period. he wo famous formulae of EOQ EPQ are reaed separaely for a uyer a endor respeciely. From he radiional poin of iew, he endor he uyer are wo indiidual eniies wih differen ojecies self-ineres. Due o rising coss, he gloalizaion rend, shrinking resources, shorened produc life cycle quicker response ime, increasing aenion has een placed on he collaoraion of he whole supply chain sysem. An effecie supply chain nework requires a cooperaie relaionship eween he endor he uyer. I assumes ha he uyer mus pay off as soon as he iems are receied. Suppliers ofen offer rade credi as a markeing sraegy o increase sales reduce on-h sock is reduced, ha leads o a reducion in he uyer s holding cos of finance. In addiion, during he ime of he credi period, uyers may earn ineres on he money. In fac, uyers, especially small usinesses which end o hae a limied numer of financing opporuniies rely on rade credi as a source of shor-erm funds. he classical inenory models hae considered dem raes which were eiher consan or depended upon a single facor only, like, sock, ime ec. Bu changing marke condiions hae rendered such a consideraion quie unfruiful, since in real life siuaion, a dem canno depend exclusiely on a single parameer. A cominaion of wo or more facors grans more auheniciy o he formulaion of he model. any deliery policies hae een proposed in lieraure for his prolem. Clark Scarf (96 presened he concep of serial muli-echelon srucures o deermine he opimal policy. Goyal (985 considered a mahemaical models wih a permissile delay in paymens o deermine he opimal order quaniies. Ha Kim (997 used a graphical mehod o analyze he inegraed endor-uyer inenory saus o derie an opimal soluion. Hwang Shinn (997 sudied effecs of permissile delay in paymens on reailer's pricing lo sizing policy for exponenially deerioraing producs. Yang Wee ( deeloped an inegraed economic ordering policy of deerioraing iems for a endor a uyer. Wang e al. ( analyzed supply chain models for perishale producs under inflaion permissile delay in paymen. 36 Page

2 Inernaional Journal of odern Engineering Research (IJER Vol., Issue. 5, Sep.-Oc. pp ISS: eng ( modified Goyal (985 model y considering he selling price, insead of purchasing cos, as he ase o calculae he ineres. Aad Jaggi (3 sudied a seller-uyer model wih a permissile delay in paymens y game heory o deermine he opimal uni price he credi period, considering ha he dem rae is a funcion of reail price. Huang, Y.F. e al. (5 considered he opimal inenory policies under permissile delay in paymens depending on he ordering quaniy. Song Cai (6 has een aken on opimal paymen ime for a reailer under permied delay of paymen y he wholesaler. Liao (7 assumed on an EPQ model for deerioraing iems under permissile delay in paymens. In he presen sudy, we hae sried o comine all he aoe menioned facors ino a single prolem. We shall underake o explore a wo echelon supply chain, comprising of a endor a uyer. he whole enironmen of usiness dealings has een assumed o e progressie credi period, which conforms o he pracical marke siuaion. he whole cominaion is ery unique ery much pracical. he ariale holding cos ariale seup has een explored numerically as well; an opimal soluion has een reached. he final oucome shows ha he model is no only economically feasile, u sale also. II. PROPOSED ASSUIOS & OAIOS. ASSUPIOS he following assumpions are used o deelop aforesaid model:. he dem rae, D(, is deerminisic, he dem funcion D( is gien y D(, a are posiie consans.. Shorages are allowed wih parial acklogging..3 If he reailer pays y, hen he supplier does no charge o he reailer. If he reailer pays afer efore ( >, he can keep he difference in he uni sale price uni purchase price in an ineres earing accoun a he rae of I e /uni/year. During [,, he supplier charges he reailer an ineres rae of Ic /uni/year on unpaid alance. If he reailer pays afer, hen supplier charges he reailer an ineres rae of Ic /uni/year (Ic > Ic on unpaid alance.. OAIOS:. P he selling price / uni.. KD he producion rae per year, where K>.3 C he uni purchase cos, wih C < P.. he firs offered credi period in seling he accoun wihou any charges..5 he second permissile credi period in seling he accoun wih ineres charge Ic on unpaid alance >..6 Ic he ineres charged per $ in sock per year y he supplier when reailer pays during [,..7 Ic he ineres charged per $ in sock per year y he supplier when reailer pays during [,. (Ic > Ic.8 Ie he ineres earned / $ / year..9 he replenishmen cycle.. r he discoun rae (r > α. he ineres earned / ime uni.. IC he ineres charged / ime uni. e.3 ( Cs he seup cos for each producion cycle for endor.. ( Cs he seup cos per order for uyer..5 ( Ch holding cos per uni ime for endor..6 ( Ch holding cos per uni ime for uyer..7 C he uni cos for endor..8 C he uni purchase cos for uyer..9 S shorage cos per uni ime for uyer.. L los sale cos per uni ime for uyer.. VC he cos of endor per uni ime.. BC he cos of uyer per uni ime..3 C( oal cos of an inenory sysem / ime uni.. B Backlogging rae..5 he deerioraion funcion (, (, < ( <<, > his is a special form of he wo parameer weiull funcion considered y Coer Philip. he funcion is some funcions of he rom ariale which range oer a space in which a p.d.f. p( is defined such ha p( d III. IDEAIOS AD EQUAIOS AHEAICAL FORULAIO he acual endor s aerage inenory leel in he inegraed wo-echelon inenory model is difference eween he endor s oal aerage inenory leel he uyer s aerage inenory leel. Since he inenory leel is depleed due o a consan deerioraion rae of he on-h sock, he uyer s inenory leel is represened y he following differenial equaion: ' I ( ( I ( e, ( ' I ( B e, ( he endor s oal inenory sysem consising of producion period non-producion period can e descried as follows: I ( ( I ( ( K e, (3 ' I ( ( I ( e, ( ' he oundary condiions are I (, (5 I (, (6 I I (7 (, I (, (8 I ( I ( (9 36 Page

3 Inernaional Journal of odern Engineering Research (IJER Vol., Issue. 5, Sep.-Oc. pp ISS: And, ( n he soluions of he aoe differenial equaions oained are ( ( 3 ( I ( I e [ e, ( 6 B I ( e e, ( 3 ( ( ( ( [, I K 6 e (3 r r ( ( (3 r ( ( r ( [ ( r r( r Respeciely. he annual oal holding cos for he uyer he endor are r HC ( Ch e I ( d 3 r ( ( (3 r Ch[ ( r r( r ( C I r h [ ( r r( r [ ( I ( [( ( (, Using he condiion ha one can ge, I ( 3 ( [ (5 6 If he produc of he deerioraion rae he replenishmen ineral is much smaller han one, he uyer s he endor s acual aerage inenory leel, I I, are r I e I( d I r [ ( r ( [ ( r 5 r ( ( r( r (3 r 8 3 I e I d e e I d I r r r ( ( (6 r ( ( (3 r 3 [( K ( ( r r ( r ( r ( I r ( r ( [ And (8 HC C e I d e C e I d I r r r ( h ( ( h ( r ( ( (3 r 3 [( K Ch { ( r r ( r } r 3 ( K ( Ch e ( r [ ( r r( r [ r 3 r ( ( ( 3 r e (3 r I [ [ 6 6 r r ( ( (3 r ( ( r ( [ ( r r( r respeciely. he annual deerioraion cos for he uyer he endor are C r DC ( e I ( d 3 3 I ( r r ( ( r [ { } { rr ( } ( r r r ( ( ( 3 r I e ( ( r [ 363 Page

4 Inernaional Journal of odern Engineering Research (IJER Vol., Issue. 5, Sep.-Oc. pp ISS: C r r r DC ( e I ( d e ( e I ( d 3 r 3 ( K ( ( r r( r e ( C[ { } { Regarding ineres charged ineres earned ased on he lengh of he cycle ime, hree cases arise: IV. FIGURES AD ABLES Regarding ineres charged ineres earned ased on he lengh of he cycle ime, hree cases arise: (3 r } ( respeciely. Case I: Inenory leel he annual se-up cos for he uyer he endor are [ ( s ( s OC C d C d Cs ( [ ( s ( s OC C d C d ( [ Cs (3 respeciely. he annual shorage cos for he uyer is Se r r SC e e e d S e e e e [ r( r r ( r r ( r ( r ( r he annual los sale cos for he uyer is r Le r LC ( B e e d ( Fig : In he firs case, reailer does no pay any ineres o he supplier. Here, reailer sells I unis during (, ime ineral paying for CI unis in full o he supplier a ime, so ineres charges are zero, i.e. IC (8 Reailers deposis he reenue in an ineres earing accoun a he rae of Ie / $ / year. herefore, ineres earned, per year is ime r L ( e B ( r ( r [ e e ( r (5 he differen coss associaed wih he sysem are se-up coss, holding coss, deerioraion cos shorage cos. Our aim is o minimize he oal cos. From (9, one can derie he following condiion: 3 ( 3 ( ( ( ( K [ 6 e [ 6 e (6 By aylor s series expansion, (.6 is deried as PI e r r [ e D( d ( e D( d 3 PIe ( r [( { ( r( } (9 3 oal cos per uni ime of an inenory sysem is C (, OC +HC + DC + SC + IC 3 r ( [ Cs Ch[ ( r r( r 6 8 K ( ( (3 r ( [ ( r r( r Page

5 Inernaional Journal of odern Engineering Research (IJER Vol., Issue. 5, Sep.-Oc. pp ISS: ChI r ( r ( I r ( r ( [ [ I ( r r ( ( r [ { } { rr ( } S e L e B r( r r ( r ( r r ( r ( r ( r r ( e e e ( r ( r [ [ e e PI ( r [( { ( ( } 3 3 e r A ( K A ( r r( r (3 [ { } r C r 3 (3 e A r { } (3 6 o minimize he oal cos per uni ime, he opimum alue of, is he soluion of following equaion. Case II: < < Inenory leel (3 Hence he mean cos Q < C > C (, p ( d (3 < C 3 > r A [ Cs Ch[ ( r r( r A (3 r [ ( A ( r r r ChI r ( r A I r ( r A [ [ IA r r A ( r [ { } { rr ( } r ( r ( r ( r r S ( e e e e Le B ( r ( r [ [ e e r( r r ( r ( r 3 PIe ( r [( { ( r( } 3 (3 Where A ( p( d (33 ime Fig: < < In he second case, supplier charges ineres a he rae Ic on unpaid alance. Ineres earned, during [, is r PI e D( d e <C > OC +HC + DC - IC 3 [ ( ( PI e r r ( ( [ Cs [( K Ch{ ( r r( r r A A (3 r ( K } [ ( r r 3 Che ( A r r A rr ( [ 5 6 A ( 3 r e (3 r I r 5 r 3 [ [ 6 6 r A ( r A [ ( r r( r Reailer pay for I unis purchased a ime a he rae of C / $ / uni o he supplier during [,. he reailer sells D (. unis a selling price P/ uni. So, he has generaed reenue of P D(. +. hen wo su cases may arise: Su Case:. Le P D(. + CI, i.e. reailer has enough money o sele his accoun for all I unis procured a ime. hen ineres charge will e IC. (36 ineres earned 365 Page

6 Inernaional Journal of odern Engineering Research (IJER Vol., Issue. 5, Sep.-Oc. pp ISS: PIe [ ( r ( r ( So, oal cos C. per uni ime of inenory sysem is <C > OC + HC + DC + SC + LC +IC.. 3 S r A [ Cs Ch[ ( r r( r 6 8 A (3 r [ ( ( r r r A ChI r ( r A I r ( r A [ [ IA r r A ( r [ { } { rr ( } o minimize he oal cos per uni ime, he opimum alue of, is he soluion of following equaion. Su Case:. Le P D(. + < CI. Here, reailer will hae o pay ineres on unpaid alance U CI (P D(. + a he rae of Ic a ime o he supplier. hen ineres paid per uni ime is gien y U Ic r IC. e I( d PI U Ic ( ( r( ( r ( [ PI 8 ( 3 3 Where, U CI (P D(. + CI 3 Ie Ie ( r 3 Ie ( r P[ ( ( ( 6 8 ( And ineres earned. S e L e B r( r r ( r ( r r ( r ( r ( r r ( e e e ( r ( r [ [ e e PIe [ ( r ( r (38 3 PIe [ ( r ( r ( 6 8 So, oal cos C. per uni ime of inenory sysem is <C > OC + HC + DC + SC + LC + IC.. <C > OC + HC + DC - IC. 3 ( [ Cs [( K Ch{ ( r r( r ( K r A A (3 r } [ ( r r 3 Che ( A r r A rr ( [ 5 6 A ( 3 r e (3 r I r 5 r 3 [ [ r A ( r A [ ( r r( r A ( K A ( r r( r (3 r [ { } C r 3 e A (3 r { } PIe [ ( r ( r ( r A [ Cs Ch[ ( r r( r 6 8 A (3 r [ ( ( r r r A ChI r ( r A I r ( r A [ [ IA r r A ( r [ { } { rr ( } S e L e B r( r r ( r ( r r ( r ( r ( r r ( e e e ( r ( r [ [ e e 3 PIe [ ( r ( r (3 6 8 <C > OC + HC + DC - IC. 3 S ( [ Cs [( K Ch{ ( r r( r Page

7 Inernaional Journal of odern Engineering Research (IJER Vol., Issue. 5, Sep.-Oc. pp ISS: r A A (3 r ( K } [ ( r A C e ( r r A rr ( [ r 3 h A ( 3 r e (3 r I r 5 r 3 [ [ 6 6 r A ( r A [ ( r r( r A ( K A ( r r( r (3 r [ { } C r 3 e A (3 r U Ic ( { } [ 6 PI 3 3 ( r( ( r ( ( 8 o minimize he oal cos per uni ime, he opimum alue of, is he soluion of following equaion. Case III: in accoun a oal money PD(.P arise: e Su Case 3. Le P D(. + CI a, here are hree su cases may his case is same as su case., here 3. designae decision ariales ojecie funcion. Su Case 3. Le P D(. + < CI PD(.( PI D( d CI ( PD(. e ( Pe ( PIe [( ( CI ( Pe his case similar o su case.. Su Case 3.3 Le P D(. + < CI ( Pe ( PIe [( ( CI ( Pe Here, reailer does no hae enough money o pay off oal purchase cos a. He will no pay money of P D(. + a PD(.( PIe[( ( a. ha s why he has o pay ineres on unpaid alance U CI (P D(. + wih Ic ineres rae during (, U U PD(.( PI D( d wih ineres rae Ic during (,. e herefore, oal ineres charged on reailer, IC 3.3 per uni ime is ( U Ic U Ic r IC3.3 e I ( d PI 3 3 UIc ( U Ic ( ( r ( [ r( r PI ( ( ( ( ( (3 r U Ic r [( P 3 3 r( ( r ( ( (5 6 Ineres earned per uni ime is 3.3 Fig 3: In he final case, reailer pays ineres a he rae of Ic o he supplier. Based on he oal purchased cos, CI, oal money P D(. + 3 PIe [ ( r ( r PIe [ ( r ( r 6 8 So, oal cos C 3.3 per uni ime of inenory sysem is <C > OC + HC + DC + SC + LC + IC (6 367 Page

8 Inernaional Journal of odern Engineering Research (IJER Vol., Issue. 5, Sep.-Oc. pp ISS: r A [ Cs Ch[ ( r r( r 6 8 A (3 r [ ( ( r r r A ChI r ( r A I r ( r A [ [ IA r r A ( r [ { } { rr ( } r ( r ( r ( r r S ( e e e e Le B ( r ( r [ [ e e r( r r ( r ( r UIc ( U Ic ( ( r ( [ r( r PI ( ( ( ( ( (3 r U Ic r [( P 3 3 r( ( r ( ( 6 3 PIe [ ( r ( r (7 6 8 C OC + HC + DC - IC ( [ Cs [( K Ch{ ( r r( r r A A (3 r ( K } [ ( r A C e ( r r A rr ( [ r 3 h UERICAL ILLUSRAIO: HE PRECEDIG HEORY CA BE ILLUSRAED BY HE FOLLOWIG UERICAL EXAPLE WHERE HE PARAEERS ARE GIVE AS FOLLOWS: Dem parameers, a 5, 5, c Selling price, P 3 Buyer s purchased cos, C 35 Buyer s percenage holding cos per year per dollar, C h. Buyer s ordering cos per order, C s 5 Buyer s shorage cos, S 5 Vendor s uni cos, C Vendor s percenage holding cos per year per dollar, C h. Vendor s seup cos per order, C s Vendor s producion rae per year, K 5 Deerioraion rae, (. Firs delay period,. Second delay period,. he ineres earned, Ie.5 he ineres charged, Ic. he ineres charged, Ic. (Ic > Ic Backlogging rae, B ale : A ( 3 r e (3 r I r 5 r 3 [ [ 6 6 r A ( r A [ ( r r( r A ( K A ( r r( r (3 r [ { } C VC BC C r 3 e A (3 r { } - 6 PIe [ ( r ( r ( o minimize he oal cos per uni ime, he opimum alue of, is he soluion of following equaion. 368 Page

9 Inernaional Journal of odern Engineering Research (IJER Vol., Issue. 5, Sep.-Oc. pp ISS: ale : VC BC C ale 3: VC BC C ale : VC BC C V. COCLUSIO Here we hae sudied a wo echelon supply chain wih some ery realisic assumpions. We sudied our model in a progressie credi period. o dou, his assumpion impars an economic iailiy o he whole sudy. In real world, i is noed ha, as a resul of progressie permissile delay in seling he replenishmen accoun, he economic replenishmen ineral order quaniy generally increase marginally, alhough he annual cos decreases consideraly. he saing in cos as a resul of permissile delay in seling he replenishmen accoun largely come he ailiy o delay paymen wihou paying any ineres. As a resul of increasing order quaniy under condiions or permissile delay in paymens, we need o order less ofen. So his EOQ model is applicale when supplier gies he rade credi o he reailer. AKOWLEGEE We are hankful o Dr.A.P.Singh of SGRR (PG College Dehradun for his consan suppor reiew of our research paper. We also hank Prof. Yudheer Singh oudgil of Uranchal Insiue of echnology for reiewing our research paper. OBSERVAIO he daa oained clearly shows ha indiidual opimal soluions are ery differen from each oher. Howeer, here exiss a soluion which ulimaely proides he minimum operaing cos o he whole supply chain. All he oseraions can e summed up as follows:. An increase in he ineres charged, increases he uyer cos BC decrease he endor cos VC of he commodiy.. Opimal soluion for he uyer is n in ale firs while for he endor, i is n5 in ale. he oerall opimal soluion which ulimaely minimizes he cos across he whole supply chain is n5 in ale REFERECES Hwang H. S. (997. A Sudy on an inenory model for iems wih Weiull amelioraing. Com. & Ind. Eng. 33, 7-7. Hwang H. S. (999. Inenory odel for Iems for oh Deerioraing Amelioraing Iems. Com. & Ind. Eng. 37, Hwang, H. S. (: A Sochasic Se-coering Locaion odel for Boh Amelioraing Deerioraing Iems. Com. & Ind. Eng. 6, Law S.. Wee H.. (6. An inegraed producion-inenory model for amelioraing deerioraing iems aking accoun of ime discouning. ah. & Comp. od., 3, ondal B., Bhunia A. K. aii. (3. An inenory sysem of amelioraing iems for price dependen dem rae. Com. & Ind. Eng. 5, oon I., Giri B. C. Ko B. (5. Economic order quaniy models for amelioraing/ deerioraing iems under inflaion ime discouning. Euro. Jour. Oper. Res. 6, Page