Integrated inventory model with controllable lead time involving investment for quality improvement in supply chain system

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1 Itratioal Joural of upply ad Opratios Maagmt IJOM May 15 Volum Issu 1 pp IN-Prit: IN-Oli: Itgratd ivtory modl with cotrollal lad tim ivolvig ivstmt for quality improvmt i supply chai systm M. Vijayashr a ad R. Uthayaumar a a partmt of Mathmatics Th Gadhigram Rural Istitut md Uivrsity Gadhigram idigul Tamil Nadu Idia. stract Th purpos of this articl is to ivstigat a two-chlo supply chai ivtory prolm cosistig of a sigl-vdor ad a sigl-uyr with cotrollal lad tim ad ivstmt for quality improvmts. This papr prsts a itgratd vdor-uyr ivtory modl i ordr to miimiz th sum of th ordrig cost holdig cost stup cost ivstmt for quality improvmt ad crashig cost y simultaously optimizig th optimal ordr quatity procss quality lad tim ad umr of dlivris. Hr th lad-tim crashig cost has assumd to a xpotially fuctio of th lad-tim lgth. Th mai cotriutio of proposd modl is a fficit itrativ algorithm dvlopd to miimiz itgratd total rlvat cost for th sigl vdor ad th sigl uyr systms with cotrollal lad tim rductio ad ivstmt for quality improvmts. It ca otaid simultaously y optimizig th optimal solutio mathmatical modllig ad solutio procdur ar mployd i this study for optimizig th ordr quatity lad tim procss quality ad th umr of dlivris from th vdor to th uyr i o productio ru with th ojctiv of miimizig total rlvat cost. Graphical rprstatio is also prstd to illustrat th proposd modl. Numrical xampls ar prstd to illustrat th procdurs ad rsults of th proposd algorithm. Matla codig is also dvlopd to driv th optimal solutio ad prst umrical xampls to illustrat th modl. Kywords: Itgratd ivtory modl; vdor uyr coordiatio; ad cotrollal lad tim crashig cost; supply chai maagmt; ivstmt for quality improvmts. orrspodig author mail addrss: 617

2 Vijayashr da Uthayaumar Itroductio om tim ago th compais ca otaid th comptitiv advatag y strgthig thir comptitivss. s a rsult w ca us th itgratd ivtory modl to otai miimum th total rlvat cost for oth vdor ad uyr. Th just-i-tim approach to productivity dmads that small lots to ru i productio. This ca oly achivd if th stup tim is rducd. Th aility to rduc stup cost ovr tim ca xplaid i th trms of th lavig curv. Ivtory cotrol is importat i supply chai maagmt. Ivtoris play a xtrmly importat rol i a atio s coomy. I t yars most ivtory prolms hav thir focus o th itgratio tw th vdor ad th uyr. For supply chai maagmt stalishig log-trm stratgic partrships tw th uyr ad th vdor is advatagous for th two partis rgardig costs ad thrfor profits sic oth partis to achiv improvd fits cooprat ad shar iformatio with ach othr. Thrfor svral rsahrs [.g. masaa () B-aya t al. () Byla () hag t al. (6) Hoqu t al. (6) Ouyag t al. (7) Pa t al. (5) Villa (1) Viswaatha (1998)] hav show that th uyr ad th vdor ca achiv thir ow miimal total cost or icras thir mutual fit through stratgic coopratio with ach othr. I th productio viromt lad tim plays a importat rol i today s logistics maagmt. fi as th tim that lapss tw th placmts of a ordr ito ivtory ilvr t al. (1988) lad tim may ifluc customr srvic ad impact ivtory costs. s th Japas xampl of just-i-tim-productio has show cosqutly rducig lad tim may icras productivity ad improv th comptitiv positio of th compay Trsi t al. (1995). lthough lad tim ca costat or varial it is oft tratd as a prscrid paramtr i most of th studis. Thrfor th lad tim crashig cost fuctio is a pricwis liar fuctio [iao t al. (1991) Ouyag t al. () Ouyag t al. (1999)]. Th umr of advatags hav associatd th fforts of cotrol of th lad tim (which is a goal of JIT ivtory maagmt philosophis that mphasizs high quality ad ps low ivtory lvl ad lad tim to a practical miimum). ad tim maagmt is a sigificat issu i productio ad opratio maagmt. I may practical situatios lad tim ca rducd usig a addd crashig cost. I othr words lad tim is cotrollal. Th crashig of lad tim maily cosists of th followig compots: ordr prparatio ordr trasit supplir lad tim ad dlivry tim Trsi (199). upply chai maagmt has ta a vry importat ad critical rol for ay compay i icrasig gloalizatio ad comptitio i th mart. upply hai Modl (M) is a twor of supplirs producrs distriutors ad customrs which sychroizs a sris of itrrlatd usiss procss i ordr to hav (1) optimal procurmt of raw matrials from atur () trasportatio of raw matrials ito a warhous () productio of goods i th productio ctr ad () distriutio of ths fiishd goods to rtailrs for sal to th customrs. With a t paradigm shift to th upply hai () th ultimat succss of a firm may dpd o its aility to li supply chai mmrs samlssly. I th currt upply hai Maagmt (M) viromt compais ar usig JIT productio to gai ad maitai a comptitiv advatag. JIT rquirs a spirit of coopratio tw th uyr ad th vdor ad it has show that formig a partrship tw th uyr ad th vdor is hlpful i achivig tagil fits for oth partis Goyal t al. (199). I t yars trpriss usiss modls ar diffrt from th past du to gloalizatio ad 618

3 It J upply Opr Maag (IJOM) iformatio. Th trpriss hav to fac may prdicamts i itratioalizatio so thr will th mrgc of th cocpt of supply chai. ompard to th traditioal usiss modl th supply chai systm ca itgrat th upstram ad dowstram compais tw trpriss ad us th rsous mor fficitly to crat mor profit. Th collaoratio cocpt has com a accptd practic i may succssful gloal usiss orgaizatios ad provids coomic advatags for oth a vdor ad a uyr. s a crtifid supplir th vdor ds to prfct productio procss y its fforts to improv th uyr s opratioal fficicy ad maitai a wi-wi rlatioship with th uyr. Th ultimat goal of JIT from th productio/ivtory maagmt stadpoit is to produc small sizs high quality products. Ivstig capital i shortig lad tim ad improvig quality ar rgardd as th most ffctiv mas of achivig this goal. With such charactristics rsahrs hav modifid traditioal ivtory modls to icorporat th implmtatio of JIT cocpts. Just-i-tim (JIT) is a philosophy of maufacturig asd o plad limiatio of all wasts ad o cotiuous improvmt of productivity. Th issu of coordiatio i supply chai maagmt (M) has ivd cosidral atttio from acadmic rsahrs ad practitiors. Traditioally oth vdors ad uyrs i th supply chai systm ma dcisios i sah of thir idividual fits. Howvr may rsahrs (.g. Parlar t al. (1997) i t al. (7) armah t al. (6) Wg (1997)] hav poitd out that coordiatio tw oth partis is importat i ordr to gai comptitiv advatags through cost rductio. Th importac of coordiatio is furthr icrasd caus vdors ad uyrs frqutly implmt th just-i-tim (JIT) cocpt i thir ow systms. t study poitd out that coordiatio is crucial to succssful JIT implmtatio for oth partis Huag t al. (). y tchiqu i succssful M is JIT applicatio to multipl dlivris hug t al. (7) showd that icrass i quality productivity ad fficicy ca achivd through JIT dlivry agrmts. t study showd that if a log-trm rlatioship has stalishd oth partis i th supply chai systm ca achiv furthr improvd fits through coopratio ad iformatio sharig hag t al. (6). Rau t al. (8) prstd a w itgratd productio-ivtory policy that showd that th prformac of itgratd cosidratio is ttr tha th prformac of ay idpdt dcisio from ithr th uyr or th vdor. I this complx viromt succssful compais hav dvotd cosidral atttio to rducig ivtory cost ad improvig quality simultaously. Th rtur o ivstmt for quality improvmt is sustatial ad may paprs hav show that improvig quality could rduc wast i othr words cut th cost. I additio th proaility of dfcts also mas a grat impact o th ivtory policy rgardig productio cycl ad lot siz so it is importat to always ta quality issus ito accout for ay usiss i a comptitiv supply chai viromt owadays. Thrfor i this prst study itgratd ivtory modl with cotrollal lad tim ivolvig ivstmt for quality improvmt i supply chai systm ad itd to proposs a simpl solutio procdur to sah th optimal productio umr of shipmts ad procss quality that ca miimis th itgratd total rlvat cost. 1. Rviw of th rlatd litratur Th sigl vdor sigl uyr itgratd productio ivtory prolm ivd a lot of atttio i t yars. This rwd itrst is motivatd y th growig focus o supply chai maagmt. Firms ar ralizig that a mor fficit maagmt of ivtoris across th tir supply chai through ttr coordiatio ad mor coopratio ar i th joit fit of all 619

4 Vijayashr da Uthayaumar partis ivolvd. uch collaoratio is facilitatd y th advacs i iformatio tchology providig fastr ad chapr commuicatio mas. I ral-lif usiss viromts it is commo to hav a supplir who provids a product to its svral rtailr clits. I this typ of supply chais maagmt is itdd to figur out th st productio-shipmt policy i ordr to miimiz th xpctd itgratd systm costs. Optimal ivtory policis hav sujct to a lot of rsah i t yars. I traditioal Ecoomic Ordr uatity (EO) ad Ecoomic Productio uatity (EP) modls most of th most of rsah tratig ivtory prolms ithr i dtrmiistic or proailistic modls th stoc out or stup costs is rgardd as prscrid costats ad qual at th optimum. Howvr th xpric of th Japas idicats that this d ot th cas. I practic stup cost may cotrolld ad rducd y virtu of various fforts such as worr traiig procdural chags ad spcializd quipmt acquisitio. Wh ivtory dcisios i supply chais ar mad idpdtly at ach stag thy ar usually asd o th local ivtory status ad local prformac ojctivs (local policis). Ths policis ar simpl to dfid ad implmtd ut igor th implicatios that dcisios at o stag ca hav o th othrs lt alo th fact that local ojctivs ar oft coflictig amog ach othr which oft lads to su optimiz th upply hai () prformac. I such cass th Ecoomic ot iz (E) of o stag may ot rsult i a optimal policy for th othr stags. To ovom this prolm rsahrs hav com up with a Joit Ecoomic ot iz (JE) modl whr th Joit Total Rlvat ost (JTR) for all stags has optimizd. Goyal (1976) first itroducd a itgratd ivtory policy for a sigl vdor ad a sigl puhasr. supply chai is a systm of facilitis ad activitis that fuctios to procur produc ad distriut goods to customrs. upply chai maagmt is asically a st of approachs utilizd to fficitly itgrat supplirs maufacturrs warhouss ad stors so that mhadis is producd ad distriutd at th right quatitis to th right locatios ad at th right tim i ordr to miimiz systm-wid costs (or maximiz profits) whil satisfyig srvic lvl rquirmts imchi-vi t al. (). I comptitiv viromt supply chai maagmt has mrgd as a popular productio ad logistics stratgy for may cotmporary firms ad th just-i-tim (JIT) puhasig plays a crucial rol i such supply chai viromts. ompais ar usig JIT puhasig to gai ad maitai a comptitiv advatag. Th fits of JIT puhasig iclud small lot sizs frqut dlivris cosistt high quality rductio i lad tims dcras i ivtory lvls lowr stup cost ad ordrig cost ad clos supplir tis. I t yars compais hav foud that thr ar sustatial fits from stalishig a log-trm sol-supplir rlatioship with supplir Martiich (1997). I th JIT viromt a clos coopratio xists tw supplir ad puhasr to solv prolms togthr ad thus maitais stal log-trm rlatioships. upply chai is th squc of usiss procsss ad activitis from supplirs through customrs that provid products srvics ad iformatio to achiv customr satisfactio i.. a chai that ca quicly rspod to customr's rquirmt. Rctly th issu of just-i-tim (JIT) maufacturig has ivd cosidral atttio ad o of th most ovl issus ar th itgratio of vdor ad uyr i th supply chai systm hag t al. (6). Itgratio of diffrt titis i th supply chai is a importat way to gai comptitiv advatag ad customr satisfactio. I currt yars rsah dalig with ivtory maagmt i supply chai systm has ivd atttio from may scholars. Goyal (1976) is amog th first rsahrs who studis itgratd ivtory modl for sigl vdor sigl uyr systm. H itroducs a modl for situatio i which vdor producs a lot asd o a ifiit 66

5 It J upply Opr Maag (IJOM) productio rat ad trasfrs it to th uyr y a lot-for-lot policy. H shows that maig ivtory dcisios joitly amog vdor ad uyr ca rsult i cost rductio compard to idividual dcisios. Th itgratio tw vdor ad uyr for improvig th prformac of ivtory cotrol has ivd a grat dal of atttio ad th itgratd approach has xamid for yars. I 1986 Barj assumd that th vdor maufacturs at a fiit rat ad cosidrd a joit coomic-lot-siz modl i which a vdor producs to ordr for a uyr o a lot-for-lot asis. Goyal (1988) rlaxd th lot-for-lot policy ad suggstd that vdors coomic productio quatity should a itgr multipl of uyr puhas quatity. s a rsult of usig th approach suggstd i th Goyal s (1988) modl sigificat rductio i ivtory cost ca achivd. Pa t al. () improvd Goyals (1988) modl y cosidrig lad tim as a cotrollal factor i th modl ad otaid a lowr joit total xpctd cost ad shortr lad tim. Ivtory modls icorporatig lad tim as a dcisio varial wr dvlopd y svral rsahrs. iao t al. (1991) addrssd a proailistic ivtory modl i which th lad tim is a dcisio varial. B-aya t al. (199) xtdd iao t al. (1991) modl y allowig oth th lad tim ad th ordr quatity as dcisio varials. atr svral rsahrs [s Ouyag t al. (1996) Moo t al. (1998) Ouyag t al. () Ouyag t al. (6) Ouyag t al. (7) Jha t al. (9)] ivstigatd various itgratd productio-ivtory modls for lad tim rductio i sigl-vdor sigl-uyr supply chai. Ha t al. (1997) proposd a itgratd lotsplittig modl of facilitatig multipl shipmts i small lots. Hoqu t al. () proposd a itgratd productio-ivtory systm ivolvig th capacity of trasport quipmt. Yag t al. () prstd a itgratd modl cosidrig coomic ordrig policy of dtrioratd itm. Niuwhuys t al. (6) foud that lot splittig policis hav fitd oth th vdor ad th uyr. Huag t. al. (1) prstd th prmissil dlay i paymt prolm i a sigl-vdor ad a sigl-uyr itgratd ivtory modl. Tasi (11) dvlopd a productio ad shipmt modl for a systm that icorporats larig ffct ad dtrioratig itms ad to driv a optimal joit total cost from th itgratd prspctiv of oth vdor ad uyr. Uthayaumar t al. (1) proposd a modl that itgrats th sigl vdor sigl uyr prolm with ordrprocssig cost rductio ad procss ma. ad tim may ifluc customr srvic ad impact ivtory costs. s th Japas xampl of just-i-tim-productio has show cosqutly rducig lad tims may icras productivity ad improv th comptitiv positio of th compay (s also Trsi t al. 1995). I most of rsahrs [ uo-el-ta t al. () Elwal t al. (6) Hadly t al. (196) Motgomry t al. (197) Posr t al. (197)Vijaya (7)] dalig with ivtory prolms ithr usig dtrmiistic or proailistic modls Th classical ivtory modls oft assum lad tim as a giv paramtr or a radom varial which is ot sujct to cotrol. Traditioally th lad tim of ivtory modl is hypothsizd as ow or with crtai proaility distriutio which thrfor is ot sujct to cotrol. But i may practical situatios lad tim ca rducd y a additioal crashig cost. That is it is cotrollal. I fact Trsi (198) thought that th lad tim usually cosists of th followig compots: ordr prparatio ordr trasit supplir lad tim dlivry tim ad st up tim. I may practical situatios lad tim ca rducd y a addd crashig cost i othr words it is cotrollal. By shortig th lad tim w ca lowr th safty stoc; rduc th loss causd y stoc out. crasig lad tim lads to th lowr safty stoc rductio of th loss sals causd y stoc out improvig th customr srvic lvl ad icrasig th comptitiv aility i usiss. 61

6 Vijayashr ad Uthayaumar otrollig ivtory is a procss ad a mthod of total ivtory maagmt. Tim-asd comptitio focuss o th rductio of ovrall systm rspos tim ad ivtory lad tim rductio has o of favourit topics for oth rsahrs ad practitiors Pa t al. (5). iao t al. (199) first prstd a stochastic ivtory modl with lad tim ig th varial. B-aya t al. (199) modifid iao t al. (199) y icludig oth lad tim ad ordr quatity as dcisio varials. Ouyag t al. (1996) xtdd B-aya t al. (199) modl y allowig shortags ad tratd th stocout tratd of acordrs ad lost sals. Ouyag t al. (1998) dvlopd a miimax distriutio fr procdur for mixd ivtory modl with varial lad tim. Pa t al. (1) modifid Ouyag t al. modl (1996) y cosidrig ac-ordr discout. Pa t al. (a) assumd th crash cost is a fuctio of oth th ordr quatity ad th rducd lad tim ad th stalishd ivtory modls with fixd ad varial lad tim crash cost. Pa t al. (5) ivstigatd a itgratd ivtory systm i which shortag is allowd ad oth lad tim ad acordrig ar gotial. hag t al. (6) proposd itgratd vdor-uyr cooprativ ivtory modls with cotrollal lad tim ad ordrig cost rductio. Vijayashr t al. (1) dvlopd a itgratd ivtory modl with cotrollal lad tim ad stup cost rductio for oth o-dfctiv itms. Priya t al. (1) dvlopd mathmatical modllig for EO ivtory systm with advac paymt ad fuzzy paramtrs. uality has highly mphasizd i modr productio/ivtory maagmt systms. lso it has vidcd that th succss of Just-I-Tim (JIT) productio is partly asd o th lif that quality is a cotrollal factor which ca improvd through various fforts such as worr traiig ad spcializd quipmt acquisitio. I th classical ivtory modl it is implicitly assumd that th quality lvl is fixd at a optimal lvl i.. all itms ar assumd to hav prfct quality. Howvr i th ral productio viromt it ca oft osrvd that thr ar dfctiv itms ig producd du to imprfct productio procsss. Th dfctiv itms must rjctd rpaird rword or if thy hav rachd th customr rfudd. I all cass sustatial costs ar icurrd. Thrfor for th systm with a imprfct productio procss th maagr may cosidr ivstig capital o quality improvmt so as to rduc th qualityrlatd costs. I th ivtory litratur Portus (1985) first itroducd th cocpt ad dvlopd a framwor for ivstig i rducig EO modl st-up cost. Th Ouyag t al. () ivstigatd th ifluc of ordrig cost rductio o modifid cotiuous rviw ivtory systms ivolvig varial lad tim with partial acordrs. Hog t al. (1995) prstd a modl icludig a udgt costrait ad othr typs of cotiuous fuctios for quality hacmt ad stup cost rductio. Ouyag t al. () ivstigatd th impact of quality improvmt o th modifid lot siz rordr poit modls ivolvig varial lad tim ad partial acordrs. Ouyag t al. () xtdd Ouyag t al. () modl y ivstig i procss quality improvmt ad stup cost rductio simultaously. atr may rsahrs [ Billgto (1987) Kim t al. (199) adurai t al. (1) oats (1996)] dvlopd EP modls with ordrig/stup cost rductio. Uthayaumar t al. (1) dvlopd supply chai modl with varial lad tim udr crdit policy. Th rlatio tw th quality ad ivtory rductio is critical for oth practitiors ad acadmics caus umrous modr productio systms advocat rductio i ivtory ad improvmt i quality. For xampl Voss (1987) claims that just-i-tim productio systms lad to icrasd quality ad rducd ivtory. I additio Kr t al. (199) shows that thr xists a gativ rlatioship tw ivtory ad quality asd o mpirical rsults. Vijayashr t al. (1) dvlopd a two-stag supply chai modl with sllig pric dpdt dmad ad ivstmt for quality improvmt. Ouyag t al. (7) dvlopd a itgratd 6

7 It J upply Opr Maag (IJOM) vdor-uyr ivtory modl with quality improvmt ad lad tim rductio. Yag t al. () dvlopd a itgratd ivtory modl ivolvig dtrmiistic varial lad tim ad quality improvmt ivstmt. W assum that crashig cost is liar fuctio ad lad tim is ws. Today s supply chai viromt rquirs a w spirit of coopratio tw th sigl-uyr ad th sigl-vdor. W cosidr itgratd ivtory modl with cotrollal lad tim ivolvig ivstmt for quality improvmt i supply chai systm. Th lad tim is idtical for all uyrs as wll as it ca shortd y payig a additioal crashig cost which is xpotially fuctio of lad tim. d lad tim xprssd i ws. Th ojctiv of this papr is to fid out a optimal ivtory stratgy that ca miimiz th valu of th itgratd total rlvat cost for oth th sigl vdor ad th sigl uyr. Fially a umrical xampl is prstd to illustrat th proposd modl. Th study also tas accouts of th followig aspcts: I sctio th fudamtal otatios ad assumptios of this study is providd. ctio dscris th modl dvlopmt. I sctio 5 a fficit algorithm is dvlopd to otai th optimal solutio. umrical xampl is providd i sctio 6 to illustrat th rsults. Fially coclusios ar show ad suggstios for futur rsah ar giv i ctio 7.. Notatios ad assumptios First of all th followig otatios ad assumptios ar usd throughout this papr to dvlop th proposd modl..1. Notatios To stalish th proposd modl th followig otatios ar usd: Buyr s xpctd dmad rat i uits pr uit tim P Vdor s productio rat i uits pr uit tim P Buyr s ordr quatity i uits Buyr ordrig cost pr ordr Vdor s stup cost pr stup Normal duratio to arriv th itms i uyr ivtoris s c v c r s Miimum duratio to arriv th itms i uyr ivtoris gth of lad tim Uit productio cost paid y th vdor Uit puhas cost paid y th uyr Th umr of shipmts i which th product is dlivrd from th vdor to th uyr i o productio cycl a positiv itgr a dcisio varial. ual ivtory holdig cost pr dollar ivstd i stocs Vdor uit dfctiv cost pr dfctiv itm. Proaility of th vdor s productio procss that ca go out-of-cotrol. Origial proaility of th vdor s productio procss that ca go out-of-cotrol. I Vdor s capital ivstmts rquir for rducig th out of cotrol proaility form to. i Vdor s fractioal opportuity cost of capital pr uit tim. TR Total Rlvat cost for th sigl vdor ad sigl th uyr. 6

8 Vijayashr ad Uthayaumar.. ssumptios To dvlop th proposd modl w adopt th followig assumptios: 1. Th systm cosists of a sigl vdor ad a sigl uyr for a sigl product i this modl ad th ivtory systm dals with oly o typ itm.. Th uyr ordrs a lot of siz ad th vdor maufacturs with a fiit productio rat P P at o st-up ut ship quatity to th uyr ovr tims. Th vdor icurs a st-up cost for ach productio ru ad th uyr icurs a ordrig cost for ach ordr of quatity.. Thr is vdor ad uyr for a sigl product i this modl.. Th dmad X durig lad tim follows a ormal distriutio with ma ad stadard dviatio. 5. Th ivtory is cotiuously rviwd. Th uyr placs th ordr wh th o had ivtory rachs th rordr poit R. 6. Th uyr placs th ordr wh th ivtory positio rachs th rordr poit R. Th rordr poit R = th xpctd dmad durig lad tim + safty stoc that is R whr is a safty factor ad is th stadard dviatio. 7. Th xtra cost icurrd y th vdor will trasfrrd to th uyr if shortd lad tim is rqustd. 8. If th uyr is ot agr to add xtra cost to cotrol th lad tim h should otai his itms at xactly ormal lad tim ( s ) ad crashig cost is zro. Hr th uyr addd crashig cost to cotrol th lad tim. Thrfor th uyr lad tim should withi this itrval s that is s. 9. Th crashig costs wr osrvd to grow with lad tim y a proportio which ca approximatd y a xpotially fuctio of lad tim. Thrfor th lad-tim crashig cost pr ordr R () is assumd to a xpotially fuctio of ad is dfid as if s R( ) whr is a positiv costat ad / if s miimum ad th ormal lad tims rspctivly. rprsts th 1. Th rlatioship tw lot siz ad quality is formulatd as follows: whil vdor is producig a lot th procss ca go out of cotrol with a giv proaility ach tim aothr uit is producd. Th procss is assumd to i cotrol i th giig of th productio procss. Oc out of cotrol th procss producs dfctiv itms ad cotius to do so util th tir lot is producd. (This assumptio is i li with Portus (1986)). 11. Th out-of-cotrol proaility is a dcisio varial ad is illustratd y a logarithmic ivstmt fuctio. Th quality improvmt ad capital ivstmt is illustratd y q ql for whr is th currt proaility that th productio procss ca go out of cotrol ad q 1 with maig th ptag dcras i pr dollar icras i q. Th applicatio of th logarithmic fuctio o capital ivstmt ad quality improvmt has proposd y may authors for xampl Portus (1986); Hog t al. (1995); Ouyag t al. () Yag t al. () ad Ouyag t al. (6). s 6

9 It J upply Opr Maag (IJOM). Modl dvlopmt.1. Itgratd Total ost IT Th joit total xpctd cost pr uit tim i Pa t al. () is th sum of th followig lmts Ordrig cost pr uit tim = Buyr s holdig cost pr uit tim is = ad tim crashig cost pr uit tim= R Vdor stup cost pr yar = Vdor s holdig cost pr uit tim: vdor s avrag ivtory is valuatd as th diff of th vdor s accumulatd ivtory ad th uyr s accumulatd ivtory (s Figur 1). That is ( 1) (1... ( 1)) P P 1 1 P P o th vdor s holdig cost pr uit tim is v 1 1 P P ccordigly th itgratd total cost pr uit tim for th sigl vdor ad th sigl uyr itgratd ivtory systm is giv y IT v P P (1) R( ) r 1 1 c c fctiv itm rwor cost pr uit tim: th xpctd umr of dfctiv itms i a ru of siz with a giv proaility of that th procss ca go out of cotrol is (s s Portus(1986) for dtail drivatio). Thus th dfctiv cost pr uit tim is giv y Hc th total cost icorporatig th dfctiv cost pr yar ca rprstd y T s () IT 65

10 Vijayashr ad Uthayaumar Figur 1. Th ivtory pattr for th uyr ad vdor 66

11 It J upply Opr Maag (IJOM).. Ivstmt for quality improvmt Basd o quatio () this study is a attmpt to study th ffct of ivstmt o quality improvmt. osqutly th ojctiv of th itgratd modl is to miimiz th sum of th ordrig/stup cost holdig cost quality improvmt ad crashig cost y simultaously dtrmiig th optimal valus of ad sujct to th costrait that. Thus th total rlvat cost pr yar is TR T iq l s R( ) r 1 1 c c v iq l P P () whr i is th fractioal opportuity cost of capital pr uit tim. Thrfor th prolm udr study ca formulatd as th followig oliar programmig modl. Miimiz TR R( ) 1 1 v s iq l P P () ujct to I ordr to fid th miimum cost for this o-liar programmig prolm igor th costrait for th momt ad miimiz th total rlvat cost fuctio ovr ad with classical optimizatio tchiqus y taig th first partial drivativs of TR with rspct to ad s as follows: TR r R 1 1 cv c s (5) P P TR s iq (6) 1 TR 1 (7) By sttig Eq. (5) ad (6) qual to zro for a giv valu of s w otai 67

12 Vijayashr ad Uthayaumar (8) r cv c 1 1 s P P iq (9) s Thortically for fixd from (8) & (9) w ca otai th valus of. Morovr it was foud that th scod ordr sufficit coditios ar satisfid as follows. For fixd th Hssia TR is positiv dfiit ad. Th proof is show i th appdix. matrix Th susqutly algorithm is proposd to fid th optimal valu of ordr quatity procss quality ad tim umr of dlivris.. lgorithm tp1. t 1. ic is itgr ad st. tp. Prform stp (.1) ad (.6) for all itgr valus of i this itrval s..1. Us to comput from quatio (8)... Us ad to comput from quatio (9)... Rpat stps (.1)-(.) util o chag occurs i th valus of ad. ot y ad rspctivly... If th th solutio is optimal for giv s. ot th solutio y..5. If th ta ad utiliz quatio (8) to dtrmi w similar to th o i (.1). Th rsult is dotd y. TR y puttig i quatio ().6. omput th corrspodig tp. t TR solutio for fixd. tp. t 1 miimum of rpat stps ()-() gt tp5. IfTR TR 1 TR th TR. is a optimal ; go to stp othrwis go to stp 6. tp6. TR TR 1 th is optimal solutios. 68

13 It J upply Opr Maag (IJOM) 5. Numrical xampl I this sctio a umrical xampl is giv to illustrat th aov solutio procdur. W cosidr th umrical xampl with th followig data 6 uit/yar P uit/yar 1 / uit $ / ordr $15 / stup r. c 7 / uit 1.1 c if 6 7 uit/w rashig ost R i.. whr 5miimum lad tim if w ormal lad tim s 1 w. pplyig th solutio procdur of th proposd algorithm th computatioal rsults ar dmostratd i Tal (1). Th optimal solutios from Tal (1) ca rad off as optimal lad tim v ws Ordr quatity 19 uits umr of shipmts procss quality. 116 ad corrspodig miimum total rlvat cost TR 657. Graphical rprstatios ar show Figurs () ad () =1 θ Tal 1. Optimal solutios for diffrt valus of lad tim whr =5 1 6 R = TR θ TR = θ TR Tal 1. otiud whr =5 1 6 R = θ TR =5 θ TR R =6 θ TR

14 Total Rlvat ost (TR) Total Rlvat ost (TR) Vijayashr ad Uthayaumar Ordr uatity () Numr of hipmts () Figur. Graphical rprstatios for optimal solutio for TR wh = = oclusio Ordr uatity () Figur. Graphical rprstatio of th optimal solutio. I this papr a ivtory modl for two-stag supply chai is ivstigatd. supply chai with sigl vdor ad sigl uyr is cosidrd. I this study w cosidr itgratd ivtory modl with cotrollal lad tim ivolvig ivstmt for quality improvmt i supply chai systm. ad tim is a importat lmt i ay ivtory maagmt systm. Idustrial uyrs oft call it as lad tim. Firm ca short dlivry tims y storig ivtory or havig xcss capacity. I may practical situatios lad tim ca rducd at a addd crashig cost; i othr words it is cotrollal. Hr th uyrs lad tim ca shortd y payig a additioal crashig cost which is xpotially fuctio of lad tim. I our modl th capital ivstmt i quality improvmt is assumd to a logarithmic fuctio. Th mai cotriutio of this proposd modl is a fficit itrativ algorithm has 66

15 It J upply Opr Maag (IJOM) dvlopd to miimiz th total rlvat cost for th sigl vdor ad sigl uyr itgratd systm with ivstmt for quality improvmt y simultaously optimizig th optimal ordr quatity lad tim procss quality ad umr of shipmts from th sigl vdor to th sigl uyr i a productio cycl. solutio procdur is dvlopd to fid th optimal solutio. computr cod usig th softwar Matla is dvlopd to driv th optimal solutio of th systm. This modl is usful particularly for itgratd ivtory systms whr th vdor ad th uyr form a stratgic alliac for profit sharig. Th umrical xampls ar giv to illustrat th fit of coordiatio tw sigl vdor ad sigl uyr. Graphical rprstatio is also prstd to illustrat th proposd modl. W propos a asy algorithm for dtrmiig th optimal solutios. I additio a umrical xampl is prstd to illustrat th proposd modl. Thr ar svral xtsios of this wor that could costitut futur rsah rlatd to this fild. O immdiat proal xtsio could to discuss th ffct of shortag. othr possil xtsio of this wor may coductd y cosidrig th vdor s provisio of a prmissil dlay i paymts i this itgratd ivtory modl. lso w ca cosidr multi-chlo supply chais such as; sigl uyr-multipl vdor multipl uyr-sigl vdor ad multipl uyr-multipl vdor systm is also proposd for th futur rsah. cowldgmt Th authors would li to covy thir hartflt thas th ditors ad th rviwrs for thir valual suggstios to improv th clarity th papr. Th first author rsah wor is supportd y T INPIRE Fllowship Miistry of cic ad Tchology Govrmt of Idia udr th grat o. T/INPIRE Fllowship/11/1 datd ad UG P partmt of Mathmatics Gadhigram Rural Istitut md Uivrsity Gadhigram Tamiladu Idia. Rfs uo-el-ta M. O. Frgay H.. ad Elwal M. F. (). Proailistic multi-itm ivtory modl with varyig ordr cost udr two rstrictios. Itratioal Joural of Productio Ecoomics Vol. 8 pp. -1. masaa K. (). Nw JIT: w maagmt tchology pricipl at Toyota. Itratioal Joural of Productio Ecoomics Vol. 8 pp adurai K Uthayaumar R. (1). Ordrig cost rductio i proailistic ivtory modl with cotrollal lad tim ad a srvic lvl. Itratioal Joural of Maagmt cic ad Egirig Maagmt Vol. 5 pp.-1. Barj. (1986). joit coomic-lot-siz modl for puhasr ad vdor. cisio cics Vol.17 pp B-aya M. Hariga M. (). Itgratd sigl vdor sigl uyr modl with stochastic dmad ad varial lad tim. Itratioal Joural of Productio Ecoomics Vol. 9 pp B-aya M. Raouf. (199). Ivtory modls ivolvig lad tim as a dcisio varial. Joural of th Opratioal Rsah ocity Vol. 5 pp

16 Vijayashr ad Uthayaumar Billgto P. (1987) Th classic coomic productio quatity modl with stup cost as a fuctio of capital xpditur. cisio cics Vol.18 pp.5-. Byla. () omptitiv ad cooprativ policis for th vdor uyr systm. Itratioal Joural of Productio Ecoomics Vol pp hag H.. Ouyag. Y. Wu ad K.. Ho. H. (6). Itgratd vdor uyr cooprativ ivtory modls with cotrollal lad tim ad ordrig cost rductio. Europa Joural of Opratioal Rsah Vol. 17 pp hug. J. ad W H. M. (7). Optimizig th coomic lot siz of a thr-stag supply chai with acordrig drivd without drivativs Europa Joural of Opratioal Rsah Vol. 18 pp. 9-9 oats E. R. (1996). Maufacturig stup cost rductio. omputrs ad Idustrial Egirig Vol. 1 pp Elwal M. F. (6). Tratmt som of th proailistic ivtory systm Ph.. srtio Faculty of cic Tata Uivrsity. Goyal. K. (1988). joit coomic-lot-siz modl for puhasr ad vdor: a commt cisio cics Vol. 19 pp Goyal. K. (1976). itgratd ivtory modl for a sigl supplir-sigl customr prolm Itratioal Joural of Productio Rsah Vol.15 pp Goyal. K. riivasa G. (199). Th idividually rsposil ad ratioal dcisio approach to coomic lot sizs for o vdor ad may puhasrs: a commt cisio cics Vol. pp Ha. Kim.. (1997). Implmtatio of JIT puhasig: a itgratd approach Productio Plaig ad otrol Vol. 8 pp Hadly G. Whiti T. M. (196). alysis of ivtory systm Prtic Hall Ic. Eglwood liffs Nw Jrsy. Hog J.. Hayya J.. (1995). Joit ivstmt i quality improvmt ad stup rductio omputrs ad Opratios Rsah Vol. pp Hoqu M.. Goyal. K. (6). huristic solutio procdur for a itgratd ivtory systm udr cotrollal lad-tim with qual or uqual siz atch shipmts tw a vdor ad a uyr. Itratioal Joural of Productio Ecoomics Vol. 1 pp Hoqu M.. Goyal. K. (). optimal policy for a sigl-vdor sigl-uyr itgratd productio-ivtory systm with capacity costrait of th trasport quipmt Itratioal Joural of Productio Ecoomics Vol. 65 pp Huag. K. (). optimal policy for a sigl-vdor sigl uyr itgratd productioivtory prolm with procss urliaility cosidratio Itratioal Joural of Productio Ecoomics Vol. 91 pp

17 It J upply Opr Maag (IJOM) Huag. K. Tsai. W. Wu J.. ad hug K. J. (1). optimal itgratd vdoruyr ivtory policy udr coditios of ordr-procssig tim rductio ad prmissil dlay i paymts Itratioal Joural of Productio Ecoomics Vol. 18 pp Jha J. K har K. (9). sigl-vdor sigl-uyr productio-ivtory modl with cotrollal lad tim ad srvic lvl costrait for dcayig itms Itratioal Joural of Productio Rsah Vol. 7 pp Kr. Muhopadhyay T. (199). Impact of lctroic data ithag tchology o quality improvmt ad ivtory rductio programs: a fild study. Itratioal Joural of Productio Ecoomics Vol. 8 pp Kim Hayya J Hog J. (199). tup rductio i th coomic productio quatity. cisio cics Vol. pp iao. J. hyu. H. (1991). aalytical dtrmiatio of lad tim with ormal dmad. Itratioal Joural of Opratios ad Productio Maagmt Vol iao. J. hyu. H. (199). aalytical dtrmiatio of lad tim with ormal dmad. Martiich J.. (1997). Productio ad Opratios Maagmt Wily Nw Yor. Motgomry.. Bazaraa M.. ad Kswai. K. (197). Ivtory modls with a mixtur of acordrs ad lost sals Naval Rsah ogistics uartrly Vol. pp Moo I. hoi. (1998). ot o lad tim ad distriutioal assumptios i cotiuous rviw ivtory modls. omputrs ad Opratios Rsah Vol. 5 pp Niuwhuys I. V. Vadal N. (6). Th impact of dlivry lot splittig o dlivry rliaility i a two-stag supply chai Itratioal Joural of Productio Ecoomics Vol. 1 pp Ouyag. Y Wu K.. ad Ho. H. (6). Th sigl-vdor sigl-uyr itgratd ivtory prolm with quality improvmt ad lad tim rductio miimax distriutio-fr approach. sia-pacific Joural of Opratios Rsah Vol. pp. 7-. Ouyag. Y. hag H.. (). Impact of ivstig i quality improvmt o ( r ) modl ivolvig imprfct productio procss Productio Plaig ad otrol Vol. 11 pp Ouyag. Y. hag H.. (). ot siz rordr poit ivtory modl with cotrollal lad tim ad stup cost rductio. Itratioal Joural of ystm cics Vol. pp Ouyag. Y. h. K. ad hag H.. (1999). ad tim ad ordrig cost rductio i cotiuous rviw ivtory systms with partial acordrs. Joural of th Opratioal Rsah ocity Vol. 5 pp Ouyag. Y. Wu K.. (1998). miimax distriutio fr procdur for mixd ivtory modl with varial lad tim. Itratioal Joural of Productio Ecoomics Vol. 56 pp

18 Vijayashr ad Uthayaumar Ouyag. Y. Wu K.. ad Ho. H. (7). itgratd vdor uyr modl with quality improvmt ad lad tim rductio. Itratioal Joural of Productio Ecoomics Vol. 18 pp Ouyag. Y. Wu K.. ad Ho. H. (). Itgratd vdor-uyr cooprativ modls with stochastic dmad i cotrollal lad tim. Itratioal Joural of Productio Ecoomics Vol. 9 pp Ouyag. Y. Yh N.. ad Wu K.. (1996). Mixtur ivtory modl with acordrs ad lost sals for varial lad tim. Joural of th Opratioal Rsah ocity Vol. 7 pp Pa. H. J. Hsiao Y.. (5). Itgratd ivtory modls with cotrollal lad tim ad acordr discout cosidratios. Itratioal Joural of Productio Ecoomics Vol. 9 9 pp Pa J.. H. Hsiao Y.. (1). Ivtory modls with ac-ordr discout ad varial lad tim. Itratioal Joural of ystms cic Vol. pp Pa J.. H. Hsiao Y.. ad. J. (a). Ivtory modls with fixd ad varial lad tim crash costs cosidratios. Joural of th Opratioal Rsah ocity Vol. 5 pp Pa J.. Yag J.. (). study of a itgratd ivtory with cotrollal lad tim. Itratioal Joural of Productio Rsah Vol. pp Parlar M. Wg Z. K. (1997). sigig a firm s coordiatd maufacturig ad supply dcisios with short product lif cycls Maagmt cic Vol. pp Portus E.. (1985). Ivstig i rducd stups i th EO modl. Maagmt cics Vol.1 pp Portus E. (1986). Optimal lot sizig procss quality improvmt ad stup cost rductio. Opratios Rsah Vol. pp Posr M. J. M. Yasoui B. (197). class of ivtory modls with customrs impatic Naval Rsah ogistics uartrly Vol. 19 pp Priya. Palaivl M. ad Uthayaumar R. (1)..Mathmatical modlig for EO ivtory systm with advac paymt ad fuzzy Paramtrs. Itratioal Joural of upply ad Opratios Maagmt Vol.1 pp i Y. Tag H. ad Guo. (7). hal coordiatio ad volum discouts with pricssitiv dmad Itratioal Joural of Productio Ecoomics Vol. 15 pp. -5. Rau H. Ouyag B.. (8). optimal atch siz for itgratd productio ivtory policy i a supply chai Europa Joural of Opratioal Rsah Vol. 185 pp armah. P. charya. ad Goyal. K. (6). Buyr-vdor coordiatio modls i supply chai maagmt Europa Joural of Opratioal Rsah Vol. 175 pp

19 It J upply Opr Maag (IJOM) ilvr E.. Py. F. ad Ptrso R. (1998). Ivtory Maagmt ad Plaig ad schdulig thirdd. WilyNw Yor. Productio imchi-vi. P. Kamisy ad imchi-vi E. (). sigig ad Maagig th upply hai Irwi McGraw-Hill: Nw Yor. Trsi R. J. (198). Pricipl of ivtory ad matrials maagmt North-Hollad Nw Yor. Trsi R. J. (199). Pricipl of Ivtory ad Matrial Maagmt. th Editio. Prtic-Hall U. Trsi R. J. Hummigird E. (1995).. ad-tim rductio th sah for comptitiv advatag. Itratioal Joural of Opratios ad Productio Maagmt Vol. 15 pp Uthayaumar R. Ramswari M. (1). itgratd ivtory modl for a sigl vdor ad sigl uyr with ordr-procssig cost rductio ad procss ma Itratioal Joural of Productio Rsah Vol. 5 pp Uthayaumar R. Ramswari. (1). upply chai modl with varial lad tim udr crdit policy Th Itratioal Joural of dvacd Maufacturig Tchology Vol. 6 pp Vijaya T. ad Kumara M. (1). Ivtory modls with a mixtur of acordrs ad lost sals udr fuzzy cost Europa Joural of Opratioal Rsah Vol. 189 pp Vijayashr M. Uthayaumar R. (1). two stag supply chai modl with sllig pric dpdt dmad ad ivstmt for quality improvmt. sia Pacific Joural of Mathmatics Vol.1 pp Vijayashr M. Uthayaumar R. (1). itgratd ivtory modl with cotrollal lad tim ad stup cost rductio for dfctiv ad o-dfctiv itm. Itratioal Joural of upply ad Opratios Maagmt Vol.1 pp Villa. (1). Itroducig som supply chai maagmt prolm. Itratioal Joural of Productio Ecoomics Vol. 7 pp. 1. Viswaatha. (1998). Optimal stratgy for th itgratd vdor uyr ivtory modl. Europa Joural of Opratioal Rsah Vol. 15 pp. 8. Wg Z. K. (1997). Pricig ad ordrig stratgis i maufacturig ad distriutio alliacs IIE Trasactios Vol. 9 pp Yag P.. W H. M. () Ecoomic ordrig policy of dtrioratd vdor ad uyr: itgratd approach Productio Plaig ad cotrol vol. 11 pp.1-7. Yag J.. Pa J.. H. (). Just-i-tim puhasig: a itgratd ivtory modl ivolvig dtrmiistic varial lad tim ad quality improvmt ivstmt Itratioal Joural Productio Rsah Vol. pp

20 Vijayashr ad Uthayaumar 66 ppdix W wat to prov th Hssia Matrix of IT at poit for fixd is positiv dfiit. W first otai th Hssia matrix H as follows TR TR TR TR TR TR TR TR TR H Whr TR iq TR TR TR TR s TR TR TR TR W procd y valuatig th pricipal mior dtrmiat of th Hssia matrix H at poit. Th first pricipal mior dtrmiat of H th coms. 11 H

21 It J upply Opr Maag (IJOM) 67 H ic c Thrfor. H. TR TR TR TR H TR TR TR TR TR.. TR TR TR TR TR..

22 Vijayashr ad Uthayaumar 68 iq - s iq + s s iq - iq + s s If iq s iq Th

23 It J upply Opr Maag (IJOM) 69 iq iq s Thrfor H. Hc for fixd th Hssia matrix is positiv ad TR is covx with rspct to.

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