An Inventory Model for Deteriorating Items with Quadratic Demand and Partial Backlogging

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Brsh Journl of Appl Sn & hnology (): - 0 SCIECEDOMAI nrnonl wwwsnomnorg An Invnory Mol for Drorng Ims wh Qur Dmn n Prl Bkloggng R Bgum S K Shu n R R Shoo Dprmn of

Frury 0 Ap 5 h Aprl 0 Onln Ry 9 h Aprl 0 ABSRAC Ams: h sop of h mol ls n s pplly n h mngmn nvnors of m-qur mn I s lso sn h lrg pl of goos sply n suprmrk wll mov h

n llowng shorgs n nvnory Suy Dsgn: hs ppr prsns n nvnory mol for rorng ms wh qur mn In h mol shorgs r llow n prlly korr h kloggng r s vrl n pnn on h wng m for h nx

h os funon s slsh o nsur h xsn of unqu pon of mnmum Conluson: h propos mol n xn n svrl wys For nsn w my xn h mn funon o sohs fluung mn prns or sok-pnn mn r Fnlly w

1 Brsh Journl of Appl Sn & hnology (): - 0 SCIECEDOMAI nrnonl wwwsnomnorg An Invnory Mol for Drorng Ims wh Qur Dmn n Prl Bkloggng R Bgum S K Shu n R R Shoo Dprmn of Mhms Pmn Collg of Engg Rourkl Osh In Dprmn of Sss Smlpur Unvrsy Jyovhr Burl Smlpur In Dprmn of Physs Synrgy Insu of hnology Bhunswr-750 In Rsrh Arl Rv h Frury 0 Ap 5 h Aprl 0 Onln Ry 9 h Aprl 0 ABSRAC Ams: h sop of h mol ls n s pplly n h mngmn nvnors of m-qur mn I s lso sn h lrg pl of goos sply n suprmrk wll mov h usomr o uy mor So h prsn of nvnory hs movonl ff on h popl roun Also hr my osonl shorgs n nvnory u o mny rsons hrfor w vlop n EOQ mol for h nvnory of rorng m kng mn r n llowng shorgs n nvnory Suy Dsgn: hs ppr prsns n nvnory mol for rorng ms wh qur mn In h mol shorgs r llow n prlly korr h kloggng r s vrl n pnn on h wng m for h nx rplnshmn A numrl xmpl s kn o llusr h mol n h snsvy nlyss s lso su Mhoology: Our purpos s o vs mhml mol on nvnory mngmn kng ll hs fors no onsron Rsuls: Convxy onon of h os funon s slsh o nsur h xsn of unqu pon of mnmum Conluson: h propos mol n xn n svrl wys For nsn w my xn h mn funon o sohs fluung mn prns or sok-pnn mn r Fnlly w oul xn h mol o norpor som mor rls furs suh s quny souns prmssl ly n pymns m vlu of mony n nflon Kywors: Invnory; qur mn; rorng ms; prl kloggng Corrsponng uhor: Eml: gum_rhn@rffmlom;

2 Brsh Journl of Appl Sn & hnology (): - 0 IRODUCIO h ff of roron of physl goos nno srgr n mny nvnory sysms In ly lf h roron of goos s ommon phnomnon Droron s fn s y spolg mg vporon osolsn plfrg loss of uly or mrgnl vlu of ommoy h rsuls n rsng usfulnss from h orgnl on For xmpl phrmuls hmls foosuff ror sgnfnly h roron r of nvnory n sok urng h sorg pro onsus n mporn for whh hs r h rsrhrs Whn (957) s h frs uhor who su n nvnory mol for fshon goos rorng h n of prsr sorg pro An xponnlly yng nvnory ws vlop y Ghr n Shrr (96) Emmons (968) slsh rplnshmn mol for rov nul gnrors Shh n Jswl (977) slsh n orr -lvl-nvnory mol for prshl ms wh onsn r of roron h roron ours s soon s h rlr rvs h ommos h hv ssum n ll nvnory mols for rorng ms Bu n rl lf suon mos of goos woul hv spn of mnnng quly or h orgnl onon Durng h pro hr ws no ourrn of roron hs ms r fsh fru m vgls lohols loo n gsoln hs phnomnon s rm s nonnsnnous roron Jyrmn n Sugpry (008) vlop n ELSP for nonnsnnous rorng ms usng pr soun In pr som usomrs woul lk o w for kloggng urng h shorg pro u h ohrs woul no Consqunly h opporuny os u o los sls shoul onsr n h molng Mny rsrhrs (Murshwr 988; Goyl l 99; Hrg 996; Chkrr n Chuhur 997; Hrg n Alyn 997) ssum h shorgs r omplly klogg Chng n Dy (999) rgu h h kloggng r shoul pnn on h lngh of h wng m hy wr h frs o gv fnon for h m-pnn prl kloggng r W (995) vlop n rl n h fl of rorng ms wh shorgs hs rvl h onom orr quny wh known mrk mn r Rsrhrs lk Sn (00) n Roy l (0 0 0) onsr h s of prl kloggng rs n hr mols Howvr n som nvnory sysm for mny soks suh s fshonl ommos h lngh of h wng m for h nx rplnshmn oms mn for for rmnng whhr h kloggng wll p or no h longr h wng m h smllr woul h kloggng r hrfor h kloggng r s vrl n s pnn on h wng m for h nx rplnshmn In hs ppr h kloggng r ws ssum o fx fron of mn r urng h shorg pro Goyl n Gr (00) prov l rvw of rorng nvnory lrurs hy n: h ssumpon of onsn mn r s no lwys ppll o mny nvnory ms lk fshonl lohs lron goos As hy xprn fluuons n h mn r Mny prous xprn pro of rsng mn urng h growh phs of hr prou lf yl m-vryng mns wr frs onsr y Slvr n Ml (969) In hs mol hy slsh h Hurs soluon prour Mny rsrh rls y Rh ( ); D n Chuhur (986); Dv ( ); Hrg (99 ); Chung n ng (99); Km (995); Jln n Chuhur (999); Ln l (000) nlyz lnr m-vryng mn Aggrwl n Bhr-Kshn (99); Hollr n Mk (98); Hrg n Bnkhrouf (99); W (995); Jln n Chuhur (999) vlop nvnory mols n whh xponnl m-vryng mn hs n kn Wh h progrss of m rsrhrs vlop nvnory mols wh m-pnn mn rs Lr Ghosh n Chuhur (00);

3 Brsh Journl of Appl Sn & hnology (): - 0 Khnr n Chuhur (00) slsh hr mols n whh qur m-vryng mn ws onsr Sn (00) onsr h s of mlorng ms n rorng ms n mul-m EOQ mol whr h mn s ng nflun y nrprss Rnly Bgum l (00) hs vlop n EOQ mol wh qur mn wh Wull sruon roron Durng h frs hr s mny mrkng rsrhrs osrv h n som rlr sysms lk suprmrk h mn of goos mgh nflun y h on-hn nvnory For xmpl Lvn l (97) pon ou h ms h prsn of nvnory hs movonl ff on h popl roun I s ommon lf h lrg pl of goos sply n suprmrk wll l h usomr o uy mor Dv n Pl (98) vlop n nvnory mol for rorng ms wh m-proporonl mn hs mol ws xn y Shn (98) o ovr h kloggng opon Donlson (977) suss h lssl no-shorg nvnory poly for lnr m-pnn mn for h frs m Sn (0 0 0) onsr h s of pr snsv mns n hs mols In h opnon of mny uhors n lrnv or prhps mor rls pproh s o onsr qur m-pnn of mn hs mn my rprsn ll yps of m-pnn pnng on h sgns of h prmrs of h m-qur mn funon Khnr l (00) onsr n EOQ mol wh sok n pr pnn mn r Sn n Chuhur (00) vlop prouon poly for rorng m wh m-pnn mn n shorgs h prsn work mps o mol h suon whr h mn r s onnuous funon of m n ms ror onsn r Hr shorgs r llow n h kloggng r s vrl n pnn on h wng m for h nx rplnshmn In h prsn ppr w hv xn h work of Shu l (007) y kng h mn r o qur funon of m h purpos of h prsn ppr s o gv nw pproh o h nvnory lrur or m-pnn mn prns Qur funon of m s h s rprsnon of lr growh (or ln) n h mn An nlyl soluon of h mol s suss n s llusr wh h hlp of numrl xmpl Snsvy of h opml soluon wh rsp o hng n ffrn prmr vlus s lso rr ou h mol ns wh sul onluson ASSUMPIOS AD OAIOS h ssumpons r s follows: h l m s zro Rplnshmn r s nfn h mn r () D ny m ' ' s gvn y ( ) 0 D( ) ( ) n r posv onsns n () whr s h nvnory lvl m Shorgs r llow For onvnn fron of mn s klogg If h wng m longr hn h kloggng r wll smllr L s ssum B() h fron whr s h wng m up o h nx rplnshmn W

4 Brsh Journl of Appl Sn & hnology (): - 0 onsr B ) ( whr s known s h kloggng prmr whh s posv onsn 5 s h lngh of m n whh h prou hs no roron ( frsh prou m) A onsn fron ( 0 ) of h on-hn nvnory rors fr hs pro n hr s no rplmn of h ror uns n r gvn onsns 6 s h lngh of m n whh h nvnory hs no shorgs h lngh of orr yl n h orr quny pr yl s gvn y n Q rspvly hus h son vrls r n Q ( s h nvnory lvl m 0 ) 7 ) oons roron ( ) hs roron ( ) ( ( s h nvnory lvl m ) prou hs shorg n whh h prou hs no s h nvnory lvl m ( ) A s h onsn orrng os pr orr s h onsn nvnory holng os pr un m s h onsn rorng os pr un s h onsn shorg os for klogg ms n whh h prou n whh h s h onsn los sl os pr un VC( ) s h ol rlvn nvnory os pr un m of nvnory sysm Q VC n s h opml vlus of lngh of m lngh of orr yl orr quny n ol rlvn nvnory os MAHEMAICAL DEVELOPME hs ppr s vlop y h onsron of h rplnshmn prolm of sngl nonnsnnous rorng m wh prl kloggng h nvnory mol runs s follows: L mx h uns of m rrv h nvnory sysm h gnnng of h yl h nvnory lvl rss o qur mn r urng h m nrvl 0 ] h [ nvnory lvl rss oh u o mn n roron ll oms zro n h nrvl h shorg nrvl kps o h n of h urrn orr yl h ol pross s rp h nvnory lvl ffrn nsns of m s shown n fgur 5

5 Brsh Journl of Appl Sn & hnology (): Fg Grphl prsnon of nvnory sysm As sr ov h nvnory lvl rss owng o qur mn r urng h m nrvl 0 h nvnory lvl s govrn y h followng ffrnl quon: 0 () hvng h ounry onon mx 0 h soluon of quon () s: mx 0 () h nvnory lvl rss no only u o h qur mn r u lso u o h roron urng h nrvl h nvnory lvl s govrn y h ffrnl quon: () hvng h ounry onon 0 Solng quon( ) for whh yls () Consrng onnuy of follows from quon ( ) n ( ) h mx

6 Brsh Journl of Appl Sn & hnology (): h mxmum nvnory lvl for h yl s gvn y mx (5) Susung quon ( 5) n quon () oms 0 (6) Durng h shorg nrvl h mn m s prlly klogg fron h ffrnl quon govrnng h moun of mn klogg s gvn y B (7) hvng h ounry onon 0 Solng quon ( 7) whh yls ln ln (8) Pung n quon ( 8) w on h mxmum moun of mn klogg pr yl s ln S (9) L Q h orr quny pr yl n s on from quon ( 5) n ( 9) s mx ln S Q (0) h ol rlvn os pr yl onsss of fv ffrn oss h ol rlvn nvnory os pr un m s h sum of ll oss pr orr yl hus h ol rlvn os pr un m s gvn y

7 Brsh Journl of Appl Sn & hnology (): sls ylu o los pr opporuny os Orrng os yl u o klog pr shorg os yl pr h roron os yl pr nvnory holng os VC OC SC DC HC A B D A 0 ln A () For onvnn 0 M 0 0 M R 0 P

8 Brsh Journl of Appl Sn & hnology (): hus quon () yls ln M P R A VC () h ol rlvn nvnory os pr un m whh s foun n quon ( ) s funon of h wo vrls n hrfor h ol rlvn os pr un m wll mnmum f 0 ) ( VC n 0 ) ( VC () From quon ( ) n ( ) w g h quon 0 () n

9 Brsh Journl of Appl Sn & hnology (): Log M R P A (5) UMERICAL EXAMPLE L A 05 n 5 0 n ppropr uns Solvng h non-lnr quons ( ) n (5); w on h opmum vlus of n r n 05 Susung h vlus of n n (0) n ( ) h opml orr quny pr yl Q n h mnmum ol rlvn os pr un m VC rspvly I s numrlly vrf h hs soluon ssfs h onvxy onon for VC whh s slsh n Appnx 5 SESIIVIY AALYSIS W now suy h ffs of hngs n h vlu of sysm prmrs A on h opml lngh of nvnory h opml lngh of orr yl h opml orr quny pr yl Q n h mnmum ol rlvn os pr un m VC of h Exmpl h snsvy nlyss s prform y hngng h of h prmr y % 5 0% %5%0% 50 n - % 50 kng on prmr m n kpng h rmnng prmrs unhng h rsuls r shown n l

10 Brsh Journl of Appl Sn & hnology (): - 0 l Eff of hngs n h prmrs of h nvnory Prmr % Chng n h prmr % Chng n Q VC A

11 Brsh Journl of Appl Sn & hnology (): - 0 l onnus On h ss of h rsuls shown n l h followng osrvons n m: ) ) ) Q rss whl h on rsuls show h vlu of A Bu Q n VC nrss wh nrs n h vlu of h prmr A VC s hghly snsv o hngs n A VC nrs wh nrs n h vlu of h prmr r morly snsv o hng n h rs wh nrs n h vlu of n Morovr s sn h n h vlu of Q VC nrss So wh nrs n prmr Q VC s nsnsv o hngs n n

12 Brsh Journl of Appl Sn & hnology (): - 0 ) Wh nrs n vlu of ; n rss whl Q VC Q r lss snsv o whr s VC s hghly snsv n ) Wh nrs n h prmr ; nrss n rss wh rs n prmr f) n Q nrs whl n h vlu of h prmr n g) Q rs whl VC Q rs whl Q rss Bu nrss VC VC rss wh nrs nrss wh nrs n h vlu of h prmr nrss n rss wh nrs n h vlu of h prmr VC r mor snsv o hngs n h vlu of 6 COCLUSIO In hs ppr n opml rplnshmn poly hs n onsr for non-nsnnous rorng ms wh qur mn Hr shorgs r llow n h kloggng r s vrl h kloggng r s pnn on h wng m for h nx rplnshmn A ompl r or onsn prl r ws us n mny sus o sr h kloggng r Bu s mor rls o ssum h kloggng r o m proporonl wh wng m of kloggng W hv onsr hr h kloggng r s whh sms o r from h xponnl kloggng r I s ommon lf h lrg pl of goos sply n suprmrk wll mov h usomr o uy mor So h prsn of nvnory hs movonl ff on h popl roun h qur m-pnn of mn of h form D( ) hs yp of mn hs r rprsnon of m-vryng mrk If w ompr h ohr wo yps of m-pnns lk lnr n xponnl s sn h lnr m-pnn mn ls o unform hng n h rl mrk A h sm nsn xponnl m-pnn mn lso sms o unrls us n xponnl r of hng s vry hgh n s n ou h h mrk mn of ny prou my unrgo hgh r of hng lk xponnl funon hus h lrnv n proly mor rls pproh s o onsr h qur m-pnn of mn whh my rprsn ll yps of m-pnn pnng on h sgns of h prmrs of h m-qur mn funon D( ) D( ) D( ) W hv n If o whh s posv only f n r of oppos sgn In hs s h mn r grully rss o mxmum n hn grully rss hs yp of mn s qu ppropr for ssonl prous lk wnr osms As h sson progrss h mn r gns o rs ns hghs lvl n h ml n hn gon wy owrs n of h sson

13 Brsh Journl of Appl Sn & hnology (): - 0 If oh n r ngv h mn r D() rss sng r whh w my ll s lr ln n mn hs usully hppns o h spr prs of n osol rrf mol or h mroompur hp of hgh hnology prous whh s ng susu y nohr If oh n r posv h mn r D() nrss nrsng r whh w my ll s lr growh n mn whh s mnly vsl n nw ompur hps spr prs of nw ropln hus w my hv ffrn yps of rls mn prns from h mn r D( ) pnng on h sgns of n Dpnng on h vlus of n w hv ll h yps of growh lk posv ngv lr ln lr growh whh mgh sul o hs mol hs vng of h m-qur mn hs mov h rsrhrs o op n h prsn mol hrfor of h vw h qur m-pnn mn s mor rls hn lnr n xponnl mpnn mn h propos mol n xn n svrl wys I us n nvnory onrol of rn non-nsnnous rorng ms suh s foo ms lron omponns fshonl ommos n ohrs For nsn w my xn h mn funon o sohs fluung mn prns or sok-pnn mn r Fnlly w oul xn h mol o norpor som mor rls furs suh s quny souns prmssl ly n pymns m vlu of mony fn r of rplnshmn nflon n prols mn ACKOWLEDGEMES h uhors r n o h Eor n h wo nonymous rvwrs for hr l ommns n vlul suggsons o mprov h xposon of hs ppr COMPEIG IERESS Auhors hv lr h no ompng nrss xs REFERECES Aggrwl V Bhr-Kshn H (99) Synhronz prouon pols for rorng ms n lnng mrk IIE rnson Bhr-Kshn H (989) Rplnshmn shul for rorng ms wh m - proporonl mn Journl of h Opronl Rsrh Soy Bgum R Shu SK Shoo RR (00) An EOQ mol for rorng ms wh Wull sruon roron un prouon os wh qur mn n shorgs Appl Mhml Sns 7 88 Chkrr Chuhur KS (997) An EOQ mol for rorng ms wh lnr rn n mn n shorgs n ll yls Inrnonl Journl of Prouon Eonoms 9 05-

14 Brsh Journl of Appl Sn & hnology (): - 0 Chng HJ Dy CY (999) An EOQ mol for rorng ms wh m ryng mn n prl kloggng Journl of h Opronl Rsrh Soy Chung KJ ng PS (99) A hurs for rplnshmn of rorng ms wh lnr rn n mn Journl of h Opronl Rsrh Soy 5- Dv U (989) A rmns lo-sz nvnory mol wh shorgs n lnr rn n mn vl Rsrh Logss Dv U (989) On hurs nvnory rplnshmn rul for ms wh lnrly nrsng mn norporng shorgs Journl of h Opronl Rsrh Soy (989) Dv U (989) A ommn on An nvnory rplnshmn poly for lnrly nrsng mn onsrng shorgs n opml soluon y Murshwr M Journl of h Opronl Rsrh Soy 0-6 Dv U Pl LK (98) On ( S) poly nvnory mol for rorng ms wh m-proporonl mn Journl of Opronl Rsrh Soy 7- D M Chuhur KS (986) An EOQ mol for ms wh fn r of prouon n rl r of roron Opsrh 75-8 Donlson WA (977) Invnory rplnshmn poly for lnr rn n mn n nlyl soluon Opronl Rsrh Qurrly Emmons H (968) A rplnshmn mol for rov nul gnrors Mngmn Sn 6-7 Ghr PM Shrr GF (96) A mol for xponnlly yng nvnors Journl of Inusrl Engnrng 8- Ghosh SK Chuhur KS (00) An orr-lvl nvnory mol for rorng m wh Wull sruon roron m-qur mn n shorgs Inrnonl Journl of Avn Mollng n Opmzon 6-5 Goswm A Chuhur KS (99) EOQ mol for n nvnory wh lnr rn n mn n fn r of rplnshmn onsrng shorgs Inrnonl Journl of Sysms Sn 8-87 Goswm A Chuhur KS (99) An EOQ mol for rorng ms wh shorgs n lnr rn n mn Journl of h Opronl Rsrh Soy 05-0 Goyl SK (986) On mprovng rplnshmn pols for lnr rn n mn Engnrng Coss n Prouon Eonoms Goyl SK (987) A smpl hurs mho for rorng onom orr nrvl for lnr rn n mn Engnrng Coss n Prouon Eonoms II 5-57 Goyl SK Morrn D P (99) h fn horzon rn nvnory rplnshmn prolm wh shorgs Journl of h Opronl Rsrh Soy 7-78 Goyl SK Gr BC (00) Rn rns n molng of rorng nvnory Europn Journl of Opronl Rsrh -6 Hrg M (99) h nvnory rplnshmn prolm wh onnuous lnr rn n mn Compurs n Inusrl Engnrs Hrg M (996) Opml EOQ mols for rorng ms wh m -vryng mn Journl of h Opronl Rsrh Soy Hrg M Alyn A (997) A lo szng hurs for rorng ms wh shorgs n growng n lnng mrks Compurs Oprons Rsrh Hrg M Bnkhrouf L (99) Opml n hurs nvnory rplnshmn mols for rorng ms wh xponnl m-vryng mn Europn Journl of Opronl Rsrh

15 Brsh Journl of Appl Sn & hnology (): - 0 Hollr RH Mk KL (98) Invnory rplnshmn pols for rorng ms n lnng mrk Inrnonl Journl of Prouon Rsrh 8-86 Jln AK Chuhur KS (999) Sruurl proprs of n nvnory sysm wh roron n rn mn Inrnonl Journl of Sysms Sn 0(6) 67-6 Jln AK Chuhur KS (999) An EOQ mol for rorng ms n lnng mrk wh SFI poly Korn Journl of Compuonl n Appl Mhms 6() 7-9 Jyrmn K Sugpry C (008) Drmnng ommon prouon yl m for n EPQ mol wh non-nsnnous rorng ms llowng pr soun usng prmssl ly n pymns ARP Journl of Engnrng n Appl Sns 6-0 Khnr S Chuhur KS (00) A no on orr lvl nvnory mol for rorng m wh m-pnn qur mn Compurs Oprons Rsrh Khnr S Sn S Chuhur KS (00) An EOQ mol for prshl m wh sok n pr pnn mn r Inrnonl Journl of Mhms n Opronl Rsrh 0-5 Km KH (995) A hurs for rplnshmn of rorng ms wh lnr rn n mn Inrnonl Journl of Prouon Eonoms Lvn PI MLughln CP Lmon RP Kos JF (97) Conmporry poly for mngng oprng sysm Prouon Oprons Mngmn MGrw- Hll: w York 7 Ln C n B L WC (000) An EOQ mol for rorng ms wh m-vryng mn n shorgs Inrnonl Journl of Sysms Sn 9-00 Murshwr M (988) Invnory rplnshmn poly for lnrly nrsng mn onsrng shorgs-n opml soluon Journl of h Opronl Rsrh Soy Rh E (980) Prl nvnory rplnshmn pols for lnr rn n mn follow y pro of sy mn Journl of Opronl Rsrh Soy Rh E (98) An EOQ for lnr nrsng mn: smpl opml soluon Journl of Opronl Rsrh Soy Rh E (985) Sok rplnshmn quns for unoun lnr nrsng mn: n nrsng onsqun of h opml poly Journl of Opronl Rsrh Soy Roy M Sn S Chuhur KS (0) An EOQ mol for mprf quly prous wh prl kloggng- omprv suy Inrnonl Journl of Srvs n Oprons Mngmn 8-0 Roy M Sn S Chuhur KS (0) An onom orr quny mol of mprf quly ms wh prl kloggng Inrnonl Journl of Sysms Sn 09-9 Roy M Sn S Chuhur KS (0) An opml shpmn srgy for mprf ms n sok-ou suon Mhml n Compur Molng In Prss Shn RS (98) On ( S ) poly nvnory mol rorng ms wh mproporonl mn Journl of Opronl Rsrh Soy Shu SK Bgum R Klm A (007) A no on n opml rplnshmn poly for rorng ms wh onsn mn n prl kloggng Inrnonl Journl of Physl Sns; Ulr Sn 9()M

16 Brsh Journl of Appl Sn & hnology (): - 0 Sn S Chuhur KS (00) On volum flxl prouon poly for rorng m wh m-pnn mn n shorgs Avn Molng n Opmzon 6() 57-7 Sn S (00) Opml sllng pr n lo-sz wh m vryng roron n prl kloggng Appl Mhms n Compuon Sn S (00) Dmn nflun y nrprss' nvs A mul-m EOQ mol of rorng n mlorng ms Mhml n Compur Molng 5(-) 8-0 Sn S (0) h sohs EOQ mol wh rnom sls pr Appl Mhms n Compuon In Prss Corr Proof Avll onln Jun Sn S (0) An EOQ mol for slsmn s nvs sok n pr snsv mn of smlr prous A ynml sysm Appl Mhms n Compuon Sn S (0) An EOQ mol of homognous prous whl mn s slsmn s nvs n sok snsv Compurs & Mhms wh Applons Sn S An EOQ mol for prshl m wh sok-pnn mn n pr soun r Amrn Journl of Mhml n Mngmn Sns (Arl n Prss) Shh YK Jswl MC (977) An orr-lvl nvnory mol for sysm wh onsn r of roron Opsrh 7-8 Slvr EA Ml HC (969) A smpl mofon of h EOQ for h s of vryng mn r Prouon of Invnory Mngmn W HM (995) A rmns lo sz nvnory mol for rorng ms wh shorgs n lnng mrk Compurs n Oprons Rsrh 5-56 Whn M (957) hory of Invnory Mngmn Prnon Unvrsy Prss Prnon J 6-7 7

17 Brsh Journl of Appl Sn & hnology (): APPEDIX Convxy onon of h os funon VC s slsh hr o nsur h xsn of unqu pon of mnmum for VC W hv ln M P R A VC 0 VC (A)

18 Brsh Journl of Appl Sn & hnology (): VC (A) 0 Log M R P A VC (A)

19 Brsh Journl of Appl Sn & hnology (): Log M R P A VC (A) VC VC (A5) ow h funon VC wll onvxy f h Hssn rmnn 0 VC VC VC VC D (A6)

20 Brsh Journl of Appl Sn & hnology (): - 0 n whr VC VC 0 VC VC n VC (A7) r gvn n (A) (A) n (A5) rspvly Consrng h rlon (A) h onon (A7) s ovous 0 Bgum l; hs s n Opn Ass rl sru unr h rms of h Crv Commons Aruon Lns ( hp://rvommonsorg/lnss/y/0) whh prms unrsr us sruon n rprouon n ny mum prov h orgnl work s proprly

Appendix. In the absence of default risk, the benefit of the tax shield due to debt financing by the firm is 1 C E C

Appendix. In the absence of default risk, the benefit of the tax shield due to debt financing by the firm is 1 C E C nx. Dvon o h n wh In h sn o ul sk h n o h x shl u o nnng y h m s s h ol ouon s h num o ssus s h oo nom x s h sonl nom x n s h v x on quy whh s wgh vg o vn n l gns x s. In hs s h o sonl nom xs on h x shl