INVENTORY MODEL FOR DETERIORATING ITEMS WITH QUADRATIC DEMAND, PARTIAL BACKLOGGING AND PARTIAL TRADE CREDIT
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1 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 NVENORY ODEL FOR DEERORNG ES WH QUDR DEND RL BKLOGGNG ND RL RDE RED D. Srmil n R.Uykumr Dprmn of mis Gnigrm Rurl nsiu Dm Univrsiy Gnigrm - 4. Dinigul mil Nu ni. SURY is ril vlops invnory mol for rioring ims wi quri mn. Sorgs r llow. W suy pril ly in pymns. ojiv is o fin opiml yl ims minimiz ol os. n is suy firs w vlop rimil mol n prour of fining mos fvorl soluion is vlop. lso soluion pross is vlop in orr o minimiz ol os. ol oss r lul y vrious prinipls. soluion pross is vlop o fin opiml soluion n sisil xmpls r prsn o monsr rsul of proposl mol. ompssion suy of mos fvorl soluion wi rsp o prmrs of sysm is rri ou n rommnions for furr rsr r provi. KEYWORDS: Quri mn riorion sorgs pril r ri. Suj lssifiion o: 9B5 *orrsponing uor l Fx: ROLOGUE nvnory mngmn is minisrion of supply sorg n onvnin of ims in orr o nsur suffiin supply wiou xssiv ovrsupply. n lso rfrr s omsi onrol - n ouning mo or sysm sign o promo ffiiny or ssur implmnion of poliy or sfgur sss or voi fru n rror. min rg of invnory orgnizion ls wi minimizion of invnory rrying oss for wi i is nssry o rmin opiml sok n opiml im of rfill for fuur. Rsrrs vlop invnory mols y ssuming quri mn for ims lik fruis vgls mrils iron goos los. sin mn r is flxil u o uyrs n sllrs. Du o usomr s oi n rrivl of nw prous my lin mn r. mping ovious f of im-vrying mn prn in rioring invnory mols yils vry mu rl im ppliion. nvnory s wo yps s on im vrying mns nmly isr n oninuous im. oninuous im invnory mols wr vlop mn prn y linrly inrsing or rsing. DO :.5/orj
2 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 rougou ir lif im ow o ly injur spoilg onsqun of or rsons. Y o pi o is f luling n minining invnory of rioring ims oms oug prolm for ssssmns. f supplirs offr usomrs prmissil ly in pymns if orr quniy is lss n prrmin quniy n su sysm is rm s pril r ri. pril r ri is on of nrl furs in supply in mngmn. n rl lif siuion is pril r ri is mor m o rilr. willingnss of usomr o wi for sorgs wi lng of wiing im. Hr sorgs r s klogging; imp on os from ly of prous n ln ou.. LERURE REVEW ukr us.l [] sri svrl rioring ims o no sr rioring immily y wr l in sok. n lso ruion of s ims s soon s y r sor woul pn on mn n wn riorion gins pn on o mn n riorion. ggrwl [] isuss n invnory mol for sysm wi rg is im pnn. mu n nrskrn [] vlop mol wi onsn riorion r n im pnn oling os. R. Bgum l. [5] vlop mol wi im pnn on quri mn r. K.nnuri n R.Uykumr [4] vlop pril r ri finning in supply in y EOQ s mol for omposing ims ogr wi sorgs n lso y isuss ou pril prmissil ly in pymns vn if orr quniy is lss n pr-rmin quniy. inny Kumr ripy.[] nor mol wi quri mn wn riorion r pns on wo prmrs n lso isuss o suj o im pn Wiull riorion y inlu sorgs n rplnismn is infini. D. [7] sui n invnory for ssuming sorgs n foun ow profi ws. Gr [8] sui ou mn wi is ying xponnil. Goyl n Giri [9] moilly rviw rn rn in rioring invnory mols. nvnory mol for mos fvorl priing n orring poliis for vnor wi r ri ws formul y Hwng n Sinn [] y v suggs oniion supplir offrs r ri o pursr if orr quniy is grr n or qul o pr rmin quniy. ir l. [] n invnory mol for onsn mn n rioring ims n lso y inlu vril oling os onsiring sorg oss. on nvnkswrlu [] vlop n EOQ mol for rioring prous y onsiring r of mn is quri wi rsp o im. Nryn Sing l. vlop r lon supply in invnory mol wi prmissil ly in pymns. Soni H l. [7] nlyz y gr o sorgs n pril r ri finning in supply in y EOQ-s mol for ying ims. Lio [4] vlop n EOQ mol wi non insnnous knowlgmn n xponnil rioring im unr wo-lvl r ri. 5
3 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 Hrik N. Soni [] rogniz n invnory mol for rioring ims unr sok- ny mn n wo-lvl r ri n oin opiml rplnismn poliy. Srkr.l [5] sri wi mos fvorl invnory rfill poliy for rioring im onsiring im-quri mn n im-pnn pril klogging wi pns on lng of wiing im for nx rplnismn ovr fini im orizon n vril rplnismn yl. r is no invnory mol y EOQ s mol for rioring ims wi pril r ri unr quri mn wi sorgs. is ppr is orgniz s follows. W sri rquir EOQ s invnory mol wi pril r ri n lso formul svrl rivions. n is ppr w onsir mn s quri form. By is form w gug ol oss. o igligs n visuliz rsuls xmpls r givn. n snsiiviy psyorpy is rri ou n som xuiv insigs r oin. Vnn n Srm [8] vlop n onomi prouion quniy invnory mol for noninsnnous rioring im sunr r ri poliy. Vnn n Srm [9] prpr n invnory mol for non- insnnous rioring ims wi quri mn r n sorgs unr r ri poliy. n is ril w suppos invnory mol for rioring ims wi quri mn pril klogging n pril r ri. invnory is ssum o rior sl r n sorg is llow. n is suy firs w vlop mmil mol n prour for fining opiml soluions n our ppro is illusr roug numril xmpl. W provi nris n yposis for propos mol in sion. rimi mol is slis in sion 4. sion 5 sris soluion pross. Sisil xmpls r provi o illusr propos mol in sion. ompssion suy of mos fvorl soluion wi rsp o prmrs of sysm is rri ou in Sion 7. Finlly w prsn rminion n fuur rsr in sion 8.. ENRES ND HYOHESS mmil mol in is ppr is vlop on sis of following nris n yposis.. Enris o xpn propos mol w p following nris D Q w Sup oss or orring oss pr orr nnul mn i.. ; ; ; orr quniy nvnory yl lng Lng of im wi no sorgs Bklogging prmr riorion r msur wi full ly pymn pl pr orr lil i of ly pymns pl y provir pr orr 5
4 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 p Uni purs os of im Sorg os for klogg im Uni os of los sls ly prio offr y sllr for sling ouns o vnor nnul inrs wi n rn nnul inrs pyl mx mx if Dfiv r in n orr lo if w if < < w.. Hyposis following yposs r us in vlopmn of mol:. r of mn is ssum o in quri form.. Sorgs r llow n klogg. longr wiing im is smllr klogging r will.l B wr is wiing im up o nx rfill wi 4. sllrs my offr pril ly in pymns vn if orr quniy is lss n pr-rmin quniy. 5. ol nnul os Orring os Holing os Opporuniy os Sorgs os u o klogging Driorion os os of inrs rgs for unsol ims fr prmissil ly in pymns os of inrs rn from ls sls uring prmissil prio. 4. RHEL ODEL n invnory mol rimil mol o fin mos fvorl rfill in yl im n o minimiz ol nnul os is vlop. 54
5 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 Figur Grpil img of invnory sysm wi sorgs L invnory im.hn iffrnil quion rprsning invnory sus is givn y Wi fronir oniion mx soluion of quion is givn y if So mximum invnory liu for yl n oin s 55
6 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 5 mx Sorg inrvl mn is prilly klogg frion B. us iffrnil quion ling moun of mn klogg is s low. ; 4 Wi fronir oniion soluion of quion 4 is givn y if 5 us w v if if By ling in quion 5 w n oin mximum moun of mn klogg pr yl s
7 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 57 Λ 7 Hn orr quniy pr yl is givn y Λ mx Q 8 W n oin im inrvl w unis r xus o zro u o o mn n riorion s log w w. f fully ly pymn is prmi n inquliy w Q iff w.f w Q < n vnor mus k lon wi inrs rg p.o py supplir pril pymn Q wn orr is fill im zro. From onsn pys rvnu rilr will l o py off lon Q im. Now ol nnul os of invnory mol for vnor is ompu using following vrious omponns.. Orring os. Holing os. Driorion os
8 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 4. Sorg os u o klogging 5. Opporuniy os u o los sls 5 4 Rgring inrs rg n inrs rn s on vlus w w v following r possil. s i i > w ii w > n iii w > > n is s r r r su - ss. Su-s i: mursmn of goos is mur n vnor srs pying pil opporuniy os for ims in sor wi r p. Hn nnul inrs pyl is p p W ssum uring im moun is no sl inrs rn from prmissil prio is pr yr. Hn 58
9 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr Su-s ii: w n is s r is no inrs pyl is rg y rilr u inrs rn is riv y m. rfor inrs pyl is zro in is s n nnul inrs rn is pr yr. Hn Su-s iii: < <w f <w n rilr mus orrow pril pymn Q im zro n py.from supplirs n n py off lon from sls rvnu im nnul inrs rg im zro o n pymn is nnul inrs pyl n nnul inrs rn us ol os for rilr n xprss s 59
10 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 < < w w if if if 9 Now w lul vlus of. Hn 4 5 p
11 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr SOLUON ROESS o minimiz ol os w rmin opiml sorg os n opiml rplnismn im. s : w Su-s i Equion is minimum wn n wi yils [ ] p
12 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 5 [ 5 p 4 From ov quion now w lul vlu of o [ ] 5
13 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 From quion 5 w v o lul vlu of Sqr o [ ] Su-s ii Equion is minimum wn n wi yils [ ] ] 7 5
14 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 4 [ 5 [ 8 From ov quion w lul vlu of [ 9 From quion 9 w g vlu of Sqr [ Su-s iii Equion is minimum wn n wi yil [ ]
15 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr [ 5 p [ [ From ov quion w lul vlu of p [ 5 From quion w lul sqr
16 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 p. SSL EXLES [ 5 4 n orr o monsr soluion mo l us onsir n invnory sysm wi following : L $5 pr orr $5 pr uni pr orr / $ pr uni $ pr uni $85 pr uni.5 pr ollr pr yr. pr ollr pr yr w n yrs p l llusrion of soluion prour for numril xmpl w w w w
17 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 7. OSSON SUDY l ompuionl ouom wi rsp o iffrn vlus of nging prmrs ng in prmrs / / /
18 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 l ompuionl rsuls wi rsp o iffrn vlus of p nging prmrs Vlus of prmrs p ERNON n is ppr w v xmin pril r ri finning in supply in y EOQ-s mol for omposing ims ogr wi sorgs. W lso ssum mn is quri. mmil is vlop for opiml soluion wi sorg poin orring quniy in orr o minimiz ol os. Sisil xmpl is prsn o monsr soluion pross n ompssion suy for vrious vlus of prmrs lso offr. propos mol n furr xn in i svrl wys. For xmpl i my xn o mulipl ims wi pril klogging n lso vloping mol o nging mn s ui. 8
19 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 KNOWLEDGEENS is rsr ws fully suppor y Nionl Bor of Higr mis Govrnmn of ni unr sm of NBH rsr proj wi /489//NBHR./R&D /D..4. REFERENES [] ukr. Bngi S.. nvnory orring poliis of ly rioring ims unr prmissil ly in pymns nrnionl Journl of rouion Eonomis:75-8. [] ggrwl S no on n orr-lvl invnory mol for sysm wi onsn r of riorion Opsr 5: [] mu R. nrskrn E.. n EOQ mol for rioring ims wi quri mn n im pnn oling os nrnionl Journl of Emrging Sin n Enginring JESE :5-. [4] nnuri K. Uykumr R.. nlysis of pril r ri finning in supply in y EOQ-s mol for ying ims wi sorgs nrnionl Journl of Emrging Sin n Enginring JESE :9 59. [5] Bgum R. Su S. K. n Rsoo R.. n EOQ mol for rioring ms wi Wiull Disriuion Driorion Uni prouion os wi Quri mn n Sorgs ppli mil Sins 4:7-88. [] iny Kumr ripy n Umkn isr. Orring oliy for Wiull Drioring ms for Quri Dmn wi rmissil Dly in ymns ppli mil Sins 4:8 9. [7] D. uuri K no on urisi for rplnismn of rn invnoris onsiring sorgs Journl of oprions rsr Soiy 8: [8] Gr.. n Srr G. F. 9. n nvnory mol for xponnilly rioring ims Journl of nusril Enginring 4:8-4. [9] Goyl S. K. Giri B...Rn rns in moling of rioring invnory E. Journl of Oprions rsr 4:-. [] Hrik N. Soni. Opiml rplnismn poliis for rioring ims wi sok snsiiv mn unr wo-lvl r ri n limi piy ppli mil olling7: [] Hwng H. Sinn S. W Rilr s priing n lo-sizing poliy for xponnilly rioring prous unr oniion of prmissil ly in pymns. ompurs n Oprions Rsr 4: [] ir. Fox J.F. n Jssjr R.R. 98. no on rmining orr quniis wi linr rn in mn. Journl of Oprions Rsr Soiy :5-. [] on R.Vnkswrlu n R. Jnury. nvnory ngmn ol wi Quri Dmn Vril Holing os wi Slvg vlu Journl of ngmn Sins :8-. [4] Lio J. J. 8. n EOQ mol wi non insnnous rip n xponnil rioring im unr wo-lvl r ri. nrnionl Journl of rouion Eonomis :85 8. [5] Srkr. Gos S. K. n uuri K.S.. n opiml invnory rplnismn poliy for rioring im wi im-quri mn n im-pnn pril klogging wi sorgs in ll yls ppli mis n ompuion 8: [] Sil S. l. Rmn l.. n invnory mol for wiull rioring ims wi linr mn sorgs unr prmissil ly in pymns n inflion nrnionl Journl of mis n Sisis nvnion :58-59 [7] Soni H. S N. H. 8. Opiml orring poliy for sok pnn mn unr progrssiv pymn sm. Europn Journl of Oprions Rsr 84:9. 9
20 Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 [8] Vnn. B. K. Srm 5. n EQ invnory mol for non-insnnous rioring im sunr r ri poliy nrnionl Journl of mil Sins n nusril ppliions JSE 9: [9] Vnn B. K. Srm 5. n invnory mol for Non- nsnnous rioring ims wi quri mn r n sorgs unr r ri poliy Journl of ppli nlysis n ompuion in prss. 7
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