A PARTIAL BACKLOGGING INVENTORY MODEL FOR NON-INSTANTANEOUS DETERIORATING ITEMS WITH STOCK-DEPENDENT CONSUMPTION RATE UNDER INFLATION
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1 Yugolav Journal of Oraion ar Volu 0 00 Nubr YJO00035C A PAA BACKOGGNG NVENOY MODE FO NON-NSANANEOUS DEEOANG EMS WH SOCK-DEPENDEN CONSUMPON AE UNDE NFAON Horng Jin CHANG Darn of Buin Ainiraion Aia Univriy aiung aiwan OC Graua niu of Managn Sin aang Univriy aui aiwan j@ail.u.u.w Wn Fng N Darn of Aviaion Manial Enginring Cina Univriy of Sin an nology aii aiwan Graua niu of Managn Sin aang Univriy aui aiwan linwn@.u.u.w iv: Jun 008 A: May 00 Abra: n i ar w riv a arial balogging invnory ol for noninananou rioraing i wi o-nn an ra unr inflaion ovr a fini lanning orizon. W roo a aaial ol an or o fin iniu oal rlvan o an oial orr quaniy. Nurial xal ar u o illura vlo ol an oluion ro. Finally a niiviy analyi of oial oluion wi r o y arar i arri ou. Kywor: Parial balogging non-inananou rioraion o-nn an inflaion.. NODUCON Drioraion i fin a ay ang aag oilag or oboln a rul in raing ufuln fro i original uro. So in of invnory rou.g. vgabl frui il an or ar ubj o rioraion. Gar an Srar 963 fir abli an onoi orr quaniy ol aving a onan
2 36 H. J. Cang W. F. in A Parial Balogging nvnory Mol ra of rioraion an onan ra of an ovr a fini lanning orizon. Covr an Pili 973 xn Gar an Srar onan rioraion ra o a woarar Wibull iribuion. Dav an Pal 98 iu an invnory ol for rioraing i wi i-roorional an wn orag wr no allow. rla analyi on invnory y wi rioraion av bn rfor by Saan 984 Bali an Bnrouf 996 W 997 Muoayay al n raliy no all in of invnory i riora a oon a y riv by railr. n fr rou i rou a no rioraion an ir original qualiy. Ouyang al. 006 na i nonon a non-inananou rioraion an y abli an invnory ol for non-inananou rioraing i wi riibl lay in ayn. n o faionabl rou o uor woul li o wai for balogging uring orag rio. Bu willingn i iiniing wi lng of waiing i for nx rlnin. longr waiing i i allr balogging ra woul b. ooruniy o u o lo al oul b onir. Cang an Dy 999 vlo an invnory ol in wi an ra i a i-oninuou funion an i riora a a onan ra wi arial balogging ra wi i riroal of a linar funion of waiing i. Paario an Souri 000 vlo an EOQ invnory ol wi i-nn arial balogging. y uo ra of balogg an inra xonnially wi waiing i for nx rlnin ra. ng al n xn balogg an o any raing funion of waiing i u o nx rlnin. rla analyi on invnory y wi arial balogging av bn rfor by ng an Yang 004 Yang 005 Dy al. 006 San Joé al. 006 ng al Many aril au a an i onan uring al rio. n o b iu. n ral lif rquirn ay b iula if r i a larg il of goo ilay on lf. vin al. 97 r a or goo ilay on lf or uor an will b gnra. Gua an Vra 986 rn an invnory ol for o-nn onuion ra on iniial o lvl rar an inananou invnory lvl. Bar an Urban 988 abli a rinii invnory y in wi an ra n on invnory lvl i rib by a olynoial funion. Wu al. 006 rn an invnory ol for non-inananou rioraing i wi o-nn. rla analyi on invnory y wi o-nn onuion ra av bn rfor by Daa an Paul 00 Bali an Bnrouf 004 Cang al n all of abov nion ol influn of inflaion an i valu of ony wr no iu. Buzao 975 fir abli an EOQ ol wi inflaion ubj o iffrn y of riing olii. Cung an in 00 follow ioun a flow aroa o inviga invnory ol wi onan an ra for rioraing i aing aoun of i valu of ony. Hou 006 abli an invnory ol wi o-nn onuion ra iulanouly onir inflaion an i valu of ony wn orag ar allow ovr a fini lanning orizon.
3 H. J. Cang W. F. in A Parial Balogging nvnory Mol 37 n i aril w vlo a arial balogging invnory ol for noninananou rioraing i wi o-nn an ra along wi ff of inflaion an i valu of ony a ar onir. W xn ol in Hou 006 o onir non-inananou an arial balogging invnory ol. r of i ar i organiz a follow. n Sion w rib auion an noaion u rougou i ar. n Sion 3 w abli aaial ol an or o fin iniu oal rlvan o an oial orr quaniy. n Sion 4 w u nurial xal o illura or an rul w roo. n Sion 5 w a a niiviy analyi o uy ff of ang in y arar on invnory ol. Finally w a a onluion an rovi uggion for fuur rar in Sion 6.. ASSUMPONS AND NOAON W giv following auion an noaion wi will b u rougou ar. Auion: Only a ingl-rou i i onir uring lanning orizon H. lnin ra i infini an la i i zro. 3 A onan fraion of on-an invnory riora r uni of i an r i no rair or rlan of riora invnory. 4 Sorag ar allow an balogg arially. balogging ra i a raing funion of waiing i. balogging ra b B wr 0 an i waiing i u o nx rlnin. 5 A Dioun Ca Flow DCF aroa i u o onir variou o a variou i. Noaion: H lanning orizon rlnin yl rlnin nubr in lanning orizon H raio of no-orag rio o uling rio in a yl lng of i in wi rou a no rioraion invnory lvl a i uring i inrval invnory lvl a i uring i inrval orag lvl a i uring i inrval 3 aoun of lo al a i uring i inrval axiu invnory lvl for a yl S axiu orag quaniy for a yl b D an ra a i. D wn > 0 an D wn 0 wr > 0 i o-nn onuion ra arar 0. onan rioraion ra
4 38 H. J. Cang W. F. in A Parial Balogging nvnory Mol o n ioun ra of inflaion orring o r orr uraing o r uni oling o r uni r uni i balogging o r uni r uni i uni o of lo al. No a if objiv i o iniiz o n >. C o rn valu of orring o in lanning orizon H C rn valu of uraing o in lanning orizon H C C rn valu of oling o in lanning orizon H rn valu of orag o in lanning orizon H C rn valu of lo al o in lanning orizon H C rn valu of oal rlvan invnory o in lanning orizon H Q oial orr quaniy in a yl 3. MAHEMACA MODE AND SOUON S b lo al H Figur.. grai rrnaion of invnory ol invnory ol i own in Fig.. lanning orizon H i ivi ino qual ar of lng H. j rlnin i a a i j j 0. axiu invnory lvl for a yl i. During i inrval j j j 0 rou a no rioraion invnory lvl i raing u o an only. During i inrval j j j 0 invnory lvl graually ru o zro owing o rioraion an an. An orag an uring i inrval j j j 0. quaniy riv a j j 3
5 H. J. Cang W. F. in A Parial Balogging nvnory Mol 39 i u arly o auula baorr in rviou yl fro i j o j wr i raio of no-orag rio o uling rio in a yl. la xra rlnin a i H i n o rlni orag gnra in la yl. objiv of invnory robl r i o rin rlnin nubr an raio in orr o iniiz oal rlvan o. n fir rlnin yl owing o o-nn onuion ra only invnory lvl a i uring i inrval 0 i govrn by following iffrnial quaion: 0 wi bounary oniion 0. oluion of Eq. an b rrn by 0 Owing o o-nn onuion ra an rioraion invnory lvl a i uring i inrval i govrn by following iffrnial quaion: 3 wi bounary oniion 0. oluion of Eq. 3 an b rrn by 4 Bau axiu invnory lvl i 5 Hn in Eq. an b rrn a 6 Sin balogging ra i a raing funion of waiing i w l balogging ra b B orag lvl a i uring i inrval i govrn by following iffrnial quaion: 3 7
6 H. J. Cang W. F. in A Parial Balogging nvnory Mol 40 wi bounary oniion 0 3. oluion of Eq. 7 an b rrn by 3 8 An aoun of lo al a i uring i inrval i τ τ 9 b S b axiu orag quaniy r yl. 3 b S 0 lnin i a a i j j 0 axiu invnory lvl for a yl i. la rlnin a i i ju o aify baorr gnra in la yl. r ar rlnin in nir i orizon H. oal rlvan invnory o involv following fiv faor. a Orring o: rn valu of orring o in nir i orizon H i 0 H H H o j j o o C b Puraing o: rn valu of uraing o in nir i orizon H i 0 H H H H H j j b j j S C Holing o: rn valu of oling o in nir i orizon H i H H j j C 0 0 3
7 H. J. Cang W. F. in A Parial Balogging nvnory Mol 4 Sorag o: rn valu of orag o in nir i orizon H i 0 3 H H H H j j C 4 o al o: rn valu of lo al o in nir i orizon H i 0 H H H H j j C 5 Hn rn valu of oal rlvan invnory o in nir i orizon H i o C C C C C C 6 l H U H V H W H W ubiu Eq. -5 ino Eq. 6 an obain W V V U C o 7 r ar wo variabl in rn valu of oal invnory o C. On i rlnin nubr wi i a ir variabl or i raio wr wi i a oninuou variabl. For a fix valu of oniion for C o b iniiz i 0 C. Conqunly w obain
8 H. J. Cang W. F. in A Parial Balogging nvnory Mol or a f < 9 r xi a uniqu oluion wr < < u a C i iniu valu of wn i givn. b f > 0 H C i iniu valu wn i givn. Proof: S Anix. Fro or w an u Nwon-aon o o fin oial valu wn rlnin nubr i givn. Howvr in ig-owr xrion of xonnial funion in C i i iffiul o ow analyi oluion of u a i a C iniiz. Following oial oluion rour roo by Mongory 98 w l no oial oluion o C an l no oial oluion o C wn i givn. f i all ingr u a C l an a valu of C in inrval 0. n w a a oial oluion o C. An w an obain axiu invnory lvl a H Alo oial orr quaniy Q i
9 H. J. Cang W. F. in A Parial Balogging nvnory Mol 43 Q S b H H 4. NUMECA EXAMPES o illura roo ol l u onir following arari aa a xal. Exal : $3 uni yar 0 $50.00 orr $5 uni $.75 uni yar $0 uni 600 uni yar H 0 yar 0.05 yar. abov aa aify or a. Following oial oluion rour roo by Mongory 98 abl ow oial rlnin nubr 3 raio 0.35 oial orr quaniy Q 464. an iniu rn valu of oal rlvan o C $599.. rlaion bwn C an unr iffrn olii in abl ar own in Figur.
10 44 H. J. Cang W. F. in A Parial Balogging nvnory Mol abl Diffrn olii wi r o oal o for xal Q C Oial oluion
11 H. J. Cang W. F. in A Parial Balogging nvnory Mol 45 C Figur laion bwn C an. oniion aifi or a. So faor a influn oal rlvan o ar own in abl. abl So ial a of invnory ol in Exal. Coniion Q C rar our xal C C C C C C C 6 rn valu of oal rlvan o C 5 i a a Hou 006 an C 6 i a a Cung an in 00. or nurial rul an a following oaraiv onluion:
12 46 H. J. Cang W. F. in A Parial Balogging nvnory Mol f inflaion an i valu of ony ar no onir 0 rn valu of oal rlvan o C i far largr an oal rlvan o C. i.. C >> C. f o-nn onuion ra i no onir 0 rn valu of oal rlvan o C i allr an oal rlvan o C. i.. C < C. 3 f rioraion in onir 0 rn valu of oal rlvan o i allr an oal rlvan o C. i.. C 3 < C. 4 f balogging i ol 0 rn valu of oal rlvan o C 3 C 4 i allr an oal rlvan o C. i.. C 4 < C. Exal : All aa ar a a xal x $0 uni an $5 uni. abov aa aify or b. Fro abl 3 w obain oial rlnin nubr raio oial orr quaniy Q an iniu rn valu of oal rlvan o C $ rlaion bwn C an unr iffrn olii in abl 3 i own in Figur 3. abl 3 Diffrn olii wi r o oal o for xal Q C Oial oluion
13 H. J. Cang W. F. in A Parial Balogging nvnory Mol 47 C Figur 3 laion bwn C an. oniion aifi or b. 5. SENSVY ANAYSS n i ion w iu y arar a influn iniu oal rlvan o C oial orr quaniy Q an rlnin nubr. or C i iniu oal rlvan o of C. f < < n C i a rily inraing funion of for 0. Proof: S Anix. or 3 C i iniu oal rlvan o of C an Q i oial orr quaniy. f < < i.. i aifi or a n C an Q ar rily raing funion of for 0. Proof: S Anix 3. or 4 Q i oial orr quaniy. Q i a rily raing funion of for 0. Proof: S Anix 4. Uing Exal aa o uy ff of ang in y arar on oial orr quaniy Q iniu oal rlvan o C an rlnin nubr. niiviy analyi i rfor by anging inraing or raing arar by 50% 0% aing a a i ing raining arar a ir original valu. ia valu of oial orr quaniy iniu oal rlvan o an rlnin nubr b Q C an
14 48 H. J. Cang W. F. in A Parial Balogging nvnory Mol rivly r valu in Exal ar Q C an. following infrn an b obrv fro niiviy analyi ba on abl 4. abl 4. Sniiviy analyi of Exal Parar Prnag of unr-iaion an ovr-iaion arar Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C oial orr quaniy inra a or 0 inra. Bu i ra a or inra.
15 H. J. Cang W. F. in A Parial Balogging nvnory Mol 49 oial orr quaniy i or niiv on ang in 0 or o or arar. 3 iniu rn valu of oal rlvan o inra a 0 or inra. Bu i ra a or inra. 4 iniu oal rlvan o i or niiv on ang in or o or arar 5 rlnin nubr inra a or inra. Bu i ra a 0 inra. 6 rlnin nubr i inniiv on ang in or o or arar 7 C i a inraing funion of for 0 i oby or. An Q i a raing funion of for 0 i oby or 4. 8 C an Q ar raing funion of for 0 i oby or CONCUSON n i aril w abli an invnory ol for non-inananou rioraing i wi o-nn onuion ra o rin oial orr quaniy iniu rn valu of oal rlvan o an rlnin nubr. ff of inflaion an i valu of ony ar alo onir. W rn oniion of uniqu oluion of oal rlvan o wn rlnin nubr i givn in or. W alo iu iniu oal rlvan o an oial orr quaniy wi r o balogging arar an non-inananou rioraing i fro or o or 4 rivly. Fro niiviy analyi oial orr quaniy i or niiv on ang in arar 0 or. iniu rn valu of oal rlvan o i or niiv on ang in arar or. l railr o a iion in iffrn rlnin olii. Finally roo ol an b xn in vral way. For xal w oul xn rinii ol o varying yl lng. Alo w oul gnraliz ol o allow for quaniy ioun or or.
16 H. J. Cang W. F. in A Parial Balogging nvnory Mol 50 APPENDEX Proof of or ar a. V C } { { 0 } > V C Clarly C i a rily inraingly funion of. Bi 0 } { > V C V C } { if < n 0 < C. Fro nria Valu or r will xi a uniqu oluion a aifi 0 C wr < <. Bau 0 > C C i a onvx funion of for a fix valu. Hn C i iniu valu of wn i givn. Proof of or ar b. if > owing o 0 > C an 0 > C C i a rily inraing
17 H. J. Cang W. F. in A Parial Balogging nvnory Mol 5 funion of in inrval H wn i givn. Conqunly iniu valu of C will an a H wn i givn. APPENDX Proof of or Now w onir rlaion bwn variabl an iniu oal rlvan o C. C { x wr 0 x an } W f x x x x f x x x x x f < < i u b f x x > 0 for 0 x. Hn f x i a rily inraing funion of x for 0 x. Bau f 0 0 w an ur f x > 0 for 0 x. i ili C > 0 w ay inraing funion of for 0. x C i a rily Proof of or 3 Coniring C an APPENDX 3 Q rivaiv o rivly. an C V Q
18 5 H. J. Cang W. F. in A Parial Balogging nvnory Mol i.. Q oniion for C > 0 an Q > 0 ar < 0 <. viola finiion woul b allr an zro in inrval or a wi jug wr iuaion C an. Hn C an < <. alo aifi xi bwn an or no. n i Q ar rily raing funion of for 0. APPENDX 4 Proof of or 4 Now w onir rlaion bwn variabl an oial orr quaniy Q. Q λ wr 0 λ an g λ λ λ. λ Bau g λ λ < 0 g λ i a rily raing funion of λ for 0 λ. Bau g 0 0 w an ur g λ < 0 for 0 λ. i ili Q < 0 w an ay Q i a rily raing funion of for 0. EFEENCES Bar.C. an Urban.. A rinii invnory y wi an invnory lvlnn an ra Journal of Oraional ar Soiy Bali Z.. an Bnrouf. On oial rlnin ul for an invnory y wi rioraing i an i-varying an an rouion ra Cour an nurial Enginring Bali Z.. an Bnrouf. On an invnory ol for rioraing i wi o nn an i-varying an ra Cour an Oraion ar Buzao J.A. Eonoi orr quaniy wi inflaion Oraional ar Quarrly Cang C.. Goyal S.K. an ng J.. On an EOQ ol for riabl i unr o-nn lling ra an i-nn arial balogging by Dy an Ouyang Euroan Journal of Oraional ar Cang H.J. an Dy C.Y. An EOQ ol for rioraing i wi i varying an an arial balogging Journal of Oraional ar Soiy Cung K.J. an in C.N. Oial invnory rlnin ol for rioraing i aing aoun of i iouning Cour & Oraion ar Covr.B. an Pili G.S. An EOQ ol wi Wibull iribuion rioraion AE ranaion
19 H. J. Cang W. F. in A Parial Balogging nvnory Mol 53 9 Daa.K. an Paul K. An invnory y wi o-nn ri-niiv an ra Prouion Planning & Conro Dav U. an Pal.K. S i oliy invnory ol for rioraing i wi i roorional an Journal of Oraional ar Soiy Dy C.Y. Cang H.J. an ng J.. A rioraing invnory ol wi i-varying an an orag-nn arial balogging Euroan Journal of Oraional ar Gar P.M. an Srar G.F. A ol for xonnial aying invnory Journal of nurial Enginring Gua. an Vra P. nvnory ol for o-nn onuion ra Oar Hou K.. An invnory ol for rioraing i wi o-nn onuion ra an orag unr inflaion an i iouning Euroan Journal of Oraional ar vin.. Mauglin C.P. aon.p. an Koa J.F. ProuionOraion Managn: Conorary Poliy for Managing Oraing Sy MGraw-Hill Nw Yor Manal B.N. an Paujar S. An invnory ol for rioraing wi o nn onuion ra Oar Mongory D.C. Eonoi ign of an X onrol ar Journal of Qualiy nology Muoayay S. Murj.N. an Cauuri K.S. Join riing an orring oliy for a rioraing invnory Cour an nurial Enginring Muoayay S. Murj.N. an Cauuri K.S. An EOQ ol wi woarar Wibull iribuion rioraion an ri-nn an nrnaional Journal of Maaial Euaion in Sin an nology Ouyang.Y. an Wu K.S. A uy on an invnory ol for non-inananou rioraing i wi riibl lay in ayn Cour & nurial Enginring Paario S. an Souri K. An oial rlnin oliy for rioraing i wi i-varying an an arial-xonnial y-balogging Oraion ar r Saan.S. On S i oliy invnory ol for rioraing i wi i roorional an Journal of Oraional ar Soiy San Joé.A.; Siilia J.; an Garía-aguna J. Analyi of an invnory y wi xonnial arial baorring nrnaional Journal of Prouion Eonoi ng J.. Cang H.J. Dy C.Y. an Hung C.H. An oial rlnin oliy for rioraing i wi i-varying an an arial balogging Oraion ar r ng J.. Yang H.. an Ouyang.Y. On an EOQ ol for rioraing i wi i-varying an an arial balogging Journal of Oraional ar Soiy ng J.. an Yang H.. Drinii onoi orr quaniy ol wi arial balogging wn an an o ar fluuaing wi i Journal of Oraional ar Soiy ng J.. Ouyang.Y. an Cn.H. A oarion bwn wo riing an lo-izing ol wi arial balogging an riora i nrnaional Journal of Prouion Eonoi W H.M. A rlnin oliy for i wi a ri-nn an an a varying ra of rioraion. Prouion Planning & Conrol
20 54 H. J. Cang W. F. in A Parial Balogging nvnory Mol 9 Wu K.S. an Ouyang.Y. An oial rlnin oliy for non-inananou rioraing i wi o-nn an an arial balogging nrnaional Journal of Prouion Eonoi Yang H.. A oarion aong variou arial balogging invnory lo-iz ol for rioraing i on bai of axiu rofi nrnaional Journal of Prouion Eonoi
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