An Inventory Model for Deteriorating Commodity under Stock Dependent Selling Rate
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- Stanley Patterson
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1 An Invnory Modl for rioring Commodiy ndr Soc ndn Slling R Wydi Soo rmn of Indril Enginring Bndng Ini of cnology Jln Gn Bndng 43 INONESIA rmn of Indril Enginring Univriy of Sl Mr Jln Ir Smi No 36 A Srr 5736 INONESIA E-mil: wydioo@dnicid or oo@ncid Snor Nr Bgi rmn of Indril Enginring Bndng Ini of cnology Jln Gn Bndng 43 INONESIA Emil: nor@miliicid Arc Economic ordr qniy EO i on of mo imorn invnory olicy v o dcidd in mnging n invnory ym rolm ddrd in i r concrn wi dciion of oiml rlnimn im for ordring n EO o lir i Modl i crd ffc of oc dndn lling r nd vrying ric W dvlod n invnory modl ndr vrying of dmnd-driorion-ric of commodiy wn rlioni of lir-grocry-conmr ocic nvironmn rlnimn md innno wi zro ld im commodiy will dcy of qliy ccording o originl condiion wi rndomiz crcriic Fir modl i ddrd o olv rolm nomnon ow long i oimm lng of cycl im n n EO of commodiy o ordrd y will drmind y modl o olv i rolm fir i dvlod mmicl modl d on rfrnc modl nd n olv modl nlyiclly Finlly n invnory modl for drioring commodiy ndr oc dndn lling r nd conidring lling ric w drivd y i rrc Kyword: driorion commodiy xcd rofi oiml rlnimn im oc dndn lling r INROUCION drioring commodiy i dfind commodiy wi dcy or lo of qliy mrginl vl rl in dcring fln from originl condiion Nmi 98; v 99 nd Rf 99 Firm lling commodiy wo qliy lvl drior ovr im ofn fc difficl dciion wn nold invnory rmin Sinc lfovr commodiy i ofn rcivd o of lowr qliy n nw commodiy crrying i ovr offr firm cond lling oorniy nd lo rdcd lling ric In i c firm old coo oiml rgy rd of ordr qniy from lir nd lling ric down olicy imc of driorion r Unfornly om commodii li fr food vgl fri nd cooing ic will ril or oild wn lifim i ovr In conr Gror fc mr onil wi ocic nvironmn for inn co r ni lif im nd dmnd of commodiy y old oin invnory olicy o mximiz ir rofi A rvy of lirr on invnory modl for drioring im w givn y Rf 99 nd Nmil 99 i rrc i rd wi concrning vriy of invnory modl from rvio rrc conidr ffc of driorion nd ril wi im dndn following cm i d o cgoriz vrio modl for inn: i dciion criri: ol co or ol rofi; conidr vrying dmnd; 3 conidr driorion vrying nd conidr co/ric vrying l Bo l 995 Giri l 996 Jml l 997 dn Go & Cdri 4 dvlod invnory modl wi im dndn drior r in drminiic of driorion vrying Mnwil Aggon l w condc rrc y ming cclr of driorion vrying i ocic roc from om f lrniv wi rndomiz crcriic Som of rcn rrc n don y Cng & y 999 Aggon l M & S 3 ng l 3 Go & Cdri 4 nd Bol l 5 y r conidr dclin of dmnd or l qniy d o dcring fln from originl condiion : Corronding Aor 5
2 AIEMS 8 rocding of 9 Ai ific Indril Enginring & Mngmn Sym Confrnc l : Smmry of rld rrc Aor lid ciion Criri Vrying mnd Vrying riorion Vrying Co/ric Bc- Ordring v 989 C YES im Linir-Incring NO Conn NO NO Bo l 995 C YES im-vrying YES Exonnil YES YES S nd J 99 YES Slling ric-dndn - YES NO Gowmi & Cdri 99 C YES im-vrying NO Conn NO YES Cng & ing 993 C YES im-vrying NO Conn NO - dmnn & Vr 995 YES Slling ric-dndn YES Exonnil YES NO Giri l 996 C YES im-vrying YES Linir YES YES Bnrof & Mmod 996 C YES im-vrying NO Conn NO - Jml l 997 C NO Conn YES Exonnil NO YES Cng & y 999 C YES im-vrying NO Conn NO YES Aggon l C YES Socic YES ocic NO NO M & S 3 C YES Exonnil NO Conn YES NO ng l 3 C YES im-vrying NO Conn NO Conn NO Go & Cdri 4 C YES im dric YES Will NO Conn NO Bol l 5 C YES Invnory Lvl NO Conn NO YES Roy 8 YES Slling ric-dndn NO Conn YES NO rn r YES Slling ric-ndn YES Exonnil YES YES In mo rvio modl ingrion mong ly dnd on lling ric wi ric olicy in ocic mr nvironmn w no condcd y In mo modl do no conidrd lling ric n minimiz ol co wr d dciion criri S & J 99 v rd o dvloing invnory modl wi mximiz rofi dciion cririon owvr modl do no rld o imc of driorion r Aggon l v condcd o ingr mong driorion wi invnory lvl in on riclr mr nvironmn wic i ocic Howvr y do no conidring co or ric vrying y imc of driorion r In rn r w v dvlod n invnory modl for drioring commodiy ndr oc dndn lling r nd conidring lling ric rformnc cririon of i modl i o mximiz rofi y imlnoly drmining lling ric HE ROBLEM FORMULAION novly w will ing ino conidrion in i rrc i dvlod n invnory modl ndr vrying of dmnd-driorion-ric of commodiy wn rlioni of lir-grocry-conmr ocic nvironmn Fir modl i ddrd o olv rolm nomnon ow long i oimm lng of cycl im n n EO of commodiy o ordrd y will drmind y modl dring lng of cycl im o g mximm rofi frmwor of ocic invnory modl for drioring commodiy w own in Figr Figr i dcrid crcriic on nrl of invnory ym I w dicd y following: i ii iii iv i ym i conrcd y ingl lir ingl gror nd mny conmr; im i ingl commodiy Ec commodiy will dcy of qliy ccording o originl condiion wi rndomiz crcriic; nd ll oc o org r lo nd no rcovrd nd xc oc i xird nd no vl Figr : Frmwor of n invnory modl Mmicl Arviion nd Symol mmicl modl i drivd wi following mion nd noion: In cycl of lnning orizon co of modl i follow: i fixd ordring co A ; ii olding co r ni in oc r ni of im roorionl wi ric nd im ; iii org co r ni of im r ni of im c nd N i ol qniy org im N Bli INONESIA cmr 3rd
3 AIEMS 8 rocding of 9 Ai ific Indril Enginring & Mngmn Sym Confrnc iv rc co r ni of im rlnimn md innno niform wi zro ld im L = 3 ingl commodiy drior wi r j cific dniy fncion 4 I i invnory lvl im 5 i ol dmnd wr dmnd r lling r im my occr long dnd on cng in invnory lvl I nd rmr of oc dndn lling i md I 6 xc of oc i xird nd no vl 7 Slling ric ciion Vril : R : i ol rvn from lling ingl commodiy C : i ol of Invnory co : i ol rofi clcld y R - C : i lng of cycl im qniy of commodiy o ordrd y EO : gror i n conomic ordr qniy * dring lng of cycl im 3 HE MAHEMAICAL MOEL W fir nlyz modl of Aggon l l rfrnc nd n dvlo nw ocic invnory modl d on riodic rviw ym Modl Aggon l dn S & J 99 mmicl modl formlion i dvlod wi following : ol of invnory co ol of invnory co i coni ol of rc co O ordring co O ; olding co O org co O nd xc co O In conqnc ol co in i invnory ym cn wrin : c O O O O O ol of rc co nd ordring co ol rc co i rc co r ni of im im qniy of commodiy o ordrd y gror ordring co i oind ingl co o lc in ordr divid y lng of cycl im ol rcing co will dnd on oimm lng of cycl im in c ordr O x O A/ 3 3 ol olding co In i invnory ym ol of vrg invnory i m of wo comonn ol of qniy o ordrd nd ol of qniy didn rvicd N min vrg ol dmnd r lng of cycl im O N 4 4 ol org co nd ol xc co ol org co i xcd ol qniy of comr ordr didn rvic im org co r ni of im dring lng of cycl im ol qniy org im r clcld y ol dmnd min ol l Any oc ov conomic im ly old old will xir ol xc co i rc co r ni of im im qniy of xc O c N 5 O if S 6 5 invnory lvl invnory lvl i dcr wi im dndn d o lling r nd driorion r Wn oc r oiiv lling r i oc dndn ooi wn oc i ngiv dmnd r i conn rfor invnory lvl dcr d o oc dndn lling wll driorion dring riod: n dring riod dmnd i cloggd Frrmor dring riod wn oc i oiiv xc of oc i xird nd no vl From dmnn & Vr 995 ic modl wi vrying r of driorion dcri i modl i givn y: I I I I 6 Clcl lvl of invnory 7 W conidr lvl of invnory im I lng of cycl im olion of qion 7 nd 8 ing Linir Fir Ordr Eqion orm Andix A for ondry condiion I = i I 9 I 8 N Bli INONESIA cmr 3rd
4 AIEMS 8 rocding of 9 Ai ific Indril Enginring & Mngmn Sym Confrnc 7 Clcl ol olding co nd ol org co o clcl ol olding co nd ol org co w rqir o drmind ow mny i ol qniy org im ol qniy org dring lng of cycl im i givn y N if nd N if roiliy of org nd mn nmr or cn rdily clcld for mlil rordr oin olicy Sinc Z i rndom vril of dmnd diriion fncion ndr mion f Z i norml diriion w v ol qniy org i N z f z dz 8 Clcl ol dmnd ol dmnd wr dmnd r dnd on cng in invnory lvl nd rmr of oc dndn lling w oind y olv qion Id n Si I y qion 9 nd w cn clcl i cn imlifid nd xrd y qion 3 [ ]d [ ] d 9 Clcl ol rofi 3 ol rofi i ql o ol rvn l n ol co ol rvn i oind y mlilicion dmnd r lng of cycl im o lling ric n rcing o qion -6 ol rofi i A N c N 4 Hnc W v rofi fncion Or ojciv i o mximiz rofi fncion y dfin From fncion 4 w cn ily xnd o mor fncion ing r ojciv vril nd xrd y qion 5: [ ] [ A ] c [ z f z dz 4 SOLUION ROCEURE 5 Uing convxiy rl w cn oind r ojciv vril: * * * ncry condiion nd fficin condiion for qion mximizing r ; ; ; ; 6 7 From qion 5 fir w cn driviv qion o / wi imli ; c f z dz 8 Hnc from qion 8 w cn dfind roiliy of oc o f z dz c 9 Scond w cn driviv qion 5 o / n w g o imlify following rlion Andix A: N Bli INONESIA cmr 3rd
5 AIEMS 8 rocding of 9 Ai ific Indril Enginring & Mngmn Sym Confrnc n n w cn dfin ing logrimic rl ln[ / ] ird w cn driviv qion 5 o / oimm vl of cn oind y olving qion 5 I i rqir wo iion conidrly mor comionl o w cn ily o olv driviv rolm L ; nd ; 3 driviv of lr ccording rodcdiffrnc rl fr om imlificion Andix A3 i A 4 c z c * f z dz z A ; f z dz 5 oimm vl of cn oind from xrion vl of / i lwy ngiv o ify fficin condiion for mximizing fficin condiion for mximm vl of rofi i 3 A 6 3 For > xrion 6 lwy ngiv oimm vl of cn oind y olving qion I rqir conidrly mor comion n non rcriv rocdr A conqnc w gg ligly modifying Hdly-Wiin lgorim follow: - S Sr wi mion =; n find xcd y olving qion 3 Com A/ d on Wilon modl - S Com roiliy of oc o y olving qion 9 - S 3 Com y m dmnd i normlly dirid z - S 4 Com ol rofi y olving qion 4 - S 5 Go o o m irion y incring i i nd com i y rforming S nd S 3 If i > l i i i n com i If i < SO irion wi n go o S 6 - S 6 o m irion y dcring i i nd com i y rforming S nd S 3 If i > l i i i n com i If i < SO irion Hnc niqn of oimm rlnimn olicy cn rovidd y cooing 5 NUMERICAL EXAMLE AN ANALYSIS o illr rn modl following xml r conidrd rolm o olvd r i Gror ll ingl commodiy i grlic c in griclrl rodcion wr o of rving i ocic nvironmn i lif im nd qliy Exml: Find oimm rlnimn olicy r w v rmr follow: =; =5; A=5; =5; c =; =; =; nd =788 Com oimm rlnimn olicy y rood lgorim ov: Com nd : Afr w clcl d on rood lgorim ov fr om imlificion n w g: + = 5 A = 5 + = 3 Ln+/ = 3497 =Ln+//+ = Find 3 = 55 = = Find α Zα R nd : l i viwd rl of irion lgorim o find oimm rlnimn olicy i N Bli INONESIA cmr 3rd
6 AIEMS 8 rocding of 9 Ai ific Indril Enginring & Mngmn Sym Confrnc l oimm rlnimn olicy Find i = Find α = Z α = fz = z = C = = = Comd rl nd oiml rlnimn olicy r own in l oiml vl of nd r * = 33 nd = i nmricl xml cr gror old oind oiml rlnimn olicy comr o oiml qniy of commodiy o ordrd n d on oiml rlnimn olicy w rood ow mny qniy of commodiy o ordrd y gror o g mximm rofi In or comrion wi rfrnc modl i i clr i modl oin lng of cycl im n w clcl qniy of commodiy o ordrd y gror rfrnc modl driv oiml oring crcriic of xcd co r ni im n rfrnc modl m roxim olicy y clcl vrg dviion o c rmr 6 CONCLUSION A ocic invnory modl for drioring commodiy ndr oc dndn lling r nd conidring lling ric w drivd y i rrc In riclr oiml rlnimn im w drivd Morovr nmricl xml wr own o vidnc fln of rood modl By ingring dmnd dnd on lling ric wn ric olicy in ocic mr nvironmn Firm or Gror cn mximiz rofi rog drmin oiml rlnimn im In rliy mny of invnory ym dling wi food im vgl nd m cn cd y rn modl; in wic oiml rlnimn im cn mrd r dy r w c d o commodiy crcriic Bid ni in oc cn mrd r g r on r oc ing ni SKU nd c For frr rrc i modl cold xndd o or crcriic of driorion rolm in gror in xml wi ccid ord conidring rmn co o ring dcy of qliy conidr rncion cm o lir ACKNOWLEGMEN or r grfl o nonymo rfr for commn nd rcommndion imrovd r AENIX Andix A: drivion of qion 7 I I I Linir Fir Ordr Eqion orm dy dx vy z vd y A z L v ; y I ndz vd d cific vl of ingrion cn xrd I d A d Siing i limi for qion ov nd g cific vl of A dfin I = n find cific vl of I roof of xrion 9: I - Andix A: roof of qion 8 L qion 8 c [ ] A z f z dz N Bli INONESIA cmr 3rd
7 W cn ily xnd o mor fncion ing rol of driviv of owr-fncion rl gnrlizd: f' y n cf y If c L conidr fncion: f' f i i lo vlid o wri: ; qion cn xrd lrnivly ; So diffrn vl of will rl in diffrn vl of driv c ] / ln[ Andix A3: roof of qion 4 From qion 5 w cn ily xnd o mor fncion ing rol of driviv of rodc of wo fncion rodc rol: v v v ' ; ' nd v of Find drivif nd ; L ' v' y n v y If driviv of lr ccording rodc-diffrnc rl i L ; nd ; ] ][ [ ' v' y v i i lo vlid o wri: * dz z f z c A REFERENCES Aggon L Bnrof L nd dj L On A Socic Invnory Modl Wi rioring Im IJMMS 5: Bnrof L nd Mmod MG 996 On n Invnory Modl for rioring Im wi Incring im-vrying mnd nd Sorg Jornl of Orionl Rrc Sociy Vol 47: 88- Bo S Gowmi A Cdri KS 995 An EO Modl for rioring Im wi Linir imdndn mnd R nd Sorg Undr Inflion nd im iconing Jornl of Orionl Rrc Sociy Bonl M dj L & Bndid Y 5 Oiml Conrol of A rodcion Sym Wi Invnory- Lvl-ndn mnd Alid Mmic E-No CngK-J nd ing -S993 Hriic for Rlnimn of rioring Im wi Linr rnd in mnd Jornl of Orionl Rrc Sociy Vol 44:35-4 Cng H-J nd y C-Y 999 An EO Modl for rioring Im wi im Vrying mnd nd ril Bclogging Jornl of Orionl AIEMS 8 rocding of 9 Ai ific Indril Enginring & Mngmn Sym Confrnc N Bli INONESIA cmr 3rd
8 AIEMS 8 rocding of 9 Ai ific Indril Enginring & Mngmn Sym Confrnc Rrc Sociy Vol 5: Yild Mngmn 76-8 v U 989 On Hriic Invnory- Rlnimn Rl for Im wi Linrly Incring mnd Incororing Sorg Jornl of Orionl Rrc Sociy Vol 4: v 99 Srvy of Lirr on Coninoly rioring Invnory Jornl of Orionl Rrc Sociy Vol 4: Go SK & Cdri K S 4 An Ordr- Lvl Invnory Modl For A rioring Im Wi Will iriion riorion im-dric mnd And Sorg AMO - Advncd Modling nd Oimizion Vol 6: -35 Giri BC Gowmi A & Cdri K S 996 An EO Modl for rioring Im wi im Vrying mnd nd Co Jornl of Orionl Rrc Sociy Vol 47: Gowmi A nd Cdri K S 99 An EO Modl for rioring Im wi Sorg nd Linr rnd in mnd Jornl of Orionl Rrc Sociy Vol 4 No 5- Jml A M M Srr B R & Wng S 997 An Ordring olicy for rioring Im wi Allowl Sorg nd rmiil ly in ymn Jornl of Orionl Rrc Sociy Vol 48: M N J & S N H 3 An invnory modl for drioring im wi xonnilly incring dmnd nd org ndr inflion nd im diconing Invigção Orcionl 3 3- Nmi S 98 ril Invnory ory: A Rviw Orion Rrc Vol 3: dmnn G nd Vr 995 EO modl for ril im ndr oc dndn lling r Eron Jornl of Orionl Rrc Rf R 99 Srvy of Lirr on Coninoly rioring Invnory Modl Jornl of Orionl Rrc Sociy Vol 4 No 7-37 Roy A 8 An invnory modl for drioring im wi ric dndn dmnd nd im-vrying olding co AMO-Advncd Modling nd Oimizion Vol : 5-37 S Y K nd J J 99 A Singl-riod Socic Invnory Modl ndr Inflnc of Mring olici Jornl of Orionl Rrc Sociy Vol 4: mnd nd ril Bclogging Jornl of Orionl Rrc Sociy Vol 54: ng J- Yng H -L & Oyng L-Y 3 On n EO Modl for rioring Im wi im-vrying AUHOR BIOGRAHIES Wydi Soo i lcrr in rmn of Indril Enginring Fcly of Enginring Univriy of Sl Mr H i cndid of from Bndng Ini of cnology nd i dirion i ly cin mngmn r Hi dy i fndd y BS colri rogrm from ircor Gnrl of Higr Edcion Miniry of Nionl Edcion Rlic of Indoni Hi mil ddr i wydioo@dnicid Snor Nr Bgi i rofor in rmn of Indril Enginring Bndng Ini of cnology H oind i dgr in Logiic Sym nd rodcion Mngmn from Univri d Aix-Mrill III Frnc Hi rrc inr i logiic nd ly cin dvlomn H lid mny r in vrl nionl nd inrnionl logiic ym jornl Hi mil ddr i nor@miliicid N Bli INONESIA cmr 3rd
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