Fractional Order EOQ Model with Linear Trend of Time-Dependent Demand

Transcription

1 I.J. Iellige Sysems d Applicios, 5,, -5 Published Olie Februry 5 i MES (hp://.mecs-press.org/ OI:.585/ijis.5..6 Frciol Order EOQ Model ih Lier red o ime-epede emd Asim Kumr s, p Kumr Roy eprme o Applied Mhemics, Idi Isiue o Egieerig Sciece d echology, Shibpur, Horh, Wes- Begl, 7, Idi Emil: sd.mh@gmil.com Absrc I his pper e iroduce he clssicl EOQ model ih lier red o ime-depede demd hvig o shorges usig he cocep o rciol clculus. he pplicio o rciol clculus hs bee lredy used i clssicl EOQ model here he demd is ssumed o be cos. I his prese ricle rciol diereil clculus c be used o describe EOQ model ih ime-depede lier red o demd o develop more geerlized EOQ model. Here, e o discuss more deeply is role s ool or describig he rdiiol clssicl EOQ model ih ime depede demd. Ide erms Frciol diereiio, Frciol Iegrio, Frciol iereil Equio, Se up os, Holdig os, Ecoomic Order Quiy. I. INROUION Frciol clculus geerlizes derivive d iegrio o ucio o o-ieger order. his geerlizio is rher old problem, s demosred by correspodece, hich lsed severl mohs i 695, beee Leibiz d L Hopil. My oher mous scieiss o he ps sudied d coribued o he developme o rciol clculus i he ield o pure mhemics[-6].i rece yers he cocep o rciol diereil clculus hs bee pplied o severl ields o egieerig, sciece d ecoomics[5],[6],[]. Some o he res here Frciol lculus hs mde impor role h re icluded viscoelsiciy d rheology, elecricl egieerig, elecrochemisry, biology, biophysics d bioegieerig, elecromgeic heory, mechics, luid mechics, sigl d imge processig heory, pricle physics, corol heory[5] d my oher ield[7], [5]. Oly recely, rciol clculus s pplied o clssicl EOQ model o geerlize his model i operio reserch. I previous pper [], e hve discussed ho he rciol clculus c uilizes o develop he clssicl EOQ model o geerlize EOQ model i operio reserch. I priculr, e hve see rciol clculus hs poeiliy o pply his cocep i y oher EOQ model. I his sese e represe he more geerlize EOQ model usig he brod cocep o rciol clculus here demd my vry ih ime, sy lierly ised o cos demd. he clssicl EOQ (Ecoomic Order Quiy [],[],[7],[9],[] model ssumes h he demd re is cos. Hoever, i he rel mre, [9] he demd or y produc co be cos. Reserchers hve pid much eio o iveory modellig ih ime depede demd. Silver d Mel [] developed heurisic pproch o deermie EOQ i he geerl cse o deermiisic ime-vryig demd per. oldso [8] discussed he clssicl o-shorge iveory policy or he cse o lier, ime depede demd. his reme s ully lyicl d much compuiol eor s eeded i order o ge he opiml soluio. Silver [], usig Silver-Mel heurisic obied pproprie soluio procedure or he cse o posiive lier red i demd o reduce he compuiol eor eeded i oldso [8]. Subseque coribuios i his ype o modellig cme rom reserchers such s Richie ([7],[8], Kics d oldso [], d ohers. Here e hve pplied he cocep o derivive/iegrls ih emphsis o puo d Riem-Liouville rciol derivives [],[] d hve some ieresig resuls d ides[] h demosre he geerlized EOQ bsed iveory model. Frciol derivives d rciol iegrls hve ieresig mhemicl properies h my be uilized o develop our moivio. I his ricle, irs e give shor descripio o geerl priciples, deiiios d severl eures o rciol derivives/iegrls d he e revie some o our ides d idigs i eplorig poeil pplicios o rciol clculus i iveory corol model. I secio II, e represe bsic cocepio o Frciol lculus d shor hisory, descripio reled o Frciol iereil lculus. I secio III, e represe he bsic cocep o lssicl EOQ model. I secio IV, e iroduce our mi or hich emphsizes o echiques d procedure or idig our opimum resuls. Filly, I secio V, e prese he coclusio o our or. II. A SHOR ESRIPION ON FRAIONAL IFFERENIAL ALULUS he origi o rciol clculus goes bc o Neo d Leibiz i he seveieh ceury. S.F Lcroi s he irs o meio i some o pges derivive o rbirry order i 7 pges e boo o 89. opyrigh 5 MES I.J. Iellige Sysems d Applicios, 5,, -5

2 Frciol Order EOQ Model ih Lier red o ime-epede emd 5 He developed he ormul or he h derivive o y m, m is posiive ieger, y m! m, (. ( m! here (m is ieger. Replcig he coril symbol by he ell-o Gmm ucio, he obied he ormul or he rciol derivive, ( ( (, (. Where, re rciol umbers. I priculr he hd, / ( / (. (. ( Agi he orml derivive o ucio is deied s, d ( ( lim h ( h h lim h lim h ( h h ( ( ( h ( h h, (. ( Ierig his operio yields epressio or he h derivive o ucio. As c be esily see d proved by iducio or y url umber, ( Where lim h h Or equivlely, r (! r r!( r! ( lim h h r (+(-rh. (.5 r ( ( rh r r. (.6 (.7 he cse o c be icluded s ell. he c h or y url umber, he clculio o h derivive is give by eplici ormul (.5 or (.7. No he geerlizio o he coril symbol (! by he gmm ucio llos! r r!( r! ( ( r ( r (.8 his is lso vlid or o-ieger vlues o. hus o usig o he ide (.8, rciol derivive leds s he limi o sum give by ( ( lim r ( ( ( r ( r rh. (.9 h h r Provided he limi eiss. Usig he ideiy ( r ( ( r ( r ( he resul (.9 becomes, h h r (. ( r ( lim ( rh (. ( ( r Whe α is ieger, he resul (.9reduce o he derivive o iegrl order s ollos i (.5. Agi i 97 Mrchud ormuled he rciol derivive o rbirry order α i he orm give by, ( ( ( ( ( ( (, Where <α< (. I 987, Smo e l hd sho h (. d (.9 re equivle. Replcig by (-m i (.7, i c be sho h ( m m m lim h ( rh h r r Where ( m ( ( m (. m m( m ( m...( m r r r! (. his observio urlly leds o he ide o geerlizio o he oios o diereiio d iegrio by lloig m i (. o be rbirry rel or eve comple umber. A. Frciol derivives d iegrls he ide o rciol derivive or rciol iegrl c be described i oher diere ys. Firs, e cosider lier o homogeeous h order ordiry diereil equio, y (, b c (.. he {,,,,..., - } is udmel se he correspodig homogeeous equio y. ( is y coiuous ucio i [b,c], he or y (b,c, y( ( (! ( (.. opyrigh 5 MES I.J. Iellige Sysems d Applicios, 5,, -5

3 6 Frciol Order EOQ Model ih Lier red o ime-epede emd is he uique soluio o he equio (.. ih he iiil d y ( (, or. Or equivlely, y( ( ( ( ( (.. Replcig by,here Re(> i he bove ormul (..,e obi he Riem-Liouville deiiio o rciol iegrl h s repored by Liouville i 8 d by Riem i 876 s Where ( ( J ( ( ( ( J ( ( ( ( (.. is he Riem-Liouville iegrl operor. Whe,(.. is he Riem deiiio o iegrl d i -, (.. represes Liouville deiiio. Iegrl o his ype ere oud o rise i heory o lier ordiry diereil equios here hey re o s Eulier rsorm o irs id. I d >,he he Lplce rsorm soluio he iiil vlue problem y((, >, y ( (, (..5 is y (s s (s (..6 Where y (s d (s re respecively he Lplce rsorm o he ucio y( d (. he iverse Lplce rsorm gives he soluio o he iiil vlue problem (..5 s y( ( Agi rom (..6 e hve hus e hve i.e L { s y( L { y( s} L { s ( s} L s ( { ( s} ( s} ( (..7 ( ( ( y( ( (..8 L { s ( s} ( ( ( his is he Riem-Liouville iegrl ormul or ieger. Replcig by rel gives he Riem- Liouville rciol iegrl (.. ih. I comple lysis he uchy iegrl ormul or he h derivive o lyic ucio (z is give by (z! i ( ( z (..9 Where is closed coour o hich (z is lyic, d z is y poi iside d z is pole. I is replced by rbirry umber d by (, he derivive o rbirry order c be deied by, (z ( ( i ( z (.. here z is o loger pole bu brch poi. I (.. is o loger pproprie coour, d i is ecessry o me brch cu log he rel is rom he poi z> o egive iiiy. hus e c deie derivive o rbirry order by loop iegrl ( i ( z (z ( (.. Where ( z ep[-(+l(-z] d l(-z is rel he -z>. Usig he clssicl mehod o coour iegrio log he brch cu coour, i c be sho h z (z ( i ( z ( ( [ ep{ pi( }] ( z ( i z ( z ( ( z (.. hich grees ih Riem-Liuville deiiio (.. ih z, d, he is replced by - B. Frciol Iegrio, Frciol iereil Equio usig Lplce rsormed Mehod: opyrigh 5 MES I.J. Iellige Sysems d Applicios, 5,, -5

4 Frciol Order EOQ Model ih Lier red o ime-epede emd 7 Oe o he very useul resuls is ormul or Lplce rsorm o he derivive o ieger order o ucio ( is give by ( L{ ( } s (s - ( s ( (.. s (s - s (s - Where ( ( ( s ( (.. s ( ( c represes he physiclly relisic give iiil codiios d (s beig he Lplce rsorm o he ucio (. Lie Lplce rsorm o ieger order derivive, i is esy o sho h he Lplce rsorm o rciol order derivive is give by L{ (} s (s - s s [ ( ] (.. s (s -, (.. c here - d c ( ] [ (..5 represes he iiil codiios hich do o hve obvious physicl ierpreio. osequely, ormul (.. hs limied pplicbiliy or idig soluios o iiil vlue problem i diereil equios. We o replce by ieger-order iegrl J ( d ( ( is used o deoe he iegrl order derivive o ucio (. I urs ou h J J I, I. (..6 his simply mes h is he le ( o he righ iverse o J. I lso ollos i (..9 ih h J ( (- ( (!, > (..7 Similrly, c lso be deied s he le iverse o J.We deie he rciol derivive o order > ih - by ( ( J ( ( [ ( ( d ( O usig (.. or, ( ( ( ] (..8 ( d ( Where is ieger d he ideiy operor I is deied by ( J ( I((, so h J I,. ue o he lc o physicl ierpreio o iiil d c i (.., puo d Mirdi doped s lerive e deiiio o rciol derivive o solve iiil vlue problems. his e deiiio s origilly iroduced by puo i he orm ( J ( ( ( d ( ( Where - d is ieger. I ollos rom (..8 d (..9 h ( J ( (..9 J ( ( (.. Uless ( d is irs (- derivives vish. Furhermore, i ollos (..9 d (.. h J J ( J J ( ( ( ( ( (.. his implies h (! [( ( ] ( ( ( ( ( ( (.. his shos h puo s rciol derivive ( icorpores he iiil vlues (, or,,,.,-. he Lplce rsorm o puo s rciol derivive (.. gives ieresig ormul L{ s ( - ( s (} (s (.. opyrigh 5 MES I.J. Iellige Sysems d Applicios, 5,, -5

5 8 Frciol Order EOQ Model ih Lier red o ime-epede emd ( rsorm o ( his is url geerlizio o he correspodig ell o ormul or he Lplce he d c be used o solve he iiil vlue problems i rciol diereil equio ih physiclly relisic iiil codiios. III. BASI ONEP ON LASSIAL EOQ MOEL he order quiy mes he quiy produced or procured i oe producio cycle or order cycle (he ime period beee plceme o o successive orders (or producio is reerred o s order cycle (or producio cycle. his is lso ermed re-order quiy he he size o order icreses, he order coss (cos o purchsig, ispecio, ec. ill decrese heres he iveory crryig coss ill icrese.hus i he producio or purchsig cse, here re o opposie coss, oe ecourges he icrese i he order size d he oher discourges. Ecoomic order quiy (EOQ is h size o order hich miimizes ol ul coss o crryig iveory d cos o orderig. Noios d Assumpios: emd re Q Order quiy U Per ui cos Holdig cos per ui Se up cos q( Soc level Orderig iervl ul vrible o i geomeric progrmmig I clssicl EOQ bsed iveory model, e lredy hve dq(, or, oherise. (. Wih he iiil codiio q(q d ih he boudry codiio q(. H q( ( Q [ Q ] ( Q (. [o usig (.] ol cos, ( Purchsig cos(p+holdig cos(h+se up cos(s UQ +. (.5 ol verge cos over [,] is give by A( [ UQ ] UQ (.6 he he clssicl EOQ model is Mi A(U+ (.7 Subjec o, >. Solvig (.7 e c sho h A( ill be miimum or d A ( U+ (.8. (.9 Fig.. evelopme o iveory level over ime By solvig he equio (., e hve q(q-, or (. Ad o usig he boudry codiio q(, e hve Q. (. Holdig cos, IV. GENERALIZE EOQ MOEL WIH LINEAR REN OF EMAN We o geerlize our discussio by ccepig he equio (. s diereil equio o rciol order ised o he lier order. i.e e here cosider h demd( vries i rciol order sy, here iseous iveory level d q( or oherise. (. here +b ;, b re coss. he e hve he equio (. s d q( ( b or (., oheris opyrigh 5 MES I.J. Iellige Sysems d Applicios, 5,, -5

6 Frciol Order EOQ Model ih Lier red o ime-epede emd 9 Wih he sme iiil d boudry codiio s described i he previous problem i equio (.. i.e q(q d ih q(. Equio (. c be rerie s q( -(+b or (. oherise. Where J derivive s described i (..9 d is he puo rciol d. o solve he iiil vlue problem o rciol order diereil equio (. e pply he Lplce rsorm mehod. So ig Lplce rsorm o he equio (., e hve { {q(} - {+b} s q( s s q( b, s s q(s beig Lplce rsorm o q(. b Q s s Q b q s s s s q(s (s s ig Lplce iversio o bove equio e hve, q( b L { q( s} Q ( ( So he iveory level y ime bsed o α ordered decresig re o demd is q ( b ( ( Q or. (. o usig he boudry codiio q( implies h Q b ( ( A. Geerlized Holdig os: No he Holdig cos o rciol order, sy i.e. H ( q( se: For d, Holdig cos is H,( q( q( ( Q b O usig (. & (.5 or α, e hve (.5 (.. b H,( ( (.. se: For, Holdig cos is H, ( q ( ( b Q ( ( [ Q ] b ( ( [ b ( ( ] (usig (.5 (.. se: For, Holdig cos o order is H ( q(, Where q( Q b No { q( } { Q b q( { } Q b ( ( ( For Holdig cos [ Q b ( ( ( } ] (.. b [( b ] ( ( ( usig (.5 or α ( { b } (..5 ( ( se : For y d, Holdig cos is ( q (, H, Where, b q ( Q { q ( } ( ( ( b ( { [ Q ]} opyrigh 5 MES I.J. Iellige Sysems d Applicios, 5,, -5

7 5 Frciol Order EOQ Model ih Lier red o ime-epede emd q ( { } Q b ( ( ( For Holdig cos Q b [ ] (..6 ( ( ( [ b b ] { } ( ( ( ( ( Usig (.5 [ { } ( ( ( b { }] ( ( ( B. Geerlized ol Averge os (..7 ol cos( Purchsig cos(p + Holdig cos(h + Se up cos(s. ol Averge os (A [ol os(] se: For α d β, Averge os A,(, b [ U( b ( Ub ( U b [ UQ H ( ] ] E + F + G Where E Ub, Here he EOQ model is, A (, Mi + (.. F ( U b & E + F + G G +, (.. subjec o, (.. c be e s priml geomeric progrmmig problem ih degree o diiculy (. ul orm o (.. M d( Subjec o, F G, (.. + +, (ormlized codiio (.. + -, (orhogol codiio (..5,,. Priml-dul relios re, F d( (..6 G d( (..7 d( (..8 Usig (..6 d (..7&(..8 e hve, F G (..9 Ad G F (.. No solve or,, rom hree sysem o olier equios (.., (..5 d (.. d obied he soluios s, d d he rom he relio (..6, e ill ble o obi or hich A is miimum. i.e e ill ble o obi (, A ( s he miimum o, d Q(. se: For y > d, Here, A i (.. (, (, UQ+ [ b ] + ( ( b U[ ] [ b ] ( ( ( ( E F G (.. U b E Where F, (, & bu G ( he ol verge cos A, (, E F G Here geerlized EOQ model is, Mi A (, ( F G, (.. E subjec o, (.. c be e s priml geomeric progrmmig problem ih degree o diiculy (. ul orm o (.. E M d( F G, (.. Subjec o, + + +, (ormlized codiio (.. opyrigh 5 MES I.J. Iellige Sysems d Applicios, 5,, -5

8 Frciol Order EOQ Model ih Lier red o ime-epede emd 5 (++ +(α- -, (orhogol codiio (..5 Priml-dul relios re, E F,,,. d( (..6 d( (..7 G d( (..8 d ( (..9 Usig (..6 d (..7,(..8 & (..9 e hve, F E (.. E G F (.. Ad G F (.. No solve or,,, rom our sysem o o lier equios (.., (..5 d (.. & (.. d obied he soluios s,, & ble o obi d he rom he relio (.., e ill or hich (, i.e e ill ble o obi ( A is miimum. A s he miimum o, A i (.. d Q(. (, se: For d or y, e hve he Holdig cos, H, ( ( { b } ( ( [rom(..5] he ol cos ( UQ+ ( { b } + ( ( Where Q b ol verge cos A (, ( [ UQ { b } ] ( ( [ U( b + ( { b } ( ( + ] (.. E F G H U Where, E bu, F, ( b G, & H ( ( So our model is mi A, ( E F G H (.. Subjec o; (.. c be e s priml geomeric progrmmig problem ih degree o diiculy (. ul orm o (.. M d( F G H Subjec o, + + (ormlized codiio (..5 ( (orhogol codiio (..6 he priml-dul relios re F d( G d(, H d ( d ( Ad O usig he bove priml dul relio e ge H G (..7 G H F (..8 Ad G F H (..9 No solve or,,, rom sysem o our o-lier equios (..5, (..6 d (..8 & (..9 d obied he soluios s,, & ble o obi d he rom he relio (..7, e ill or hich, ( A is miimum. i.e e ill ble o obi A ( s he miimum o (,, A i (.. d Q(. se: For y α > d y >, e hve he Holdig cos s H, ( opyrigh 5 MES I.J. Iellige Sysems d Applicios, 5,, -5

9 5 Frciol Order EOQ Model ih Lier red o ime-epede emd b [rom(..7] he ol cos, ( UQ+ { } ( ( ( { } ( ( ( { } ( ( ( + b { } ( ( ( Where Q is give i(.5 (.. ol verge cos is give by A, ( { UQ+ ( ( ( ( + } b ( ( ( ( E + F G H (sy (.. Ub Where E (, F U, ( G ( ( (, H b ( ( ( So our model is mi A, ( E + F G H subjec o; No o miimize A, (, e pply geomeric progrmmig mehod, d he degree o diiculy( is. E M d( Subjec o, F G H (ormlized codiio (.. (- + ( ( (orhogol codiio (.. 5 5,,,,. 5 Agi he priml-dul vrible relios re give by E d( F d( G d ( H d ( 5 d( O usig he bove priml dul relio e ge H G (.. E G F H (..5 E F (..6 H 5 FG (..7 No solve or,,,, 5 rom bove ive sysem o o-lier equios (.., (.. d (..5, (..6 &(..7 d obied he soluios s,,,, 5 d he rom he relio (.., e ill ble o obi or hich A, ( is miimum. i.e e ill ble o obi A, ( s he miimum o A, ( i (.. d Q( V. ONLUSION I his pper, e hve developed clssicl EOQ model o geerlized EOQ model usig he cocep o rciol order diereil clculus o he ssumpio h he demd o be lierly icresig ucio o ime d o shorge o be lloed. Alhough rciol clculus is much more compliced, sill i hs poeiliy o describe y oher clssicl model o more geerl model precisely. Here i is sho h clssicl EOQ model is he priculr cse o geerlized EOQ model. I uure or, rciol diereil clculus c be used o develop y oher EOQ model i is more geerlized orm. REFERENES [] Aser. S, Iveory orol, secod ediio, chper,pp. 5-6.Librry o ogress orol Number:6987, ISBN-: (HB, 6 by Spriger Sciece +Busiess Medi, LL. opyrigh 5 MES I.J. Iellige Sysems d Applicios, 5,, -5

10 Frciol Order EOQ Model ih Lier red o ime-epede emd 5 [] Bechohr. U, Hmi. S, Nouys. S.K, Boudry vlue problem or diereil equio ih rciol order, ISSN 8-698(elecroic, 8-765(pri, Volume (8, -. [] hu o., A EOQ model ih deeriorig iems uder ilio he supplier credis lied o order quiy Ieriol Jourl o Producio Ecoomics, ; Volume 88, Issue, 8; Pges 7-6. [] s. A.K, Roy..K, Role o Frciol lculus o he Geerlized Iveory Model, ISSN-9-7X, Volume 5, No-, Februry.JGRS. [5] s. S, (8, Fuciol Frciol lculus or sysem Ideiicio d orols, ISBN Spriger Berli Heidelberg Ne Yor 8. [6] ebh. L (, Rece Applicio o Frciol lculus o Sciece d Egieerig, IJMS ; 5, -. [7] ebh. L (, Frciol Iegrl d Frciol iereil equio i Fluid Mechics, o pper i Frc. lc. Al.,. [8] oldso, W.A., Iveory repleishme policy or lier red i demd - lyicl soluio, Operiol Reserch Qurerly, 8 ( [9] Geues. J, She. J.Z, Romeij. H.E, Ecoomic orderig ecisio ih Mre hoice Fleibiliy, OI./v.9, Jue. [] Hiler. R, Applicio o rciol clculus i Physics, orld scieiic, Sigpore,, Zbl [] Kics. P, d oldso, W.A., Irregulr demd: ssessig rough d redy lo size ormul, Jourl o Operiol Reserch Sociey, ( [] Kleiz. M d Osler..J, A child grde o Frciol erivives, he college Mhemics Jourl, Mrch, Volume, Number, pp [] Miller. K.S d Ross. B, A Iroducio o he Frciol lculus d Frciol iereil Equios, opyrigh 99 by Joh Wiley & Sos, A Wiley-Iroducio Publicio, Ide, ISBN (cid ree [] Oldhm. K.B, Spier.J, he Frciol lculus, opyrigh 97 by Acdemic Press IN.(LONON L. [5] Podluby.I, he Lplce rsorm mehod or lier diereil equios o he rciol order, Slov Acdemy o Sciece Isiue o Eperimel Physics, Jue 99. [6] Podluby. I, Geomeric d Physicl ierpreio o Frciol Iegrl d Frciol iereiio, Volume 5,Number (, A ieriol Jourl o heory d Applicio, ISSN -5. [7] Richie. E., Prcicl iveory repleishme policies or lier red i demd olloed by period o sedy demd, Jourl o Operiol Reserch Sociey, ( [8] Richie. E., he EOQ or lier icresig demd: simple opiml soluio Jourl o Operiol Reserch Sociey, 5 ( [9] Roch. B, Origis o Ecoomic Order Quiy ormul, Wsh Bur Uiversiy school o busiess orig pper series, Number 7, Jury 5. [] Silver.E.A A simple iveory repleishme decisio rule or lier red i demd, Jourl o Operiol Reserch Sociey, ( [] Silver. E.A., d Mel. H.., A simple modiicio o he EOQ or he cse o vryig demd re, Producio d Iveory Mgeme, ( ( [] h. A.H, Operios Reserch: A Iroducio, chper/8 h ediio, ISBN [] X. Zhg, Some resuls o lier rciol order imedely sysem, Appl. Mh.omp. (8 7-. Auhors Proile Asim Kumr s, Grdued i Mhemics rom Uiversiy o lcu, lcu, Idi (5, Mser i Applied Mhemics rom uiversiy o lcu, lcu, Idi (7. He is currely reserch scholr o Idi Isiue o Egieerig Sciece d echology (IIES uder he guidce o Pro. p Kumr Roy. He is ieresed i he ield o severl pplicio o rciol clculus i operio reserch model d uzzy mhemics. Pro. p Kumr Roy, proessor o Applied Mhemics Idi Isiue o Egieerig Sciece d echology (IIES,Idi. His reserch re i Fuzzy d Iuiioisic Fuzzy se heory, Iveory, rsporio, Relibiliy Opimizio, Porolio Opimizio, Fuzzy d Sochsic Opimizio, ec. He hs lmos publicio i severl ield o Applied Mhemics i severl ieriol jourl. N.B: Formlly he isiuio s med s Begl Egieerig d Sciece Uiversiy. Presely i is remed s Idi Isiue o Egieerig Sciece d echology. Ho o cie his pper: Asim Kumr s, p Kumr Roy,"Frciol Order EOQ Model ih Lier red o ime- epede emd", Ieriol Jourl o Iellige Sysems d Applicios (IJISA, vol.7, o., pp.-5, 5. OI:.585/ijis.5..6 opyrigh 5 MES I.J. Iellige Sysems d Applicios, 5,, -5