An optimal EOQ model for perishable products with varying demand pattern

Transcription

1 Ffh Inernonl Workshop on Compuonl Inellgence & Applcons IEEE SMC Hroshm Chper, Hroshm Unversy, Jpn, Novemer, &, 9 An opml EOQ model for pershle producs wh vryng demnd pern Adul, Irheem nd Asuo Mur Deprmen of Inellgen Mechncl Sysems, Dvson of Indusrl Innovon Scences, Grdue School of Nurl Scence nd echnology, Okym Unversy, Okym, 3--, sushmnk, Okym, Jpn rheem@mssysokym-ucjp mur@mssysokym-ucjp Asrc he demnd pern for mos pershle producs vres durng her lfe cycle n he mrke hese vrons mus e properly refleced n nvenory mngemen n order o preven unnecessry sock-ou or excess nvenory wh ssoced ncrese n cos In hs pper, mul-perod economc order quny (EOQ model for mngng he nvenory of pershle ems hvng vryng demnd pern s presened he model ws formuled usng generl rmp-ype demnd funcon h llows hree-phse vron n demnd pern hese phses represen he growh, he sedy nd he declne phses commonly experenced y he demnd for mos producs durng her lfe cycle n he mrke he model generes replenshmen polces h gurnees opml nvenory cos for ll he phses Numercl expermens nd sensvy nlyss were crred ou o demonsre he suly of he model for wde rnge of sesonl producs Resul of he expermens reveled h he pons whch demnd pern chnges re crcl pons n mngng nvenory of producs wh rmp ype demnd I INRODUCION Pershle ems re hose ems hvng mxmum usle lfeme eg foodsuff, humn lood, phoogrphc flms, ec Fresh produce, mes nd oher foodsuffs deerore grdully nd ecome unusle fer cern mes In drugs sores, medcnes hve fxed shelf lves whle whole uns of lood, phoogrphc flms re ypcl exmples of ems wh lmed useful lfe mes he mxmum usle lfeme my e fxed, n whch cse he ems ecome unusle he end of fxed perod eg humn lood Lfeme for some producs s ssumed o e rndom vrle wh s proly dsruon represened y gmm, Weull, exponenl or ny oher dsruon pern Due o s mporn connecon wh commonly used ems n dly lfe, nvenory modelng for pershle ems connues o receve consderle enon mong reserchers Demnd s very mporn componen of he nvenory sysem, s he nvenory prolem wll no exs whou he nure of he demnd deermnes he nure of model h wll e developed o solve n nvenory prolem he demnd my e sc or dynmc hroughou he lfeme of he produc or he sysem Sc demnd re of rre occurrence n prcce s demnd for producs ofen vry wh severl fcors lke me, prce, sock ec Some pershle producs re lso sesonl n nure nd demnd for hem exh vrous pern durng he seson Ghre nd Shrder [] ws he frs o exend he clsscl EOQ formul o pershle ems, wheren consn frcon of on hnd nvenory s ssumed los due o deeroron Susequen works ncorpored me vryng demnd perns n modelng Dve nd Pel [] developed he frs pershle nvenory model wh dynmc demnd n form of lner funcon of me Schn [3] nd ohers mproved Dve s model y relxng he no shorge ssumpon nd ssume complee ckloggng of shorge nd equl replenshmen perod Ler Hrg [4], Chng e l [5], nd ohers nroduced generl connuous log-concve demnd funcon n plce of he lner rend o cer for ems wh oher forms of me dependen demnd funcons However, he demnd pern n hese models s undreconl nd no sule for producs whose demnd pern chnges wh me An Invenory model h cers for vryng demnd perns ws frs proposed y Hll [6] he model comprsed me dependen demnd pern h s comnon of wo dfferen ypes of demnd n wo successve me perods over he enre me horzon hs pern, clled rmp-ype pern, ws oserved y Wu [7] o e common n he cse of new rnd of consumer goods comng o he mrke he demnd re for such ems ncreses wh me up o cern me nd hen ulmely slzes nd ecomes consn Wu [7] developed sngle-perod EOQ model nvenory for ems h deerores Weull re, usng rmp ype demnd re he rmp ype pern ws lso found o e useful for sesonl producs whose demnd vres wh he seson henceforh, severl reserchers hve worked o develop nvenory modelng for ems hvng rmp ype demnd perns he works of Mndl nd Pl [8], Wu [7], nd Deng e l [9] re nole conruons n hs drecon Skour e l [] exended he froners of nvenory models wh rmp ype demnd y nroducng generl rmp ype demnd pern whose vrle pr s ny posve funcon of me All he rmp ype models menoned ove, however, consdered only wo-phse vron n demnd perns hs represens only he growh nd he sle phses of demnd 58

2 In rel lfe, he demnd for some producs does no remn sle forever I usully egns wh ncresng rend, ns pek nd ecomes sedy he mddle of s lfe cycle n he mrke, nd fnlly decreses wh me owrds he end he cycle hs pern s rmp-ype n nure nd hs een oserved o pply o mny pershle ems s well s sesonl ems lke frus, fsh, wner cosmecs, ec (see Cheng nd Wng [] Recenly, Pnd e l [] oserved h demnd of sesonl producs (frus, eg, mngo, ornge, ec, se fsh over he enre me seson s hree folded A he egnnng of he seson ncreses, n he md of he seson ecomes sedy nd owrds he end of he seson decreses nd ecomes sympoc An nvenory model for deerorng sesonl producs wh hree me perods clssfed s me dependen rmpype funcon ws developed y Pnd e l [] Unlke prevous rmp ype models, hs model llows hree-phse vron n demnd perns over me horzon I however resrced he vron of demnd durng he growh nd he declne phse o e of he sme ype I lso ssumed h demnd vron s n exponenl funcon of me hese resrcons re no relsc n mny nvenory suons he demnd pern durng he growh nd he declne phse my e dfferen Lkewse he exponenl ncrese/decrese n demnd hs een noed o e oo hgh nd my no e relsc n some rel mrke suons In hs pper, we develop mul-perod nvenory model for pershle ems usng generl rmp-ype pern h llows hree-phse vron n demnd he model relxes prevous unrelsc resrcons y llowng vrons n demnd perns durng he growh nd he declne phse of demnd A generl connuously ncresng/decresng funcon of me s used for he growh/declne phse of he rmp-ype demnd pern o mke he model sule for dfferen clsses of pershle ems wh me dependen demnd he model seek o exend he works of Deng e l [9] nd ohers y consderng hree-phse vron n demnd nsed of wo-phse vron h neglecs he declne phse of demnd I lso exends he model of Pnd e l [] y llowng vron n demnd pern durng he growh nd declne phse of demnd hese re necessry exensons o enle he model su some rel lfe demnd suons II MODEL ASSUMPIONS AND NOAIONS he followng ssumpons nd noons re used n formulng he models: A consn frcon ( of on-hnd nvenory deerores per un me Replenshmen re s nfne 3 Shorges re no llowed 4 No repr or replcemen of deerored ems durng he perod under revew 5 Invenory holdng cos (H, replenshmen cos (S, nd cos of deerored ems (P re known nd consn durng he horzon 6 he nvenory level ny me ( durng he h replenshmen perod s I ( 7 he Lengh of he h replenshmen perod nd order quny (, nd Q vres long he cycle 8 he sysem hs severl replenshmen perods durng he horzon 9 Demnd re f ( s generl me dependen rmpype funcon of he form: g,, f ( = g(,, h(, g(, h(,, g( = h he funcon g( cn e ny connuously ncresng funcon of me, whle h( s ny connuously decresng funcon of me n he gven nervl Prmeers nd represen he prmeer of he rmp ype demnd funcon he pern f ( s s depced n Fg f( Fgure he Rmp ype demnd pern III MODEL FORMULAION he nvenory sysem consss severl replenshmen perods Ech perod egns wh full nvenory nd ends wh zero nvenory level Durng he h replenshmen perod, consumpon due o demnd nd deeroron rngs he nvenory level o zero he me Replenshmen occurs me nd he cycle repes self he ojecve s o deermne he replenshmen schedules nd order qunes (e, nd Q for he frs nd ll oher susequen perods y mnmzng he ol nvenory cos per un me for ech replenshmen perod Fg shows ypcl schedule Invenory level 3 me Fgure Vron of nvenory level wh me Durng replenshmen perod he demnd pern wll fll no ny of he followng cegores: Consn demnd pern: In hs cse demnd pern does no chnge durng he perod he demnd 59

3 funcon s sngle funcon hroughou he perod Demnd slzon pern: Demnd pern chnges from ncresng o consn pern durng he perod hs s depced n Fg pon 3 Declnng demnd pern: Demnd pern chnges from consn pern o declnng pern hs s depced n Fg pon 4 Sngle perod pern: In hs cse oh slzon nd declne of demnd occurs durng he sme perod hs s commonly encounered when sngle replenshmen s o e mde o cover he enre horzon he Demnd pern chnges wce durng he perod he nlyss of he sysem under hese cses s consdered elow Cse : Consn demnd pern he demnd funcon, f (, n hs cse my e consn or n ncresng/decresng funcon of me he equon of he nvenory sysem for ny replenshmen perod under hs cse s gven y: di ( + I ( + f ( =, ( d Snce he nvenory level s zero me, hen I ( = he soluon o Eq ( ove s gven y Eq ( hus: I ( = e ( e f ( x, ( ol numer of uns n nvenory durng he h replenshmen perod s gven y I I = I ( d, whle ol numer of uns h deerore s gven y I D = I I he ol nvenory cos per un me under hs cse (C s gven y: C = ( S + PI D + HI I = ( S + ( P + H I I (3 Usng Eq ( nd he expresson for I I ove n Eq (3, gves he generl expresson for he ol nvenory cos per un me n hs cse C = ( S + ( P + H ( e ( e f x d (4 he frs condon for mnmzng ol nvenory cos per un me (C s gven y dc d = Applyng hs condon o Eq (4 ove gves: dr S R = d ( P + H (5 R ( e ( e f ( x = d Solvng Eq (5 wh he ppropre demnd funcon gves he opml lengh of he h replenshmen perod,, d C provded > he mnmum pon d he opml order quny (Q, oned usng Eq (, s: ( Q = e f d (6 Cse : Demnd slzon pern he equon of he nvenory sysem n hs cse s s gven elow: di + I ( + g( =,, d (7 di ( + I + g =, d ( Soluon o Eq (7 wh he oundry condon I ( = nd I ( - = I ( + s gven y: I I (, g ( e, ( = e e g( x + e g( ( =, he ol numer of uns crred n nvenory durng he perod s gven n Eq (9 elow I I d I = + I d (9 ol numer of uns h deerore s gven y I D = I I ( he ol nvenory cos per un me (C s gven y: C = ( S + PI D + HI I = ( S + ( P + H I I ( Susung he expresson for I I n Eq ( followed y some smplfcons gves he expresson for he ol cos per un me elow C = g ( ( e e ( e ( e g( x d + ( P + H + S (8 ( Smplfyng Eq ( nd dfferenng wh respec o gves Eq (3 elow: dc = { ( P H / ( e g( ( R ( P H S } d (3 R g ( ( e d + e e = ( e g( x he necessry nd suffcen condon for mnmzng ol dc nvenory cos per un me, C, s =, provded d d C > he mnmum pon Applyng hs condon, d usng Eq (3 gves: P + H e g( ( R ( P + H + S = / (4 6

4 Dfferenng Eq (3 wh respec o, nd usng he frs opmly condon sed n Eq (4 ove, we hve d C d ( P + H e g = (5 Bu g( >, nd e > for ll I herefore follows d C from Eq (5 ove h > All condons for d mnmum vlue of C re hus, ssfed nd Eq (4 gves he opml vlue of he opml order quny, oned usng Eq (8, s shown elow ( d e g( Q = e g + d (6 Cse 3: Declnng demnd pern Demnd pern for he produc chnges from consn vlue nd egn o decrese wh me durng hs phse Equon of he sysem s represened n Eq (7 elow di d di d ( ( + I + I ( + g( ( + h( =,, =, (7 he oundry condons re I ( = nd I ( - = I ( + he soluon o Eq (7 s gven y: I I,, e h x, ( = e e g( + e h( x ( = e (8 he ol numer of uns crred n nvenory durng he perod s gven n Eq (9 elow I I ( d I = + I ( d (9 Usng smlr procedure s n cse, he expresson for he ol cos per un me s gven elow C 3 = + ( e ( e g( + e h( x ( e ( e h( x d d ( P + H + S ( hs expresson s presened n smplfed form n Eq ( ( ( + If ( ( = e e g e h x d R ( + e e h x d C3 ( = S + P H R + ( Applyng he frs condon for mnmzng ol nvenory cos per un me (C 3 o Eq ( gves: dr S R = d ( P + H (3 Solvng Eq (3 gves he opml lengh of he h replenshmen perod ( d C3, provded > he d mnmum pon he opml order quny (Q 3, oned from Eq (8 s s gven elow Q3 = e g( d + e h d (4 Cse 4: Sngle perod pern hs s when chnge n demnd pern occurs wce durng replenshmen cycle he equon of he sysem n hs cse s gven y Eq (5 elow: di d di d di d ( ( + I + I + I ( + g( ( + g( =, =,,, ( + h( =, (5 he soluon o Eq (5 wh he oundry condons; I ( =, I ( - = I ( +, nd I ( - = I ( + s gven y: I I I [ + F ] + F,, [ e g + F ],, [ ] e h x, ( = e e g( x ( = e ( = e x x F, = Where, = e h( x F e g( (6 he ol numer of uns crred n nvenory durng he perod s gven y: I I d I d I = I d + + (7 he ol nvenory cos per un me (C 4 s gven y: ( e ( F + F + F e ( F + F d 3 C 4 = + ( 4 ( ( 5 + e F d d ( P + H + S x 3, 4 5 = (8 Where F = e g( x F = e g(, F e h( x As n Cse 3 ove, he expresson n Eq (8 cn e smplfed s shown n Eq (9 elow C4 ( = S + P H R 3 + (9 ( e ( F + F + F ( e ( F + F4 d e ( F d 3 d Where, R 3 = + + ( 5 Applyng he frs condon for mnmzng ol nvenory cos per un me (C 4 o Eq (9 gves: dr3 S R 3 = d ( P + H (3 6

5 Solvng Eq (3 gves he opml lengh of he h replenshmen perod ( d C4, provded > he d mnmum pon he opml order quny (Q 4, oned from Eq (6 s s gven elow ( d + e g( d e h( Q4 = e g d + (3 Wh he ove equons, he opml lengh of ll replenshmen perods nd her correspondng cos nd order qunes cn e oned hroughou he horzon A summry of he resul s presened elow: ( Replenshmen perod wh no chnge n demnd pern: Opml lengh of perod,, s gven y Eq (5 Opml nvenory cos durng perod s C (Eq (4 Opml order quny s Q (Eq (6 ( Replenshmen perod wh chnge n demnd pern (ncresng o consn: Opml lengh of perod,, s gven y Eq (4 Opml nvenory cos durng perod s C (Eq ( Opml order quny s Q (Eq (6 (c Replenshmen perod wh chnge n demnd pern (consn o decresng: Opml lengh of perod,, s gven y Eq (3 Opml nvenory cos durng perod s C 3 (Eq ( Opml order quny s Q 3 (Eq (4 (d Replenshmen perod wh doule chnge n demnd pern (ncresng - consn - decresng: Opml lengh of perod,, s gven y Eq (3 Opml nvenory cos durng perod s C 4 (Eq (8 Opml order quny s Q 4 (Eq (3 IV SOLUION ALGORIHM o fnd he opml replenshmen schedules, coss, nd order qunes over he enre me horzon, he followng smple lgorhm oulne he procedure o follow All replenshmen perods re solved usng he procedure for Cse excep he chnge pons when demnd pern chnges Sep : Deermne ll he opml vlues (, C, Q for he frs nd susequen replenshmens usng Eq (5, Eq (4 nd Eq (6 wh ppropre demnd funcon Sep : Check for chnge pons y comprng wo successve vlues of (e nd + (Noe: + = +, = Sep 3: If < nd < + Move sep ckwrd nd reclcule he opml vlues (, C, Q for he perod usng Eq (4, Eq ( nd Eq (6 Sep 4: If < nd + >, Move sep ckwrd nd reclcule he opml vlues (, C 3, Q 3 for he perod usng Eq (3, Eq ( nd Eq (4 Sep 5: If < nd + >, Move sep ckwrd nd reclcule he opml vlues (, C 4, Q 4 for he perod usng Eq (3, Eq (8 nd Eq (3 V NUMERICAL EXPERIMENS wo ses of expermens were conduced usng numercl d o nlyze he model ehvor, nd demonsre s suly for pplcon o sesonl producs he numercl d for he expermens, presened elow, were dped from he models of Pnd e l [] nd Cheng nd Wng [] nvolvng pershle sesonl producs Numercl d for Expermen : f = A exp[ { ( μ H (, μ ( γ H (, γ }] μ = yers ; γ = 3yers; = 3; A = 3; S = $8 per order P = $ per un; H = $ per un; = H (,μ nd H (,γ, re Hevsde funcons Numercl d for Expermen : + 4, < μ, f ( = 8, μ μ, μ μ = weeks ; μ = 4weeks; 4 = S = $ per order; P = $3 per un; H = $ per un he demnd funcons, f (, ove represens sngle cses of he generl rmp ype model presened n hs pper he vrle pr of he generl rmp funcon s represened y exponenl funcon n expermen, nd y lner funcon n expermen he numerc d were ppled o he model one fer he oher, followng he lgorhm oulned erler, nd he resul genered re shown n le nd le le Resul of Expermen n C Q le Resul of Expermen n C Q

6 VI DISCUSSION AND SENSIVIY ANALYSIS Expermen demonsred he pplcon of he model o pershle sesonl producs wh exponenl rmp-ype demnd whle expermen shows he suly of he model for producs wh rpezodl demnd From he les of resul ove, cn e oserved h he vlues of opml coss nd order qunes for replenshmen perods re generlly close excep pons where here s chnge n demnd pern (eg, n=3 nd 7 n expermen, nd n=3 nd 5 n expermen hs shows h hese pons re crcl pons h mus e well noed n mngng nvenory of pershle producs wh rmp ype demnd he resul shown n le s he sme wh h oned y Pnd e l [] when he me horzon s no fxed excep one of he chnge pons (n=7 he dfference n resul hs pon s eleved o e due o compuonl error on he pr of Pnd e l [] For furher nlyss of he model ehvor, he resul of sensvy nlyss crred ou sed on he mos crcl replenshmen perod n he frs expermen s shown n le 3 elow le 3 Sensvy nlyss sed on Expermen Prmeers % Chnge n vlue of prmeers % Chnge n Cos % Chnge n Order Quny A P H S VII CONCLUSION he EOQ model developed n hs pper generes opml replenshmen schedules nd order quny for pershle producs hvng rmp ype demnd he model exends he works of Deng e l [9] nd ohers y consderng hreephse vron n demnd nsed of wo-phse vron I lso exends he model of Pnd e l [] y llowng vron n demnd pern durng he growh nd declne phse of demnd Applcon of he model o pershle sesonl producs wh dfferen demnd perns showed h s sule for wde rnge pershle nd sesonl producs he sensvy nlyss showed h he model s sensve o chnges n demnd re (A, nvenory holdng cos (H, nd replenshmen cos (S whle s sensvy o deeroron re, nd cos of deeroron (P s low hs ndces, mong oher hngs, h he demnd re s n mporn fcor n he model I ws lso shown h he chnge pons of demnd pern re crcl pons h mus e well noed y mngers n usng he model Opml replenshmen polcy genered y he model wll sss nvenory mngers n ensurng mnmum nvenory coss for wde rnge of pershle ems Consderon of shorges nd prl ckloggng of demnd re recommended fuure works o furher exend he pplcon of he model REFERENCES [] P M Ghre, G F Schrder A model for exponenl decyng nvenory, Journl of Indusrl Engneerng, Vol 4, pp 38-43, 963 [] U Dve, L K Pel (, S polcy nvenory model for deerorng ems wh me proporonl demnd, Journl of he Operonl Reserch Socey, Vol 3, pp 37 4, 98 [3] R S Schn On (,S nvenory polcy model for deerorng ems wh me proporonl demnd, Journl of he Operonl Reserch Socey, Vol 35 (, pp 3 9, 984 [4] M A Hrg Opml EOQ model for deerorng ems wh mevryng demnd, Journl of he Operonl Reserch Socey, Vol 47, pp 8-46, 996 [5] H-J Chng, J- eng, L-Y Ouyng, C-Y Dye Reler's opml prcng nd lo-szng polces for deerorng ems wh prl ckloggng, Europen Journl of Operonl Reserch, Vol 68, pp 5 64, 6 [6] R M Hll Invenory model for ncresng demnd followed y level demnd, Journl of he Operonl Reserch Socey, Vol 46, pp 5 59, 995 [7] K-S Wu An EOQ nvenory model for ems wh Weull dsruon deeroron, rmp ype demnd re nd prl ckloggng, Producon Plnnng nd Conrol, Vol (8, pp , [8] B, Mndl, A K Pl Order level nvenory sysem wh rmp ype demnd re for deerorng ems, Journl of Inerdscplnry Mhemcs, Vol, pp 49 66, 998 [9] P S Deng, R H-J Ln, P Chu A noe on he nvenory models for deerorng ems wh rmp ype demnd re, Europen Journl of Operonl Reserch 7, Vol 78, pp, 7 [] K Skour, I Konsnrs, S, Ppchrsos, I Gns Invenory models wh rmp ype demnd re, prl ckloggng nd Weull deeroron re, Europen Journl of Operonl Reserch, Vol 9, pp 79 9, 9 [] M Cheng, G Wng A noe on he nvenory model for deerorng ems wh rpezodl ype demnd re, Compuers nd Indusrl Engneerng, Vol 56, pp 96-3, 9 [] S Pnd, S Senp, M Bsu Opml replenshmen polcy for pershle sesonl producs n seson wh rmp-ype me dependen demnd, Compuers nd Indusrl Engneerng, Vol 54, pp 3 34, 8 63